research article dynamic response of bridge subjected to...

Research ArticleDynamic Response of Bridge Subjected to Eccentrically MovingFlexible Vehicle A Semianalytical Approach

R Lalthlamuana and S Talukdar

Department of Civil Engineering Indian Institute of Technology Guwahati Guwahati 781039 India

Correspondence should be addressed to S Talukdar stalukiitgernetin

Received 2 July 2013 Revised 8 January 2014 Accepted 8 March 2014 Published 5 May 2014

Academic Editor Peijun Xu

Copyright copy 2014 R Lalthlamuana and S Talukdar This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Dynamic response of a single span bridge subjected to moving flexible vehicles has been studied using a semianalytical approachThe eccentricity of vehicle path giving rise to torsional motion of the bridge has been incorporated in the approach The bridgesurface irregularity has been considered as the nonhomogeneous process in spatial domain A closed form expression has beenderived to generate response samples corresponding to each input of roughness profile to form an ensemble Thereafter averagingacross the ensemble has been carried out at each time step to determinemean and standard deviation of bridge and vehicle responseFurther dynamic amplification factor (DAF) of the bridge response has been obtained for several combinations of bridge-vehicleparametersThe study reveals that structural bendingmodes of vehicle can significantly reduce dynamic response of the bridgeTheeccentricity of vehicle path and flexuraltorsional rigidity ratios plays a significant role in dynamic amplification of bridge response

1 Introduction

The dynamic effect resulting from the passage of vehicles isan important problem generally encountered in the bridgedesignThe irregularity or unevenness of the bridge pavementsurface is the main cause of exciting the vehicle which in turnimposes a time varying load as it travels along the span ofthe bridge Starting from the year 1922 various theoreticaland experimental studies have been conducted to understandthe dynamic behavior of bridge subjected to moving loadA review of literature on the said topic starting from basicformulation with moving mass has been published by Fryba[1] and Yang et al [2] in their books with detailed discussionon the formulation and its limitations Earlier researchershave considered vehicle as a moving mass on the bridgeeither neglecting their inertia effect or incorporating thesame Although researchers have revealed various dynamiccharacteristics for practical applications modern bridges ofslender cross-section and larger span do not actually reflecttrue behavior whenmovingmass problems have been solvedThe deformation of bridge can cause significant change indynamic forces at the contact point of the vehicle wheel

Realizing these facts numerous studies have been conductedconsidering bridge-vehicle a coupled system with consid-eration of stiffness and damping parameters of suspensionsystems Biggs [3] Fryba [4] and Wen [5] are some of theearlier authors who considered vehicle as single lumpedmasssystem having only bounce motion or a rigid system withbounce and pitch motion A three-dimensional heave-pitch-roll model has been investigated by Yadav and Upadhyay [6]to find the response of railway tracks on elastic subgradeVehicle models with seven and twelve degrees of freedomwere developed by Wang et al [7] according to H20-44 andHS20-44 which are major design vehicles in the AmericanAssociation of State Highway and Transportation Officials(AASHTO) However analyses of vehicle motions wereconfined to rigidmodes only Kou and deWolf [8] Cheung etal [9] andMarchesiello et al [10] have studied the vibrationalbehavior of multispan continuous bridge by adoption ofbeam or isotropic plate model The effect of moving massor rigid standard vehicle models on the bridge responsewas examined Yin et al [11] investigated lateral vibrationof high pier bridge subjected to moving vehicle In additionto employment of three-dimensional rigid vehicle model

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 546156 24 pageshttpdxdoiorg1011552014546156

2 Shock and Vibration

special attention was given to the modeling of tyre with athree-dimensional linear spring and a rectangular contactsurface Other notable works in bridge-vehicle interactiondynamics that include various dynamic properties of rigidvehicle have been reported by Chen and Cai [12] Law andZhu [13] Zhang et al [14] and Green and Cebon [15]Marchesiello et al [16] studied the interaction of multispancontinuous bridges modeled by isotropic plates with multi-degree of freedom vehicles moving at constant speed The-oretical modes of plate vibration have been obtained withthe help of Rayleigh-Ritz approximate method Prestressedconcrete single track railway bridges consisting of severalbox girders were studied by Chu et al [17] in which theyconsidered each girder to share equal loads and modelled asa beam Rail irregularities are modeled using power spectraldensity function Variousworks on bridge-vehicle interactiononmultispan bridges have been reported by Huang et al [18]Wang [19] and Ichikawa et al [20] Numerical techniques forsolving the finite element model with rigid vehicle have beenadopted Other important works which include complexity inbridge model are that of horizontally curved girder [21 22]Analytical methods were employed to solve bridge dynamicproblem subject to moving mass taking effect of centrifugalforce Effect of vehicle braking has been included in studyof vehicle-bridge interaction problem by Kishan and Traill-Nash [23] and Radley and Traill-Nash [24]

Interests in research on bridge-vehicle interactiondynamics are still growing because of several complexities inmodeling and uncertainties in excitation In the last decaderesearchers have undertaken more and more complexproblems and attempted to solve by newer methodologytaking into consideration support flexibility and nonuniformcross-section [25] A vehicle-track-bridge interactionelement considering vehiclersquos pitching effect has beendeveloped by Lou [26] Experimental results and theircomparison with finite element model analysis of vehicle-bridge interaction problem have been presented by Bradyet al [27] Rigid model of vehicle has been considered inthe study to determine set of critical velocity associated withpeaks of dynamic amplification factor Multiple resonanceresponse of railway bridge has been investigated by Yau andYang [28] A vehicle-bridge interaction process has beensimulatedwithMATLABSimulink byHarris et al [29] to findout bridge-friendly damping control strategy with a tractorsemitrailer vehicle model Very recently some interestingstudies of bridge-vehicle or road-vehicle coupled dynamicsinclude stochastic numerics and discrete integration schemesfor digital simulation of road-vehicle system by Wedig [30]application of spectral stochastic finite element by Wu andLaw [31] and use of spectral matrix operator for directsolution of stochastic coupled differential equations by Kozarand Malic [32] Those studies have significantly improvedthe understanding of complex problem in vehicle-bridgeinteractions However vehicle model has been assumed aslumped masses with rigid link connected by suspensionelements exhibiting various discrete degrees of freedomIn most of the studies numerical integration and MonteCarlo simulation techniques have been used In the moderndays characteristics of vehicles have greatly changed due

to incorporation of larger pay load and for increasingdemand of traffic In the past vehicles have been modeledby a rigid 2D or 3D system having degrees of freedom inbounce pitch and roll However due to long and slendervehicle plying frequently over the bridges there is a needto consider flexibility in the vehicle model and to examinethe effect of flexible modes of vehicle on the dynamics ofbridge This aspect of bridge-vehicle dynamic interactionis not adequately addressed in the literature With this inmind present study has been conducted to find out theresponse statistics of single span bridge due to movement offlexible vehicle along an eccentric path By the term ldquoflexiblevehiclerdquo it is understood here that flexural modes havebeen included in addition to rigid body modes The bridgehas been considered to be under independent transversebending as well as under torsional excitation arising outof the eccentric path of the vehicle Nonhomogeneousprofile of deck surface has been incorporated in the studyby considering a deterministic mean surface superimposedby zero mean random process Such formulation can takecare of defects in surface finishing construction jointspotholes bump approach slab settlement and so forthBridge-vehicle system equations are expressed in state-space form and decoupled at each time step using complexeigenvalue analysis Closed form expression has been derivedfor state vectors for each sample input of bridge profile toform the ensemble of response Finally averaging acrossthe ensemble has been carried out to determine mean andstandard deviation of the response quantities In presentapproach numerical integration can be avoided to generateresponse samples which can save considerable amountof computational time The results obtained from presentapproach have been validated by numerical simulation andavailable experimental results from the literature The effectof vehicle velocity on the response and combined effect ofseveral vehicle-bridge parameters on dynamic amplificationfactor (DAF) have been examined

2 Mathematical Model

The bridge-vehicle model has been shown in Figure 1 Thebridge has been modeled as a uniform beam with simplysupported end conditions The mass stiffness and dampingare assumed to be uniform along the span of bridge Dueto eccentricity of the vehicle path the bridge is subjected toflexure as well as torsionThe bridge deck is unevenwhich hasbeen realized as nonhomogeneous process in spatial domainThis is represented by a function ℎ(119909)

21 Equation of Motion of Vehicle Vehicle body has beenidealized as Euler-Bernoulli beam of length 119897V The behaviorof suspension systems consisting of spring and dashpot isassumed as linear The governing differential equation ofmotion of the vehicle deflection can be expressed as

119864V119868V1205974119911 (119906 119905)

1205971199064+ 119862V

120597119911 (119906 119905)

120597119905+ 119898V

1205972119911 (119906 119905)

1205971199052= 119891V (119906 119905) (1)

Shock and Vibration 3

c1c2 z(u t)

mw1mw2

ct1ct2y(x t)

kt1kt2

z1(t)z2(t)

Figure 1 Bridge-vehicle model

in which 119898V 119864V119868V and 119862V denote the mass per unit lengthflexural rigidity and viscous damping per unit length of thevehicle body and 119911(119906 119905) represents vertical deflection of thevehicle body measured at location 119906 from the reference point(taken at the left end of the vehicle) at time instant 119905 Theimpressed vertical force on the vehicle body is given by

119891V (119906 119905)

= [119896V1 119911 (119906 119905) minus 1199111 (119905) + 119888V1 (119906 119905) minus 1 (119905)]

times 120575 (119906 minus 1199061)

+ [119896V2 119911 (119906 119905) minus 1199112 (119905) + 119888V2 (119906 119905) minus 2 (119905)]

times 120575 (119906 minus 1199062)

where 1199061and 119906

2represent the location of the attachment

point of vehicle suspension from the reference point 1199111

and 1199112denote the vertical displacement of front and rear

wheel masses respectively 119896V1 and 119896V2 are the front and rearvehicle suspension stiffness respectively 119888V1 and 119888V2 representdamping for vehicle front and rear suspension respectively 120575represents Dirac delta function with the property

minusinfin

119891 (119909) 120575 (119909 minus 1199091) 119889119909 = 119891 (119909

1) (3)

The equation of motion for the front unsprung mass is givenby

1198981(119905)1+ 11989611990511199111(119905) minus 119910 (119909

1 119905) minus ℎ (119909

+ 119896V1 1199111 (119905) minus 119911 (1199061 119905) + 119888V1 1 (119905) minus (1199061 119905)

+ 1198881199051[1(119905) minus

119863

119863119905119910 (1199091 119905) + ℎ (119909

1)] = 0

The equation of motion for the rear unsprung mass is givenby

11989822(119905) + 119896

11990521199112(119905) minus 119910 (119909

2 119905) minus ℎ (119909

+ 119896V2 1199112 (119905) minus 119911 (1199062 119905) + 119888V2 2 (119905) minus (1199062 119905)

+ 1198881199052[2(119905) minus

119863

119863119905119910 (1199092 119905) + ℎ (119909

2)] = 0

where1198981and119898

2are front and rear wheel mass respectively

1198961199051 1198961199052are front and rear suspension stiffness respectively

and 1198881199051 1198881199052are front and rear suspension damping respec-

tively ℎ(1199091) and ℎ(119909

2) represent the nonhomogeneous deck

profile under the front and rear wheels respectively 119910(1199091 119905)

and 119910(1199092 119905) are bridge displacements under front and rear

wheels respectively at any instant of time 119905 119911(1199061 119905) and

119911(1199062 119905) represent vehicle body deflection at the front and rear

wheels position at any instant of time 119905 1199061and 119906

2are the

location of wheel from the end of the vehicle body Coriolisforces that arise due to rolling of wheel on the deflectedprofile of the bridge have been considered in the equation ofmotion using total derivative operator 119863119863119905 (with 119863119910119863119905 =(120597119910120597119909)(120597119909120597119905) + 120597119910120597119905) [1 33]

22 Equation of Motion of Bridge It is assumed that forsymmetrical cross-section (symmetrical about vertical axis)bending and torsion of the bridge would be independentunder vertically applied live loadThus governing differentialequation of motion of the bridge in flexure can be expressedas

119864119887119868119887

1205974119910 (119909 119905)

1205971199094+ 119862119887

120597119910 (119909 119905)

120597119905+ 119898119887

1205972119910 (119909 119905)

1205971199052= 119891119887(119909 119905) (6)

in which119898119887 119864119887119868119887 and 119862

119887represent the mass per unit length

flexural rigidity and viscous damping per unit length of

bridgeThe impressed vertical force119891119887(119909 119905) on the bridge due

to vehicle interaction is given by

119891119887(119909 119905)

= minus [11989611990511199111(119905) minus 119910 (119909 119905) minus ℎ (119909)

+11988811990511(119905) minus

119863

119863119905[119910 (119909 119905) + ℎ (119909)]]

times 120575 (119909 minus 1199091)

minus [11989611990521199112(119905) minus 119910 (119909 119905) minus ℎ (119909)

+11988811990522(119905) minus

119863

119863119905[119910 (119909 119905) + ℎ (119909)]] 120575 (119909 minus 119909

minus 1198981+1

2119898V119897V119892120575 (119909 minus 1199091) minus 1198982 +

2119898V119897V119892

times 120575 (119909 minus 1199092)

+ 1198981

1198632

1198631199052[119910 (119909 119905) + ℎ (119909)] 120575 (119909 minus 119909

+ 1198982

1198632

1198631199052[119910 (119909 119905) + ℎ (119909)] 120575 (119909 minus 119909

where 119892 is the acceleration due to gravity The governingdifferential equation of the bridge in torsion can be writtenas

119866119887119869119887

1205972120574 (119909 119905)

1205971199092minus 119862119887119879

120597120574 (119909 119905)

120597119905minus 119868119887

1205972120574 (119909 119905)

1205971199052= 119891119879(119909 119905)

in which 119868119887119866119887119869119887119862119887119905 and 120574(119909 119905) represent the mass moment

of inertia per unit length torsional rigidity distributedviscous damping to rotational motion and torsional functionof bridge respectively 119869

119887is torsional constant and 119866

119887is

the shear modulus of beam material 119891119879(119909 119905) is the torque

produced in the bridge cross-section due to eccentric loadingwhich is given by

119891119879(119909 119905)

+11988811990511(119905) minus

119863

119863119905[119910 (119909 119905) + ℎ (119909)]]

times 119890119909120575 (119909 minus 119909

+11988811990522(119905) minus

119863

119863119905[119910 (119909 119905) + ℎ (119909)]]

times 119890119909120575 (119909 minus 119909

minus 1198981+1

2119898V119897V119892119890119909120575 (119909 minus 1199091)

minus 1198982+1

2119898V119897V119892119890119909120575 (119909 minus 1199092)

+ 1198981

1198632

1198631199052[119910 (119909 119905) + ℎ (119909)] 119890

119909120575 (119909 minus 119909

+ 1198982

1198632

1198631199052[119910 (119909 119905) + ℎ (119909)] 119890

119909120575(119909 minus 119909

2)119909120575 (119909 minus 119909

The parameter 119890119909in (9) denotes the eccentricity of vehicle

wheels from the centre line of bridge deck

23 Bridge Deck Roughness In the present study we intro-duce a roughness which is nonhomogeneous in space eventhough vehicle velocity is constant by adopting the followingrelation

ℎ (119909) = ℎ119898(119909) +

119873

119904=1

120589119904cos (2120587Ω

119904119909 + 120579119904) (10)

where ℎ119898(119909) is a deterministic mean which represents

construction defects expansion joints created pot holesapproach slab settlement expansion joints development ofcorrugation and so forth 120589

119904is the amplitude of cosine wave

and Ω119904is the spatial frequency (radm) within the interval

[Ω119871 Ω119880] in which power spectral density is defined Ω

119871

and Ω119880are lower and upper cut-off frequencies The deck

roughness is a Gaussian process [34] with a random phaseangle 120579

119904uniformly distributed from 0 to 2120587119873 is the number

of terms used to build up the road surface roughness Theparameters 120589

119904andΩ

119904are computed as

120589119904= radic2119878 (Ω

119904) ΔΩ

Ω119904= Ω119871+ (119904 minus

2)ΔΩ

ΔΩ =(Ω119880minus Ω119871)

119873

in which 119878(Ω119904) is the power spectral density function

(m3rad) taken from [35] modifying the same with additionof one term in denominator so that the function exists whenΩ rarr 0

119878 (Ω) = 119878 (Ω0) times

Ω119904

2+ Ω2119871

In the above equation Ω0= 12120587 radm has been taken

The spatial frequency Ω (radm) and temporal frequency 120596(rads) for the surface profile are related to the vehicle speed119881 (ms) as 120596 = Ω119881 In the present study vehicle forwardvelocity has been assumed as constant

24 Discretization of Flexible Vehicle Equation of Motion Asmentioned earlier vehicle body has beenmodeled as free-free

beam which has two rigid modes and 119899V number of elasticmodes It can be shown that when the translation of the masscentroid and the rotational motion about the mass centroidare considered the two motions are orthogonal with respectto each other and with respect to the elastic modes [36]Thustotal displacement of these rigid body degrees of freedom andelastic modes can be described by

119911 (119906 119905) =

119899V

119895 =minus1

120601V119895 (119906) 120578119895 (119905) (13)

where 120601V119895(119906) is the vehicle mode shapes the subscript Vdenotes vehicle 120578

119895(119905) is the time dependent generalized

coordinate 119895 is the mode number 119895 = minus1 0 are takento denote rigid body translatory and pitching mode 119895 =

1 2 3 119899V represent elastic mode sequence of free-freebeam and 119899V is the number of significant modes of flexiblevehicle body Thus two rigid modes can be written as

120601minus1= 1

1206010= 119906 minus 119863

where 1198632is a distance of vehicle centre of gravity from the

trailing edge as given in Figure 1The elastic bending modes of free-free beam for 119895 =

1 2 3 are given by [37]

120601V119895 = sin (120572119895119906) + sinh (120572

119895119906)

+ 120573119895[cos (120572

119895119906) + cosh (120572

119895119906)]

120573119895=cos (120572

119895119897V) + cosh (120572

119895119897V)

sin (120572119895119897V) + sinh (120572

119895119897V)

where corresponding nondimensional frequency parameters120572119895119897V can be related to circular natural frequency as

120596V119895 = 1205722

119895radic

119864V119868V

119898V1198974

V (16)

Substituting (13) in (1) and multiplying both sides of theequation by 120601V119896(119906) and then integrating with respect to 119906

from 0 to 119897V alongwith orthogonality conditions the equationof motion can be discretized as

120578119895(119905) + 2120585V119895120596V119895 120578119895 (119905) + 120596

V119895120578119895 (119905) = 119876V119895 (119905) (17)

where 119895 = minus1 0 1 2 Generalized force 119876V119895(119905) in the 119895th mode acting on the

vehicle is given as

119876V119895 (119905) =1

119872V119895int

119897119881

119891V (119906 119905) 120601119895 (119906) 119889119906 (18)

in which generalized mass119872V119895 in the 119895th mode is given by

119872V119895 = int

119897V

119898V1206012

V119895 (119906) 119889119906 (19)

Making use of (2) and (13) in (17) and integrating theexpression using the property of Dirac delta function one hasthe following expression for generalized force

119876V119895 (119905)

119872V119895

119896V1

1199111(119905) minus

119899V

119895 =minus1

120601119895(1199061) 120578 (119905)

120601119895(1199061)

+ 119888V1

1(119905) minus

119899V

119895 =minus1

120601119895(1199061) 120578 (119905)

120601119895(1199061)

+ 119896V2

1199112(119905) minus

119899V

119895 =minus1

120601119895(1199062) 120578 (119905)

120601119895(1199062)

+119888V2

2(119905) minus

119899V

119895 =minus1

120601119895(1199062) 120578 (119905)

120601119895(1199062)]

It may be mentioned that infinite number of modes arepossible in continuous system considered in the presentstudy However for practical implementation only first 119899Vmodes of vehicle body have been included

25 Discretization of Bridge Equation of Motion Using first119899119887number of bridge bending modes the bridge deflection in

flexure be written as

119910 (119909 119905) =

119899119887

119896=1

120601119887119896(119909) 119902119896(119905) (21)

where subscript 119887 represents bridge 120601119887119896(119909) is the flexural

mode of the beam for simply supported boundary conditioncorresponding to natural frequency and 120596

119887119896and 119902

119896(119905) are

generalized coordinates in 119896th mode [37]Now substituting (21) in (6) and multiplying both sides

of the equation by 120601119887119895(119909) and then integrating with respect to

119909 from 0 to 119871 with the use of orthogonality conditions theequation of motion can be discretized in normal coordinatesas

119902119896(119905) + 2120585

119887119896120596119887119896

119902119896(119905) + 120596

119887119896119902119896(119905) = 119876

119887119896(119905) (22)

where 119896 = 1 2 3 119899119887

The generalized force119876119887119896(119905) in the 119896th mode of bridge in

flexure is given as

119876119887119896(119905) =

119872119887119896

119871

119891119887(119909 119905) 120601

119887119896(119909) 119889119909 (23)

in which generalized mass119872119887119896in the 119896th mode is given by

119872119887119896= int

119871

1198981198871206012

119887119896(119909) 119889119909 (24)

The generalized force in the 119896th of mode of bridge transversevibration has been worked out as

119876119887119896(119905)

= minus1

119872119887119896

[11989611990511199111(119905) minus

119899119887

119896=1

120601119887119896(1199091) 119902119896(119905) minus ℎ (119909

times 120601119887119896(1199091)

+ 11989611990521199112(119905) minus

119899119887

119896=1

120601119887119896(1199092) 119902119896(119905) minus ℎ (119909

times 120601119887119896(1199092)

+ 11988811990511(119905) minus 119881

119899119887

119896=1

1206011015840

119887119896(1199091) 119902119896(119905)

minus119881

119899119887

119896=1

120601119887119896(1199091) 119902119896(119905) minus 119881ℎ

1015840(1199091)

times 120601119887119896(1199091)

+ 11988811990522(119905) minus 119881

119899119887

119896=1

1206011015840

119887119896(1199092) 119902119896(119905)

minus119881

119899119887

119896=1

120601119887119896(1199092) 119902119896(119905) minus 119881ℎ

1015840(1199092)

times 120601119887119896(1199092)

minus 1198981+1

2119898V119897V119892120601119887119896 (1199091)

minus 1198982+1

2119898V119897V119892120601119887119896 (1199092)

times 1198981

119899119887

119896=1

120601119887119896(1199091) 119902119896(119905)

+ 2119881

119899119887

119896=1

1206011015840

119887119896(1199091) 119902119896(119905)

+1198812

119899119887

119896=1

12060110158401015840

119887119896(1199091) 119902119896(119905) + 4119881ℎ

10158401015840(1199091)

times 120601119887119896(1199091)

+ 1198982

119899119887

119896=1

120601119887119896(1199092) 119902119896(119905)

+ 2119881

119899119887

119896=1

1206011015840

119887119896(1199092) 119902119896(119905)

+ 1198812

119899119887

119896=1

12060110158401015840

119887119896(1199092) 119902119896(119905)

+4119881ℎ10158401015840(1199092)

times 120601119887119896(1199092) ]

in which (sdot) denotes time derivative Repeating the similarsteps the discretized bridge equation for torsion in normalcoordinates can be expressed as

120574119897(119905) + 2120585

119879119897120596119879119897

120574119897(119905) + 120596

119879119897120574119897(119905) = 119876

119879119897(119905)

(119897 = 1 2 3 119899119879)

where 119899119879represents number of bridge torsional modes con-

sidered and 120596119879119897and 120585119879119897are the natural frequency and modal

damping coefficient of the 119897th mode in torsion respectivelyThe generalized torque in the 119897th mode is given by

119876119879119897(119905) =

119872119879119897

119871

119891119879(119909 119905) 120601

119879119897(119909) 119889119909 (27)

The torsional natural frequency120596119879119897and correspondingmode

120601119879119897

for the given simply supported boundary conditionsfor no warping restrains have been taken from [37] Thegeneralized mass moment of inertia 119872

119879119897in the 119897th mode is

given by

119872119879119897= int

119871

1198681198871206012

119879119897(119909) 119889119909 (28)

The generalized torque in the 119897th mode can be expressed as

119876119879119897(119905)

= minus119890119909

119872119879119897

[11989611990511199111(119905) minus

119899119879

119897=1

120601119879119897(1199091) 120574119897(119905) minus ℎ (119909

times 120601119879119897(1199091)

+ 11989611990521199112(119905) minus

119899119879

119897=1

120601119879119897(1199092) 120574119897(119905) minus ℎ (119909

times 120601119879119897(1199092)

+ 11988811990511(119905) minus 119881

119899119879

119897=1

1206011015840

119879119897(1199091) 120574119897(119905)

minus119881

119899119879

119897=1

120601119879119897(1199091) 120574119897(119905) minus 119881ℎ

1015840(1199091)

times 120601119879119897(1199091)

+ 11988811990522(119905) minus 119881

119899119879

119897=1

1206011015840

119879119897(1199092) 120574119897(119905)

minus119881

119899119879

119897=1

120601119879119897(1199092) 120574119897(119905) minus 119881ℎ

1015840(1199092)

times 120601119879119897(1199092)

minus 1198981+1

2119898V119897V119892120601119879119897 (1199091) minus 1198982 +

2119898V119897V

times 119892120601119879119897(1199092)

times 1198981

119899119879

119897=1

120601119879119897(1199091) 120574119897(119905) + 2119881

119899119879

119897=1

1206011015840

119879119897(1199091) 120574119897(119905)

+1198812

119899119879

119897=1

12060110158401015840

119879119897(1199091) 120574119897(119905) + 4119881ℎ

10158401015840(1199091)

times 120601119879119897(1199091)

+ 1198982

119899119879

119897=1

120601119879119897(1199092) 120574119897(119905) + 2119881

119899119879

119897=1

1206011015840

119879119896(1199092) 120574119897(119905)

+1198812

119899119879

119896=1

12060110158401015840

119879119896(1199092) 120574119897(119905) + 4119881ℎ

10158401015840(1199092)

times 120601119879119897(1199092) ]

3 Method of Solution

The system of (4) (5) (17) (22) and (26) is coupled secondorder ordinary differential equations In general for continu-ous system like the ones (vehicle and bridge) presented in thepaper infinite number of modes exist However for practicalapplications modes have to be truncated to a finite size Let119899V 119899119887 and 119899

119879be number of significant modes of vehicle

motion bridge flexural and torsional vibration respectivelyThe number of coupled equations becomes 119899 = 2 + 119899V + 119899

119887+

119899119879The system equations can be expressed inmatrix notation

[119872] 119903 (119905) + [119862] 119903 (119905) + [119870] 119903 (119905) = 119865 (119905) (30)

where 119903(119905) = 1205781(119905) 1205782(119905) 120578

119899V(119905) 1199111(119905) 1199112(119905) 1199021(119905)1199022(119905) 119902

119899119887(119905) 1205741(119905) 1205742(119905) 120574

119899119879(119905)119879 is the response vec-

tor 119865(119905) is the generalized force vector and [119872] [C]and [119870] are system mass damping and stiffness matrixrespectively

The system equation can be cast into a 2119899-dimensionalfirst order differential equation of the following form

(119905) + [119860] 119901 (119905) = 119875 (119905) (31)

119901 (119905) = 119903 (119905)

119903 (119905) 119875 (119905) =

[119872]minus1119865 (119905)

[119860] = [119872]minus1[119862]

minus [119868]

[119872]minus1[119870]

This form is suitable for bridge-vehicle interaction problemssince suspension damping is not small and diagonalizationof damping matrix as in case of Rayleighrsquos damping may notbe fully convincing [119868] is an identity matrix and 0 a nullvector or matrix Let the eigenvalues of the state matrix A be1205721 1205722 1205723 120572

2119899and the corresponding eigenvectors u1

u2 u3 u2119899 The modal matrix is defined as[119880] = [119906

11199062 119906

2119899] (33)

The eigenvalues are assumed to be distinct Thus one has

[119880]minus1[119860] [119880] = diagonal [120572

1 1205722 1205723 120572

2119899] (34)

Let us assume119901 = [119880] V (35)

Using linear transformation in (35) the state-space equationsin decoupled form can be written as

V119895(119905) + 120572

119895V119895(119905) = 119877

119895(119905) 119895 = 1 2 3 2119899 (36)

119877119895=

119899

119896=1

1199061015840

119895119904

119899

119904=1

1198981015840

119904119896119865119896(119905) (37)

1199061015840

119895119904denotes elements in the inverse of the matrix [119880] and1198981015840

119904119896

the elements in the inverse of matrix [119872]The general solution of (36) in Stieltjes integral [38] form

may be expressed as

V119895(119905) = 119883

0119895exp (minus120572

119895119905) + int

minusinfin

119867119895(120596 119905) 119889119878 [119865

119896(120596)] (38)

where 1198830119895

are constants of integration to be determinedfrom the initial conditions119867

119895(120596 119905) is the transient frequency

response function given by [38]

119867119895(120596 119905) =

minus119894120596 + 120572119895

[exp (minus119894120596119905) minus exp minus120572119895(119905 minus 1199050)]

(39)It can be seen that as 119905

0rarr minusinfin for the positive real

part of 120572119895 119867119895(120596 119905) approaches the limiting value (1 minus

119894120596 + 120572119895) exp(minus119894120596119905) This is the characteristics of stable

dynamic system Using (37) and (39) in (38) the response ingeneralized normal coordinate may be expressed as

119903119898(119905) =

2119899

119895=1

119906119898+119899119895

1198830119895exp (minus120572

119895119905)

2119899

119895=1

119906119898+119899119895

119899

119904=1

1199061015840

119895119904

119899

119896=1

1198981015840

119904119896int

minusinfin

119867119895(120596 119905) 119889119878 (119865

119896(120596))

119898 = 1 2 3 119899

The first term of (40) represents homogeneous solution ofthe system equation due to initial condition and the secondterm of the equation may be written using the limiting valueof transient frequency response function

119903119875119898

(119905) =

2119899

119895=1

119906119898+119899119895

119899

119904=1

1199061015840

119895119904

119899

119896=1

1198981015840

119904119896int

minusinfin

exp (minus119894120596119905)minus119894120596 + 120572

119895

119865119896(120596) 119889120596

Using Fourier integral [39] and changing the order of integra-tion (41) can be written as

119903119875119898

(119905)

2119899

119895=1

119906119898+119899119895

119899

119904=1

1199061015840

119895119904

119899

119896=1

1198981015840

119904119896

times int

minusinfin

119865119896(120591) [

2120587int

minusinfin

exp [119894120596 (120591 minus 119905)]minus119894120596 + 120572

119895

119889120596]119889120591

Using Cauchyrsquos residue theorem [39] the particular solutioncan be expressed as

119903119875119898

(119905) =

2119899

119895=1

119906119898+119899119895

119899

119904=1

1199061015840

119895119904

119899

119896=1

1198981015840

119904119896119868119895119896 (43)

119868119895119896= int

119905

exp [120572119895(120591 minus 119905)] 119865

119896(120591) 119889120591 (44)

in which 119905 is the bridge loading time 119868119895119896

can be split up forconvenience as

119868119895119896= 119868119895119908+ 119868119895119909+ 119868119895119910+ 119868119895119911 (45)

where 119908 = 1 2 3 119899V 119909 = 119899V + 1 119899V + 2 119910 = 119899V + 3119899V+4 119899V+2+119899119887 119911 = 119899V+119899119887+3 119899V+119899119887+4 119899V+119899119887+119899119879+2

Closed form expressions for the components of the aboveintegral which generates each of the response samples havebeen developed and given in the appendixThe response sam-ples thus form complete ensemble of the process Averagingacross the ensemble at each time step yields mean 120583

119884(119905119896) and

standard deviation 120590119884(119905119896) of a response process 119884

4 Results and Discussions

41 Model Validation The results obtained by present ana-lytical method have been first compared with numericallysimulated results and published results from the literatureNewmark integration scheme [40] has been adopted fornumerical simulation Time step for integration purpose hasbeen selected as 0005 sec Two-axle vehicle model data andbridge of single span with simple supports have been usedfor comparison purpose For this purpose bridge and vehicleparameters have been selected from study conducted byDengand Cai [41] in which they have also presented experimental

0 05 1 15 2 25

Time (s)

Numerically simulated resultPresent analytical resultExperimental result [41]

50E minus 4

00E + 0

minus50E minus 4

minus10E minus 3

minus15E minus 3

minus20E minus 3

Figure 2 Comparison of analytical numerical and experimentalresults

data Following are the inputs taken in both analytical modeland numerical integration scheme

span (119871) 16764m mass per unit length (119898119887) 108 times

104 kgm flexural rigidity (119864

119887119868119887) 248 times 10

12Nm2 torsionalrigidity (119866

119887119869119887) 248 times 10

12Nm2 mass moment of inertia(119868119887) 187 times 10

5 kgm2m 119864119887= 32GPa deck = 2512GPa

vehicle mass (119898V) 248 times 104 kg mass of front and rear

wheel (1198981199081 1198981199082) 7254 kg eachmassmoment of inertia (119868V)

172times105 kgm2 front and rear suspension stiffness (119896V1 119896V2)

728 times 105Nm 197 times 10

6Nm front and rear suspensiondamping (119888V1 119888V2) 21896Nsm 71818Nsm and front andrear tyre stiffness (119896

1199051 1198961199052) 197 times 10

6Nm 474 times 106Nm

In Figure 2 the analytically obtained response sample hasbeen compared with numerically simulated response sampleand experimental results found in [41] The comparison ofresponse sample exhibits close agreement between analyticaland numerically simulated results The experimental data inthe initial portion has shown considerable deviation fromtheoretical results which however becomes closer in thelater part of the response time historyThe peakmagnitude ofthe displacement in analytical numerical and experimentalresults agrees well

42 Parametric Study The following system data have beenadopted to generate numerical results and to conduct para-metric study a RC slab-girder bridge of span (119871) 20mthree longitudinal girders along the span and three cross-girders at midspan and at supports are provided in bridgethe lane width 86m deck thickness 200mm and concretecharacteristic strength 25Nmm2 The cross-section of thebridge is shown in Figure 3 A finite element (FE) model ofbridge in SAP2000 commercial software is first developedusing the above details of the bridge so as to match thefundamental natural frequency andfirstmodal damping ratioof the simply supported beam model of T-beam bridgeThe sectional properties of FE model are then used inthe present analytical program of the bridge The followingphysical parameters are finally selected for the beam modelto represent a T-beam RC concrete bridge

mass (119898119887) 1115times103 kgm flexural rigidity (119864

119887119868119887) 37times

1010N-m2 and torsional rigidity (119866

119887119869119887) 1695 times 10

10N-m2

56 MM THKAC wearing coat

063 063 063

300 300

Figure 3 Cross-section of T-beam bridge (all dimensions are in meter)

265100

Figure 4 Longitudinal section of vehicle (all dimensions are in meter)

The sectional parameters to be given in the beam model ofbridge have been found from the FE model of bridge in SAP2000 commercial software to match the fundamental naturalfrequency and first modal damping of both the beam and theFE model of bridge

Vehicle parameters are as follows a long vehicle carryingheavy load often crossing the bridge has been chosen toillustrate the present approach The standards of vehicle aredifferent from the live load prescribed by bridge code Inthe present study we use a vehicle type TATA 3516C-EX asshown in Figures 4 and 5 This has been idealized as Euler-Bernoulli beam adopted in present formulation Followingare the important physical parameters pertaining to vehiclelength (119897V) 12m flexural rigidity (119864V119868V) 53times10

6N-m2 massper unit length (119898V) 1500 kgm front and rear wheel masses(1198981199081 1198981199082) 800 kg each suspension stiffness front and rear

(119896V1 119896V2) 36 times 107Nm suspension damping front and rear

(119888V1 119888V2) 72 times 104N-secm front and rear tyre stiffness

(1198961199051 1198961199052) 09 times 10

7Nm and front and rear tyre damping(1198961199051 1198961199052) 07 times 104N-secm

421 Bridge Response Statistics Bridge mean responses atmidspan displacement (120583

119889119910) velocity (120583V119910) and acceleration

(120583119886119910) subjected to three vehicle speed have been shown in

Figures 6 7 and 8 The roughness of the bridge deck hasbeen assumed to be of poor category as per ISO classifi-cation corresponding to roughness coefficient 120589

119904= 16 times

10minus6m2(mcycle) [42] The mean peak central deflection is

found to increase with the vehicle speed The correspondingstandard deviation of each response displacement (120590

119889119911)

velocity (120590V119911) and acceleration (120590119886119911) has been shown in

Figures 9 10 and 11 It may be noted that bridge response fre-quency ismodified by change in vehicle velocity and obtainedbehavior has appropriate physical sense that higher velocityincreases the temporal frequency of road excitation This isobvious from left shift of the peak when velocity increasesfrom lower to higher The peak magnitude is also higher incase of vehicle travelling with higher velocity No definitepattern is discernible in the standard deviation Althoughthe magnitudes of standard deviation are insignificant forpractical purpose magnitudes of the coefficient of variationof peak displacement velocity and accelerations are found tobe 011 0183 and 0105 respectively Itmay bementioned thatdynamic loads do not lead to major bridge damage except inresonance but they contribute to continuous degradation ofbridge increasing necessity of bridge maintenance

075 120 075

Figure 5 Vehicle cross-section (all dimensions are in meter)

0 05 1 15 2 25 3 35 4

Time (s)10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

120583dy

50kmh60kmh80kmh

Figure 6 Mean displacement of bridge at the midspan

0 05 1 15 2 25 3 35

Time (s)

minus80E minus 3

40E minus 3

minus10E minus 18

minus40E minus 3

minus80E minus 3

minus12E minus 2

120583y

50kmh60kmh80kmh

Figure 7 Mean velocity of bridge at the midspan

0 05 1 15 2 25 3 35

Time (s)40E minus 2

00E + 0

minus40E minus 2

minus80E minus 2

minus12E minus 1

minus16E minus 1

120583ay

50kmh60kmh80kmh

Figure 8 Mean acceleration of bridge at the midspan

0 05 1 15 2 25 3 35 4Time (s)

40E minus 5

30E minus 5

20E minus 5

10E minus 5

00E + 0

minus10E minus 5

120590dy

50kmh60kmh80kmh

Figure 9 Standard deviation of bridge midspan deflection

422 Vehicle Response Statistics The mean vehicle bodyresponses (at centroid)mdashdisplacement (120583

119889119911) velocity (120583V119911)

and acceleration (120583119886119911) time historymdashincluding residual

response have been shown in Figures 12 13 and 14 Thecorresponding standard deviation of responses displacement(120590119889119911) velocity (120590V119911) and acceleration (120590

119886119911) is shown in

Figures 15 16 and 17 It has been seen that the peak response isincreased with increase in vehicle speed Standard deviationhas no definite trend in the present case for vehicle response

423 Effect of Vehicle Flexibility on Mean and Standard Devi-ation The contribution of significant number of structuralmodes of vehicle to bridge response has been examined ina bar diagram presented in Figures 18 and 19 by comparingmaximum mean and standard deviation respectively It maybe seen that inclusion of bendingmodes of the vehicle reducesthe response magnitude of the bridge The same figure alsoreveals that when flexiblemodes of the vehicle are consideredsummation of first five modes for finding the response isadequate Figures 20 and 21 show the comparison of mean

0 05 1 15 2 25 3 35Time (s)

16E minus 3

12E minus 3

80E minus 4

40E minus 4

00E + 0

minus40E minus 4

120590y

50kmh60kmh80kmh

Figure 10 Standard deviation of bridge midspan velocity

0 05 1 15 2 25 3 35Time (s)

16E minus 2

12E minus 2

80E minus 3

40E minus 3

00E + 0

120590ay

50kmh60kmh80kmh

Figure 11 Standard deviation of bridge midspan acceleration

0 05 1 15 2 25 3Time (s)

80E minus 3

40E minus 3

00E + 0

minus40E minus 3

minus80E minus 3

minus12E minus 2

minus16E minus 2

120583dz

50kmh60kmh80kmh

Figure 12 Mean displacement of vehicle CG

0 05 1 15 2 25 3Time (s)

12E minus 1

80E minus 2

40E minus 2

00E + 0

minus40E minus 2

minus80E minus 2

120583z

50kmh60kmh80kmh

Figure 13 Mean velocity of vehicle at CG

0 1 2 3Time (s)

80E minus 1

40E minus 1

00E + 0

minus40E minus 1

minus80E minus 1

50kmh60kmh80kmh

120583az

Figure 14 Mean acceleration of vehicle at CG

0 05 1 15 2 25 3Time (s)

10E minus 3

80E minus 4

60E minus 4

40E minus 4

20E minus 4

00E + 0

120590dz (m

50kmh60kmh80kmh

Figure 15 Standard deviation of vehicle displacement at CG

0 1 2 3Time (s)

80E minus 3

60E minus 3

40E minus 3

20E minus 3

00E + 0

120590z

50kmh60kmh80kmh

Figure 16 Standard deviation of vehicle velocity at CG

0 05 1 15 2 25 3Time (s)

40E minus 2

30E minus 2

20E minus 2

10E minus 2

00E + 0

120590az

50kmh60kmh80kmh

Figure 17 Standard deviation of vehicle acceleration at CG

A B C D

50E minus 04

40E minus 04

30E minus 04

20E minus 04

10E minus 04

00E + 00

120583dy

Figure 18 Maximummean displacement of bridge at midspan (A)rigid vehicle (B) flexible vehicle only with first structural mode (C)flexible vehicle with first three structural modes and (D) flexiblevehicle with first five structural modes

A B C D

40E minus 05

30E minus 05

20E minus 05

10E minus 05

00E + 00

120590dy

Figure 19 Maximum standard deviation of displacement atmidspan (A) rigid vehicle (B) flexible vehicle only with firststructural mode (C) flexible vehicle with first three structuralmodes and (D) flexible vehicle with first five structural modes

0 025 05 075 1 125 15 175Time (s)

Flexible vehicleRigid vehicle

30E + 5

25E + 5

20E + 5

15E + 5

10E + 5

50E + 4

00E + 0

120583F

Figure 20 Mean of imposed force time history of the bridge due topassage of rigid and flexible vehicle

0 025 05 075 1 125 15 175Time (s)

25E + 4

20E + 4

15E + 4

10E + 4

50E + 3

00E + 0

minus50 + 3

120590F

Figure 21 Standard deviation of imposed force time history of thebridge due to passage of rigid and flexible vehicle

0 05 1 15 2 25 3 35

Time (s)

10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

minus40E minus 4

120583dy

Figure 22 Effect of flexibility on mean response of bridge (A)119864V119868V = 53 times 10

6N-m2 (B) 119864V119868V = 64 times 107 N-m2 and (C) 119864V119868V =

82 times 1010 N-m2

0 05 1 15 2 25 3 35Time (s)

20E minus 5

16E minus 5

12E minus 5

80E minus 6

40E minus 6

00E + 0

120590dy

Figure 23 Effect of flexibility on standard deviation of bridgeresponse (A) 119864V119868V = 53 times 10

6N-m2 (B) 119864V119868V = 64 times 107N-m2

and (C) 119864V119868V = 82 times 1010N-m2

and standard deviation of imposed force on the bridge dueto passage of a rigid vehicle with those of flexible vehicle Itreveals that when structural bending modes are consideredthe imposed force on the bridge is lessThismay be attributedto the fact that part total strain energy has been utilized inbending of elastic vehicle body as compared to rigid beamwhich reduces imposed force on the bridge

The comparison of mean response of bridge at midspanwith different values of vehicle flexural rigidity has beenpresented in Figure 22 Corresponding standard deviation ispresented in Figure 23 It may be noted that vehicle withlower flexural rigidity generates less response in the bridgeIn generating numerical results under this section vehiclevelocity is taken as 60 kmh

0 05 1 15 2 25 3 35

Time (s)

10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

120583dy

Figure 24 Mean of bridge midspan displacement due to differenteccentricity of vehicle path (119890

119909) (A) 119890

119909= 05m (B) 119890

119909= 1m and

(C) 119890119909= 15m

0 05 1 15 2 25 3 35Time (s)

30E minus 5

25E minus 5

20E minus 5

15E minus 5

10E minus 5

50E minus 6

00E + 0

120590dy

Figure 25 Standard deviation of bridge midspan displacement dueto different eccentricity of vehicle path (119890

119909) (A) 119890

119909= 05m (B) 119890

119909=

1m and (C) 119890119909= 15m

424 Effect of Eccentricity of Vehicle Path on Mean andStandard Deviation The effect of eccentricity on the bridgeresponse has been studied by varying the loading positionfrom the centre of the bridge width In the result shown inFigures 24 and 25 6 to 8 increment in the bridge dynamicresponses has been observed as the eccentricity varies from05m to 15m The vehicle stiffness for comparing the resultsin Figures 24 and 25 is taken uniform (119864V119868V = 53times10

6Nsdotm2)In generating numerical results under this section vehiclevelocity is taken as 60 kmh

43 Dynamic Amplification Factor (DAF) Considering anyresponse variable 119884 as random process the maximumdynamic response can be written as

119884dynamic =1003816100381610038161003816120583119884 (119909119896 119905119896) + 120590119884 (119909119896 119905119896)

1003816100381610038161003816 (46)

0 0002 0004 0006 0008 001

GbJbEbIb

Figure 26 Dynamic amplification factor (DAF) with the ratio ofbridge torsional rigidity to flexural rigidity (119866

119887119869119887119864119887119868119887) for different

vehicle flexural rigidity (A) 119864V119868V = 53 times 106N-m2 (B) 119864V119868V = 64 times

107 N-m2 and (C) 119864V119868V = 82 times 10

10 N-m2

where 119884dynamic denotes the maximum response due to fluctu-ating load imposed on bridge due to vibratory motion of thevehicle excited by road unevenness

Thus dynamic amplification factor (DAF) in this study isdefined as

DAF =119884static + 119884dynamic

119884static (47)

where119884static refers to the response of the bridge at themidspanlocation for adverse position of static wheel loads

431 Effect of Bridge Torsional Rigidity on Dynamic Ampli-fication Factor (DAF) The effect of torsional rigidity ofbridge has been studied by obtaining DAF for differentratios of torsional rigidity (119866

119887119869119887) to flexural rigidity (119864

119887119868119887)

of bridge with various values of vehicle flexural rigidity(119864V119868V) and presented in Figure 26 The results reveal thatdynamic amplification factor decreases by an amount of 23to 34 when the ratio of torsional rigidity to flexural rigidityof bridge increases from 0002 to 001 This indicates thattorsionally stiffer bridge produces less dynamic amplificationof static live load response

432 Effect of VehicleMass and Speed onDAF Since dynamicamplification factor depends on several variables we chooseto represent it by surface plot rather than a two-dimensionalplot The effect of vehicle mass as well as speed on thedynamic amplification factor (DAF) has been shown inFigure 27 Individual effect on earlier classical studies [43]has shown that effect of increasing mass has reducing effecton DAF However a surface plot shown in Figure 18 shows

that combined effect has an increasing trend Earlier authorsargued that reduction of dynamic amplification factor withincreasing vehicle weight is due to increase of static deflec-tion However the coupled dynamic interaction may be thecause of increased inertia force in addition to suspensionforce imposed by moving mass at greater velocity whichincreases dynamic component of the response

433 Effect of Bridge Surface Roughness and Speed Thesurface roughness and speed of the vehicle are two mostinfluential factors that can cause increased dynamic ampli-fication factor and rapid degradation of the bridge Bridgedynamic amplification factor has been found by changingbridge surface roughness coefficient for good condition topoor condition as mentioned in ISO specification [42] withchange in vehicle speed Figure 28 shows that poor conditionof road induces more deflection on the bridge when vehiclemoves over it and also can be catalyzed by vehicle speed

434 Effect of Bridge Span and Velocity The span of thebridge is an important factor which decides the impact factorinmost of the bridge design codes Combined effect of bridgespan and speed of the vehicle on DAF is not fully knownFigure 29 shows the DAF with variation of bridge span andvelocity It has been found thatwhen the bridge span increasesfrom 14m to 30m dynamic amplification factor decreases byan amount of 12 to 15 irrespective of increase of vehiclevelocity Although increasing span shows a decreasing trendin DAF similar to the earlier studies the increment found inthe present case is not very significant for the range of span14ndash30m

5 Conclusions

In the present study an analytical solution for coupledvehicle-bridge interaction problem considering eccentricallymoving flexible vehicle as well as randomdeck surface rough-ness has been obtained Response samples are generatedfrom the expressions derived in the present study consideringnonhomogeneity of deck roughness The results obtainedfrom analytical approach have been validated with numericalsimulation and experimental data available in the literatureIndividual and combined effects of several bridge-vehicleparameters have been considered to find out the responsestatistics The increased effect of vehicle speed has beenfoundmore significant in changing the frequency of imposedoscillation rather than noticeable increase in response peaksCombined effect of increasing vehicle weight and speed hasbeen found to increase the dynamic amplification factor Thestudy reveals that flexibility of long vehicle is an importantconsideration in obtaining bridge response and due tochange in load carrying vehicle configuration it is nowimperative to address this issue in bridge design codesTorsion of the bridge is activated by eccentric movement ofvehicle and dynamic amplification factor is largely depen-dent on the ratio of torsional rigidity to flexural rigidity ofbridge cross-section

1520 25 30 35 40

45 5055

404550556065707580146148

15152154156158

16162164

Vehicle mass (kg)Vehicle speed (kmh)

times103

Figure 27 Dynamic amplification factor with change in vehicle speed and vehicle mass

8013514

Vehicle speed (kmh)

Roughness coefficient (m2cyclem)

18e minus 6 16e minus 6 14e minus 612e minus 6

10e minus 6 8e minus 66e minus 6

2e minus 64e minus 6

Figure 28 Dynamic amplification factor with vehicle speed and bridge surface roughness

14161820222426283013

Vehicle speed (kmh)Bridge span (m)

Figure 29 Dynamic amplification factor with change in bridge span and vehicle speed

Appendix

Expression of the Integral 119868119895119896

forGenerating Response Sample

The vector 119865(119905) needed to perform the integration is givenbelow

119865119895(119905) = 0 for119895 = 1 2 3 119899V

= 119896119905119895minus119899V

ℎ (119909119895minus119899V

) + 119888119905119895minus119899V

ℎ (119909119895minus119899V

For119895 = 119899V + 1 119899V + 2

= 1198961199051ℎ (1199091) 120601119887119895minus(119899V+2) (1199091) + 1198961199052ℎ (1199092) 120601119887119895minus(119899V+2) (1199092)

+ 1198881199051ℎ (1199091) 120601119887119895minus(119899V+2) (1199091) + 1198881199052ℎ (1199092) 120601119887119895minus(119899V+2) (1199092)

For 119895 = 119899V + 3 119899V + 4 119899V + 2 + 119899119887

= 1198901199091198961199051ℎ (1199091) 120601119905119895minus(119899V+2+119899119887)

(1199091)

+ 1198961199052ℎ (1199092) 120601119905119895minus(119899V+2+119899119887)

(1199092)

+ 1198881199051ℎ (1199091) 120601119905119895minus(119899V+2+119899119887)

(1199091)

+1198881199052ℎ1015840(1199092) 120601119905119895minus(119899V+2+119899119887)

(1199092)

For 119895 = 119899V + 119899119887 + 3

119899V + 119899119887 + 4 119899V + 119899119887 + 119899119905 + 2

Themean surface profile has been taken as shallow parabolicthe equation of which with respect to one end of the bridge isgiven by

ℎ119898(119909) =

1198712119909 (119871 minus 119909) (A2)

For one trial a generation of a set of random phase angles 120579119904

(119904 = 1 2 119873) is employed to express a Gaussian process is

ℎ119903(119909) =

119873

119904=1

119860119904cos (2120587Ω

119904119909 + 120579119904) (A3)

Further ℎ1015840(119909) = 119889ℎ119889119909 and ℎ(119909) = 119889ℎ119889119905 = 119881(119889ℎ119889119909) where119881 is the speed of vehicle

For convenience of computation

119868119895119896= 119868119895119908+ 119868119895119909+ 119868119895119910+ 119868119895119911 (A4)

where 119908 = 1 2 3 119899V 119909 = 119899V + 1 119899V + 2 119910 = 119899V + 2 + 1119899V + 2 + 2 119899V + 2 + 119899

119887 119911 = 119899V + 2 + 119899

119887+ 1 119899V + 2 + 119899

119887+

2 119899V + 2 + 119899119887 + 119899119879Thefirst term 119868

119895119908(119908 = 1 2 119899V) of (A4) can bewritten

1198681198951= 1198681198952= sdot sdot sdot = 119868

119895119899V= 0 (A5)

The second term of (A4) can be written as

119868119895119909= 1198681198951199091

+ 1198681198951199092 (A6)

Splitting the first term of integral (A6) again into two we get

1198681198951199091

= 11986811989511990911

+ 11986811989511990912

Introducing 119860119904= radic2119878(119881Ω

119904)minus2 and 119861

119904= 2120587Ω

119871+ (119904 minus

05)(Ω119880minus Ω119871)119873 the components can be written as

11986811989511990911

= 1198961199051[

119873

119904=1

119860119904

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2minus120572119895cos (120579

119904) minus 119861119904119881 sin (120579

119904)

+ exp (120572119895119905)

times [120572119895cos (119861

119904119881119905 + 120579

119904)

+119861119904119881 cos (119861

119904119881119905 + 120579

119904)]

+4ℎ0119881

12057231198951198712

120572119895119871 (120572119895119905 + exp (minus120572

119895119905) minus 1)

minus119881

12057231198951198712

120572119895119905 (120572119895119905 minus 2) minus 2 exp (minus120572

119895119905) + 2 ]

11986811989511990912

= 1198881199051[

119873

119904=1

119881119860119904119861119904

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2minus120572119895sin (120579119904) minus 119861119904119881 cos (120579

119904)

+ exp (120572119895119905)

times [120572119895sin (119861

119904119881119905 + 120579

119904)

minus119861119904119881 sin (119861

119904119881119905 + 120579

119904)]

minus4ℎ0119881 exp (minus120572

119895119905)

12057231198951198712

times 2119881 [exp (minus120572119895119905) (120572119895119905 minus 1) + 1]

minus1

120572119895119871exp (120572

119895119905) minus 2]

Similarly splitting the second term of integral (A6) into twowe get

1198681198951199092

= 11986811989511990921

+ 11986811989511990922

11986811989511990921

= 1198961199052[

119873

119904=1

119860119904

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2

times minus 120572119895cos (119861

119904119897V minus 120579119904) minus 119861119904119881 sin (119861

119904119897V minus 120579119904)

+ exp (120572119895119905) [120572119895cos (119861

119904(119897 minus 119881119905) minus 120579

119904)

+119861119904119881 sin (119861

119904(119897V minus 119881119905) + 120579119904)]

+1198812[2 minus exp (120572

119895119905) 120572119895119905 (120572119895119905 minus 2) + 2]

+4ℎ0

12057231198951198712

minus1205722

119895119897V (119897V + 119871) (exp (120572119895119905) minus 1)

+120572119895119881 (2119897V + 119871) [exp (120572119895119905) (120572119895119905 minus 1) + 1] ]

11986811989511990922

= 1198881199052[

119873

119904=1

119860119904119861119904119881

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2120572119895sin (119861

119904119897V minus 120579119904)

+ 119861119904119881 cos (119861

119904119897V minus 120579119904)

minus exp (120572119895119905) [120572119895sin (119861

119904(119897 minus 119881119905) minus 120579

119904)

+119861119904119881 cos (119861

119904(119897V minus 119881119905) + 120579119904)]

+4ℎ0119881 exp (minus120572

119895119905)

12057221198951198712

times 2119881 [exp (120572119895119905) (120572119895119905 minus 1) + 1]

minus120572119895(2119897V + 119871) (exp (120572119895119905) minus 1)]

The third term of (A4) is written as

119868119895119910=

119899119887

119896=1

119868119895119899V+2+119896

The 119896th term of the above series is composed of ten differentterms as given below

119868119895119899V+2+119896

119886=1

119868(119895119899V+2+119896)119886

(119896 = 1 2 3 119899119887) (A11)

The components of (A11) are as followsIntroducing the following parameters

119876119904= 119861119904119881 119877

119896=119896120587119881

119871

119878119896=119896120587

119871 119885

119904= 119861119904119897V

119868(119895119899V+2+119896)

21198961199051

119873

119904=1

119860119904 ((2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) cos (120579

119904)

+2120572119895119876119904sin (120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times cos (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119876

119904+ 119877119896) 119905 + 120579

119904

minus (119877119896+ 119876119904)

times cos (119876119904+ 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

21198961199051[

4ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198762119904)3

times 119871 (1205722

119895+ 1198762

119904) [exp (120572

119895119905) sin (119876

119904119905)

times 1198762

119904(120572119895119905 + 1)

+1205722

119895(120572119895119905 minus 1)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904119905 + 120572119895(120572119895119905 minus 2)] minus 2120572

119895119876119904

+ 119881 (21198763

119904+ 119876119904exp (120572

119895119905)) cos (119876

119904119905)

times 1198764

1199041199052+ 21198762

119904(1205722

1198951199052minus 2120572119895119905 minus 1)

+1205722

119895(1205722

1198951199052minus 4120572119895119905 + 6)

minus exp (120572119895119905) sin (119876

119904119905)

times 1198764

119904(120572119895119905 + 2) + 120572

1198951198762

119904(1205722

1198951199052minus 3)

+1205723

119895(1205722

1198951199052minus 2120572119895119905 + 2) minus 6120572

119895119876119904]

119868(119895119899V+2+119896)

21198961199052

119873

119904=1

119860119904[ exp (minus120572

119895119905)

times (((119877119896+ 119876119904) cos (119878

119904minus 119885119904minus 120579119904)

+120572119895sin (119878119896minus 119885119904minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

times (((119877119896minus 119876119904) cos (119878

119896minus 119885119904+ 120579119904)

+120572119895sin (119878119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 119878

119896minus 119885119904+ 120579119904

minus (119877119896+ 119876119904) cos (119876

119904minus 119877119896) 119905

+119878119896minus 119885119904+ 120579119904)

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

+ ((120572119895sin 119878119896+ 119885119904minus (119876119904+ 119877) 119905 minus 120579

119904

+ (119877119896+ 119876119904) cos 119878

119896+ 119885119904

+ (119876119904+ 119877119896) 119905

minus120579119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1

21198961199052[

4ℎ0119881

1198712(1205722119895+ 1198812)

times sin (119881119905 minus 119897V) [1205721198951198972

V(1205722

119895+ 1198812)2

minus 119897V (1205722

119895+ 1198812)

times minus1205723

119895119871 minus 1205721198951198711198812

+ 21198813(120572119895119905 + 1)

+21205722

119895119881(120572119895119905 minus 1)

+ 119881 [119881 21205723

119895+ 120572119895119905(1205722

119895+ 1198812)2

+2119905 (1205724

119895minus 1198814) minus 6120572

1198951198812

minus 119871 (1205722

119895+ 1198812)

times 1205722

119895(120572119895119905 minus 1)+ 119881

2(120572119895119905 +1)]

+ 119881 cos (119897V minus 119881119905) [1198814(1198972

V + 119897V119871

+21205722

1198951199052minus 4120572119895119905 minus 2)

+ 1205722

1198951198812(21198972

V + 2119897V119871

+41205722

1198951199052+ 6)

+ 1205724

119895119897V (119897V + 119871)

minus 1198815119905 (2119897V + 119871)

times (120572119895119905 minus 1)

minus 1205723

119895119881 (2119897V + 119871)

times (120572119895119905 minus 2) + 119897

V1198816]]

119868(119895119899V+2+119896)

21198881199051

119873

119904=1

119860119904119876119904( minus (2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) sin (120579

119904)

minus2120572119895119876119904cos (120579

119904))

times (1205722

119895+ (119876119904minus 119877)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times sin (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos (119876

119904+ 119877119896) 119905 + 120579

119904

+ (119877119896+ 119876119904)

times sin (119876119904+ 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199051

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) cos (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) sin (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

21198881199052

119873

119904=1

119860119904119876119904

times [ exp (minus120572119895119905 ( ( (119877

119896+ 119876119904) sin (119878

119896+ 119885119904minus 120579119904)

minus120572119895cos (119878

119896+ 119885119904minus 120579119904))

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

times ( ( (119876119904minus 119877119896) sin (119878

119896minus 119885119904+ 120579119904)

+120572119895cos (119878

119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos 119878

119896+ 119885119904minus (119876119904+ 119877119896) 119905 minus 120579

119904

minus (119877119896+ 119876119904) sin 119878

119896+ 119885119904

minus (119876119904+ 119877119896) 119905 minus 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos 119878

119896minus 119885119904+ (119876119904minus 119877119896) 119905 + 120579

119904

+ (119876119904minus 119877119896) sin 119878

119896minus 119885119904

+ (119876119904minus 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199052

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) sin (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

= minusexp (minus120572

119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881 + exp (120572119895119905) 119896120587119881 cos(119896120587119881119905

119871)

+119871120572119895sin(119896120587119881119905

119871)]

minusexp (minus120572

119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881cos(119896120587119863119871

minus exp (120572119895119905) cos(119896120587119863 minus 119881119905

119871)

minus 119871120572119895sin(119896120587119863

119871)

minus exp (120572119895119905) sin(119896120587119863 minus 119881119905

119871)]

119868(119895119899V+2+119896)

= minus21198711198812(1198981199081

+ 1198981199082)

119873

119904=1

1198601199041198612

119904[(((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881 cos [(119876

119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus exp (minus120572119895119905) ((minus (119896120587 + 119861

119904119871)119881 cos 120579

119904

+119871120572119895sin 120579119904)

times (1198612

11990411987121198812

+ 21198961205871198611199041198711198812

+ 119896212058721198812

+11987121205722

119895)minus1

+ ((119896120587 minus 119861119904119871)119881 cos 120579

119904

minus119871120572119895sin 120579119904)

times ((119896120587 minus 119861119904119871)2

1198812

+11987121205722

119895)minus1

minus8ℎ01198812(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

times [119896120587119881

times exp (minus120572119895119905) minus cos(119896120587119881119905

119871)

+119871120572119895sin(119896120587119881119905

119871)]

The fourth term of (A4) is written as

119868119895119911=

119899119905

119896=1

119868119895119899V+2+119899119887+119896

The 119896th term of the above series can be split up into ten partsas given below

119868119895119899V+2+119899119887+119897

119886=1

119868(119895119899V+2+119899119887+119897)119886

(119896 = 1 2 3 119899119879) (A14)

The components of (A14) are as follows

119868(119895119899V+2+119899119887+119897)1

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895cos (120579

119904) + (119876

119904minus 119877119896) sin (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895cos (120579

119904) + (119876

119904+ 119877119896) sin (120579

119904)

1205722119895+ (119876119904+ 119877119896)2

+ ((120572119895cos [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119876119904minus 119877119896) sin [(119876

119904minus 119877119896) 119905 minus 120579

119904])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [(119876

119904+ 119877119896) 119905 + 120579

119904]

+ (119876119904+ 119877119896) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)2

=41198961199051119890119909ℎ0119871

119881(1205722119895+ 119877119896

times [ (1205722

119895+ 119877119896

2) exp (minus120572

119895119905)

times 1205722

119895+ exp (minus120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)] minus 119877

119896

+ 2120572119895exp (minus120572

119895119905) (1205722

119895+ 3119877119896

+ 119877119896sin (119877

119896119905) 6120572

119895+ 1199052(1205722

119895+ 119877119896

+4120572119895119905 (1205722

119895+ 119877119896

2) minus 2119877

119896

+ cos (119877119896119905) 2119905 (119877

119896

4minus 1205724

119895)

+ 2120572119895(1205722

119895minus 3119877119896

times 1205721198951199052(1205722

119895+ 119877119896

119868(119895119899V+2+119899119887+119897)3

= minus1

21198961199052119890119909

119873

119904=1

[119860119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119877119896minus 119876119904) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) sin (119861

119904119897V + 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos [(119877

119896minus 119876119904) 119905 + (119861

119904119897V minus 119878119896 minus 120579119904)]

+ (119877119896minus 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [minus (119877

119896+ 119876119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

minus (119877119896+ 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)4

=41198961199052119890119909ℎ0

1198712[((119897V (119897V + 119871) exp (minus120572119895119905)

times 120572119895cos (119878

119896minus 119877119896119905)

minus119877119896sin (119878119896minus 119877119896119905)

+119877119896sin (119878) minus 120572

119895cos (119878

119896))

times (1205722

119895+ 119877119896

2)minus1

+2119871119881 exp (minus120572

119895119905)

(1205722119895+ 119877119896

times (1205722

119895+ 119877119896

2) cos (119878

119896)

+ exp (120572119895119905) cos (119878

119896minus 119877119896119905)

times [1198772(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896[119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

times sin (119878119896minus 119877119896119905) minus 2120572

119895119877119896sin (119878119896)

minus1198812

(1205722119895+ 119877119896

times119877119896sin (119878119896minus 119877119896119905)[61205722

119895+1199052(1205722

119895+ 119877119896

minus 4120572119895119905 (1205722

119895+ 119877119896

minus2119877119896

+ 2 exp (minus120572119895119905)

times [(1205723

119895minus 3120572119895119877119896

2) cos (119878

119896)

+ 119877119896(119877119896

2minus 31205722

119895)

times sin (119878119896) ]

minus [2119905 (119877119896

4minus 1205724

119895) + 2120572

119895(1205722

119895minus 3119877119896

+1205721198951199052(1205722

119895+119877119896

]cos (119878119896minus 119877119896119905)]

119868(119895119899V+2+119899119887+119897)5

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895sin (120579119904) + (119876

119904minus 119877119896) cos (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895sin (120579119904) minus (119876

119904+ 119877119896) cos (120579

119904)

1205722119895+ (119876119904+ 119877)2

+ ((120572119895sin [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119877119896minus 119876119904) cos [(119876

119904minus 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895sin [(119876

119904+ 119877119896) 119905 + 120579

119904]

minus (119876119904+ 119877) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)6

=41198881199051119890119909ℎ0119871

(1205722119895+ 119877119896

2)3(119881

119871)

times [(1205722

119895+ 119877119896

2) 120572119895cos (119877

119896119905)

+ 119877119896sin (119877

119896119905) minus 120572

119895exp (minus120572

119895119905)

+ 2 exp (minus120572119895119905) 1205722

119895+ exp (120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

minus119877119896

119868(119895119899V+2+119899119887+119897)7

=41198881199052119890119909ℎ0

119881(1205722119895+ 119877119896

times [ minus 119877119896sin (119878119896minus 119877119896119905)

times 119897V (1205722

119895+ 119877119896

2) + 119871 (120572

119895+ 119877119896

minus2119881 [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

+ cos (119878119896minus 119877119896119905)

times 119877119896

2[120572119895(119897V + 119871) minus 2119881 (120572

119895119905 + 1)]

+1205722

119895[120572119895(119897V + 119871) + 2119881 (1 minus 120572

119895119905)]

+ exp (minus120572119895119905) 119877119896sin (119878119896)

times [119897V (1205722

119895+ 119877119896

+119871 (1205722

119895+ 119877119896

2)+4120572119895119881]

+ cos (119878119896)

times [119877119896

22119881 minus 120572

119895(119897V + 119871)

minus 1205722

119895120572119895(119897V + 119871)

+2119881]]

119868(119895119899V+2+119899119887+119897)8

21198881199052119890119909

119873

119904=1

[119860119904119876119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119876119904minus 119877119896) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) cos (119861

119904119897V + 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin [(119877 minus 119876

119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)]

+ (119876119904minus 119877119896) cos [(119877

119896minus 119876119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)])

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

minus ((120572119895sin [minus (119877 + 119876

119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

+ (119877119896+ 119876119904) cos [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)9

= minus119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [exp (120572119895119905) 119896120587119881 sin(119896120587119881119905

119871)

minus119871120572119895+ exp 119871120572

119895(120572119895119905) cos(119896120587119881119905

119871)]

minus 119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881sin(119896120587119863119871

119871)

minus 119890119909119871120572119895cos(119896120587119863

119871)

minus exp (120572119895119905)

times cos(119896120587119863 minus 119881119905

119871)]

119868(119895119899V+2+119899119887+119897)10

= minus21198711198812(1198981199081

+ 1198981199082) 119890119909

119873

119904=1

1198601199041198612

119904[ (((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881

times cos [(119876119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus exp (minus120572119895119905)

times ((minus (119896120587 +119861119904119871)119881 cos 120579

119904+119871120572119895sin 120579119904)

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

+ (((119896120587 minus 119861119904119871)119881 cos 120579

119904minus 119871120572119895sin 120579119904)

times((119896120587 minus 119861119904119871)2

1198812+ 11987121205722

119895)minus1

minus8ℎ01198812119890119909(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

times [119896120587119881exp (minus120572119895119905) minus cos(119896120587119881119905

119871)

+119871120572119895sin(119896120587119881119905

119871)]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] L FrybaVibration of Solids and Structures underMoving LoadsNoordhoff International Groningen The Netherlands 1968

[2] Y B Yang J D Yau and Y S Wu Vehicle Bridge InteractionDynamics with Application to High Speed Railways WorldScientific 2004

[3] J M Biggs Introduction to Structural Dynamics McGraw-HillNewYork NY USA 1964

[4] L FrybaDynamics of Railway BridgesThomas Telford PragueCzech Republic 1996

[5] R K Wen ldquoDynamic response of beams traversed by two axleloadsrdquo Journal of EngineeringMechanics Division ASCE vol 86pp 91ndash111 1960

[6] D Yadav and H C Upadhyay ldquoHeave-pitch-roll dynamics ofa vehicle with a variable velocity over a non-homogeneouslyprofiled flexible trackrdquo Journal of Sound and Vibration vol 164no 2 pp 337ndash348 1993

[7] T L Wang M Shahawy and D Z Huang ldquoDynamic responseof highway trucks due to road surface roughnessrdquo Journals ofComputer and Structures vol 49 no 6 pp 1055ndash1067 1993

[8] J-W Kou and J T deWolf ldquoVibrational behavior of continuousspan highway bridge-influencing variablesrdquo Journal of Struc-tural Engineering ASCE vol 123 no 3 pp 333ndash344 1997

[9] Y K Cheung F T K Au D Y Zheng and Y S ChengldquoVibration of multi-span non-uniform bridges under movingvehicles and trains by usingmodified beamvibration functionsrdquoJournal of Sound andVibration vol 228 no 3 pp 611ndash628 1999

[10] S Marchesiello A Fasana L Garibaldi and B A D PiomboldquoDynamics of multi-span continuous straight bridges subject tomulti-degrees of freedommoving vehicle excitationrdquo Journal ofSound and Vibration vol 224 no 3 pp 541ndash561 1999

[11] X Yin Z Fang and C S Cai ldquoLateral vibration of high-pier bridges under moving vehicular loadsrdquo Journal of BridgeEngineering vol 16 no 3 pp 400ndash412 2011

[12] S R Chen and C S Cai ldquoAccident assessment of vehicles onlong-span bridges in windy environmentsrdquo Journal of Wind

Engineering and Industrial Aerodynamics vol 92 no 12 pp991ndash1024 2004

[13] S S Law and X Q Zhu ldquoBridge dynamic responses due to roadsurface roughness and braking of vehiclerdquo Journal of Sound andVibration vol 282 no 3ndash5 pp 805ndash830 2005

[14] Y Zhang C S Cai X Shi and C Wang ldquoVehicle-induceddynamic performance of FRP versus concrete slab bridgerdquoJournal of Bridge Engineering vol 11 no 4 pp 410ndash419 2006

[15] M F Green and D Cebon ldquoDynamic interaction betweenheavy vehicles and highway bridgesrdquoComputers and Structuresvol 62 no 2 pp 253ndash264 1997

[17] K H Chu V K Garg and T L Wang ldquoImpact in railway pre-stressed concrete bridgesrdquo Journals of Structural EngineeringASCE vol 112 no 5 pp 1036ndash1051 1986

[18] D Huang T-L Wang and M Shahawy ldquoImpact analysisof continuous multigirder bridges due to moving vehiclesrdquoJournals of Structural Engineering ASCE vol 118 no 12 pp3427ndash3443 1992

[19] T-L Wang ldquoDynamic response of multigirder bridgesrdquo Jour-nals of Structural Engineering ASCE vol 118 no 8 pp 2222ndash2238 1992

[20] M Ichikawa Y Miyakawa and A Matsuda ldquoVibration analysisof the continuous beam subjected to a moving massrdquo Journal ofSound and Vibration vol 230 no 3 pp 493ndash506 2000

[21] N H Galdos D R Schelling and M A Sahin ldquoMethodologyfor impact factor of horizontally curved box bridgesrdquo Journalsof Structural Engineering ASCE vol 119 no 6 pp 1917ndash19341993

[22] Y B Yang C M Wu and J D Yau ldquoDynamic response of ahorizontally curved beam subjected to vertical and horizontalmoving loadsrdquo Journal of Sound and Vibration vol 242 no 3pp 519ndash537 2001

[23] H Kishan and R W Traill-Nash ldquoA model method for calcu-lation of highway bridge response with vehicle brakingrdquo CivilEngineering Transactions Institute Engineers Australia vol 19no 1 pp 44ndash50 1977

[24] K G Radley and R W Traill-Nash ldquoDynamic loading ofhighway bridgesrdquo Journal of Engineering Mechanics ASCE vol106 no 4 pp 641ndash658 1980

[26] P Lou ldquoA vehicle-track-bridge interaction element consideringvehiclersquos pitching effectrdquo Finite Elements in Analysis and Designvol 41 no 4 pp 397ndash427 2005

[27] S P Brady E J OrsquoBrien and A Znidaric ldquoEffect of vehiclevelocity on the dynamic amplification of a vehicle crossing asimply supported bridgerdquo Journal of Bridge Engineering ASCEvol 11 no 2 pp 241ndash249 2006

[28] J-D Yau and Y-B Yang ldquoA wideband MTMD system forreducing the dynamic response of continuous truss bridges tomoving train loadsrdquo Engineering Structures vol 26 no 12 pp1795ndash1807 2004

[29] N K Harris E J Obrien and A Gonzalez ldquoReduction ofbridge dynamic amplification through adjustment of vehiclesuspension dampingrdquo Journal of Sound and Vibration vol 302no 3 pp 471ndash485 2007

[30] W V Wedig ldquoDigital simulation of roadvehicle systemsrdquoProbabilistic Engineering Mechanics vol 27 no 1 pp 82ndash872012

[31] S Q Wu and S S Law ldquoEvaluating the response statistics of anuncertain bridgevehicle systemrdquoMechanical Systems and SignalProcessing vol 27 no 1 pp 576ndash589 2012

[32] I Kozar and N TMalic ldquoSpectral method in realistic modelingof bridges under moving vehiclesrdquo Engineering Structures vol50 pp 149ndash157 2013

[33] HANasrellah andC SManohar ldquoAparticle filtering approachfor structural system identification in vehicle-structure interac-tion problemsrdquo Journal of Sound and Vibration vol 329 no 9pp 1289ndash1309 2010

[34] M Shinozuka ldquoSimulation of multivariate and multidimen-sional random processrdquo Journal of Accoustical Society of Amer-ica vol 49 pp 357ndash367 1971

[35] D Huang and T-L Wang ldquoImpact analysis of cable-stayedbridgesrdquo Computers and Structures vol 43 no 5 pp 897ndash9081992

[36] D H Hodges and G A Pierce Introduction to StructuralDynamics and Aero-Elasticity Cambridge Aerospace Series2002

[37] D J Inman Engineering Vibration Prentice Hall 2001[38] N C Nigam Introduction to Random Vibrations The MIT

Press Cambridge Mass USA 1983[39] M C Potter and J Goldberg Mathematical Methods Prentice

Hall New Delhi India 1991[40] S S RaoMechanical Vibrations Pearson 4th edition 2004[41] L Deng and C S Cai ldquoIdentification of parameters of vehicles

moving on bridgesrdquo Journal of Engineering Structures vol 31no 10 pp 2474ndash2485 2009

[42] ISO 86061995 Mechanical vibration-Road surface profiles-reporting measured data

[43] E S Hwang and S A Nowak ldquoSimulation of dynamic load forbridgesrdquo Journal of Structural Engineering ASCE vol 117 pp1413ndash1434 1991

International Journal of

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Control Scienceand Engineering

Journal of

RotatingMachinery

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Shock and Vibration

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

Navigation and Observation

DistributedSensor Networks

special attention was given to the modeling of tyre with athree-dimensional linear spring and a rectangular contactsurface Other notable works in bridge-vehicle interactiondynamics that include various dynamic properties of rigidvehicle have been reported by Chen and Cai [12] Law andZhu [13] Zhang et al [14] and Green and Cebon [15]Marchesiello et al [16] studied the interaction of multispancontinuous bridges modeled by isotropic plates with multi-degree of freedom vehicles moving at constant speed The-oretical modes of plate vibration have been obtained withthe help of Rayleigh-Ritz approximate method Prestressedconcrete single track railway bridges consisting of severalbox girders were studied by Chu et al [17] in which theyconsidered each girder to share equal loads and modelled asa beam Rail irregularities are modeled using power spectraldensity function Variousworks on bridge-vehicle interactiononmultispan bridges have been reported by Huang et al [18]Wang [19] and Ichikawa et al [20] Numerical techniques forsolving the finite element model with rigid vehicle have beenadopted Other important works which include complexity inbridge model are that of horizontally curved girder [21 22]Analytical methods were employed to solve bridge dynamicproblem subject to moving mass taking effect of centrifugalforce Effect of vehicle braking has been included in studyof vehicle-bridge interaction problem by Kishan and Traill-Nash [23] and Radley and Traill-Nash [24]

Interests in research on bridge-vehicle interactiondynamics are still growing because of several complexities inmodeling and uncertainties in excitation In the last decaderesearchers have undertaken more and more complexproblems and attempted to solve by newer methodologytaking into consideration support flexibility and nonuniformcross-section [25] A vehicle-track-bridge interactionelement considering vehiclersquos pitching effect has beendeveloped by Lou [26] Experimental results and theircomparison with finite element model analysis of vehicle-bridge interaction problem have been presented by Bradyet al [27] Rigid model of vehicle has been considered inthe study to determine set of critical velocity associated withpeaks of dynamic amplification factor Multiple resonanceresponse of railway bridge has been investigated by Yau andYang [28] A vehicle-bridge interaction process has beensimulatedwithMATLABSimulink byHarris et al [29] to findout bridge-friendly damping control strategy with a tractorsemitrailer vehicle model Very recently some interestingstudies of bridge-vehicle or road-vehicle coupled dynamicsinclude stochastic numerics and discrete integration schemesfor digital simulation of road-vehicle system by Wedig [30]application of spectral stochastic finite element by Wu andLaw [31] and use of spectral matrix operator for directsolution of stochastic coupled differential equations by Kozarand Malic [32] Those studies have significantly improvedthe understanding of complex problem in vehicle-bridgeinteractions However vehicle model has been assumed aslumped masses with rigid link connected by suspensionelements exhibiting various discrete degrees of freedomIn most of the studies numerical integration and MonteCarlo simulation techniques have been used In the moderndays characteristics of vehicles have greatly changed due

to incorporation of larger pay load and for increasingdemand of traffic In the past vehicles have been modeledby a rigid 2D or 3D system having degrees of freedom inbounce pitch and roll However due to long and slendervehicle plying frequently over the bridges there is a needto consider flexibility in the vehicle model and to examinethe effect of flexible modes of vehicle on the dynamics ofbridge This aspect of bridge-vehicle dynamic interactionis not adequately addressed in the literature With this inmind present study has been conducted to find out theresponse statistics of single span bridge due to movement offlexible vehicle along an eccentric path By the term ldquoflexiblevehiclerdquo it is understood here that flexural modes havebeen included in addition to rigid body modes The bridgehas been considered to be under independent transversebending as well as under torsional excitation arising outof the eccentric path of the vehicle Nonhomogeneousprofile of deck surface has been incorporated in the studyby considering a deterministic mean surface superimposedby zero mean random process Such formulation can takecare of defects in surface finishing construction jointspotholes bump approach slab settlement and so forthBridge-vehicle system equations are expressed in state-space form and decoupled at each time step using complexeigenvalue analysis Closed form expression has been derivedfor state vectors for each sample input of bridge profile toform the ensemble of response Finally averaging acrossthe ensemble has been carried out to determine mean andstandard deviation of the response quantities In presentapproach numerical integration can be avoided to generateresponse samples which can save considerable amountof computational time The results obtained from presentapproach have been validated by numerical simulation andavailable experimental results from the literature The effectof vehicle velocity on the response and combined effect ofseveral vehicle-bridge parameters on dynamic amplificationfactor (DAF) have been examined

2 Mathematical Model

The bridge-vehicle model has been shown in Figure 1 Thebridge has been modeled as a uniform beam with simplysupported end conditions The mass stiffness and dampingare assumed to be uniform along the span of bridge Dueto eccentricity of the vehicle path the bridge is subjected toflexure as well as torsionThe bridge deck is unevenwhich hasbeen realized as nonhomogeneous process in spatial domainThis is represented by a function ℎ(119909)

21 Equation of Motion of Vehicle Vehicle body has beenidealized as Euler-Bernoulli beam of length 119897V The behaviorof suspension systems consisting of spring and dashpot isassumed as linear The governing differential equation ofmotion of the vehicle deflection can be expressed as

119864V119868V1205974119911 (119906 119905)

1205971199064+ 119862V

120597119911 (119906 119905)

120597119905+ 119898V

1205972119911 (119906 119905)

1205971199052= 119891V (119906 119905) (1)

c1c2 z(u t)

mw1mw2

ct1ct2y(x t)

kt1kt2

z1(t)z2(t)

119891V (119906 119905)

= [119896V1 119911 (119906 119905) minus 1199111 (119905) + 119888V1 (119906 119905) minus 1 (119905)]

times 120575 (119906 minus 1199061)

+ [119896V2 119911 (119906 119905) minus 1199112 (119905) + 119888V2 (119906 119905) minus 2 (119905)]

times 120575 (119906 minus 1199062)

where 1199061and 119906

minusinfin

119891 (119909) 120575 (119909 minus 1199091) 119889119909 = 119891 (119909

1) (3)

1198981(119905)1+ 11989611990511199111(119905) minus 119910 (119909

1 119905) minus ℎ (119909

+ 119896V1 1199111 (119905) minus 119911 (1199061 119905) + 119888V1 1 (119905) minus (1199061 119905)

+ 1198881199051[1(119905) minus

119863

119863119905119910 (1199091 119905) + ℎ (119909

1)] = 0

11989822(119905) + 119896

11990521199112(119905) minus 119910 (119909

2 119905) minus ℎ (119909

+ 119896V2 1199112 (119905) minus 119911 (1199062 119905) + 119888V2 2 (119905) minus (1199062 119905)

+ 1198881199052[2(119905) minus

119863

119863119905119910 (1199092 119905) + ℎ (119909

2)] = 0

where1198981and119898

tively ℎ(1199091) and ℎ(119909

2are the

119864119887119868119887

1205974119910 (119909 119905)

1205971199094+ 119862119887

120597119910 (119909 119905)

120597119905+ 119898119887

1205972119910 (119909 119905)

1205971199052= 119891119887(119909 119905) (6)

in which119898119887 119864119887119868119887 and 119862

119891119887(119909 119905)

+11988811990511(119905) minus

119863

119863119905[119910 (119909 119905) + ℎ (119909)]]

times 120575 (119909 minus 1199091)

+11988811990522(119905) minus

119863

119863119905[119910 (119909 119905) + ℎ (119909)]] 120575 (119909 minus 119909

minus 1198981+1

2119898V119897V119892120575 (119909 minus 1199091) minus 1198982 +

2119898V119897V119892

times 120575 (119909 minus 1199092)

+ 1198981

1198632

1198631199052[119910 (119909 119905) + ℎ (119909)] 120575 (119909 minus 119909

+ 1198982

1198632

1198631199052[119910 (119909 119905) + ℎ (119909)] 120575 (119909 minus 119909

119866119887119869119887

1205972120574 (119909 119905)

1205971199092minus 119862119887119879

120597120574 (119909 119905)

120597119905minus 119868119887

1205972120574 (119909 119905)

1205971199052= 119891119879(119909 119905)

119887is

119891119879(119909 119905)

+11988811990511(119905) minus

119863

119863119905[119910 (119909 119905) + ℎ (119909)]]

times 119890119909120575 (119909 minus 119909

+11988811990522(119905) minus

119863

119863119905[119910 (119909 119905) + ℎ (119909)]]

times 119890119909120575 (119909 minus 119909

minus 1198981+1

2119898V119897V119892119890119909120575 (119909 minus 1199091)

minus 1198982+1

2119898V119897V119892119890119909120575 (119909 minus 1199092)

+ 1198981

1198632

1198631199052[119910 (119909 119905) + ℎ (119909)] 119890

119909120575 (119909 minus 119909

+ 1198982

1198632

1198631199052[119910 (119909 119905) + ℎ (119909)] 119890

119909120575(119909 minus 119909

2)119909120575 (119909 minus 119909

ℎ (119909) = ℎ119898(119909) +

119873

119904=1

120589119904cos (2120587Ω

119904119909 + 120579119904) (10)

119871

119904andΩ

120589119904= radic2119878 (Ω

119904) ΔΩ

Ω119904= Ω119871+ (119904 minus

2)ΔΩ

ΔΩ =(Ω119880minus Ω119871)

119873

119878 (Ω) = 119878 (Ω0) times

Ω119904

2+ Ω2119871

119911 (119906 119905) =

119899V

119895 =minus1

120601V119895 (119906) 120578119895 (119905) (13)

120601minus1= 1

1206010= 119906 minus 119863

120601V119895 = sin (120572119895119906) + sinh (120572

119895119906)

+ 120573119895[cos (120572

119895119906) + cosh (120572

119895119906)]

120573119895=cos (120572

119895119897V) + cosh (120572

119895119897V)

sin (120572119895119897V) + sinh (120572

119895119897V)

120596V119895 = 1205722

119895radic

119864V119868V

119898V1198974

V (16)

120578119895(119905) + 2120585V119895120596V119895 120578119895 (119905) + 120596

V119895120578119895 (119905) = 119876V119895 (119905) (17)

vehicle is given as

119876V119895 (119905) =1

119872V119895int

119897119881

119891V (119906 119905) 120601119895 (119906) 119889119906 (18)

119872V119895 = int

119897V

119898V1206012

V119895 (119906) 119889119906 (19)

119876V119895 (119905)

119872V119895

119896V1

1199111(119905) minus

119899V

119895 =minus1

120601119895(1199061) 120578 (119905)

120601119895(1199061)

+ 119888V1

1(119905) minus

119899V

119895 =minus1

120601119895(1199061) 120578 (119905)

120601119895(1199061)

+ 119896V2

1199112(119905) minus

119899V

119895 =minus1

120601119895(1199062) 120578 (119905)

120601119895(1199062)

+119888V2

2(119905) minus

119899V

119895 =minus1

120601119895(1199062) 120578 (119905)

120601119895(1199062)]

119910 (119909 119905) =

119899119887

119896=1

120601119887119896(119909) 119902119896(119905) (21)

119887119896and 119902

119896(119905) are

119902119896(119905) + 2120585

119887119896120596119887119896

119902119896(119905) + 120596

119887119896119902119896(119905) = 119876

119887119896(119905) (22)

where 119896 = 1 2 3 119899119887

flexure is given as

119876119887119896(119905) =

119872119887119896

119871

119891119887(119909 119905) 120601

119887119896(119909) 119889119909 (23)

119872119887119896= int

119871

1198981198871206012

119887119896(119909) 119889119909 (24)

119876119887119896(119905)

= minus1

119872119887119896

[11989611990511199111(119905) minus

119899119887

119896=1

120601119887119896(1199091) 119902119896(119905) minus ℎ (119909

times 120601119887119896(1199091)

+ 11989611990521199112(119905) minus

119899119887

119896=1

120601119887119896(1199092) 119902119896(119905) minus ℎ (119909

times 120601119887119896(1199092)

+ 11988811990511(119905) minus 119881

119899119887

119896=1

1206011015840

119887119896(1199091) 119902119896(119905)

minus119881

119899119887

119896=1

120601119887119896(1199091) 119902119896(119905) minus 119881ℎ

1015840(1199091)

times 120601119887119896(1199091)

+ 11988811990522(119905) minus 119881

119899119887

119896=1

1206011015840

119887119896(1199092) 119902119896(119905)

minus119881

119899119887

119896=1

120601119887119896(1199092) 119902119896(119905) minus 119881ℎ

1015840(1199092)

times 120601119887119896(1199092)

minus 1198981+1

2119898V119897V119892120601119887119896 (1199091)

minus 1198982+1

2119898V119897V119892120601119887119896 (1199092)

times 1198981

119899119887

119896=1

120601119887119896(1199091) 119902119896(119905)

+ 2119881

119899119887

119896=1

1206011015840

119887119896(1199091) 119902119896(119905)

+1198812

119899119887

119896=1

12060110158401015840

119887119896(1199091) 119902119896(119905) + 4119881ℎ

10158401015840(1199091)

times 120601119887119896(1199091)

+ 1198982

119899119887

119896=1

120601119887119896(1199092) 119902119896(119905)

+ 2119881

119899119887

119896=1

1206011015840

119887119896(1199092) 119902119896(119905)

+ 1198812

119899119887

119896=1

12060110158401015840

119887119896(1199092) 119902119896(119905)

+4119881ℎ10158401015840(1199092)

times 120601119887119896(1199092) ]

120574119897(119905) + 2120585

119879119897120596119879119897

120574119897(119905) + 120596

119879119897120574119897(119905) = 119876

119879119897(119905)

(119897 = 1 2 3 119899119879)

119876119879119897(119905) =

119872119879119897

119871

119891119879(119909 119905) 120601

119879119897(119909) 119889119909 (27)

120601119879119897

119879119897in the 119897th mode is

given by

119872119879119897= int

119871

1198681198871206012

119879119897(119909) 119889119909 (28)

119876119879119897(119905)

= minus119890119909

119872119879119897

[11989611990511199111(119905) minus

119899119879

119897=1

120601119879119897(1199091) 120574119897(119905) minus ℎ (119909

times 120601119879119897(1199091)

+ 11989611990521199112(119905) minus

119899119879

119897=1

120601119879119897(1199092) 120574119897(119905) minus ℎ (119909

times 120601119879119897(1199092)

+ 11988811990511(119905) minus 119881

119899119879

119897=1

1206011015840

119879119897(1199091) 120574119897(119905)

minus119881

119899119879

119897=1

120601119879119897(1199091) 120574119897(119905) minus 119881ℎ

1015840(1199091)

times 120601119879119897(1199091)

+ 11988811990522(119905) minus 119881

119899119879

119897=1

1206011015840

119879119897(1199092) 120574119897(119905)

minus119881

119899119879

119897=1

120601119879119897(1199092) 120574119897(119905) minus 119881ℎ

1015840(1199092)

times 120601119879119897(1199092)

minus 1198981+1

2119898V119897V119892120601119879119897 (1199091) minus 1198982 +

2119898V119897V

times 119892120601119879119897(1199092)

times 1198981

119899119879

119897=1

120601119879119897(1199091) 120574119897(119905) + 2119881

119899119879

119897=1

1206011015840

119879119897(1199091) 120574119897(119905)

+1198812

119899119879

119897=1

12060110158401015840

119879119897(1199091) 120574119897(119905) + 4119881ℎ

10158401015840(1199091)

times 120601119879119897(1199091)

+ 1198982

119899119879

119897=1

120601119879119897(1199092) 120574119897(119905) + 2119881

119899119879

119897=1

1206011015840

119879119896(1199092) 120574119897(119905)

+1198812

119899119879

119896=1

12060110158401015840

119879119896(1199092) 120574119897(119905) + 4119881ℎ

10158401015840(1199092)

times 120601119879119897(1199092) ]

119887+

[119872] 119903 (119905) + [119862] 119903 (119905) + [119870] 119903 (119905) = 119865 (119905) (30)

where 119903(119905) = 1205781(119905) 1205782(119905) 120578

119899V(119905) 1199111(119905) 1199112(119905) 1199021(119905)1199022(119905) 119902

119899119887(119905) 1205741(119905) 1205742(119905) 120574

(119905) + [119860] 119901 (119905) = 119875 (119905) (31)

119901 (119905) = 119903 (119905)

119903 (119905) 119875 (119905) =

[119872]minus1119865 (119905)

[119860] = [119872]minus1[119862]

minus [119868]

[119872]minus1[119870]

11199062 119906

2119899] (33)

[119880]minus1[119860] [119880] = diagonal [120572

1 1205722 1205723 120572

2119899] (34)

Let us assume119901 = [119880] V (35)

V119895(119905) + 120572

119895V119895(119905) = 119877

119895(119905) 119895 = 1 2 3 2119899 (36)

119877119895=

119899

119896=1

1199061015840

119895119904

119899

119904=1

1198981015840

119904119896119865119896(119905) (37)

1199061015840

119904119896

may be expressed as

V119895(119905) = 119883

0119895exp (minus120572

119895119905) + int

minusinfin

119867119895(120596 119905) 119889119878 [119865

119896(120596)] (38)

where 1198830119895

119867119895(120596 119905) =

minus119894120596 + 120572119895

119903119898(119905) =

2119899

119895=1

119906119898+119899119895

1198830119895exp (minus120572

119895119905)

2119899

119895=1

119906119898+119899119895

119899

119904=1

1199061015840

119895119904

119899

119896=1

1198981015840

119904119896int

minusinfin

119867119895(120596 119905) 119889119878 (119865

119896(120596))

119898 = 1 2 3 119899

119903119875119898

(119905) =

2119899

119895=1

119906119898+119899119895

119899

119904=1

1199061015840

119895119904

119899

119896=1

1198981015840

119904119896int

minusinfin

exp (minus119894120596119905)minus119894120596 + 120572

119895

119865119896(120596) 119889120596

119903119875119898

(119905)

2119899

119895=1

119906119898+119899119895

119899

119904=1

1199061015840

119895119904

119899

119896=1

1198981015840

119904119896

times int

minusinfin

119865119896(120591) [

2120587int

minusinfin

exp [119894120596 (120591 minus 119905)]minus119894120596 + 120572

119895

119889120596]119889120591

119903119875119898

(119905) =

2119899

119895=1

119906119898+119899119895

119899

119904=1

1199061015840

119895119904

119899

119896=1

1198981015840

119904119896119868119895119896 (43)

119868119895119896= int

119905

exp [120572119895(120591 minus 119905)] 119865

119896(120591) 119889120591 (44)

119868119895119896= 119868119895119908+ 119868119895119909+ 119868119895119910+ 119868119895119911 (45)

where 119908 = 1 2 3 119899V 119909 = 119899V + 1 119899V + 2 119910 = 119899V + 3119899V+4 119899V+2+119899119887 119911 = 119899V+119899119887+3 119899V+119899119887+4 119899V+119899119887+119899119879+2

119884(119905119896) and

0 05 1 15 2 25

Time (s)

50E minus 4

00E + 0

minus50E minus 4

minus10E minus 3

minus15E minus 3

minus20E minus 3

119887119868119887) 248 times 10

119887119869119887) 248 times 10

5 kgm2m 119864119887= 32GPa deck = 2512GPa

1199051 1198961199052) 197 times 10

6Nm 474 times 106Nm

119887119868119887) 37times

119887119869119887) 1695 times 10

10N-m2

063 063 063

300 300

265100

(1198961199051 1198961199052) 09 times 10

119904= 16 times

119889119911)

075 120 075

0 05 1 15 2 25 3 35 4

Time (s)10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

120583dy

50kmh60kmh80kmh

0 05 1 15 2 25 3 35

Time (s)

minus80E minus 3

40E minus 3

minus10E minus 18

minus40E minus 3

minus80E minus 3

minus12E minus 2

120583y

50kmh60kmh80kmh

0 05 1 15 2 25 3 35

Time (s)40E minus 2

00E + 0

minus40E minus 2

minus80E minus 2

minus12E minus 1

minus16E minus 1

120583ay

50kmh60kmh80kmh

0 05 1 15 2 25 3 35 4Time (s)

40E minus 5

30E minus 5

20E minus 5

10E minus 5

00E + 0

minus10E minus 5

120590dy

50kmh60kmh80kmh

119889119911) velocity (120583V119911)

119886119911) is shown in

0 05 1 15 2 25 3 35Time (s)

16E minus 3

12E minus 3

80E minus 4

40E minus 4

00E + 0

minus40E minus 4

120590y

50kmh60kmh80kmh

0 05 1 15 2 25 3 35Time (s)

16E minus 2

12E minus 2

80E minus 3

40E minus 3

00E + 0

120590ay

50kmh60kmh80kmh

0 05 1 15 2 25 3Time (s)

80E minus 3

40E minus 3

00E + 0

minus40E minus 3

minus80E minus 3

minus12E minus 2

minus16E minus 2

120583dz

50kmh60kmh80kmh

0 05 1 15 2 25 3Time (s)

12E minus 1

80E minus 2

40E minus 2

00E + 0

minus40E minus 2

minus80E minus 2

120583z

50kmh60kmh80kmh

0 1 2 3Time (s)

80E minus 1

40E minus 1

00E + 0

minus40E minus 1

minus80E minus 1

50kmh60kmh80kmh

120583az

0 05 1 15 2 25 3Time (s)

10E minus 3

80E minus 4

60E minus 4

40E minus 4

20E minus 4

00E + 0

120590dz (m

50kmh60kmh80kmh

0 1 2 3Time (s)

80E minus 3

60E minus 3

40E minus 3

20E minus 3

00E + 0

120590z

50kmh60kmh80kmh

0 05 1 15 2 25 3Time (s)

40E minus 2

30E minus 2

20E minus 2

10E minus 2

00E + 0

120590az

50kmh60kmh80kmh

A B C D

50E minus 04

40E minus 04

30E minus 04

20E minus 04

10E minus 04

00E + 00

120583dy

A B C D

40E minus 05

30E minus 05

20E minus 05

10E minus 05

00E + 00

120590dy

0 025 05 075 1 125 15 175Time (s)

30E + 5

25E + 5

20E + 5

15E + 5

10E + 5

50E + 4

00E + 0

120583F

0 025 05 075 1 125 15 175Time (s)

25E + 4

20E + 4

15E + 4

10E + 4

50E + 3

00E + 0

minus50 + 3

120590F

0 05 1 15 2 25 3 35

Time (s)

10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

minus40E minus 4

120583dy

82 times 1010 N-m2

0 05 1 15 2 25 3 35Time (s)

20E minus 5

16E minus 5

12E minus 5

80E minus 6

40E minus 6

00E + 0

120590dy

6N-m2 (B) 119864V119868V = 64 times 107N-m2

and (C) 119864V119868V = 82 times 1010N-m2

0 05 1 15 2 25 3 35

Time (s)

10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

120583dy

119909) (A) 119890

119909= 05m (B) 119890

119909= 1m and

(C) 119890119909= 15m

0 05 1 15 2 25 3 35Time (s)

30E minus 5

25E minus 5

20E minus 5

15E minus 5

10E minus 5

50E minus 6

00E + 0

120590dy

119909) (A) 119890

119909= 05m (B) 119890

119909=

1m and (C) 119890119909= 15m

119884dynamic =1003816100381610038161003816120583119884 (119909119896 119905119896) + 120590119884 (119909119896 119905119896)

1003816100381610038161003816 (46)

0 0002 0004 0006 0008 001

GbJbEbIb

119887119869119887119864119887119868119887) for different

107 N-m2 and (C) 119864V119868V = 82 times 10

10 N-m2

119884static (47)

119887119868119887)

5 Conclusions

1520 25 30 35 40

45 5055

404550556065707580146148

15152154156158

16162164

times103

8013514

Vehicle speed (kmh)

14161820222426283013

Appendix

119865119895(119905) = 0 for119895 = 1 2 3 119899V

= 119896119905119895minus119899V

ℎ (119909119895minus119899V

) + 119888119905119895minus119899V

ℎ (119909119895minus119899V

For119895 = 119899V + 1 119899V + 2

= 1198961199051ℎ (1199091) 120601119887119895minus(119899V+2) (1199091) + 1198961199052ℎ (1199092) 120601119887119895minus(119899V+2) (1199092)

+ 1198881199051ℎ (1199091) 120601119887119895minus(119899V+2) (1199091) + 1198881199052ℎ (1199092) 120601119887119895minus(119899V+2) (1199092)

For 119895 = 119899V + 3 119899V + 4 119899V + 2 + 119899119887

= 1198901199091198961199051ℎ (1199091) 120601119905119895minus(119899V+2+119899119887)

(1199091)

+ 1198961199052ℎ (1199092) 120601119905119895minus(119899V+2+119899119887)

(1199092)

+ 1198881199051ℎ (1199091) 120601119905119895minus(119899V+2+119899119887)

(1199091)

+1198881199052ℎ1015840(1199092) 120601119905119895minus(119899V+2+119899119887)

(1199092)

For 119895 = 119899V + 119899119887 + 3

119899V + 119899119887 + 4 119899V + 119899119887 + 119899119905 + 2

ℎ119898(119909) =

1198712119909 (119871 minus 119909) (A2)

ℎ119903(119909) =

119873

119904=1

119860119904cos (2120587Ω

119904119909 + 120579119904) (A3)

119868119895119896= 119868119895119908+ 119868119895119909+ 119868119895119910+ 119868119895119911 (A4)

where 119908 = 1 2 3 119899V 119909 = 119899V + 1 119899V + 2 119910 = 119899V + 2 + 1119899V + 2 + 2 119899V + 2 + 119899

119887 119911 = 119899V + 2 + 119899

119887+ 1 119899V + 2 + 119899

119887+

2 119899V + 2 + 119899119887 + 119899119879Thefirst term 119868

1198681198951= 1198681198952= sdot sdot sdot = 119868

119895119899V= 0 (A5)

119868119895119909= 1198681198951199091

+ 1198681198951199092 (A6)

1198681198951199091

= 11986811989511990911

+ 11986811989511990912

119904)minus2 and 119861

119904= 2120587Ω

119871+ (119904 minus

11986811989511990911

= 1198961199051[

119873

119904=1

119860119904

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2minus120572119895cos (120579

119904) minus 119861119904119881 sin (120579

119904)

+ exp (120572119895119905)

times [120572119895cos (119861

119904119881119905 + 120579

119904)

+119861119904119881 cos (119861

119904119881119905 + 120579

119904)]

+4ℎ0119881

12057231198951198712

120572119895119871 (120572119895119905 + exp (minus120572

119895119905) minus 1)

minus119881

12057231198951198712

119895119905) + 2 ]

11986811989511990912

= 1198881199051[

119873

119904=1

119881119860119904119861119904

exp (minus120572119895119905)

119904)

+ exp (120572119895119905)

times [120572119895sin (119861

119904119881119905 + 120579

119904)

minus119861119904119881 sin (119861

119904119881119905 + 120579

119904)]

119895119905)

12057231198951198712

minus1

120572119895119871exp (120572

119895119905) minus 2]

1198681198951199092

= 11986811989511990921

+ 11986811989511990922

11986811989511990921

= 1198961199052[

119873

119904=1

119860119904

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2

times minus 120572119895cos (119861

119904119897V minus 120579119904) minus 119861119904119881 sin (119861

119904119897V minus 120579119904)

+ exp (120572119895119905) [120572119895cos (119861

119904(119897 minus 119881119905) minus 120579

119904)

+119861119904119881 sin (119861

119904(119897V minus 119881119905) + 120579119904)]

+1198812[2 minus exp (120572

119895119905) 120572119895119905 (120572119895119905 minus 2) + 2]

+4ℎ0

12057231198951198712

minus1205722

119895119897V (119897V + 119871) (exp (120572119895119905) minus 1)

+120572119895119881 (2119897V + 119871) [exp (120572119895119905) (120572119895119905 minus 1) + 1] ]

11986811989511990922

= 1198881199052[

119873

119904=1

119860119904119861119904119881

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2120572119895sin (119861

119904119897V minus 120579119904)

+ 119861119904119881 cos (119861

119904119897V minus 120579119904)

minus exp (120572119895119905) [120572119895sin (119861

119904(119897 minus 119881119905) minus 120579

119904)

+119861119904119881 cos (119861

119904(119897V minus 119881119905) + 120579119904)]

+4ℎ0119881 exp (minus120572

119895119905)

12057221198951198712

times 2119881 [exp (120572119895119905) (120572119895119905 minus 1) + 1]

minus120572119895(2119897V + 119871) (exp (120572119895119905) minus 1)]

119868119895119910=

119899119887

119896=1

119868119895119899V+2+119896

119886=1

119868(119895119899V+2+119896)119886

(119896 = 1 2 3 119899119887) (A11)

119876119904= 119861119904119881 119877

119896=119896120587119881

119871

119878119896=119896120587

119871 119885

119904= 119861119904119897V

119868(119895119899V+2+119896)

21198961199051

119873

119904=1

119860119904 ((2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) cos (120579

119904)

+2120572119895119876119904sin (120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times cos (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119876

119904+ 119877119896) 119905 + 120579

119904

minus (119877119896+ 119876119904)

times cos (119876119904+ 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

21198961199051[

4ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198762119904)3

times 119871 (1205722

119895+ 1198762

119904) [exp (120572

119895119905) sin (119876

119904119905)

times 1198762

119904(120572119895119905 + 1)

+1205722

119895(120572119895119905 minus 1)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904119905 + 120572119895(120572119895119905 minus 2)] minus 2120572

119895119876119904

+ 119881 (21198763

119904+ 119876119904exp (120572

119895119905)) cos (119876

119904119905)

times 1198764

1199041199052+ 21198762

119904(1205722

1198951199052minus 2120572119895119905 minus 1)

+1205722

119895(1205722

1198951199052minus 4120572119895119905 + 6)

minus exp (120572119895119905) sin (119876

119904119905)

times 1198764

119904(120572119895119905 + 2) + 120572

1198951198762

119904(1205722

1198951199052minus 3)

+1205723

119895(1205722

1198951199052minus 2120572119895119905 + 2) minus 6120572

119895119876119904]

119868(119895119899V+2+119896)

21198961199052

119873

119904=1

119860119904[ exp (minus120572

119895119905)

times (((119877119896+ 119876119904) cos (119878

119904minus 119885119904minus 120579119904)

+120572119895sin (119878119896minus 119885119904minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

times (((119877119896minus 119876119904) cos (119878

119896minus 119885119904+ 120579119904)

+120572119895sin (119878119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 119878

119896minus 119885119904+ 120579119904

minus (119877119896+ 119876119904) cos (119876

119904minus 119877119896) 119905

+119878119896minus 119885119904+ 120579119904)

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

+ ((120572119895sin 119878119896+ 119885119904minus (119876119904+ 119877) 119905 minus 120579

119904

+ (119877119896+ 119876119904) cos 119878

119896+ 119885119904

+ (119876119904+ 119877119896) 119905

minus120579119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1

21198961199052[

4ℎ0119881

1198712(1205722119895+ 1198812)

times sin (119881119905 minus 119897V) [1205721198951198972

V(1205722

119895+ 1198812)2

minus 119897V (1205722

119895+ 1198812)

times minus1205723

119895119871 minus 1205721198951198711198812

+ 21198813(120572119895119905 + 1)

+21205722

119895119881(120572119895119905 minus 1)

+ 119881 [119881 21205723

119895+ 120572119895119905(1205722

119895+ 1198812)2

+2119905 (1205724

119895minus 1198814) minus 6120572

1198951198812

minus 119871 (1205722

119895+ 1198812)

times 1205722

119895(120572119895119905 minus 1)+ 119881

2(120572119895119905 +1)]

+ 119881 cos (119897V minus 119881119905) [1198814(1198972

V + 119897V119871

+21205722

1198951199052minus 4120572119895119905 minus 2)

+ 1205722

1198951198812(21198972

V + 2119897V119871

+41205722

1198951199052+ 6)

+ 1205724

119895119897V (119897V + 119871)

minus 1198815119905 (2119897V + 119871)

times (120572119895119905 minus 1)

minus 1205723

119895119881 (2119897V + 119871)

times (120572119895119905 minus 2) + 119897

V1198816]]

119868(119895119899V+2+119896)

21198881199051

119873

119904=1

119860119904119876119904( minus (2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) sin (120579

119904)

minus2120572119895119876119904cos (120579

119904))

times (1205722

119895+ (119876119904minus 119877)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times sin (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos (119876

119904+ 119877119896) 119905 + 120579

119904

+ (119877119896+ 119876119904)

times sin (119876119904+ 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199051

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) cos (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) sin (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

21198881199052

119873

119904=1

119860119904119876119904

times [ exp (minus120572119895119905 ( ( (119877

119896+ 119876119904) sin (119878

119896+ 119885119904minus 120579119904)

minus120572119895cos (119878

119896+ 119885119904minus 120579119904))

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

times ( ( (119876119904minus 119877119896) sin (119878

119896minus 119885119904+ 120579119904)

+120572119895cos (119878

119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos 119878

119896+ 119885119904minus (119876119904+ 119877119896) 119905 minus 120579

119904

minus (119877119896+ 119876119904) sin 119878

119896+ 119885119904

minus (119876119904+ 119877119896) 119905 minus 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos 119878

119896minus 119885119904+ (119876119904minus 119877119896) 119905 + 120579

119904

+ (119876119904minus 119877119896) sin 119878

119896minus 119885119904

+ (119876119904minus 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199052

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) sin (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881 + exp (120572119895119905) 119896120587119881 cos(119896120587119881119905

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881cos(119896120587119863119871

119871)

minus 119871120572119895sin(119896120587119863

119871)

119871)]

119868(119895119899V+2+119896)

= minus21198711198812(1198981199081

+ 1198981199082)

119873

119904=1

1198601199041198612

119904[(((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881 cos [(119876

119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

119904119871)119881 cos 120579

119904

+119871120572119895sin 120579119904)

times (1198612

11990411987121198812

+ 21198961205871198611199041198711198812

+ 119896212058721198812

+11987121205722

119895)minus1

+ ((119896120587 minus 119861119904119871)119881 cos 120579

119904

minus119871120572119895sin 120579119904)

times ((119896120587 minus 119861119904119871)2

1198812

+11987121205722

119895)minus1

minus8ℎ01198812(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

times [119896120587119881

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119868119895119911=

119899119905

119896=1

119868119895119899V+2+119899119887+119896

119868119895119899V+2+119899119887+119897

119886=1

119868(119895119899V+2+119899119887+119897)119886

(119896 = 1 2 3 119899119879) (A14)

119868(119895119899V+2+119899119887+119897)1

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895cos (120579

119904) + (119876

119904minus 119877119896) sin (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895cos (120579

119904) + (119876

119904+ 119877119896) sin (120579

119904)

1205722119895+ (119876119904+ 119877119896)2

+ ((120572119895cos [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119876119904minus 119877119896) sin [(119876

119904minus 119877119896) 119905 minus 120579

119904])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [(119876

119904+ 119877119896) 119905 + 120579

119904]

+ (119876119904+ 119877119896) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)2

=41198961199051119890119909ℎ0119871

119881(1205722119895+ 119877119896

times [ (1205722

119895+ 119877119896

2) exp (minus120572

119895119905)

times 1205722

119895+ exp (minus120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)] minus 119877

119896

+ 2120572119895exp (minus120572

119895119905) (1205722

119895+ 3119877119896

+ 119877119896sin (119877

119896119905) 6120572

119895+ 1199052(1205722

119895+ 119877119896

+4120572119895119905 (1205722

119895+ 119877119896

2) minus 2119877

119896

+ cos (119877119896119905) 2119905 (119877

119896

4minus 1205724

119895)

+ 2120572119895(1205722

119895minus 3119877119896

times 1205721198951199052(1205722

119895+ 119877119896

119868(119895119899V+2+119899119887+119897)3

= minus1

21198961199052119890119909

119873

119904=1

[119860119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119877119896minus 119876119904) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) sin (119861

119904119897V + 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos [(119877

119896minus 119876119904) 119905 + (119861

119904119897V minus 119878119896 minus 120579119904)]

+ (119877119896minus 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [minus (119877

119896+ 119876119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

minus (119877119896+ 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)4

=41198961199052119890119909ℎ0

1198712[((119897V (119897V + 119871) exp (minus120572119895119905)

times 120572119895cos (119878

119896minus 119877119896119905)

minus119877119896sin (119878119896minus 119877119896119905)

+119877119896sin (119878) minus 120572

119895cos (119878

119896))

times (1205722

119895+ 119877119896

2)minus1

+2119871119881 exp (minus120572

119895119905)

(1205722119895+ 119877119896

times (1205722

119895+ 119877119896

2) cos (119878

119896)

+ exp (120572119895119905) cos (119878

119896minus 119877119896119905)

times [1198772(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896[119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

119895119877119896sin (119878119896)

minus1198812

(1205722119895+ 119877119896

times119877119896sin (119878119896minus 119877119896119905)[61205722

119895+1199052(1205722

119895+ 119877119896

minus 4120572119895119905 (1205722

119895+ 119877119896

minus2119877119896

+ 2 exp (minus120572119895119905)

times [(1205723

119895minus 3120572119895119877119896

2) cos (119878

119896)

+ 119877119896(119877119896

2minus 31205722

119895)

times sin (119878119896) ]

minus [2119905 (119877119896

4minus 1205724

119895) + 2120572

119895(1205722

119895minus 3119877119896

+1205721198951199052(1205722

119895+119877119896

]cos (119878119896minus 119877119896119905)]

119868(119895119899V+2+119899119887+119897)5

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895sin (120579119904) + (119876

119904minus 119877119896) cos (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895sin (120579119904) minus (119876

119904+ 119877119896) cos (120579

119904)

1205722119895+ (119876119904+ 119877)2

+ ((120572119895sin [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119877119896minus 119876119904) cos [(119876

119904minus 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895sin [(119876

119904+ 119877119896) 119905 + 120579

119904]

minus (119876119904+ 119877) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)6

=41198881199051119890119909ℎ0119871

(1205722119895+ 119877119896

2)3(119881

119871)

times [(1205722

119895+ 119877119896

2) 120572119895cos (119877

119896119905)

+ 119877119896sin (119877

119896119905) minus 120572

119895exp (minus120572

119895119905)

+ 2 exp (minus120572119895119905) 1205722

119895+ exp (120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

minus119877119896

119868(119895119899V+2+119899119887+119897)7

=41198881199052119890119909ℎ0

119881(1205722119895+ 119877119896

times 119897V (1205722

119895+ 119877119896

2) + 119871 (120572

119895+ 119877119896

minus2119881 [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

+ cos (119878119896minus 119877119896119905)

times 119877119896

2[120572119895(119897V + 119871) minus 2119881 (120572

119895119905 + 1)]

+1205722

119895[120572119895(119897V + 119871) + 2119881 (1 minus 120572

119895119905)]

+ exp (minus120572119895119905) 119877119896sin (119878119896)

times [119897V (1205722

119895+ 119877119896

+119871 (1205722

119895+ 119877119896

2)+4120572119895119881]

+ cos (119878119896)

times [119877119896

22119881 minus 120572

119895(119897V + 119871)

minus 1205722

119895120572119895(119897V + 119871)

+2119881]]

119868(119895119899V+2+119899119887+119897)8

21198881199052119890119909

119873

119904=1

[119860119904119876119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119876119904minus 119877119896) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) cos (119861

119904119897V + 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin [(119877 minus 119876

119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)]

+ (119876119904minus 119877119896) cos [(119877

119896minus 119876119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)])

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

minus ((120572119895sin [minus (119877 + 119876

119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

+ (119877119896+ 119876119904) cos [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)9

= minus119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [exp (120572119895119905) 119896120587119881 sin(119896120587119881119905

119871)

minus119871120572119895+ exp 119871120572

119895(120572119895119905) cos(119896120587119881119905

119871)]

minus 119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881sin(119896120587119863119871

119871)

minus 119890119909119871120572119895cos(119896120587119863

119871)

minus exp (120572119895119905)

times cos(119896120587119863 minus 119881119905

119871)]

119868(119895119899V+2+119899119887+119897)10

= minus21198711198812(1198981199081

+ 1198981199082) 119890119909

119873

119904=1

1198601199041198612

119904[ (((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881

times cos [(119876119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus exp (minus120572119895119905)

times ((minus (119896120587 +119861119904119871)119881 cos 120579

119904+119871120572119895sin 120579119904)

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

+ (((119896120587 minus 119861119904119871)119881 cos 120579

119904minus 119871120572119895sin 120579119904)

times((119896120587 minus 119861119904119871)2

1198812+ 11987121205722

119895)minus1

minus8ℎ01198812119890119909(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

119871)

+119871120572119895sin(119896120587119881119905

119871)]

References

RoboticsJournal of

Journal of

RotatingMachinery

VLSI Design

Shock and Vibration

Journal of

Volume 2014

SensorsJournal of

Propagation

c1c2 z(u t)

mw1mw2

ct1ct2y(x t)

kt1kt2

z1(t)z2(t)

119891V (119906 119905)

= [119896V1 119911 (119906 119905) minus 1199111 (119905) + 119888V1 (119906 119905) minus 1 (119905)]

times 120575 (119906 minus 1199061)

+ [119896V2 119911 (119906 119905) minus 1199112 (119905) + 119888V2 (119906 119905) minus 2 (119905)]

times 120575 (119906 minus 1199062)

where 1199061and 119906

minusinfin

119891 (119909) 120575 (119909 minus 1199091) 119889119909 = 119891 (119909

1) (3)

1198981(119905)1+ 11989611990511199111(119905) minus 119910 (119909

1 119905) minus ℎ (119909

+ 119896V1 1199111 (119905) minus 119911 (1199061 119905) + 119888V1 1 (119905) minus (1199061 119905)

+ 1198881199051[1(119905) minus

119863

119863119905119910 (1199091 119905) + ℎ (119909

1)] = 0

11989822(119905) + 119896

11990521199112(119905) minus 119910 (119909

2 119905) minus ℎ (119909

+ 119896V2 1199112 (119905) minus 119911 (1199062 119905) + 119888V2 2 (119905) minus (1199062 119905)

+ 1198881199052[2(119905) minus

119863

119863119905119910 (1199092 119905) + ℎ (119909

2)] = 0

where1198981and119898

tively ℎ(1199091) and ℎ(119909

2are the

119864119887119868119887

1205974119910 (119909 119905)

1205971199094+ 119862119887

120597119910 (119909 119905)

120597119905+ 119898119887

1205972119910 (119909 119905)

1205971199052= 119891119887(119909 119905) (6)

in which119898119887 119864119887119868119887 and 119862

119891119887(119909 119905)

+11988811990511(119905) minus

119863

119863119905[119910 (119909 119905) + ℎ (119909)]]

times 120575 (119909 minus 1199091)

+11988811990522(119905) minus

119863

119863119905[119910 (119909 119905) + ℎ (119909)]] 120575 (119909 minus 119909

minus 1198981+1

2119898V119897V119892120575 (119909 minus 1199091) minus 1198982 +

2119898V119897V119892

times 120575 (119909 minus 1199092)

+ 1198981

1198632

1198631199052[119910 (119909 119905) + ℎ (119909)] 120575 (119909 minus 119909

+ 1198982

1198632

1198631199052[119910 (119909 119905) + ℎ (119909)] 120575 (119909 minus 119909

119866119887119869119887

1205972120574 (119909 119905)

1205971199092minus 119862119887119879

120597120574 (119909 119905)

120597119905minus 119868119887

1205972120574 (119909 119905)

1205971199052= 119891119879(119909 119905)

119887is

119891119879(119909 119905)

+11988811990511(119905) minus

119863

119863119905[119910 (119909 119905) + ℎ (119909)]]

times 119890119909120575 (119909 minus 119909

+11988811990522(119905) minus

119863

119863119905[119910 (119909 119905) + ℎ (119909)]]

times 119890119909120575 (119909 minus 119909

minus 1198981+1

2119898V119897V119892119890119909120575 (119909 minus 1199091)

minus 1198982+1

2119898V119897V119892119890119909120575 (119909 minus 1199092)

+ 1198981

1198632

1198631199052[119910 (119909 119905) + ℎ (119909)] 119890

119909120575 (119909 minus 119909

+ 1198982

1198632

1198631199052[119910 (119909 119905) + ℎ (119909)] 119890

119909120575(119909 minus 119909

2)119909120575 (119909 minus 119909

ℎ (119909) = ℎ119898(119909) +

119873

119904=1

120589119904cos (2120587Ω

119904119909 + 120579119904) (10)

119871

119904andΩ

120589119904= radic2119878 (Ω

119904) ΔΩ

Ω119904= Ω119871+ (119904 minus

2)ΔΩ

ΔΩ =(Ω119880minus Ω119871)

119873

119878 (Ω) = 119878 (Ω0) times

Ω119904

2+ Ω2119871

119911 (119906 119905) =

119899V

119895 =minus1

120601V119895 (119906) 120578119895 (119905) (13)

120601minus1= 1

1206010= 119906 minus 119863

120601V119895 = sin (120572119895119906) + sinh (120572

119895119906)

+ 120573119895[cos (120572

119895119906) + cosh (120572

119895119906)]

120573119895=cos (120572

119895119897V) + cosh (120572

119895119897V)

sin (120572119895119897V) + sinh (120572

119895119897V)

120596V119895 = 1205722

119895radic

119864V119868V

119898V1198974

V (16)

120578119895(119905) + 2120585V119895120596V119895 120578119895 (119905) + 120596

V119895120578119895 (119905) = 119876V119895 (119905) (17)

vehicle is given as

119876V119895 (119905) =1

119872V119895int

119897119881

119891V (119906 119905) 120601119895 (119906) 119889119906 (18)

119872V119895 = int

119897V

119898V1206012

V119895 (119906) 119889119906 (19)

119876V119895 (119905)

119872V119895

119896V1

1199111(119905) minus

119899V

119895 =minus1

120601119895(1199061) 120578 (119905)

120601119895(1199061)

+ 119888V1

1(119905) minus

119899V

119895 =minus1

120601119895(1199061) 120578 (119905)

120601119895(1199061)

+ 119896V2

1199112(119905) minus

119899V

119895 =minus1

120601119895(1199062) 120578 (119905)

120601119895(1199062)

+119888V2

2(119905) minus

119899V

119895 =minus1

120601119895(1199062) 120578 (119905)

120601119895(1199062)]

119910 (119909 119905) =

119899119887

119896=1

120601119887119896(119909) 119902119896(119905) (21)

119887119896and 119902

119896(119905) are

119902119896(119905) + 2120585

119887119896120596119887119896

119902119896(119905) + 120596

119887119896119902119896(119905) = 119876

119887119896(119905) (22)

where 119896 = 1 2 3 119899119887

flexure is given as

119876119887119896(119905) =

119872119887119896

119871

119891119887(119909 119905) 120601

119887119896(119909) 119889119909 (23)

119872119887119896= int

119871

1198981198871206012

119887119896(119909) 119889119909 (24)

119876119887119896(119905)

= minus1

119872119887119896

[11989611990511199111(119905) minus

119899119887

119896=1

120601119887119896(1199091) 119902119896(119905) minus ℎ (119909

times 120601119887119896(1199091)

+ 11989611990521199112(119905) minus

119899119887

119896=1

120601119887119896(1199092) 119902119896(119905) minus ℎ (119909

times 120601119887119896(1199092)

+ 11988811990511(119905) minus 119881

119899119887

119896=1

1206011015840

119887119896(1199091) 119902119896(119905)

minus119881

119899119887

119896=1

120601119887119896(1199091) 119902119896(119905) minus 119881ℎ

1015840(1199091)

times 120601119887119896(1199091)

+ 11988811990522(119905) minus 119881

119899119887

119896=1

1206011015840

119887119896(1199092) 119902119896(119905)

minus119881

119899119887

119896=1

120601119887119896(1199092) 119902119896(119905) minus 119881ℎ

1015840(1199092)

times 120601119887119896(1199092)

minus 1198981+1

2119898V119897V119892120601119887119896 (1199091)

minus 1198982+1

2119898V119897V119892120601119887119896 (1199092)

times 1198981

119899119887

119896=1

120601119887119896(1199091) 119902119896(119905)

+ 2119881

119899119887

119896=1

1206011015840

119887119896(1199091) 119902119896(119905)

+1198812

119899119887

119896=1

12060110158401015840

119887119896(1199091) 119902119896(119905) + 4119881ℎ

10158401015840(1199091)

times 120601119887119896(1199091)

+ 1198982

119899119887

119896=1

120601119887119896(1199092) 119902119896(119905)

+ 2119881

119899119887

119896=1

1206011015840

119887119896(1199092) 119902119896(119905)

+ 1198812

119899119887

119896=1

12060110158401015840

119887119896(1199092) 119902119896(119905)

+4119881ℎ10158401015840(1199092)

times 120601119887119896(1199092) ]

120574119897(119905) + 2120585

119879119897120596119879119897

120574119897(119905) + 120596

119879119897120574119897(119905) = 119876

119879119897(119905)

(119897 = 1 2 3 119899119879)

119876119879119897(119905) =

119872119879119897

119871

119891119879(119909 119905) 120601

119879119897(119909) 119889119909 (27)

120601119879119897

119879119897in the 119897th mode is

given by

119872119879119897= int

119871

1198681198871206012

119879119897(119909) 119889119909 (28)

119876119879119897(119905)

= minus119890119909

119872119879119897

[11989611990511199111(119905) minus

119899119879

119897=1

120601119879119897(1199091) 120574119897(119905) minus ℎ (119909

times 120601119879119897(1199091)

+ 11989611990521199112(119905) minus

119899119879

119897=1

120601119879119897(1199092) 120574119897(119905) minus ℎ (119909

times 120601119879119897(1199092)

+ 11988811990511(119905) minus 119881

119899119879

119897=1

1206011015840

119879119897(1199091) 120574119897(119905)

minus119881

119899119879

119897=1

120601119879119897(1199091) 120574119897(119905) minus 119881ℎ

1015840(1199091)

times 120601119879119897(1199091)

+ 11988811990522(119905) minus 119881

119899119879

119897=1

1206011015840

119879119897(1199092) 120574119897(119905)

minus119881

119899119879

119897=1

120601119879119897(1199092) 120574119897(119905) minus 119881ℎ

1015840(1199092)

times 120601119879119897(1199092)

minus 1198981+1

2119898V119897V119892120601119879119897 (1199091) minus 1198982 +

2119898V119897V

times 119892120601119879119897(1199092)

times 1198981

119899119879

119897=1

120601119879119897(1199091) 120574119897(119905) + 2119881

119899119879

119897=1

1206011015840

119879119897(1199091) 120574119897(119905)

+1198812

119899119879

119897=1

12060110158401015840

119879119897(1199091) 120574119897(119905) + 4119881ℎ

10158401015840(1199091)

times 120601119879119897(1199091)

+ 1198982

119899119879

119897=1

120601119879119897(1199092) 120574119897(119905) + 2119881

119899119879

119897=1

1206011015840

119879119896(1199092) 120574119897(119905)

+1198812

119899119879

119896=1

12060110158401015840

119879119896(1199092) 120574119897(119905) + 4119881ℎ

10158401015840(1199092)

times 120601119879119897(1199092) ]

119887+

[119872] 119903 (119905) + [119862] 119903 (119905) + [119870] 119903 (119905) = 119865 (119905) (30)

where 119903(119905) = 1205781(119905) 1205782(119905) 120578

119899V(119905) 1199111(119905) 1199112(119905) 1199021(119905)1199022(119905) 119902

119899119887(119905) 1205741(119905) 1205742(119905) 120574

(119905) + [119860] 119901 (119905) = 119875 (119905) (31)

119901 (119905) = 119903 (119905)

119903 (119905) 119875 (119905) =

[119872]minus1119865 (119905)

[119860] = [119872]minus1[119862]

minus [119868]

[119872]minus1[119870]

11199062 119906

2119899] (33)

[119880]minus1[119860] [119880] = diagonal [120572

1 1205722 1205723 120572

2119899] (34)

Let us assume119901 = [119880] V (35)

V119895(119905) + 120572

119895V119895(119905) = 119877

119895(119905) 119895 = 1 2 3 2119899 (36)

119877119895=

119899

119896=1

1199061015840

119895119904

119899

119904=1

1198981015840

119904119896119865119896(119905) (37)

1199061015840

119904119896

may be expressed as

V119895(119905) = 119883

0119895exp (minus120572

119895119905) + int

minusinfin

119867119895(120596 119905) 119889119878 [119865

119896(120596)] (38)

where 1198830119895

119867119895(120596 119905) =

minus119894120596 + 120572119895

119903119898(119905) =

2119899

119895=1

119906119898+119899119895

1198830119895exp (minus120572

119895119905)

2119899

119895=1

119906119898+119899119895

119899

119904=1

1199061015840

119895119904

119899

119896=1

1198981015840

119904119896int

minusinfin

119867119895(120596 119905) 119889119878 (119865

119896(120596))

119898 = 1 2 3 119899

119903119875119898

(119905) =

2119899

119895=1

119906119898+119899119895

119899

119904=1

1199061015840

119895119904

119899

119896=1

1198981015840

119904119896int

minusinfin

exp (minus119894120596119905)minus119894120596 + 120572

119895

119865119896(120596) 119889120596

119903119875119898

(119905)

2119899

119895=1

119906119898+119899119895

119899

119904=1

1199061015840

119895119904

119899

119896=1

1198981015840

119904119896

times int

minusinfin

119865119896(120591) [

2120587int

minusinfin

exp [119894120596 (120591 minus 119905)]minus119894120596 + 120572

119895

119889120596]119889120591

119903119875119898

(119905) =

2119899

119895=1

119906119898+119899119895

119899

119904=1

1199061015840

119895119904

119899

119896=1

1198981015840

119904119896119868119895119896 (43)

119868119895119896= int

119905

exp [120572119895(120591 minus 119905)] 119865

119896(120591) 119889120591 (44)

119868119895119896= 119868119895119908+ 119868119895119909+ 119868119895119910+ 119868119895119911 (45)

where 119908 = 1 2 3 119899V 119909 = 119899V + 1 119899V + 2 119910 = 119899V + 3119899V+4 119899V+2+119899119887 119911 = 119899V+119899119887+3 119899V+119899119887+4 119899V+119899119887+119899119879+2

119884(119905119896) and

0 05 1 15 2 25

Time (s)

50E minus 4

00E + 0

minus50E minus 4

minus10E minus 3

minus15E minus 3

minus20E minus 3

119887119868119887) 248 times 10

119887119869119887) 248 times 10

5 kgm2m 119864119887= 32GPa deck = 2512GPa

1199051 1198961199052) 197 times 10

6Nm 474 times 106Nm

119887119868119887) 37times

119887119869119887) 1695 times 10

10N-m2

063 063 063

300 300

265100

(1198961199051 1198961199052) 09 times 10

119904= 16 times

119889119911)

075 120 075

0 05 1 15 2 25 3 35 4

Time (s)10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

120583dy

50kmh60kmh80kmh

0 05 1 15 2 25 3 35

Time (s)

minus80E minus 3

40E minus 3

minus10E minus 18

minus40E minus 3

minus80E minus 3

minus12E minus 2

120583y

50kmh60kmh80kmh

0 05 1 15 2 25 3 35

Time (s)40E minus 2

00E + 0

minus40E minus 2

minus80E minus 2

minus12E minus 1

minus16E minus 1

120583ay

50kmh60kmh80kmh

0 05 1 15 2 25 3 35 4Time (s)

40E minus 5

30E minus 5

20E minus 5

10E minus 5

00E + 0

minus10E minus 5

120590dy

50kmh60kmh80kmh

119889119911) velocity (120583V119911)

119886119911) is shown in

0 05 1 15 2 25 3 35Time (s)

16E minus 3

12E minus 3

80E minus 4

40E minus 4

00E + 0

minus40E minus 4

120590y

50kmh60kmh80kmh

0 05 1 15 2 25 3 35Time (s)

16E minus 2

12E minus 2

80E minus 3

40E minus 3

00E + 0

120590ay

50kmh60kmh80kmh

0 05 1 15 2 25 3Time (s)

80E minus 3

40E minus 3

00E + 0

minus40E minus 3

minus80E minus 3

minus12E minus 2

minus16E minus 2

120583dz

50kmh60kmh80kmh

0 05 1 15 2 25 3Time (s)

12E minus 1

80E minus 2

40E minus 2

00E + 0

minus40E minus 2

minus80E minus 2

120583z

50kmh60kmh80kmh

0 1 2 3Time (s)

80E minus 1

40E minus 1

00E + 0

minus40E minus 1

minus80E minus 1

50kmh60kmh80kmh

120583az

0 05 1 15 2 25 3Time (s)

10E minus 3

80E minus 4

60E minus 4

40E minus 4

20E minus 4

00E + 0

120590dz (m

50kmh60kmh80kmh

0 1 2 3Time (s)

80E minus 3

60E minus 3

40E minus 3

20E minus 3

00E + 0

120590z

50kmh60kmh80kmh

0 05 1 15 2 25 3Time (s)

40E minus 2

30E minus 2

20E minus 2

10E minus 2

00E + 0

120590az

50kmh60kmh80kmh

A B C D

50E minus 04

40E minus 04

30E minus 04

20E minus 04

10E minus 04

00E + 00

120583dy

A B C D

40E minus 05

30E minus 05

20E minus 05

10E minus 05

00E + 00

120590dy

0 025 05 075 1 125 15 175Time (s)

30E + 5

25E + 5

20E + 5

15E + 5

10E + 5

50E + 4

00E + 0

120583F

0 025 05 075 1 125 15 175Time (s)

25E + 4

20E + 4

15E + 4

10E + 4

50E + 3

00E + 0

minus50 + 3

120590F

0 05 1 15 2 25 3 35

Time (s)

10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

minus40E minus 4

120583dy

82 times 1010 N-m2

0 05 1 15 2 25 3 35Time (s)

20E minus 5

16E minus 5

12E minus 5

80E minus 6

40E minus 6

00E + 0

120590dy

6N-m2 (B) 119864V119868V = 64 times 107N-m2

and (C) 119864V119868V = 82 times 1010N-m2

0 05 1 15 2 25 3 35

Time (s)

10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

120583dy

119909) (A) 119890

119909= 05m (B) 119890

119909= 1m and

(C) 119890119909= 15m

0 05 1 15 2 25 3 35Time (s)

30E minus 5

25E minus 5

20E minus 5

15E minus 5

10E minus 5

50E minus 6

00E + 0

120590dy

119909) (A) 119890

119909= 05m (B) 119890

119909=

1m and (C) 119890119909= 15m

119884dynamic =1003816100381610038161003816120583119884 (119909119896 119905119896) + 120590119884 (119909119896 119905119896)

1003816100381610038161003816 (46)

0 0002 0004 0006 0008 001

GbJbEbIb

119887119869119887119864119887119868119887) for different

107 N-m2 and (C) 119864V119868V = 82 times 10

10 N-m2

119884static (47)

119887119868119887)

5 Conclusions

1520 25 30 35 40

45 5055

404550556065707580146148

15152154156158

16162164

times103

8013514

Vehicle speed (kmh)

14161820222426283013

Appendix

119865119895(119905) = 0 for119895 = 1 2 3 119899V

= 119896119905119895minus119899V

ℎ (119909119895minus119899V

) + 119888119905119895minus119899V

ℎ (119909119895minus119899V

For119895 = 119899V + 1 119899V + 2

= 1198961199051ℎ (1199091) 120601119887119895minus(119899V+2) (1199091) + 1198961199052ℎ (1199092) 120601119887119895minus(119899V+2) (1199092)

+ 1198881199051ℎ (1199091) 120601119887119895minus(119899V+2) (1199091) + 1198881199052ℎ (1199092) 120601119887119895minus(119899V+2) (1199092)

For 119895 = 119899V + 3 119899V + 4 119899V + 2 + 119899119887

= 1198901199091198961199051ℎ (1199091) 120601119905119895minus(119899V+2+119899119887)

(1199091)

+ 1198961199052ℎ (1199092) 120601119905119895minus(119899V+2+119899119887)

(1199092)

+ 1198881199051ℎ (1199091) 120601119905119895minus(119899V+2+119899119887)

(1199091)

+1198881199052ℎ1015840(1199092) 120601119905119895minus(119899V+2+119899119887)

(1199092)

For 119895 = 119899V + 119899119887 + 3

119899V + 119899119887 + 4 119899V + 119899119887 + 119899119905 + 2

ℎ119898(119909) =

1198712119909 (119871 minus 119909) (A2)

ℎ119903(119909) =

119873

119904=1

119860119904cos (2120587Ω

119904119909 + 120579119904) (A3)

119868119895119896= 119868119895119908+ 119868119895119909+ 119868119895119910+ 119868119895119911 (A4)

where 119908 = 1 2 3 119899V 119909 = 119899V + 1 119899V + 2 119910 = 119899V + 2 + 1119899V + 2 + 2 119899V + 2 + 119899

119887 119911 = 119899V + 2 + 119899

119887+ 1 119899V + 2 + 119899

119887+

2 119899V + 2 + 119899119887 + 119899119879Thefirst term 119868

1198681198951= 1198681198952= sdot sdot sdot = 119868

119895119899V= 0 (A5)

119868119895119909= 1198681198951199091

+ 1198681198951199092 (A6)

1198681198951199091

= 11986811989511990911

+ 11986811989511990912

119904)minus2 and 119861

119904= 2120587Ω

119871+ (119904 minus

11986811989511990911

= 1198961199051[

119873

119904=1

119860119904

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2minus120572119895cos (120579

119904) minus 119861119904119881 sin (120579

119904)

+ exp (120572119895119905)

times [120572119895cos (119861

119904119881119905 + 120579

119904)

+119861119904119881 cos (119861

119904119881119905 + 120579

119904)]

+4ℎ0119881

12057231198951198712

120572119895119871 (120572119895119905 + exp (minus120572

119895119905) minus 1)

minus119881

12057231198951198712

119895119905) + 2 ]

11986811989511990912

= 1198881199051[

119873

119904=1

119881119860119904119861119904

exp (minus120572119895119905)

119904)

+ exp (120572119895119905)

times [120572119895sin (119861

119904119881119905 + 120579

119904)

minus119861119904119881 sin (119861

119904119881119905 + 120579

119904)]

119895119905)

12057231198951198712

minus1

120572119895119871exp (120572

119895119905) minus 2]

1198681198951199092

= 11986811989511990921

+ 11986811989511990922

11986811989511990921

= 1198961199052[

119873

119904=1

119860119904

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2

times minus 120572119895cos (119861

119904119897V minus 120579119904) minus 119861119904119881 sin (119861

119904119897V minus 120579119904)

+ exp (120572119895119905) [120572119895cos (119861

119904(119897 minus 119881119905) minus 120579

119904)

+119861119904119881 sin (119861

119904(119897V minus 119881119905) + 120579119904)]

+1198812[2 minus exp (120572

119895119905) 120572119895119905 (120572119895119905 minus 2) + 2]

+4ℎ0

12057231198951198712

minus1205722

119895119897V (119897V + 119871) (exp (120572119895119905) minus 1)

+120572119895119881 (2119897V + 119871) [exp (120572119895119905) (120572119895119905 minus 1) + 1] ]

11986811989511990922

= 1198881199052[

119873

119904=1

119860119904119861119904119881

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2120572119895sin (119861

119904119897V minus 120579119904)

+ 119861119904119881 cos (119861

119904119897V minus 120579119904)

minus exp (120572119895119905) [120572119895sin (119861

119904(119897 minus 119881119905) minus 120579

119904)

+119861119904119881 cos (119861

119904(119897V minus 119881119905) + 120579119904)]

+4ℎ0119881 exp (minus120572

119895119905)

12057221198951198712

times 2119881 [exp (120572119895119905) (120572119895119905 minus 1) + 1]

minus120572119895(2119897V + 119871) (exp (120572119895119905) minus 1)]

119868119895119910=

119899119887

119896=1

119868119895119899V+2+119896

119886=1

119868(119895119899V+2+119896)119886

(119896 = 1 2 3 119899119887) (A11)

119876119904= 119861119904119881 119877

119896=119896120587119881

119871

119878119896=119896120587

119871 119885

119904= 119861119904119897V

119868(119895119899V+2+119896)

21198961199051

119873

119904=1

119860119904 ((2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) cos (120579

119904)

+2120572119895119876119904sin (120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times cos (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119876

119904+ 119877119896) 119905 + 120579

119904

minus (119877119896+ 119876119904)

times cos (119876119904+ 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

21198961199051[

4ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198762119904)3

times 119871 (1205722

119895+ 1198762

119904) [exp (120572

119895119905) sin (119876

119904119905)

times 1198762

119904(120572119895119905 + 1)

+1205722

119895(120572119895119905 minus 1)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904119905 + 120572119895(120572119895119905 minus 2)] minus 2120572

119895119876119904

+ 119881 (21198763

119904+ 119876119904exp (120572

119895119905)) cos (119876

119904119905)

times 1198764

1199041199052+ 21198762

119904(1205722

1198951199052minus 2120572119895119905 minus 1)

+1205722

119895(1205722

1198951199052minus 4120572119895119905 + 6)

minus exp (120572119895119905) sin (119876

119904119905)

times 1198764

119904(120572119895119905 + 2) + 120572

1198951198762

119904(1205722

1198951199052minus 3)

+1205723

119895(1205722

1198951199052minus 2120572119895119905 + 2) minus 6120572

119895119876119904]

119868(119895119899V+2+119896)

21198961199052

119873

119904=1

119860119904[ exp (minus120572

119895119905)

times (((119877119896+ 119876119904) cos (119878

119904minus 119885119904minus 120579119904)

+120572119895sin (119878119896minus 119885119904minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

times (((119877119896minus 119876119904) cos (119878

119896minus 119885119904+ 120579119904)

+120572119895sin (119878119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 119878

119896minus 119885119904+ 120579119904

minus (119877119896+ 119876119904) cos (119876

119904minus 119877119896) 119905

+119878119896minus 119885119904+ 120579119904)

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

+ ((120572119895sin 119878119896+ 119885119904minus (119876119904+ 119877) 119905 minus 120579

119904

+ (119877119896+ 119876119904) cos 119878

119896+ 119885119904

+ (119876119904+ 119877119896) 119905

minus120579119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1

21198961199052[

4ℎ0119881

1198712(1205722119895+ 1198812)

times sin (119881119905 minus 119897V) [1205721198951198972

V(1205722

119895+ 1198812)2

minus 119897V (1205722

119895+ 1198812)

times minus1205723

119895119871 minus 1205721198951198711198812

+ 21198813(120572119895119905 + 1)

+21205722

119895119881(120572119895119905 minus 1)

+ 119881 [119881 21205723

119895+ 120572119895119905(1205722

119895+ 1198812)2

+2119905 (1205724

119895minus 1198814) minus 6120572

1198951198812

minus 119871 (1205722

119895+ 1198812)

times 1205722

119895(120572119895119905 minus 1)+ 119881

2(120572119895119905 +1)]

+ 119881 cos (119897V minus 119881119905) [1198814(1198972

V + 119897V119871

+21205722

1198951199052minus 4120572119895119905 minus 2)

+ 1205722

1198951198812(21198972

V + 2119897V119871

+41205722

1198951199052+ 6)

+ 1205724

119895119897V (119897V + 119871)

minus 1198815119905 (2119897V + 119871)

times (120572119895119905 minus 1)

minus 1205723

119895119881 (2119897V + 119871)

times (120572119895119905 minus 2) + 119897

V1198816]]

119868(119895119899V+2+119896)

21198881199051

119873

119904=1

119860119904119876119904( minus (2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) sin (120579

119904)

minus2120572119895119876119904cos (120579

119904))

times (1205722

119895+ (119876119904minus 119877)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times sin (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos (119876

119904+ 119877119896) 119905 + 120579

119904

+ (119877119896+ 119876119904)

times sin (119876119904+ 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199051

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) cos (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) sin (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

21198881199052

119873

119904=1

119860119904119876119904

times [ exp (minus120572119895119905 ( ( (119877

119896+ 119876119904) sin (119878

119896+ 119885119904minus 120579119904)

minus120572119895cos (119878

119896+ 119885119904minus 120579119904))

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

times ( ( (119876119904minus 119877119896) sin (119878

119896minus 119885119904+ 120579119904)

+120572119895cos (119878

119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos 119878

119896+ 119885119904minus (119876119904+ 119877119896) 119905 minus 120579

119904

minus (119877119896+ 119876119904) sin 119878

119896+ 119885119904

minus (119876119904+ 119877119896) 119905 minus 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos 119878

119896minus 119885119904+ (119876119904minus 119877119896) 119905 + 120579

119904

+ (119876119904minus 119877119896) sin 119878

119896minus 119885119904

+ (119876119904minus 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199052

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) sin (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881 + exp (120572119895119905) 119896120587119881 cos(119896120587119881119905

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881cos(119896120587119863119871

119871)

minus 119871120572119895sin(119896120587119863

119871)

119871)]

119868(119895119899V+2+119896)

= minus21198711198812(1198981199081

+ 1198981199082)

119873

119904=1

1198601199041198612

119904[(((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881 cos [(119876

119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

119904119871)119881 cos 120579

119904

+119871120572119895sin 120579119904)

times (1198612

11990411987121198812

+ 21198961205871198611199041198711198812

+ 119896212058721198812

+11987121205722

119895)minus1

+ ((119896120587 minus 119861119904119871)119881 cos 120579

119904

minus119871120572119895sin 120579119904)

times ((119896120587 minus 119861119904119871)2

1198812

+11987121205722

119895)minus1

minus8ℎ01198812(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

times [119896120587119881

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119868119895119911=

119899119905

119896=1

119868119895119899V+2+119899119887+119896

119868119895119899V+2+119899119887+119897

119886=1

119868(119895119899V+2+119899119887+119897)119886

(119896 = 1 2 3 119899119879) (A14)

119868(119895119899V+2+119899119887+119897)1

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895cos (120579

119904) + (119876

119904minus 119877119896) sin (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895cos (120579

119904) + (119876

119904+ 119877119896) sin (120579

119904)

1205722119895+ (119876119904+ 119877119896)2

+ ((120572119895cos [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119876119904minus 119877119896) sin [(119876

119904minus 119877119896) 119905 minus 120579

119904])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [(119876

119904+ 119877119896) 119905 + 120579

119904]

+ (119876119904+ 119877119896) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)2

=41198961199051119890119909ℎ0119871

119881(1205722119895+ 119877119896

times [ (1205722

119895+ 119877119896

2) exp (minus120572

119895119905)

times 1205722

119895+ exp (minus120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)] minus 119877

119896

+ 2120572119895exp (minus120572

119895119905) (1205722

119895+ 3119877119896

+ 119877119896sin (119877

119896119905) 6120572

119895+ 1199052(1205722

119895+ 119877119896

+4120572119895119905 (1205722

119895+ 119877119896

2) minus 2119877

119896

+ cos (119877119896119905) 2119905 (119877

119896

4minus 1205724

119895)

+ 2120572119895(1205722

119895minus 3119877119896

times 1205721198951199052(1205722

119895+ 119877119896

119868(119895119899V+2+119899119887+119897)3

= minus1

21198961199052119890119909

119873

119904=1

[119860119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119877119896minus 119876119904) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) sin (119861

119904119897V + 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos [(119877

119896minus 119876119904) 119905 + (119861

119904119897V minus 119878119896 minus 120579119904)]

+ (119877119896minus 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [minus (119877

119896+ 119876119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

minus (119877119896+ 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)4

=41198961199052119890119909ℎ0

1198712[((119897V (119897V + 119871) exp (minus120572119895119905)

times 120572119895cos (119878

119896minus 119877119896119905)

minus119877119896sin (119878119896minus 119877119896119905)

+119877119896sin (119878) minus 120572

119895cos (119878

119896))

times (1205722

119895+ 119877119896

2)minus1

+2119871119881 exp (minus120572

119895119905)

(1205722119895+ 119877119896

times (1205722

119895+ 119877119896

2) cos (119878

119896)

+ exp (120572119895119905) cos (119878

119896minus 119877119896119905)

times [1198772(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896[119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

119895119877119896sin (119878119896)

minus1198812

(1205722119895+ 119877119896

times119877119896sin (119878119896minus 119877119896119905)[61205722

119895+1199052(1205722

119895+ 119877119896

minus 4120572119895119905 (1205722

119895+ 119877119896

minus2119877119896

+ 2 exp (minus120572119895119905)

times [(1205723

119895minus 3120572119895119877119896

2) cos (119878

119896)

+ 119877119896(119877119896

2minus 31205722

119895)

times sin (119878119896) ]

minus [2119905 (119877119896

4minus 1205724

119895) + 2120572

119895(1205722

119895minus 3119877119896

+1205721198951199052(1205722

119895+119877119896

]cos (119878119896minus 119877119896119905)]

119868(119895119899V+2+119899119887+119897)5

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895sin (120579119904) + (119876

119904minus 119877119896) cos (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895sin (120579119904) minus (119876

119904+ 119877119896) cos (120579

119904)

1205722119895+ (119876119904+ 119877)2

+ ((120572119895sin [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119877119896minus 119876119904) cos [(119876

119904minus 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895sin [(119876

119904+ 119877119896) 119905 + 120579

119904]

minus (119876119904+ 119877) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)6

=41198881199051119890119909ℎ0119871

(1205722119895+ 119877119896

2)3(119881

119871)

times [(1205722

119895+ 119877119896

2) 120572119895cos (119877

119896119905)

+ 119877119896sin (119877

119896119905) minus 120572

119895exp (minus120572

119895119905)

+ 2 exp (minus120572119895119905) 1205722

119895+ exp (120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

minus119877119896

119868(119895119899V+2+119899119887+119897)7

=41198881199052119890119909ℎ0

119881(1205722119895+ 119877119896

times 119897V (1205722

119895+ 119877119896

2) + 119871 (120572

119895+ 119877119896

minus2119881 [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

+ cos (119878119896minus 119877119896119905)

times 119877119896

2[120572119895(119897V + 119871) minus 2119881 (120572

119895119905 + 1)]

+1205722

119895[120572119895(119897V + 119871) + 2119881 (1 minus 120572

119895119905)]

+ exp (minus120572119895119905) 119877119896sin (119878119896)

times [119897V (1205722

119895+ 119877119896

+119871 (1205722

119895+ 119877119896

2)+4120572119895119881]

+ cos (119878119896)

times [119877119896

22119881 minus 120572

119895(119897V + 119871)

minus 1205722

119895120572119895(119897V + 119871)

+2119881]]

119868(119895119899V+2+119899119887+119897)8

21198881199052119890119909

119873

119904=1

[119860119904119876119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119876119904minus 119877119896) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) cos (119861

119904119897V + 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin [(119877 minus 119876

119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)]

+ (119876119904minus 119877119896) cos [(119877

119896minus 119876119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)])

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

minus ((120572119895sin [minus (119877 + 119876

119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

+ (119877119896+ 119876119904) cos [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)9

= minus119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [exp (120572119895119905) 119896120587119881 sin(119896120587119881119905

119871)

minus119871120572119895+ exp 119871120572

119895(120572119895119905) cos(119896120587119881119905

119871)]

minus 119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881sin(119896120587119863119871

119871)

minus 119890119909119871120572119895cos(119896120587119863

119871)

minus exp (120572119895119905)

times cos(119896120587119863 minus 119881119905

119871)]

119868(119895119899V+2+119899119887+119897)10

= minus21198711198812(1198981199081

+ 1198981199082) 119890119909

119873

119904=1

1198601199041198612

119904[ (((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881

times cos [(119876119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus exp (minus120572119895119905)

times ((minus (119896120587 +119861119904119871)119881 cos 120579

119904+119871120572119895sin 120579119904)

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

+ (((119896120587 minus 119861119904119871)119881 cos 120579

119904minus 119871120572119895sin 120579119904)

times((119896120587 minus 119861119904119871)2

1198812+ 11987121205722

119895)minus1

minus8ℎ01198812119890119909(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

119871)

+119871120572119895sin(119896120587119881119905

119871)]

References

RoboticsJournal of

Journal of

RotatingMachinery

VLSI Design

Shock and Vibration

Journal of

Volume 2014

SensorsJournal of

Propagation

119891119887(119909 119905)

+11988811990511(119905) minus

119863

119863119905[119910 (119909 119905) + ℎ (119909)]]

times 120575 (119909 minus 1199091)

+11988811990522(119905) minus

119863

119863119905[119910 (119909 119905) + ℎ (119909)]] 120575 (119909 minus 119909

minus 1198981+1

2119898V119897V119892120575 (119909 minus 1199091) minus 1198982 +

2119898V119897V119892

times 120575 (119909 minus 1199092)

+ 1198981

1198632

1198631199052[119910 (119909 119905) + ℎ (119909)] 120575 (119909 minus 119909

+ 1198982

1198632

1198631199052[119910 (119909 119905) + ℎ (119909)] 120575 (119909 minus 119909

119866119887119869119887

1205972120574 (119909 119905)

1205971199092minus 119862119887119879

120597120574 (119909 119905)

120597119905minus 119868119887

1205972120574 (119909 119905)

1205971199052= 119891119879(119909 119905)

119887is

119891119879(119909 119905)

+11988811990511(119905) minus

119863

119863119905[119910 (119909 119905) + ℎ (119909)]]

times 119890119909120575 (119909 minus 119909

+11988811990522(119905) minus

119863

119863119905[119910 (119909 119905) + ℎ (119909)]]

times 119890119909120575 (119909 minus 119909

minus 1198981+1

2119898V119897V119892119890119909120575 (119909 minus 1199091)

minus 1198982+1

2119898V119897V119892119890119909120575 (119909 minus 1199092)

+ 1198981

1198632

1198631199052[119910 (119909 119905) + ℎ (119909)] 119890

119909120575 (119909 minus 119909

+ 1198982

1198632

1198631199052[119910 (119909 119905) + ℎ (119909)] 119890

119909120575(119909 minus 119909

2)119909120575 (119909 minus 119909

ℎ (119909) = ℎ119898(119909) +

119873

119904=1

120589119904cos (2120587Ω

119904119909 + 120579119904) (10)

119871

119904andΩ

120589119904= radic2119878 (Ω

119904) ΔΩ

Ω119904= Ω119871+ (119904 minus

2)ΔΩ

ΔΩ =(Ω119880minus Ω119871)

119873

119878 (Ω) = 119878 (Ω0) times

Ω119904

2+ Ω2119871

119911 (119906 119905) =

119899V

119895 =minus1

120601V119895 (119906) 120578119895 (119905) (13)

120601minus1= 1

1206010= 119906 minus 119863

120601V119895 = sin (120572119895119906) + sinh (120572

119895119906)

+ 120573119895[cos (120572

119895119906) + cosh (120572

119895119906)]

120573119895=cos (120572

119895119897V) + cosh (120572

119895119897V)

sin (120572119895119897V) + sinh (120572

119895119897V)

120596V119895 = 1205722

119895radic

119864V119868V

119898V1198974

V (16)

120578119895(119905) + 2120585V119895120596V119895 120578119895 (119905) + 120596

V119895120578119895 (119905) = 119876V119895 (119905) (17)

vehicle is given as

119876V119895 (119905) =1

119872V119895int

119897119881

119891V (119906 119905) 120601119895 (119906) 119889119906 (18)

119872V119895 = int

119897V

119898V1206012

V119895 (119906) 119889119906 (19)

119876V119895 (119905)

119872V119895

119896V1

1199111(119905) minus

119899V

119895 =minus1

120601119895(1199061) 120578 (119905)

120601119895(1199061)

+ 119888V1

1(119905) minus

119899V

119895 =minus1

120601119895(1199061) 120578 (119905)

120601119895(1199061)

+ 119896V2

1199112(119905) minus

119899V

119895 =minus1

120601119895(1199062) 120578 (119905)

120601119895(1199062)

+119888V2

2(119905) minus

119899V

119895 =minus1

120601119895(1199062) 120578 (119905)

120601119895(1199062)]

119910 (119909 119905) =

119899119887

119896=1

120601119887119896(119909) 119902119896(119905) (21)

119887119896and 119902

119896(119905) are

119902119896(119905) + 2120585

119887119896120596119887119896

119902119896(119905) + 120596

119887119896119902119896(119905) = 119876

119887119896(119905) (22)

where 119896 = 1 2 3 119899119887

flexure is given as

119876119887119896(119905) =

119872119887119896

119871

119891119887(119909 119905) 120601

119887119896(119909) 119889119909 (23)

119872119887119896= int

119871

1198981198871206012

119887119896(119909) 119889119909 (24)

119876119887119896(119905)

= minus1

119872119887119896

[11989611990511199111(119905) minus

119899119887

119896=1

120601119887119896(1199091) 119902119896(119905) minus ℎ (119909

times 120601119887119896(1199091)

+ 11989611990521199112(119905) minus

119899119887

119896=1

120601119887119896(1199092) 119902119896(119905) minus ℎ (119909

times 120601119887119896(1199092)

+ 11988811990511(119905) minus 119881

119899119887

119896=1

1206011015840

119887119896(1199091) 119902119896(119905)

minus119881

119899119887

119896=1

120601119887119896(1199091) 119902119896(119905) minus 119881ℎ

1015840(1199091)

times 120601119887119896(1199091)

+ 11988811990522(119905) minus 119881

119899119887

119896=1

1206011015840

119887119896(1199092) 119902119896(119905)

minus119881

119899119887

119896=1

120601119887119896(1199092) 119902119896(119905) minus 119881ℎ

1015840(1199092)

times 120601119887119896(1199092)

minus 1198981+1

2119898V119897V119892120601119887119896 (1199091)

minus 1198982+1

2119898V119897V119892120601119887119896 (1199092)

times 1198981

119899119887

119896=1

120601119887119896(1199091) 119902119896(119905)

+ 2119881

119899119887

119896=1

1206011015840

119887119896(1199091) 119902119896(119905)

+1198812

119899119887

119896=1

12060110158401015840

119887119896(1199091) 119902119896(119905) + 4119881ℎ

10158401015840(1199091)

times 120601119887119896(1199091)

+ 1198982

119899119887

119896=1

120601119887119896(1199092) 119902119896(119905)

+ 2119881

119899119887

119896=1

1206011015840

119887119896(1199092) 119902119896(119905)

+ 1198812

119899119887

119896=1

12060110158401015840

119887119896(1199092) 119902119896(119905)

+4119881ℎ10158401015840(1199092)

times 120601119887119896(1199092) ]

120574119897(119905) + 2120585

119879119897120596119879119897

120574119897(119905) + 120596

119879119897120574119897(119905) = 119876

119879119897(119905)

(119897 = 1 2 3 119899119879)

119876119879119897(119905) =

119872119879119897

119871

119891119879(119909 119905) 120601

119879119897(119909) 119889119909 (27)

120601119879119897

119879119897in the 119897th mode is

given by

119872119879119897= int

119871

1198681198871206012

119879119897(119909) 119889119909 (28)

119876119879119897(119905)

= minus119890119909

119872119879119897

[11989611990511199111(119905) minus

119899119879

119897=1

120601119879119897(1199091) 120574119897(119905) minus ℎ (119909

times 120601119879119897(1199091)

+ 11989611990521199112(119905) minus

119899119879

119897=1

120601119879119897(1199092) 120574119897(119905) minus ℎ (119909

times 120601119879119897(1199092)

+ 11988811990511(119905) minus 119881

119899119879

119897=1

1206011015840

119879119897(1199091) 120574119897(119905)

minus119881

119899119879

119897=1

120601119879119897(1199091) 120574119897(119905) minus 119881ℎ

1015840(1199091)

times 120601119879119897(1199091)

+ 11988811990522(119905) minus 119881

119899119879

119897=1

1206011015840

119879119897(1199092) 120574119897(119905)

minus119881

119899119879

119897=1

120601119879119897(1199092) 120574119897(119905) minus 119881ℎ

1015840(1199092)

times 120601119879119897(1199092)

minus 1198981+1

2119898V119897V119892120601119879119897 (1199091) minus 1198982 +

2119898V119897V

times 119892120601119879119897(1199092)

times 1198981

119899119879

119897=1

120601119879119897(1199091) 120574119897(119905) + 2119881

119899119879

119897=1

1206011015840

119879119897(1199091) 120574119897(119905)

+1198812

119899119879

119897=1

12060110158401015840

119879119897(1199091) 120574119897(119905) + 4119881ℎ

10158401015840(1199091)

times 120601119879119897(1199091)

+ 1198982

119899119879

119897=1

120601119879119897(1199092) 120574119897(119905) + 2119881

119899119879

119897=1

1206011015840

119879119896(1199092) 120574119897(119905)

+1198812

119899119879

119896=1

12060110158401015840

119879119896(1199092) 120574119897(119905) + 4119881ℎ

10158401015840(1199092)

times 120601119879119897(1199092) ]

119887+

[119872] 119903 (119905) + [119862] 119903 (119905) + [119870] 119903 (119905) = 119865 (119905) (30)

where 119903(119905) = 1205781(119905) 1205782(119905) 120578

119899V(119905) 1199111(119905) 1199112(119905) 1199021(119905)1199022(119905) 119902

119899119887(119905) 1205741(119905) 1205742(119905) 120574

(119905) + [119860] 119901 (119905) = 119875 (119905) (31)

119901 (119905) = 119903 (119905)

119903 (119905) 119875 (119905) =

[119872]minus1119865 (119905)

[119860] = [119872]minus1[119862]

minus [119868]

[119872]minus1[119870]

11199062 119906

2119899] (33)

[119880]minus1[119860] [119880] = diagonal [120572

1 1205722 1205723 120572

2119899] (34)

Let us assume119901 = [119880] V (35)

V119895(119905) + 120572

119895V119895(119905) = 119877

119895(119905) 119895 = 1 2 3 2119899 (36)

119877119895=

119899

119896=1

1199061015840

119895119904

119899

119904=1

1198981015840

119904119896119865119896(119905) (37)

1199061015840

119904119896

may be expressed as

V119895(119905) = 119883

0119895exp (minus120572

119895119905) + int

minusinfin

119867119895(120596 119905) 119889119878 [119865

119896(120596)] (38)

where 1198830119895

119867119895(120596 119905) =

minus119894120596 + 120572119895

119903119898(119905) =

2119899

119895=1

119906119898+119899119895

1198830119895exp (minus120572

119895119905)

2119899

119895=1

119906119898+119899119895

119899

119904=1

1199061015840

119895119904

119899

119896=1

1198981015840

119904119896int

minusinfin

119867119895(120596 119905) 119889119878 (119865

119896(120596))

119898 = 1 2 3 119899

119903119875119898

(119905) =

2119899

119895=1

119906119898+119899119895

119899

119904=1

1199061015840

119895119904

119899

119896=1

1198981015840

119904119896int

minusinfin

exp (minus119894120596119905)minus119894120596 + 120572

119895

119865119896(120596) 119889120596

119903119875119898

(119905)

2119899

119895=1

119906119898+119899119895

119899

119904=1

1199061015840

119895119904

119899

119896=1

1198981015840

119904119896

times int

minusinfin

119865119896(120591) [

2120587int

minusinfin

exp [119894120596 (120591 minus 119905)]minus119894120596 + 120572

119895

119889120596]119889120591

119903119875119898

(119905) =

2119899

119895=1

119906119898+119899119895

119899

119904=1

1199061015840

119895119904

119899

119896=1

1198981015840

119904119896119868119895119896 (43)

119868119895119896= int

119905

exp [120572119895(120591 minus 119905)] 119865

119896(120591) 119889120591 (44)

119868119895119896= 119868119895119908+ 119868119895119909+ 119868119895119910+ 119868119895119911 (45)

where 119908 = 1 2 3 119899V 119909 = 119899V + 1 119899V + 2 119910 = 119899V + 3119899V+4 119899V+2+119899119887 119911 = 119899V+119899119887+3 119899V+119899119887+4 119899V+119899119887+119899119879+2

119884(119905119896) and

0 05 1 15 2 25

Time (s)

50E minus 4

00E + 0

minus50E minus 4

minus10E minus 3

minus15E minus 3

minus20E minus 3

119887119868119887) 248 times 10

119887119869119887) 248 times 10

5 kgm2m 119864119887= 32GPa deck = 2512GPa

1199051 1198961199052) 197 times 10

6Nm 474 times 106Nm

119887119868119887) 37times

119887119869119887) 1695 times 10

10N-m2

063 063 063

300 300

265100

(1198961199051 1198961199052) 09 times 10

119904= 16 times

119889119911)

075 120 075

0 05 1 15 2 25 3 35 4

Time (s)10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

120583dy

50kmh60kmh80kmh

0 05 1 15 2 25 3 35

Time (s)

minus80E minus 3

40E minus 3

minus10E minus 18

minus40E minus 3

minus80E minus 3

minus12E minus 2

120583y

50kmh60kmh80kmh

0 05 1 15 2 25 3 35

Time (s)40E minus 2

00E + 0

minus40E minus 2

minus80E minus 2

minus12E minus 1

minus16E minus 1

120583ay

50kmh60kmh80kmh

0 05 1 15 2 25 3 35 4Time (s)

40E minus 5

30E minus 5

20E minus 5

10E minus 5

00E + 0

minus10E minus 5

120590dy

50kmh60kmh80kmh

119889119911) velocity (120583V119911)

119886119911) is shown in

0 05 1 15 2 25 3 35Time (s)

16E minus 3

12E minus 3

80E minus 4

40E minus 4

00E + 0

minus40E minus 4

120590y

50kmh60kmh80kmh

0 05 1 15 2 25 3 35Time (s)

16E minus 2

12E minus 2

80E minus 3

40E minus 3

00E + 0

120590ay

50kmh60kmh80kmh

0 05 1 15 2 25 3Time (s)

80E minus 3

40E minus 3

00E + 0

minus40E minus 3

minus80E minus 3

minus12E minus 2

minus16E minus 2

120583dz

50kmh60kmh80kmh

0 05 1 15 2 25 3Time (s)

12E minus 1

80E minus 2

40E minus 2

00E + 0

minus40E minus 2

minus80E minus 2

120583z

50kmh60kmh80kmh

0 1 2 3Time (s)

80E minus 1

40E minus 1

00E + 0

minus40E minus 1

minus80E minus 1

50kmh60kmh80kmh

120583az

0 05 1 15 2 25 3Time (s)

10E minus 3

80E minus 4

60E minus 4

40E minus 4

20E minus 4

00E + 0

120590dz (m

50kmh60kmh80kmh

0 1 2 3Time (s)

80E minus 3

60E minus 3

40E minus 3

20E minus 3

00E + 0

120590z

50kmh60kmh80kmh

0 05 1 15 2 25 3Time (s)

40E minus 2

30E minus 2

20E minus 2

10E minus 2

00E + 0

120590az

50kmh60kmh80kmh

A B C D

50E minus 04

40E minus 04

30E minus 04

20E minus 04

10E minus 04

00E + 00

120583dy

A B C D

40E minus 05

30E minus 05

20E minus 05

10E minus 05

00E + 00

120590dy

0 025 05 075 1 125 15 175Time (s)

30E + 5

25E + 5

20E + 5

15E + 5

10E + 5

50E + 4

00E + 0

120583F

0 025 05 075 1 125 15 175Time (s)

25E + 4

20E + 4

15E + 4

10E + 4

50E + 3

00E + 0

minus50 + 3

120590F

0 05 1 15 2 25 3 35

Time (s)

10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

minus40E minus 4

120583dy

82 times 1010 N-m2

0 05 1 15 2 25 3 35Time (s)

20E minus 5

16E minus 5

12E minus 5

80E minus 6

40E minus 6

00E + 0

120590dy

6N-m2 (B) 119864V119868V = 64 times 107N-m2

and (C) 119864V119868V = 82 times 1010N-m2

0 05 1 15 2 25 3 35

Time (s)

10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

120583dy

119909) (A) 119890

119909= 05m (B) 119890

119909= 1m and

(C) 119890119909= 15m

0 05 1 15 2 25 3 35Time (s)

30E minus 5

25E minus 5

20E minus 5

15E minus 5

10E minus 5

50E minus 6

00E + 0

120590dy

119909) (A) 119890

119909= 05m (B) 119890

119909=

1m and (C) 119890119909= 15m

119884dynamic =1003816100381610038161003816120583119884 (119909119896 119905119896) + 120590119884 (119909119896 119905119896)

1003816100381610038161003816 (46)

0 0002 0004 0006 0008 001

GbJbEbIb

119887119869119887119864119887119868119887) for different

107 N-m2 and (C) 119864V119868V = 82 times 10

10 N-m2

119884static (47)

119887119868119887)

5 Conclusions

1520 25 30 35 40

45 5055

404550556065707580146148

15152154156158

16162164

times103

8013514

Vehicle speed (kmh)

14161820222426283013

Appendix

119865119895(119905) = 0 for119895 = 1 2 3 119899V

= 119896119905119895minus119899V

ℎ (119909119895minus119899V

) + 119888119905119895minus119899V

ℎ (119909119895minus119899V

For119895 = 119899V + 1 119899V + 2

= 1198961199051ℎ (1199091) 120601119887119895minus(119899V+2) (1199091) + 1198961199052ℎ (1199092) 120601119887119895minus(119899V+2) (1199092)

+ 1198881199051ℎ (1199091) 120601119887119895minus(119899V+2) (1199091) + 1198881199052ℎ (1199092) 120601119887119895minus(119899V+2) (1199092)

For 119895 = 119899V + 3 119899V + 4 119899V + 2 + 119899119887

= 1198901199091198961199051ℎ (1199091) 120601119905119895minus(119899V+2+119899119887)

(1199091)

+ 1198961199052ℎ (1199092) 120601119905119895minus(119899V+2+119899119887)

(1199092)

+ 1198881199051ℎ (1199091) 120601119905119895minus(119899V+2+119899119887)

(1199091)

+1198881199052ℎ1015840(1199092) 120601119905119895minus(119899V+2+119899119887)

(1199092)

For 119895 = 119899V + 119899119887 + 3

119899V + 119899119887 + 4 119899V + 119899119887 + 119899119905 + 2

ℎ119898(119909) =

1198712119909 (119871 minus 119909) (A2)

ℎ119903(119909) =

119873

119904=1

119860119904cos (2120587Ω

119904119909 + 120579119904) (A3)

119868119895119896= 119868119895119908+ 119868119895119909+ 119868119895119910+ 119868119895119911 (A4)

where 119908 = 1 2 3 119899V 119909 = 119899V + 1 119899V + 2 119910 = 119899V + 2 + 1119899V + 2 + 2 119899V + 2 + 119899

119887 119911 = 119899V + 2 + 119899

119887+ 1 119899V + 2 + 119899

119887+

2 119899V + 2 + 119899119887 + 119899119879Thefirst term 119868

1198681198951= 1198681198952= sdot sdot sdot = 119868

119895119899V= 0 (A5)

119868119895119909= 1198681198951199091

+ 1198681198951199092 (A6)

1198681198951199091

= 11986811989511990911

+ 11986811989511990912

119904)minus2 and 119861

119904= 2120587Ω

119871+ (119904 minus

11986811989511990911

= 1198961199051[

119873

119904=1

119860119904

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2minus120572119895cos (120579

119904) minus 119861119904119881 sin (120579

119904)

+ exp (120572119895119905)

times [120572119895cos (119861

119904119881119905 + 120579

119904)

+119861119904119881 cos (119861

119904119881119905 + 120579

119904)]

+4ℎ0119881

12057231198951198712

120572119895119871 (120572119895119905 + exp (minus120572

119895119905) minus 1)

minus119881

12057231198951198712

119895119905) + 2 ]

11986811989511990912

= 1198881199051[

119873

119904=1

119881119860119904119861119904

exp (minus120572119895119905)

119904)

+ exp (120572119895119905)

times [120572119895sin (119861

119904119881119905 + 120579

119904)

minus119861119904119881 sin (119861

119904119881119905 + 120579

119904)]

119895119905)

12057231198951198712

minus1

120572119895119871exp (120572

119895119905) minus 2]

1198681198951199092

= 11986811989511990921

+ 11986811989511990922

11986811989511990921

= 1198961199052[

119873

119904=1

119860119904

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2

times minus 120572119895cos (119861

119904119897V minus 120579119904) minus 119861119904119881 sin (119861

119904119897V minus 120579119904)

+ exp (120572119895119905) [120572119895cos (119861

119904(119897 minus 119881119905) minus 120579

119904)

+119861119904119881 sin (119861

119904(119897V minus 119881119905) + 120579119904)]

+1198812[2 minus exp (120572

119895119905) 120572119895119905 (120572119895119905 minus 2) + 2]

+4ℎ0

12057231198951198712

minus1205722

119895119897V (119897V + 119871) (exp (120572119895119905) minus 1)

+120572119895119881 (2119897V + 119871) [exp (120572119895119905) (120572119895119905 minus 1) + 1] ]

11986811989511990922

= 1198881199052[

119873

119904=1

119860119904119861119904119881

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2120572119895sin (119861

119904119897V minus 120579119904)

+ 119861119904119881 cos (119861

119904119897V minus 120579119904)

minus exp (120572119895119905) [120572119895sin (119861

119904(119897 minus 119881119905) minus 120579

119904)

+119861119904119881 cos (119861

119904(119897V minus 119881119905) + 120579119904)]

+4ℎ0119881 exp (minus120572

119895119905)

12057221198951198712

times 2119881 [exp (120572119895119905) (120572119895119905 minus 1) + 1]

minus120572119895(2119897V + 119871) (exp (120572119895119905) minus 1)]

119868119895119910=

119899119887

119896=1

119868119895119899V+2+119896

119886=1

119868(119895119899V+2+119896)119886

(119896 = 1 2 3 119899119887) (A11)

119876119904= 119861119904119881 119877

119896=119896120587119881

119871

119878119896=119896120587

119871 119885

119904= 119861119904119897V

119868(119895119899V+2+119896)

21198961199051

119873

119904=1

119860119904 ((2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) cos (120579

119904)

+2120572119895119876119904sin (120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times cos (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119876

119904+ 119877119896) 119905 + 120579

119904

minus (119877119896+ 119876119904)

times cos (119876119904+ 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

21198961199051[

4ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198762119904)3

times 119871 (1205722

119895+ 1198762

119904) [exp (120572

119895119905) sin (119876

119904119905)

times 1198762

119904(120572119895119905 + 1)

+1205722

119895(120572119895119905 minus 1)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904119905 + 120572119895(120572119895119905 minus 2)] minus 2120572

119895119876119904

+ 119881 (21198763

119904+ 119876119904exp (120572

119895119905)) cos (119876

119904119905)

times 1198764

1199041199052+ 21198762

119904(1205722

1198951199052minus 2120572119895119905 minus 1)

+1205722

119895(1205722

1198951199052minus 4120572119895119905 + 6)

minus exp (120572119895119905) sin (119876

119904119905)

times 1198764

119904(120572119895119905 + 2) + 120572

1198951198762

119904(1205722

1198951199052minus 3)

+1205723

119895(1205722

1198951199052minus 2120572119895119905 + 2) minus 6120572

119895119876119904]

119868(119895119899V+2+119896)

21198961199052

119873

119904=1

119860119904[ exp (minus120572

119895119905)

times (((119877119896+ 119876119904) cos (119878

119904minus 119885119904minus 120579119904)

+120572119895sin (119878119896minus 119885119904minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

times (((119877119896minus 119876119904) cos (119878

119896minus 119885119904+ 120579119904)

+120572119895sin (119878119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 119878

119896minus 119885119904+ 120579119904

minus (119877119896+ 119876119904) cos (119876

119904minus 119877119896) 119905

+119878119896minus 119885119904+ 120579119904)

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

+ ((120572119895sin 119878119896+ 119885119904minus (119876119904+ 119877) 119905 minus 120579

119904

+ (119877119896+ 119876119904) cos 119878

119896+ 119885119904

+ (119876119904+ 119877119896) 119905

minus120579119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1

21198961199052[

4ℎ0119881

1198712(1205722119895+ 1198812)

times sin (119881119905 minus 119897V) [1205721198951198972

V(1205722

119895+ 1198812)2

minus 119897V (1205722

119895+ 1198812)

times minus1205723

119895119871 minus 1205721198951198711198812

+ 21198813(120572119895119905 + 1)

+21205722

119895119881(120572119895119905 minus 1)

+ 119881 [119881 21205723

119895+ 120572119895119905(1205722

119895+ 1198812)2

+2119905 (1205724

119895minus 1198814) minus 6120572

1198951198812

minus 119871 (1205722

119895+ 1198812)

times 1205722

119895(120572119895119905 minus 1)+ 119881

2(120572119895119905 +1)]

+ 119881 cos (119897V minus 119881119905) [1198814(1198972

V + 119897V119871

+21205722

1198951199052minus 4120572119895119905 minus 2)

+ 1205722

1198951198812(21198972

V + 2119897V119871

+41205722

1198951199052+ 6)

+ 1205724

119895119897V (119897V + 119871)

minus 1198815119905 (2119897V + 119871)

times (120572119895119905 minus 1)

minus 1205723

119895119881 (2119897V + 119871)

times (120572119895119905 minus 2) + 119897

V1198816]]

119868(119895119899V+2+119896)

21198881199051

119873

119904=1

119860119904119876119904( minus (2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) sin (120579

119904)

minus2120572119895119876119904cos (120579

119904))

times (1205722

119895+ (119876119904minus 119877)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times sin (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos (119876

119904+ 119877119896) 119905 + 120579

119904

+ (119877119896+ 119876119904)

times sin (119876119904+ 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199051

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) cos (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) sin (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

21198881199052

119873

119904=1

119860119904119876119904

times [ exp (minus120572119895119905 ( ( (119877

119896+ 119876119904) sin (119878

119896+ 119885119904minus 120579119904)

minus120572119895cos (119878

119896+ 119885119904minus 120579119904))

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

times ( ( (119876119904minus 119877119896) sin (119878

119896minus 119885119904+ 120579119904)

+120572119895cos (119878

119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos 119878

119896+ 119885119904minus (119876119904+ 119877119896) 119905 minus 120579

119904

minus (119877119896+ 119876119904) sin 119878

119896+ 119885119904

minus (119876119904+ 119877119896) 119905 minus 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos 119878

119896minus 119885119904+ (119876119904minus 119877119896) 119905 + 120579

119904

+ (119876119904minus 119877119896) sin 119878

119896minus 119885119904

+ (119876119904minus 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199052

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) sin (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881 + exp (120572119895119905) 119896120587119881 cos(119896120587119881119905

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881cos(119896120587119863119871

119871)

minus 119871120572119895sin(119896120587119863

119871)

119871)]

119868(119895119899V+2+119896)

= minus21198711198812(1198981199081

+ 1198981199082)

119873

119904=1

1198601199041198612

119904[(((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881 cos [(119876

119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

119904119871)119881 cos 120579

119904

+119871120572119895sin 120579119904)

times (1198612

11990411987121198812

+ 21198961205871198611199041198711198812

+ 119896212058721198812

+11987121205722

119895)minus1

+ ((119896120587 minus 119861119904119871)119881 cos 120579

119904

minus119871120572119895sin 120579119904)

times ((119896120587 minus 119861119904119871)2

1198812

+11987121205722

119895)minus1

minus8ℎ01198812(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

times [119896120587119881

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119868119895119911=

119899119905

119896=1

119868119895119899V+2+119899119887+119896

119868119895119899V+2+119899119887+119897

119886=1

119868(119895119899V+2+119899119887+119897)119886

(119896 = 1 2 3 119899119879) (A14)

119868(119895119899V+2+119899119887+119897)1

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895cos (120579

119904) + (119876

119904minus 119877119896) sin (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895cos (120579

119904) + (119876

119904+ 119877119896) sin (120579

119904)

1205722119895+ (119876119904+ 119877119896)2

+ ((120572119895cos [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119876119904minus 119877119896) sin [(119876

119904minus 119877119896) 119905 minus 120579

119904])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [(119876

119904+ 119877119896) 119905 + 120579

119904]

+ (119876119904+ 119877119896) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)2

=41198961199051119890119909ℎ0119871

119881(1205722119895+ 119877119896

times [ (1205722

119895+ 119877119896

2) exp (minus120572

119895119905)

times 1205722

119895+ exp (minus120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)] minus 119877

119896

+ 2120572119895exp (minus120572

119895119905) (1205722

119895+ 3119877119896

+ 119877119896sin (119877

119896119905) 6120572

119895+ 1199052(1205722

119895+ 119877119896

+4120572119895119905 (1205722

119895+ 119877119896

2) minus 2119877

119896

+ cos (119877119896119905) 2119905 (119877

119896

4minus 1205724

119895)

+ 2120572119895(1205722

119895minus 3119877119896

times 1205721198951199052(1205722

119895+ 119877119896

119868(119895119899V+2+119899119887+119897)3

= minus1

21198961199052119890119909

119873

119904=1

[119860119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119877119896minus 119876119904) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) sin (119861

119904119897V + 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos [(119877

119896minus 119876119904) 119905 + (119861

119904119897V minus 119878119896 minus 120579119904)]

+ (119877119896minus 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [minus (119877

119896+ 119876119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

minus (119877119896+ 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)4

=41198961199052119890119909ℎ0

1198712[((119897V (119897V + 119871) exp (minus120572119895119905)

times 120572119895cos (119878

119896minus 119877119896119905)

minus119877119896sin (119878119896minus 119877119896119905)

+119877119896sin (119878) minus 120572

119895cos (119878

119896))

times (1205722

119895+ 119877119896

2)minus1

+2119871119881 exp (minus120572

119895119905)

(1205722119895+ 119877119896

times (1205722

119895+ 119877119896

2) cos (119878

119896)

+ exp (120572119895119905) cos (119878

119896minus 119877119896119905)

times [1198772(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896[119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

119895119877119896sin (119878119896)

minus1198812

(1205722119895+ 119877119896

times119877119896sin (119878119896minus 119877119896119905)[61205722

119895+1199052(1205722

119895+ 119877119896

minus 4120572119895119905 (1205722

119895+ 119877119896

minus2119877119896

+ 2 exp (minus120572119895119905)

times [(1205723

119895minus 3120572119895119877119896

2) cos (119878

119896)

+ 119877119896(119877119896

2minus 31205722

119895)

times sin (119878119896) ]

minus [2119905 (119877119896

4minus 1205724

119895) + 2120572

119895(1205722

119895minus 3119877119896

+1205721198951199052(1205722

119895+119877119896

]cos (119878119896minus 119877119896119905)]

119868(119895119899V+2+119899119887+119897)5

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895sin (120579119904) + (119876

119904minus 119877119896) cos (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895sin (120579119904) minus (119876

119904+ 119877119896) cos (120579

119904)

1205722119895+ (119876119904+ 119877)2

+ ((120572119895sin [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119877119896minus 119876119904) cos [(119876

119904minus 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895sin [(119876

119904+ 119877119896) 119905 + 120579

119904]

minus (119876119904+ 119877) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)6

=41198881199051119890119909ℎ0119871

(1205722119895+ 119877119896

2)3(119881

119871)

times [(1205722

119895+ 119877119896

2) 120572119895cos (119877

119896119905)

+ 119877119896sin (119877

119896119905) minus 120572

119895exp (minus120572

119895119905)

+ 2 exp (minus120572119895119905) 1205722

119895+ exp (120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

minus119877119896

119868(119895119899V+2+119899119887+119897)7

=41198881199052119890119909ℎ0

119881(1205722119895+ 119877119896

times 119897V (1205722

119895+ 119877119896

2) + 119871 (120572

119895+ 119877119896

minus2119881 [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

+ cos (119878119896minus 119877119896119905)

times 119877119896

2[120572119895(119897V + 119871) minus 2119881 (120572

119895119905 + 1)]

+1205722

119895[120572119895(119897V + 119871) + 2119881 (1 minus 120572

119895119905)]

+ exp (minus120572119895119905) 119877119896sin (119878119896)

times [119897V (1205722

119895+ 119877119896

+119871 (1205722

119895+ 119877119896

2)+4120572119895119881]

+ cos (119878119896)

times [119877119896

22119881 minus 120572

119895(119897V + 119871)

minus 1205722

119895120572119895(119897V + 119871)

+2119881]]

119868(119895119899V+2+119899119887+119897)8

21198881199052119890119909

119873

119904=1

[119860119904119876119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119876119904minus 119877119896) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) cos (119861

119904119897V + 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin [(119877 minus 119876

119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)]

+ (119876119904minus 119877119896) cos [(119877

119896minus 119876119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)])

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

minus ((120572119895sin [minus (119877 + 119876

119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

+ (119877119896+ 119876119904) cos [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)9

= minus119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [exp (120572119895119905) 119896120587119881 sin(119896120587119881119905

119871)

minus119871120572119895+ exp 119871120572

119895(120572119895119905) cos(119896120587119881119905

119871)]

minus 119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881sin(119896120587119863119871

119871)

minus 119890119909119871120572119895cos(119896120587119863

119871)

minus exp (120572119895119905)

times cos(119896120587119863 minus 119881119905

119871)]

119868(119895119899V+2+119899119887+119897)10

= minus21198711198812(1198981199081

+ 1198981199082) 119890119909

119873

119904=1

1198601199041198612

119904[ (((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881

times cos [(119876119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus exp (minus120572119895119905)

times ((minus (119896120587 +119861119904119871)119881 cos 120579

119904+119871120572119895sin 120579119904)

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

+ (((119896120587 minus 119861119904119871)119881 cos 120579

119904minus 119871120572119895sin 120579119904)

times((119896120587 minus 119861119904119871)2

1198812+ 11987121205722

119895)minus1

minus8ℎ01198812119890119909(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

119871)

+119871120572119895sin(119896120587119881119905

119871)]

References

RoboticsJournal of

Journal of

RotatingMachinery

VLSI Design

Shock and Vibration

Journal of

Volume 2014

SensorsJournal of

Propagation

119911 (119906 119905) =

119899V

119895 =minus1

120601V119895 (119906) 120578119895 (119905) (13)

120601minus1= 1

1206010= 119906 minus 119863

120601V119895 = sin (120572119895119906) + sinh (120572

119895119906)

+ 120573119895[cos (120572

119895119906) + cosh (120572

119895119906)]

120573119895=cos (120572

119895119897V) + cosh (120572

119895119897V)

sin (120572119895119897V) + sinh (120572

119895119897V)

120596V119895 = 1205722

119895radic

119864V119868V

119898V1198974

V (16)

120578119895(119905) + 2120585V119895120596V119895 120578119895 (119905) + 120596

V119895120578119895 (119905) = 119876V119895 (119905) (17)

vehicle is given as

119876V119895 (119905) =1

119872V119895int

119897119881

119891V (119906 119905) 120601119895 (119906) 119889119906 (18)

119872V119895 = int

119897V

119898V1206012

V119895 (119906) 119889119906 (19)

119876V119895 (119905)

119872V119895

119896V1

1199111(119905) minus

119899V

119895 =minus1

120601119895(1199061) 120578 (119905)

120601119895(1199061)

+ 119888V1

1(119905) minus

119899V

119895 =minus1

120601119895(1199061) 120578 (119905)

120601119895(1199061)

+ 119896V2

1199112(119905) minus

119899V

119895 =minus1

120601119895(1199062) 120578 (119905)

120601119895(1199062)

+119888V2

2(119905) minus

119899V

119895 =minus1

120601119895(1199062) 120578 (119905)

120601119895(1199062)]

119910 (119909 119905) =

119899119887

119896=1

120601119887119896(119909) 119902119896(119905) (21)

119887119896and 119902

119896(119905) are

119902119896(119905) + 2120585

119887119896120596119887119896

119902119896(119905) + 120596

119887119896119902119896(119905) = 119876

119887119896(119905) (22)

where 119896 = 1 2 3 119899119887

flexure is given as

119876119887119896(119905) =

119872119887119896

119871

119891119887(119909 119905) 120601

119887119896(119909) 119889119909 (23)

119872119887119896= int

119871

1198981198871206012

119887119896(119909) 119889119909 (24)

119876119887119896(119905)

= minus1

119872119887119896

[11989611990511199111(119905) minus

119899119887

119896=1

120601119887119896(1199091) 119902119896(119905) minus ℎ (119909

times 120601119887119896(1199091)

+ 11989611990521199112(119905) minus

119899119887

119896=1

120601119887119896(1199092) 119902119896(119905) minus ℎ (119909

times 120601119887119896(1199092)

+ 11988811990511(119905) minus 119881

119899119887

119896=1

1206011015840

119887119896(1199091) 119902119896(119905)

minus119881

119899119887

119896=1

120601119887119896(1199091) 119902119896(119905) minus 119881ℎ

1015840(1199091)

times 120601119887119896(1199091)

+ 11988811990522(119905) minus 119881

119899119887

119896=1

1206011015840

119887119896(1199092) 119902119896(119905)

minus119881

119899119887

119896=1

120601119887119896(1199092) 119902119896(119905) minus 119881ℎ

1015840(1199092)

times 120601119887119896(1199092)

minus 1198981+1

2119898V119897V119892120601119887119896 (1199091)

minus 1198982+1

2119898V119897V119892120601119887119896 (1199092)

times 1198981

119899119887

119896=1

120601119887119896(1199091) 119902119896(119905)

+ 2119881

119899119887

119896=1

1206011015840

119887119896(1199091) 119902119896(119905)

+1198812

119899119887

119896=1

12060110158401015840

119887119896(1199091) 119902119896(119905) + 4119881ℎ

10158401015840(1199091)

times 120601119887119896(1199091)

+ 1198982

119899119887

119896=1

120601119887119896(1199092) 119902119896(119905)

+ 2119881

119899119887

119896=1

1206011015840

119887119896(1199092) 119902119896(119905)

+ 1198812

119899119887

119896=1

12060110158401015840

119887119896(1199092) 119902119896(119905)

+4119881ℎ10158401015840(1199092)

times 120601119887119896(1199092) ]

120574119897(119905) + 2120585

119879119897120596119879119897

120574119897(119905) + 120596

119879119897120574119897(119905) = 119876

119879119897(119905)

(119897 = 1 2 3 119899119879)

119876119879119897(119905) =

119872119879119897

119871

119891119879(119909 119905) 120601

119879119897(119909) 119889119909 (27)

120601119879119897

119879119897in the 119897th mode is

given by

119872119879119897= int

119871

1198681198871206012

119879119897(119909) 119889119909 (28)

119876119879119897(119905)

= minus119890119909

119872119879119897

[11989611990511199111(119905) minus

119899119879

119897=1

120601119879119897(1199091) 120574119897(119905) minus ℎ (119909

times 120601119879119897(1199091)

+ 11989611990521199112(119905) minus

119899119879

119897=1

120601119879119897(1199092) 120574119897(119905) minus ℎ (119909

times 120601119879119897(1199092)

+ 11988811990511(119905) minus 119881

119899119879

119897=1

1206011015840

119879119897(1199091) 120574119897(119905)

minus119881

119899119879

119897=1

120601119879119897(1199091) 120574119897(119905) minus 119881ℎ

1015840(1199091)

times 120601119879119897(1199091)

+ 11988811990522(119905) minus 119881

119899119879

119897=1

1206011015840

119879119897(1199092) 120574119897(119905)

minus119881

119899119879

119897=1

120601119879119897(1199092) 120574119897(119905) minus 119881ℎ

1015840(1199092)

times 120601119879119897(1199092)

minus 1198981+1

2119898V119897V119892120601119879119897 (1199091) minus 1198982 +

2119898V119897V

times 119892120601119879119897(1199092)

times 1198981

119899119879

119897=1

120601119879119897(1199091) 120574119897(119905) + 2119881

119899119879

119897=1

1206011015840

119879119897(1199091) 120574119897(119905)

+1198812

119899119879

119897=1

12060110158401015840

119879119897(1199091) 120574119897(119905) + 4119881ℎ

10158401015840(1199091)

times 120601119879119897(1199091)

+ 1198982

119899119879

119897=1

120601119879119897(1199092) 120574119897(119905) + 2119881

119899119879

119897=1

1206011015840

119879119896(1199092) 120574119897(119905)

+1198812

119899119879

119896=1

12060110158401015840

119879119896(1199092) 120574119897(119905) + 4119881ℎ

10158401015840(1199092)

times 120601119879119897(1199092) ]

119887+

[119872] 119903 (119905) + [119862] 119903 (119905) + [119870] 119903 (119905) = 119865 (119905) (30)

where 119903(119905) = 1205781(119905) 1205782(119905) 120578

119899V(119905) 1199111(119905) 1199112(119905) 1199021(119905)1199022(119905) 119902

119899119887(119905) 1205741(119905) 1205742(119905) 120574

(119905) + [119860] 119901 (119905) = 119875 (119905) (31)

119901 (119905) = 119903 (119905)

119903 (119905) 119875 (119905) =

[119872]minus1119865 (119905)

[119860] = [119872]minus1[119862]

minus [119868]

[119872]minus1[119870]

11199062 119906

2119899] (33)

[119880]minus1[119860] [119880] = diagonal [120572

1 1205722 1205723 120572

2119899] (34)

Let us assume119901 = [119880] V (35)

V119895(119905) + 120572

119895V119895(119905) = 119877

119895(119905) 119895 = 1 2 3 2119899 (36)

119877119895=

119899

119896=1

1199061015840

119895119904

119899

119904=1

1198981015840

119904119896119865119896(119905) (37)

1199061015840

119904119896

may be expressed as

V119895(119905) = 119883

0119895exp (minus120572

119895119905) + int

minusinfin

119867119895(120596 119905) 119889119878 [119865

119896(120596)] (38)

where 1198830119895

119867119895(120596 119905) =

minus119894120596 + 120572119895

119903119898(119905) =

2119899

119895=1

119906119898+119899119895

1198830119895exp (minus120572

119895119905)

2119899

119895=1

119906119898+119899119895

119899

119904=1

1199061015840

119895119904

119899

119896=1

1198981015840

119904119896int

minusinfin

119867119895(120596 119905) 119889119878 (119865

119896(120596))

119898 = 1 2 3 119899

119903119875119898

(119905) =

2119899

119895=1

119906119898+119899119895

119899

119904=1

1199061015840

119895119904

119899

119896=1

1198981015840

119904119896int

minusinfin

exp (minus119894120596119905)minus119894120596 + 120572

119895

119865119896(120596) 119889120596

119903119875119898

(119905)

2119899

119895=1

119906119898+119899119895

119899

119904=1

1199061015840

119895119904

119899

119896=1

1198981015840

119904119896

times int

minusinfin

119865119896(120591) [

2120587int

minusinfin

exp [119894120596 (120591 minus 119905)]minus119894120596 + 120572

119895

119889120596]119889120591

119903119875119898

(119905) =

2119899

119895=1

119906119898+119899119895

119899

119904=1

1199061015840

119895119904

119899

119896=1

1198981015840

119904119896119868119895119896 (43)

119868119895119896= int

119905

exp [120572119895(120591 minus 119905)] 119865

119896(120591) 119889120591 (44)

119868119895119896= 119868119895119908+ 119868119895119909+ 119868119895119910+ 119868119895119911 (45)

where 119908 = 1 2 3 119899V 119909 = 119899V + 1 119899V + 2 119910 = 119899V + 3119899V+4 119899V+2+119899119887 119911 = 119899V+119899119887+3 119899V+119899119887+4 119899V+119899119887+119899119879+2

119884(119905119896) and

0 05 1 15 2 25

Time (s)

50E minus 4

00E + 0

minus50E minus 4

minus10E minus 3

minus15E minus 3

minus20E minus 3

119887119868119887) 248 times 10

119887119869119887) 248 times 10

5 kgm2m 119864119887= 32GPa deck = 2512GPa

1199051 1198961199052) 197 times 10

6Nm 474 times 106Nm

119887119868119887) 37times

119887119869119887) 1695 times 10

10N-m2

063 063 063

300 300

265100

(1198961199051 1198961199052) 09 times 10

119904= 16 times

119889119911)

075 120 075

0 05 1 15 2 25 3 35 4

Time (s)10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

120583dy

50kmh60kmh80kmh

0 05 1 15 2 25 3 35

Time (s)

minus80E minus 3

40E minus 3

minus10E minus 18

minus40E minus 3

minus80E minus 3

minus12E minus 2

120583y

50kmh60kmh80kmh

0 05 1 15 2 25 3 35

Time (s)40E minus 2

00E + 0

minus40E minus 2

minus80E minus 2

minus12E minus 1

minus16E minus 1

120583ay

50kmh60kmh80kmh

0 05 1 15 2 25 3 35 4Time (s)

40E minus 5

30E minus 5

20E minus 5

10E minus 5

00E + 0

minus10E minus 5

120590dy

50kmh60kmh80kmh

119889119911) velocity (120583V119911)

119886119911) is shown in

0 05 1 15 2 25 3 35Time (s)

16E minus 3

12E minus 3

80E minus 4

40E minus 4

00E + 0

minus40E minus 4

120590y

50kmh60kmh80kmh

0 05 1 15 2 25 3 35Time (s)

16E minus 2

12E minus 2

80E minus 3

40E minus 3

00E + 0

120590ay

50kmh60kmh80kmh

0 05 1 15 2 25 3Time (s)

80E minus 3

40E minus 3

00E + 0

minus40E minus 3

minus80E minus 3

minus12E minus 2

minus16E minus 2

120583dz

50kmh60kmh80kmh

0 05 1 15 2 25 3Time (s)

12E minus 1

80E minus 2

40E minus 2

00E + 0

minus40E minus 2

minus80E minus 2

120583z

50kmh60kmh80kmh

0 1 2 3Time (s)

80E minus 1

40E minus 1

00E + 0

minus40E minus 1

minus80E minus 1

50kmh60kmh80kmh

120583az

0 05 1 15 2 25 3Time (s)

10E minus 3

80E minus 4

60E minus 4

40E minus 4

20E minus 4

00E + 0

120590dz (m

50kmh60kmh80kmh

0 1 2 3Time (s)

80E minus 3

60E minus 3

40E minus 3

20E minus 3

00E + 0

120590z

50kmh60kmh80kmh

0 05 1 15 2 25 3Time (s)

40E minus 2

30E minus 2

20E minus 2

10E minus 2

00E + 0

120590az

50kmh60kmh80kmh

A B C D

50E minus 04

40E minus 04

30E minus 04

20E minus 04

10E minus 04

00E + 00

120583dy

A B C D

40E minus 05

30E minus 05

20E minus 05

10E minus 05

00E + 00

120590dy

0 025 05 075 1 125 15 175Time (s)

30E + 5

25E + 5

20E + 5

15E + 5

10E + 5

50E + 4

00E + 0

120583F

0 025 05 075 1 125 15 175Time (s)

25E + 4

20E + 4

15E + 4

10E + 4

50E + 3

00E + 0

minus50 + 3

120590F

0 05 1 15 2 25 3 35

Time (s)

10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

minus40E minus 4

120583dy

82 times 1010 N-m2

0 05 1 15 2 25 3 35Time (s)

20E minus 5

16E minus 5

12E minus 5

80E minus 6

40E minus 6

00E + 0

120590dy

6N-m2 (B) 119864V119868V = 64 times 107N-m2

and (C) 119864V119868V = 82 times 1010N-m2

0 05 1 15 2 25 3 35

Time (s)

10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

120583dy

119909) (A) 119890

119909= 05m (B) 119890

119909= 1m and

(C) 119890119909= 15m

0 05 1 15 2 25 3 35Time (s)

30E minus 5

25E minus 5

20E minus 5

15E minus 5

10E minus 5

50E minus 6

00E + 0

120590dy

119909) (A) 119890

119909= 05m (B) 119890

119909=

1m and (C) 119890119909= 15m

119884dynamic =1003816100381610038161003816120583119884 (119909119896 119905119896) + 120590119884 (119909119896 119905119896)

1003816100381610038161003816 (46)

0 0002 0004 0006 0008 001

GbJbEbIb

119887119869119887119864119887119868119887) for different

107 N-m2 and (C) 119864V119868V = 82 times 10

10 N-m2

119884static (47)

119887119868119887)

5 Conclusions

1520 25 30 35 40

45 5055

404550556065707580146148

15152154156158

16162164

times103

8013514

Vehicle speed (kmh)

14161820222426283013

Appendix

119865119895(119905) = 0 for119895 = 1 2 3 119899V

= 119896119905119895minus119899V

ℎ (119909119895minus119899V

) + 119888119905119895minus119899V

ℎ (119909119895minus119899V

For119895 = 119899V + 1 119899V + 2

= 1198961199051ℎ (1199091) 120601119887119895minus(119899V+2) (1199091) + 1198961199052ℎ (1199092) 120601119887119895minus(119899V+2) (1199092)

+ 1198881199051ℎ (1199091) 120601119887119895minus(119899V+2) (1199091) + 1198881199052ℎ (1199092) 120601119887119895minus(119899V+2) (1199092)

For 119895 = 119899V + 3 119899V + 4 119899V + 2 + 119899119887

= 1198901199091198961199051ℎ (1199091) 120601119905119895minus(119899V+2+119899119887)

(1199091)

+ 1198961199052ℎ (1199092) 120601119905119895minus(119899V+2+119899119887)

(1199092)

+ 1198881199051ℎ (1199091) 120601119905119895minus(119899V+2+119899119887)

(1199091)

+1198881199052ℎ1015840(1199092) 120601119905119895minus(119899V+2+119899119887)

(1199092)

For 119895 = 119899V + 119899119887 + 3

119899V + 119899119887 + 4 119899V + 119899119887 + 119899119905 + 2

ℎ119898(119909) =

1198712119909 (119871 minus 119909) (A2)

ℎ119903(119909) =

119873

119904=1

119860119904cos (2120587Ω

119904119909 + 120579119904) (A3)

119868119895119896= 119868119895119908+ 119868119895119909+ 119868119895119910+ 119868119895119911 (A4)

where 119908 = 1 2 3 119899V 119909 = 119899V + 1 119899V + 2 119910 = 119899V + 2 + 1119899V + 2 + 2 119899V + 2 + 119899

119887 119911 = 119899V + 2 + 119899

119887+ 1 119899V + 2 + 119899

119887+

2 119899V + 2 + 119899119887 + 119899119879Thefirst term 119868

1198681198951= 1198681198952= sdot sdot sdot = 119868

119895119899V= 0 (A5)

119868119895119909= 1198681198951199091

+ 1198681198951199092 (A6)

1198681198951199091

= 11986811989511990911

+ 11986811989511990912

119904)minus2 and 119861

119904= 2120587Ω

119871+ (119904 minus

11986811989511990911

= 1198961199051[

119873

119904=1

119860119904

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2minus120572119895cos (120579

119904) minus 119861119904119881 sin (120579

119904)

+ exp (120572119895119905)

times [120572119895cos (119861

119904119881119905 + 120579

119904)

+119861119904119881 cos (119861

119904119881119905 + 120579

119904)]

+4ℎ0119881

12057231198951198712

120572119895119871 (120572119895119905 + exp (minus120572

119895119905) minus 1)

minus119881

12057231198951198712

119895119905) + 2 ]

11986811989511990912

= 1198881199051[

119873

119904=1

119881119860119904119861119904

exp (minus120572119895119905)

119904)

+ exp (120572119895119905)

times [120572119895sin (119861

119904119881119905 + 120579

119904)

minus119861119904119881 sin (119861

119904119881119905 + 120579

119904)]

119895119905)

12057231198951198712

minus1

120572119895119871exp (120572

119895119905) minus 2]

1198681198951199092

= 11986811989511990921

+ 11986811989511990922

11986811989511990921

= 1198961199052[

119873

119904=1

119860119904

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2

times minus 120572119895cos (119861

119904119897V minus 120579119904) minus 119861119904119881 sin (119861

119904119897V minus 120579119904)

+ exp (120572119895119905) [120572119895cos (119861

119904(119897 minus 119881119905) minus 120579

119904)

+119861119904119881 sin (119861

119904(119897V minus 119881119905) + 120579119904)]

+1198812[2 minus exp (120572

119895119905) 120572119895119905 (120572119895119905 minus 2) + 2]

+4ℎ0

12057231198951198712

minus1205722

119895119897V (119897V + 119871) (exp (120572119895119905) minus 1)

+120572119895119881 (2119897V + 119871) [exp (120572119895119905) (120572119895119905 minus 1) + 1] ]

11986811989511990922

= 1198881199052[

119873

119904=1

119860119904119861119904119881

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2120572119895sin (119861

119904119897V minus 120579119904)

+ 119861119904119881 cos (119861

119904119897V minus 120579119904)

minus exp (120572119895119905) [120572119895sin (119861

119904(119897 minus 119881119905) minus 120579

119904)

+119861119904119881 cos (119861

119904(119897V minus 119881119905) + 120579119904)]

+4ℎ0119881 exp (minus120572

119895119905)

12057221198951198712

times 2119881 [exp (120572119895119905) (120572119895119905 minus 1) + 1]

minus120572119895(2119897V + 119871) (exp (120572119895119905) minus 1)]

119868119895119910=

119899119887

119896=1

119868119895119899V+2+119896

119886=1

119868(119895119899V+2+119896)119886

(119896 = 1 2 3 119899119887) (A11)

119876119904= 119861119904119881 119877

119896=119896120587119881

119871

119878119896=119896120587

119871 119885

119904= 119861119904119897V

119868(119895119899V+2+119896)

21198961199051

119873

119904=1

119860119904 ((2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) cos (120579

119904)

+2120572119895119876119904sin (120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times cos (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119876

119904+ 119877119896) 119905 + 120579

119904

minus (119877119896+ 119876119904)

times cos (119876119904+ 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

21198961199051[

4ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198762119904)3

times 119871 (1205722

119895+ 1198762

119904) [exp (120572

119895119905) sin (119876

119904119905)

times 1198762

119904(120572119895119905 + 1)

+1205722

119895(120572119895119905 minus 1)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904119905 + 120572119895(120572119895119905 minus 2)] minus 2120572

119895119876119904

+ 119881 (21198763

119904+ 119876119904exp (120572

119895119905)) cos (119876

119904119905)

times 1198764

1199041199052+ 21198762

119904(1205722

1198951199052minus 2120572119895119905 minus 1)

+1205722

119895(1205722

1198951199052minus 4120572119895119905 + 6)

minus exp (120572119895119905) sin (119876

119904119905)

times 1198764

119904(120572119895119905 + 2) + 120572

1198951198762

119904(1205722

1198951199052minus 3)

+1205723

119895(1205722

1198951199052minus 2120572119895119905 + 2) minus 6120572

119895119876119904]

119868(119895119899V+2+119896)

21198961199052

119873

119904=1

119860119904[ exp (minus120572

119895119905)

times (((119877119896+ 119876119904) cos (119878

119904minus 119885119904minus 120579119904)

+120572119895sin (119878119896minus 119885119904minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

times (((119877119896minus 119876119904) cos (119878

119896minus 119885119904+ 120579119904)

+120572119895sin (119878119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 119878

119896minus 119885119904+ 120579119904

minus (119877119896+ 119876119904) cos (119876

119904minus 119877119896) 119905

+119878119896minus 119885119904+ 120579119904)

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

+ ((120572119895sin 119878119896+ 119885119904minus (119876119904+ 119877) 119905 minus 120579

119904

+ (119877119896+ 119876119904) cos 119878

119896+ 119885119904

+ (119876119904+ 119877119896) 119905

minus120579119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1

21198961199052[

4ℎ0119881

1198712(1205722119895+ 1198812)

times sin (119881119905 minus 119897V) [1205721198951198972

V(1205722

119895+ 1198812)2

minus 119897V (1205722

119895+ 1198812)

times minus1205723

119895119871 minus 1205721198951198711198812

+ 21198813(120572119895119905 + 1)

+21205722

119895119881(120572119895119905 minus 1)

+ 119881 [119881 21205723

119895+ 120572119895119905(1205722

119895+ 1198812)2

+2119905 (1205724

119895minus 1198814) minus 6120572

1198951198812

minus 119871 (1205722

119895+ 1198812)

times 1205722

119895(120572119895119905 minus 1)+ 119881

2(120572119895119905 +1)]

+ 119881 cos (119897V minus 119881119905) [1198814(1198972

V + 119897V119871

+21205722

1198951199052minus 4120572119895119905 minus 2)

+ 1205722

1198951198812(21198972

V + 2119897V119871

+41205722

1198951199052+ 6)

+ 1205724

119895119897V (119897V + 119871)

minus 1198815119905 (2119897V + 119871)

times (120572119895119905 minus 1)

minus 1205723

119895119881 (2119897V + 119871)

times (120572119895119905 minus 2) + 119897

V1198816]]

119868(119895119899V+2+119896)

21198881199051

119873

119904=1

119860119904119876119904( minus (2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) sin (120579

119904)

minus2120572119895119876119904cos (120579

119904))

times (1205722

119895+ (119876119904minus 119877)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times sin (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos (119876

119904+ 119877119896) 119905 + 120579

119904

+ (119877119896+ 119876119904)

times sin (119876119904+ 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199051

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) cos (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) sin (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

21198881199052

119873

119904=1

119860119904119876119904

times [ exp (minus120572119895119905 ( ( (119877

119896+ 119876119904) sin (119878

119896+ 119885119904minus 120579119904)

minus120572119895cos (119878

119896+ 119885119904minus 120579119904))

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

times ( ( (119876119904minus 119877119896) sin (119878

119896minus 119885119904+ 120579119904)

+120572119895cos (119878

119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos 119878

119896+ 119885119904minus (119876119904+ 119877119896) 119905 minus 120579

119904

minus (119877119896+ 119876119904) sin 119878

119896+ 119885119904

minus (119876119904+ 119877119896) 119905 minus 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos 119878

119896minus 119885119904+ (119876119904minus 119877119896) 119905 + 120579

119904

+ (119876119904minus 119877119896) sin 119878

119896minus 119885119904

+ (119876119904minus 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199052

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) sin (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881 + exp (120572119895119905) 119896120587119881 cos(119896120587119881119905

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881cos(119896120587119863119871

119871)

minus 119871120572119895sin(119896120587119863

119871)

119871)]

119868(119895119899V+2+119896)

= minus21198711198812(1198981199081

+ 1198981199082)

119873

119904=1

1198601199041198612

119904[(((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881 cos [(119876

119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

119904119871)119881 cos 120579

119904

+119871120572119895sin 120579119904)

times (1198612

11990411987121198812

+ 21198961205871198611199041198711198812

+ 119896212058721198812

+11987121205722

119895)minus1

+ ((119896120587 minus 119861119904119871)119881 cos 120579

119904

minus119871120572119895sin 120579119904)

times ((119896120587 minus 119861119904119871)2

1198812

+11987121205722

119895)minus1

minus8ℎ01198812(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

times [119896120587119881

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119868119895119911=

119899119905

119896=1

119868119895119899V+2+119899119887+119896

119868119895119899V+2+119899119887+119897

119886=1

119868(119895119899V+2+119899119887+119897)119886

(119896 = 1 2 3 119899119879) (A14)

119868(119895119899V+2+119899119887+119897)1

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895cos (120579

119904) + (119876

119904minus 119877119896) sin (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895cos (120579

119904) + (119876

119904+ 119877119896) sin (120579

119904)

1205722119895+ (119876119904+ 119877119896)2

+ ((120572119895cos [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119876119904minus 119877119896) sin [(119876

119904minus 119877119896) 119905 minus 120579

119904])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [(119876

119904+ 119877119896) 119905 + 120579

119904]

+ (119876119904+ 119877119896) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)2

=41198961199051119890119909ℎ0119871

119881(1205722119895+ 119877119896

times [ (1205722

119895+ 119877119896

2) exp (minus120572

119895119905)

times 1205722

119895+ exp (minus120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)] minus 119877

119896

+ 2120572119895exp (minus120572

119895119905) (1205722

119895+ 3119877119896

+ 119877119896sin (119877

119896119905) 6120572

119895+ 1199052(1205722

119895+ 119877119896

+4120572119895119905 (1205722

119895+ 119877119896

2) minus 2119877

119896

+ cos (119877119896119905) 2119905 (119877

119896

4minus 1205724

119895)

+ 2120572119895(1205722

119895minus 3119877119896

times 1205721198951199052(1205722

119895+ 119877119896

119868(119895119899V+2+119899119887+119897)3

= minus1

21198961199052119890119909

119873

119904=1

[119860119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119877119896minus 119876119904) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) sin (119861

119904119897V + 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos [(119877

119896minus 119876119904) 119905 + (119861

119904119897V minus 119878119896 minus 120579119904)]

+ (119877119896minus 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [minus (119877

119896+ 119876119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

minus (119877119896+ 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)4

=41198961199052119890119909ℎ0

1198712[((119897V (119897V + 119871) exp (minus120572119895119905)

times 120572119895cos (119878

119896minus 119877119896119905)

minus119877119896sin (119878119896minus 119877119896119905)

+119877119896sin (119878) minus 120572

119895cos (119878

119896))

times (1205722

119895+ 119877119896

2)minus1

+2119871119881 exp (minus120572

119895119905)

(1205722119895+ 119877119896

times (1205722

119895+ 119877119896

2) cos (119878

119896)

+ exp (120572119895119905) cos (119878

119896minus 119877119896119905)

times [1198772(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896[119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

119895119877119896sin (119878119896)

minus1198812

(1205722119895+ 119877119896

times119877119896sin (119878119896minus 119877119896119905)[61205722

119895+1199052(1205722

119895+ 119877119896

minus 4120572119895119905 (1205722

119895+ 119877119896

minus2119877119896

+ 2 exp (minus120572119895119905)

times [(1205723

119895minus 3120572119895119877119896

2) cos (119878

119896)

+ 119877119896(119877119896

2minus 31205722

119895)

times sin (119878119896) ]

minus [2119905 (119877119896

4minus 1205724

119895) + 2120572

119895(1205722

119895minus 3119877119896

+1205721198951199052(1205722

119895+119877119896

]cos (119878119896minus 119877119896119905)]

119868(119895119899V+2+119899119887+119897)5

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895sin (120579119904) + (119876

119904minus 119877119896) cos (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895sin (120579119904) minus (119876

119904+ 119877119896) cos (120579

119904)

1205722119895+ (119876119904+ 119877)2

+ ((120572119895sin [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119877119896minus 119876119904) cos [(119876

119904minus 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895sin [(119876

119904+ 119877119896) 119905 + 120579

119904]

minus (119876119904+ 119877) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)6

=41198881199051119890119909ℎ0119871

(1205722119895+ 119877119896

2)3(119881

119871)

times [(1205722

119895+ 119877119896

2) 120572119895cos (119877

119896119905)

+ 119877119896sin (119877

119896119905) minus 120572

119895exp (minus120572

119895119905)

+ 2 exp (minus120572119895119905) 1205722

119895+ exp (120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

minus119877119896

119868(119895119899V+2+119899119887+119897)7

=41198881199052119890119909ℎ0

119881(1205722119895+ 119877119896

times 119897V (1205722

119895+ 119877119896

2) + 119871 (120572

119895+ 119877119896

minus2119881 [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

+ cos (119878119896minus 119877119896119905)

times 119877119896

2[120572119895(119897V + 119871) minus 2119881 (120572

119895119905 + 1)]

+1205722

119895[120572119895(119897V + 119871) + 2119881 (1 minus 120572

119895119905)]

+ exp (minus120572119895119905) 119877119896sin (119878119896)

times [119897V (1205722

119895+ 119877119896

+119871 (1205722

119895+ 119877119896

2)+4120572119895119881]

+ cos (119878119896)

times [119877119896

22119881 minus 120572

119895(119897V + 119871)

minus 1205722

119895120572119895(119897V + 119871)

+2119881]]

119868(119895119899V+2+119899119887+119897)8

21198881199052119890119909

119873

119904=1

[119860119904119876119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119876119904minus 119877119896) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) cos (119861

119904119897V + 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin [(119877 minus 119876

119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)]

+ (119876119904minus 119877119896) cos [(119877

119896minus 119876119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)])

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

minus ((120572119895sin [minus (119877 + 119876

119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

+ (119877119896+ 119876119904) cos [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)9

= minus119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [exp (120572119895119905) 119896120587119881 sin(119896120587119881119905

119871)

minus119871120572119895+ exp 119871120572

119895(120572119895119905) cos(119896120587119881119905

119871)]

minus 119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881sin(119896120587119863119871

119871)

minus 119890119909119871120572119895cos(119896120587119863

119871)

minus exp (120572119895119905)

times cos(119896120587119863 minus 119881119905

119871)]

119868(119895119899V+2+119899119887+119897)10

= minus21198711198812(1198981199081

+ 1198981199082) 119890119909

119873

119904=1

1198601199041198612

119904[ (((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881

times cos [(119876119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus exp (minus120572119895119905)

times ((minus (119896120587 +119861119904119871)119881 cos 120579

119904+119871120572119895sin 120579119904)

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

+ (((119896120587 minus 119861119904119871)119881 cos 120579

119904minus 119871120572119895sin 120579119904)

times((119896120587 minus 119861119904119871)2

1198812+ 11987121205722

119895)minus1

minus8ℎ01198812119890119909(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

119871)

+119871120572119895sin(119896120587119881119905

119871)]

References

RoboticsJournal of

Journal of

RotatingMachinery

VLSI Design

Shock and Vibration

Journal of

Volume 2014

SensorsJournal of

Propagation

119876119887119896(119905)

= minus1

119872119887119896

[11989611990511199111(119905) minus

119899119887

119896=1

120601119887119896(1199091) 119902119896(119905) minus ℎ (119909

times 120601119887119896(1199091)

+ 11989611990521199112(119905) minus

119899119887

119896=1

120601119887119896(1199092) 119902119896(119905) minus ℎ (119909

times 120601119887119896(1199092)

+ 11988811990511(119905) minus 119881

119899119887

119896=1

1206011015840

119887119896(1199091) 119902119896(119905)

minus119881

119899119887

119896=1

120601119887119896(1199091) 119902119896(119905) minus 119881ℎ

1015840(1199091)

times 120601119887119896(1199091)

+ 11988811990522(119905) minus 119881

119899119887

119896=1

1206011015840

119887119896(1199092) 119902119896(119905)

minus119881

119899119887

119896=1

120601119887119896(1199092) 119902119896(119905) minus 119881ℎ

1015840(1199092)

times 120601119887119896(1199092)

minus 1198981+1

2119898V119897V119892120601119887119896 (1199091)

minus 1198982+1

2119898V119897V119892120601119887119896 (1199092)

times 1198981

119899119887

119896=1

120601119887119896(1199091) 119902119896(119905)

+ 2119881

119899119887

119896=1

1206011015840

119887119896(1199091) 119902119896(119905)

+1198812

119899119887

119896=1

12060110158401015840

119887119896(1199091) 119902119896(119905) + 4119881ℎ

10158401015840(1199091)

times 120601119887119896(1199091)

+ 1198982

119899119887

119896=1

120601119887119896(1199092) 119902119896(119905)

+ 2119881

119899119887

119896=1

1206011015840

119887119896(1199092) 119902119896(119905)

+ 1198812

119899119887

119896=1

12060110158401015840

119887119896(1199092) 119902119896(119905)

+4119881ℎ10158401015840(1199092)

times 120601119887119896(1199092) ]

120574119897(119905) + 2120585

119879119897120596119879119897

120574119897(119905) + 120596

119879119897120574119897(119905) = 119876

119879119897(119905)

(119897 = 1 2 3 119899119879)

119876119879119897(119905) =

119872119879119897

119871

119891119879(119909 119905) 120601

119879119897(119909) 119889119909 (27)

120601119879119897

119879119897in the 119897th mode is

given by

119872119879119897= int

119871

1198681198871206012

119879119897(119909) 119889119909 (28)

119876119879119897(119905)

= minus119890119909

119872119879119897

[11989611990511199111(119905) minus

119899119879

119897=1

120601119879119897(1199091) 120574119897(119905) minus ℎ (119909

times 120601119879119897(1199091)

+ 11989611990521199112(119905) minus

119899119879

119897=1

120601119879119897(1199092) 120574119897(119905) minus ℎ (119909

times 120601119879119897(1199092)

+ 11988811990511(119905) minus 119881

119899119879

119897=1

1206011015840

119879119897(1199091) 120574119897(119905)

minus119881

119899119879

119897=1

120601119879119897(1199091) 120574119897(119905) minus 119881ℎ

1015840(1199091)

times 120601119879119897(1199091)

+ 11988811990522(119905) minus 119881

119899119879

119897=1

1206011015840

119879119897(1199092) 120574119897(119905)

minus119881

119899119879

119897=1

120601119879119897(1199092) 120574119897(119905) minus 119881ℎ

1015840(1199092)

times 120601119879119897(1199092)

minus 1198981+1

2119898V119897V119892120601119879119897 (1199091) minus 1198982 +

2119898V119897V

times 119892120601119879119897(1199092)

times 1198981

119899119879

119897=1

120601119879119897(1199091) 120574119897(119905) + 2119881

119899119879

119897=1

1206011015840

119879119897(1199091) 120574119897(119905)

+1198812

119899119879

119897=1

12060110158401015840

119879119897(1199091) 120574119897(119905) + 4119881ℎ

10158401015840(1199091)

times 120601119879119897(1199091)

+ 1198982

119899119879

119897=1

120601119879119897(1199092) 120574119897(119905) + 2119881

119899119879

119897=1

1206011015840

119879119896(1199092) 120574119897(119905)

+1198812

119899119879

119896=1

12060110158401015840

119879119896(1199092) 120574119897(119905) + 4119881ℎ

10158401015840(1199092)

times 120601119879119897(1199092) ]

119887+

[119872] 119903 (119905) + [119862] 119903 (119905) + [119870] 119903 (119905) = 119865 (119905) (30)

where 119903(119905) = 1205781(119905) 1205782(119905) 120578

119899V(119905) 1199111(119905) 1199112(119905) 1199021(119905)1199022(119905) 119902

119899119887(119905) 1205741(119905) 1205742(119905) 120574

(119905) + [119860] 119901 (119905) = 119875 (119905) (31)

119901 (119905) = 119903 (119905)

119903 (119905) 119875 (119905) =

[119872]minus1119865 (119905)

[119860] = [119872]minus1[119862]

minus [119868]

[119872]minus1[119870]

11199062 119906

2119899] (33)

[119880]minus1[119860] [119880] = diagonal [120572

1 1205722 1205723 120572

2119899] (34)

Let us assume119901 = [119880] V (35)

V119895(119905) + 120572

119895V119895(119905) = 119877

119895(119905) 119895 = 1 2 3 2119899 (36)

119877119895=

119899

119896=1

1199061015840

119895119904

119899

119904=1

1198981015840

119904119896119865119896(119905) (37)

1199061015840

119904119896

may be expressed as

V119895(119905) = 119883

0119895exp (minus120572

119895119905) + int

minusinfin

119867119895(120596 119905) 119889119878 [119865

119896(120596)] (38)

where 1198830119895

119867119895(120596 119905) =

minus119894120596 + 120572119895

119903119898(119905) =

2119899

119895=1

119906119898+119899119895

1198830119895exp (minus120572

119895119905)

2119899

119895=1

119906119898+119899119895

119899

119904=1

1199061015840

119895119904

119899

119896=1

1198981015840

119904119896int

minusinfin

119867119895(120596 119905) 119889119878 (119865

119896(120596))

119898 = 1 2 3 119899

119903119875119898

(119905) =

2119899

119895=1

119906119898+119899119895

119899

119904=1

1199061015840

119895119904

119899

119896=1

1198981015840

119904119896int

minusinfin

exp (minus119894120596119905)minus119894120596 + 120572

119895

119865119896(120596) 119889120596

119903119875119898

(119905)

2119899

119895=1

119906119898+119899119895

119899

119904=1

1199061015840

119895119904

119899

119896=1

1198981015840

119904119896

times int

minusinfin

119865119896(120591) [

2120587int

minusinfin

exp [119894120596 (120591 minus 119905)]minus119894120596 + 120572

119895

119889120596]119889120591

119903119875119898

(119905) =

2119899

119895=1

119906119898+119899119895

119899

119904=1

1199061015840

119895119904

119899

119896=1

1198981015840

119904119896119868119895119896 (43)

119868119895119896= int

119905

exp [120572119895(120591 minus 119905)] 119865

119896(120591) 119889120591 (44)

119868119895119896= 119868119895119908+ 119868119895119909+ 119868119895119910+ 119868119895119911 (45)

where 119908 = 1 2 3 119899V 119909 = 119899V + 1 119899V + 2 119910 = 119899V + 3119899V+4 119899V+2+119899119887 119911 = 119899V+119899119887+3 119899V+119899119887+4 119899V+119899119887+119899119879+2

119884(119905119896) and

0 05 1 15 2 25

Time (s)

50E minus 4

00E + 0

minus50E minus 4

minus10E minus 3

minus15E minus 3

minus20E minus 3

119887119868119887) 248 times 10

119887119869119887) 248 times 10

5 kgm2m 119864119887= 32GPa deck = 2512GPa

1199051 1198961199052) 197 times 10

6Nm 474 times 106Nm

119887119868119887) 37times

119887119869119887) 1695 times 10

10N-m2

063 063 063

300 300

265100

(1198961199051 1198961199052) 09 times 10

119904= 16 times

119889119911)

075 120 075

0 05 1 15 2 25 3 35 4

Time (s)10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

120583dy

50kmh60kmh80kmh

0 05 1 15 2 25 3 35

Time (s)

minus80E minus 3

40E minus 3

minus10E minus 18

minus40E minus 3

minus80E minus 3

minus12E minus 2

120583y

50kmh60kmh80kmh

0 05 1 15 2 25 3 35

Time (s)40E minus 2

00E + 0

minus40E minus 2

minus80E minus 2

minus12E minus 1

minus16E minus 1

120583ay

50kmh60kmh80kmh

0 05 1 15 2 25 3 35 4Time (s)

40E minus 5

30E minus 5

20E minus 5

10E minus 5

00E + 0

minus10E minus 5

120590dy

50kmh60kmh80kmh

119889119911) velocity (120583V119911)

119886119911) is shown in

0 05 1 15 2 25 3 35Time (s)

16E minus 3

12E minus 3

80E minus 4

40E minus 4

00E + 0

minus40E minus 4

120590y

50kmh60kmh80kmh

0 05 1 15 2 25 3 35Time (s)

16E minus 2

12E minus 2

80E minus 3

40E minus 3

00E + 0

120590ay

50kmh60kmh80kmh

0 05 1 15 2 25 3Time (s)

80E minus 3

40E minus 3

00E + 0

minus40E minus 3

minus80E minus 3

minus12E minus 2

minus16E minus 2

120583dz

50kmh60kmh80kmh

0 05 1 15 2 25 3Time (s)

12E minus 1

80E minus 2

40E minus 2

00E + 0

minus40E minus 2

minus80E minus 2

120583z

50kmh60kmh80kmh

0 1 2 3Time (s)

80E minus 1

40E minus 1

00E + 0

minus40E minus 1

minus80E minus 1

50kmh60kmh80kmh

120583az

0 05 1 15 2 25 3Time (s)

10E minus 3

80E minus 4

60E minus 4

40E minus 4

20E minus 4

00E + 0

120590dz (m

50kmh60kmh80kmh

0 1 2 3Time (s)

80E minus 3

60E minus 3

40E minus 3

20E minus 3

00E + 0

120590z

50kmh60kmh80kmh

0 05 1 15 2 25 3Time (s)

40E minus 2

30E minus 2

20E minus 2

10E minus 2

00E + 0

120590az

50kmh60kmh80kmh

A B C D

50E minus 04

40E minus 04

30E minus 04

20E minus 04

10E minus 04

00E + 00

120583dy

A B C D

40E minus 05

30E minus 05

20E minus 05

10E minus 05

00E + 00

120590dy

0 025 05 075 1 125 15 175Time (s)

30E + 5

25E + 5

20E + 5

15E + 5

10E + 5

50E + 4

00E + 0

120583F

0 025 05 075 1 125 15 175Time (s)

25E + 4

20E + 4

15E + 4

10E + 4

50E + 3

00E + 0

minus50 + 3

120590F

0 05 1 15 2 25 3 35

Time (s)

10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

minus40E minus 4

120583dy

82 times 1010 N-m2

0 05 1 15 2 25 3 35Time (s)

20E minus 5

16E minus 5

12E minus 5

80E minus 6

40E minus 6

00E + 0

120590dy

6N-m2 (B) 119864V119868V = 64 times 107N-m2

and (C) 119864V119868V = 82 times 1010N-m2

0 05 1 15 2 25 3 35

Time (s)

10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

120583dy

119909) (A) 119890

119909= 05m (B) 119890

119909= 1m and

(C) 119890119909= 15m

0 05 1 15 2 25 3 35Time (s)

30E minus 5

25E minus 5

20E minus 5

15E minus 5

10E minus 5

50E minus 6

00E + 0

120590dy

119909) (A) 119890

119909= 05m (B) 119890

119909=

1m and (C) 119890119909= 15m

119884dynamic =1003816100381610038161003816120583119884 (119909119896 119905119896) + 120590119884 (119909119896 119905119896)

1003816100381610038161003816 (46)

0 0002 0004 0006 0008 001

GbJbEbIb

119887119869119887119864119887119868119887) for different

107 N-m2 and (C) 119864V119868V = 82 times 10

10 N-m2

119884static (47)

119887119868119887)

5 Conclusions

1520 25 30 35 40

45 5055

404550556065707580146148

15152154156158

16162164

times103

8013514

Vehicle speed (kmh)

14161820222426283013

Appendix

119865119895(119905) = 0 for119895 = 1 2 3 119899V

= 119896119905119895minus119899V

ℎ (119909119895minus119899V

) + 119888119905119895minus119899V

ℎ (119909119895minus119899V

For119895 = 119899V + 1 119899V + 2

= 1198961199051ℎ (1199091) 120601119887119895minus(119899V+2) (1199091) + 1198961199052ℎ (1199092) 120601119887119895minus(119899V+2) (1199092)

+ 1198881199051ℎ (1199091) 120601119887119895minus(119899V+2) (1199091) + 1198881199052ℎ (1199092) 120601119887119895minus(119899V+2) (1199092)

For 119895 = 119899V + 3 119899V + 4 119899V + 2 + 119899119887

= 1198901199091198961199051ℎ (1199091) 120601119905119895minus(119899V+2+119899119887)

(1199091)

+ 1198961199052ℎ (1199092) 120601119905119895minus(119899V+2+119899119887)

(1199092)

+ 1198881199051ℎ (1199091) 120601119905119895minus(119899V+2+119899119887)

(1199091)

+1198881199052ℎ1015840(1199092) 120601119905119895minus(119899V+2+119899119887)

(1199092)

For 119895 = 119899V + 119899119887 + 3

119899V + 119899119887 + 4 119899V + 119899119887 + 119899119905 + 2

ℎ119898(119909) =

1198712119909 (119871 minus 119909) (A2)

ℎ119903(119909) =

119873

119904=1

119860119904cos (2120587Ω

119904119909 + 120579119904) (A3)

119868119895119896= 119868119895119908+ 119868119895119909+ 119868119895119910+ 119868119895119911 (A4)

where 119908 = 1 2 3 119899V 119909 = 119899V + 1 119899V + 2 119910 = 119899V + 2 + 1119899V + 2 + 2 119899V + 2 + 119899

119887 119911 = 119899V + 2 + 119899

119887+ 1 119899V + 2 + 119899

119887+

2 119899V + 2 + 119899119887 + 119899119879Thefirst term 119868

1198681198951= 1198681198952= sdot sdot sdot = 119868

119895119899V= 0 (A5)

119868119895119909= 1198681198951199091

+ 1198681198951199092 (A6)

1198681198951199091

= 11986811989511990911

+ 11986811989511990912

119904)minus2 and 119861

119904= 2120587Ω

119871+ (119904 minus

11986811989511990911

= 1198961199051[

119873

119904=1

119860119904

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2minus120572119895cos (120579

119904) minus 119861119904119881 sin (120579

119904)

+ exp (120572119895119905)

times [120572119895cos (119861

119904119881119905 + 120579

119904)

+119861119904119881 cos (119861

119904119881119905 + 120579

119904)]

+4ℎ0119881

12057231198951198712

120572119895119871 (120572119895119905 + exp (minus120572

119895119905) minus 1)

minus119881

12057231198951198712

119895119905) + 2 ]

11986811989511990912

= 1198881199051[

119873

119904=1

119881119860119904119861119904

exp (minus120572119895119905)

119904)

+ exp (120572119895119905)

times [120572119895sin (119861

119904119881119905 + 120579

119904)

minus119861119904119881 sin (119861

119904119881119905 + 120579

119904)]

119895119905)

12057231198951198712

minus1

120572119895119871exp (120572

119895119905) minus 2]

1198681198951199092

= 11986811989511990921

+ 11986811989511990922

11986811989511990921

= 1198961199052[

119873

119904=1

119860119904

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2

times minus 120572119895cos (119861

119904119897V minus 120579119904) minus 119861119904119881 sin (119861

119904119897V minus 120579119904)

+ exp (120572119895119905) [120572119895cos (119861

119904(119897 minus 119881119905) minus 120579

119904)

+119861119904119881 sin (119861

119904(119897V minus 119881119905) + 120579119904)]

+1198812[2 minus exp (120572

119895119905) 120572119895119905 (120572119895119905 minus 2) + 2]

+4ℎ0

12057231198951198712

minus1205722

119895119897V (119897V + 119871) (exp (120572119895119905) minus 1)

+120572119895119881 (2119897V + 119871) [exp (120572119895119905) (120572119895119905 minus 1) + 1] ]

11986811989511990922

= 1198881199052[

119873

119904=1

119860119904119861119904119881

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2120572119895sin (119861

119904119897V minus 120579119904)

+ 119861119904119881 cos (119861

119904119897V minus 120579119904)

minus exp (120572119895119905) [120572119895sin (119861

119904(119897 minus 119881119905) minus 120579

119904)

+119861119904119881 cos (119861

119904(119897V minus 119881119905) + 120579119904)]

+4ℎ0119881 exp (minus120572

119895119905)

12057221198951198712

times 2119881 [exp (120572119895119905) (120572119895119905 minus 1) + 1]

minus120572119895(2119897V + 119871) (exp (120572119895119905) minus 1)]

119868119895119910=

119899119887

119896=1

119868119895119899V+2+119896

119886=1

119868(119895119899V+2+119896)119886

(119896 = 1 2 3 119899119887) (A11)

119876119904= 119861119904119881 119877

119896=119896120587119881

119871

119878119896=119896120587

119871 119885

119904= 119861119904119897V

119868(119895119899V+2+119896)

21198961199051

119873

119904=1

119860119904 ((2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) cos (120579

119904)

+2120572119895119876119904sin (120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times cos (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119876

119904+ 119877119896) 119905 + 120579

119904

minus (119877119896+ 119876119904)

times cos (119876119904+ 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

21198961199051[

4ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198762119904)3

times 119871 (1205722

119895+ 1198762

119904) [exp (120572

119895119905) sin (119876

119904119905)

times 1198762

119904(120572119895119905 + 1)

+1205722

119895(120572119895119905 minus 1)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904119905 + 120572119895(120572119895119905 minus 2)] minus 2120572

119895119876119904

+ 119881 (21198763

119904+ 119876119904exp (120572

119895119905)) cos (119876

119904119905)

times 1198764

1199041199052+ 21198762

119904(1205722

1198951199052minus 2120572119895119905 minus 1)

+1205722

119895(1205722

1198951199052minus 4120572119895119905 + 6)

minus exp (120572119895119905) sin (119876

119904119905)

times 1198764

119904(120572119895119905 + 2) + 120572

1198951198762

119904(1205722

1198951199052minus 3)

+1205723

119895(1205722

1198951199052minus 2120572119895119905 + 2) minus 6120572

119895119876119904]

119868(119895119899V+2+119896)

21198961199052

119873

119904=1

119860119904[ exp (minus120572

119895119905)

times (((119877119896+ 119876119904) cos (119878

119904minus 119885119904minus 120579119904)

+120572119895sin (119878119896minus 119885119904minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

times (((119877119896minus 119876119904) cos (119878

119896minus 119885119904+ 120579119904)

+120572119895sin (119878119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 119878

119896minus 119885119904+ 120579119904

minus (119877119896+ 119876119904) cos (119876

119904minus 119877119896) 119905

+119878119896minus 119885119904+ 120579119904)

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

+ ((120572119895sin 119878119896+ 119885119904minus (119876119904+ 119877) 119905 minus 120579

119904

+ (119877119896+ 119876119904) cos 119878

119896+ 119885119904

+ (119876119904+ 119877119896) 119905

minus120579119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1

21198961199052[

4ℎ0119881

1198712(1205722119895+ 1198812)

times sin (119881119905 minus 119897V) [1205721198951198972

V(1205722

119895+ 1198812)2

minus 119897V (1205722

119895+ 1198812)

times minus1205723

119895119871 minus 1205721198951198711198812

+ 21198813(120572119895119905 + 1)

+21205722

119895119881(120572119895119905 minus 1)

+ 119881 [119881 21205723

119895+ 120572119895119905(1205722

119895+ 1198812)2

+2119905 (1205724

119895minus 1198814) minus 6120572

1198951198812

minus 119871 (1205722

119895+ 1198812)

times 1205722

119895(120572119895119905 minus 1)+ 119881

2(120572119895119905 +1)]

+ 119881 cos (119897V minus 119881119905) [1198814(1198972

V + 119897V119871

+21205722

1198951199052minus 4120572119895119905 minus 2)

+ 1205722

1198951198812(21198972

V + 2119897V119871

+41205722

1198951199052+ 6)

+ 1205724

119895119897V (119897V + 119871)

minus 1198815119905 (2119897V + 119871)

times (120572119895119905 minus 1)

minus 1205723

119895119881 (2119897V + 119871)

times (120572119895119905 minus 2) + 119897

V1198816]]

119868(119895119899V+2+119896)

21198881199051

119873

119904=1

119860119904119876119904( minus (2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) sin (120579

119904)

minus2120572119895119876119904cos (120579

119904))

times (1205722

119895+ (119876119904minus 119877)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times sin (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos (119876

119904+ 119877119896) 119905 + 120579

119904

+ (119877119896+ 119876119904)

times sin (119876119904+ 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199051

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) cos (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) sin (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

21198881199052

119873

119904=1

119860119904119876119904

times [ exp (minus120572119895119905 ( ( (119877

119896+ 119876119904) sin (119878

119896+ 119885119904minus 120579119904)

minus120572119895cos (119878

119896+ 119885119904minus 120579119904))

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

times ( ( (119876119904minus 119877119896) sin (119878

119896minus 119885119904+ 120579119904)

+120572119895cos (119878

119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos 119878

119896+ 119885119904minus (119876119904+ 119877119896) 119905 minus 120579

119904

minus (119877119896+ 119876119904) sin 119878

119896+ 119885119904

minus (119876119904+ 119877119896) 119905 minus 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos 119878

119896minus 119885119904+ (119876119904minus 119877119896) 119905 + 120579

119904

+ (119876119904minus 119877119896) sin 119878

119896minus 119885119904

+ (119876119904minus 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199052

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) sin (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881 + exp (120572119895119905) 119896120587119881 cos(119896120587119881119905

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881cos(119896120587119863119871

119871)

minus 119871120572119895sin(119896120587119863

119871)

119871)]

119868(119895119899V+2+119896)

= minus21198711198812(1198981199081

+ 1198981199082)

119873

119904=1

1198601199041198612

119904[(((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881 cos [(119876

119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

119904119871)119881 cos 120579

119904

+119871120572119895sin 120579119904)

times (1198612

11990411987121198812

+ 21198961205871198611199041198711198812

+ 119896212058721198812

+11987121205722

119895)minus1

+ ((119896120587 minus 119861119904119871)119881 cos 120579

119904

minus119871120572119895sin 120579119904)

times ((119896120587 minus 119861119904119871)2

1198812

+11987121205722

119895)minus1

minus8ℎ01198812(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

times [119896120587119881

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119868119895119911=

119899119905

119896=1

119868119895119899V+2+119899119887+119896

119868119895119899V+2+119899119887+119897

119886=1

119868(119895119899V+2+119899119887+119897)119886

(119896 = 1 2 3 119899119879) (A14)

119868(119895119899V+2+119899119887+119897)1

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895cos (120579

119904) + (119876

119904minus 119877119896) sin (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895cos (120579

119904) + (119876

119904+ 119877119896) sin (120579

119904)

1205722119895+ (119876119904+ 119877119896)2

+ ((120572119895cos [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119876119904minus 119877119896) sin [(119876

119904minus 119877119896) 119905 minus 120579

119904])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [(119876

119904+ 119877119896) 119905 + 120579

119904]

+ (119876119904+ 119877119896) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)2

=41198961199051119890119909ℎ0119871

119881(1205722119895+ 119877119896

times [ (1205722

119895+ 119877119896

2) exp (minus120572

119895119905)

times 1205722

119895+ exp (minus120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)] minus 119877

119896

+ 2120572119895exp (minus120572

119895119905) (1205722

119895+ 3119877119896

+ 119877119896sin (119877

119896119905) 6120572

119895+ 1199052(1205722

119895+ 119877119896

+4120572119895119905 (1205722

119895+ 119877119896

2) minus 2119877

119896

+ cos (119877119896119905) 2119905 (119877

119896

4minus 1205724

119895)

+ 2120572119895(1205722

119895minus 3119877119896

times 1205721198951199052(1205722

119895+ 119877119896

119868(119895119899V+2+119899119887+119897)3

= minus1

21198961199052119890119909

119873

119904=1

[119860119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119877119896minus 119876119904) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) sin (119861

119904119897V + 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos [(119877

119896minus 119876119904) 119905 + (119861

119904119897V minus 119878119896 minus 120579119904)]

+ (119877119896minus 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [minus (119877

119896+ 119876119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

minus (119877119896+ 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)4

=41198961199052119890119909ℎ0

1198712[((119897V (119897V + 119871) exp (minus120572119895119905)

times 120572119895cos (119878

119896minus 119877119896119905)

minus119877119896sin (119878119896minus 119877119896119905)

+119877119896sin (119878) minus 120572

119895cos (119878

119896))

times (1205722

119895+ 119877119896

2)minus1

+2119871119881 exp (minus120572

119895119905)

(1205722119895+ 119877119896

times (1205722

119895+ 119877119896

2) cos (119878

119896)

+ exp (120572119895119905) cos (119878

119896minus 119877119896119905)

times [1198772(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896[119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

119895119877119896sin (119878119896)

minus1198812

(1205722119895+ 119877119896

times119877119896sin (119878119896minus 119877119896119905)[61205722

119895+1199052(1205722

119895+ 119877119896

minus 4120572119895119905 (1205722

119895+ 119877119896

minus2119877119896

+ 2 exp (minus120572119895119905)

times [(1205723

119895minus 3120572119895119877119896

2) cos (119878

119896)

+ 119877119896(119877119896

2minus 31205722

119895)

times sin (119878119896) ]

minus [2119905 (119877119896

4minus 1205724

119895) + 2120572

119895(1205722

119895minus 3119877119896

+1205721198951199052(1205722

119895+119877119896

]cos (119878119896minus 119877119896119905)]

119868(119895119899V+2+119899119887+119897)5

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895sin (120579119904) + (119876

119904minus 119877119896) cos (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895sin (120579119904) minus (119876

119904+ 119877119896) cos (120579

119904)

1205722119895+ (119876119904+ 119877)2

+ ((120572119895sin [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119877119896minus 119876119904) cos [(119876

119904minus 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895sin [(119876

119904+ 119877119896) 119905 + 120579

119904]

minus (119876119904+ 119877) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)6

=41198881199051119890119909ℎ0119871

(1205722119895+ 119877119896

2)3(119881

119871)

times [(1205722

119895+ 119877119896

2) 120572119895cos (119877

119896119905)

+ 119877119896sin (119877

119896119905) minus 120572

119895exp (minus120572

119895119905)

+ 2 exp (minus120572119895119905) 1205722

119895+ exp (120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

minus119877119896

119868(119895119899V+2+119899119887+119897)7

=41198881199052119890119909ℎ0

119881(1205722119895+ 119877119896

times 119897V (1205722

119895+ 119877119896

2) + 119871 (120572

119895+ 119877119896

minus2119881 [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

+ cos (119878119896minus 119877119896119905)

times 119877119896

2[120572119895(119897V + 119871) minus 2119881 (120572

119895119905 + 1)]

+1205722

119895[120572119895(119897V + 119871) + 2119881 (1 minus 120572

119895119905)]

+ exp (minus120572119895119905) 119877119896sin (119878119896)

times [119897V (1205722

119895+ 119877119896

+119871 (1205722

119895+ 119877119896

2)+4120572119895119881]

+ cos (119878119896)

times [119877119896

22119881 minus 120572

119895(119897V + 119871)

minus 1205722

119895120572119895(119897V + 119871)

+2119881]]

119868(119895119899V+2+119899119887+119897)8

21198881199052119890119909

119873

119904=1

[119860119904119876119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119876119904minus 119877119896) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) cos (119861

119904119897V + 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin [(119877 minus 119876

119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)]

+ (119876119904minus 119877119896) cos [(119877

119896minus 119876119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)])

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

minus ((120572119895sin [minus (119877 + 119876

119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

+ (119877119896+ 119876119904) cos [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)9

= minus119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [exp (120572119895119905) 119896120587119881 sin(119896120587119881119905

119871)

minus119871120572119895+ exp 119871120572

119895(120572119895119905) cos(119896120587119881119905

119871)]

minus 119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881sin(119896120587119863119871

119871)

minus 119890119909119871120572119895cos(119896120587119863

119871)

minus exp (120572119895119905)

times cos(119896120587119863 minus 119881119905

119871)]

119868(119895119899V+2+119899119887+119897)10

= minus21198711198812(1198981199081

+ 1198981199082) 119890119909

119873

119904=1

1198601199041198612

119904[ (((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881

times cos [(119876119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus exp (minus120572119895119905)

times ((minus (119896120587 +119861119904119871)119881 cos 120579

119904+119871120572119895sin 120579119904)

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

+ (((119896120587 minus 119861119904119871)119881 cos 120579

119904minus 119871120572119895sin 120579119904)

times((119896120587 minus 119861119904119871)2

1198812+ 11987121205722

119895)minus1

minus8ℎ01198812119890119909(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

119871)

+119871120572119895sin(119896120587119881119905

119871)]

References

RoboticsJournal of

Journal of

RotatingMachinery

VLSI Design

Shock and Vibration

Journal of

Volume 2014

SensorsJournal of

Propagation

times 120601119879119897(1199091)

+ 11988811990522(119905) minus 119881

119899119879

119897=1

1206011015840

119879119897(1199092) 120574119897(119905)

minus119881

119899119879

119897=1

120601119879119897(1199092) 120574119897(119905) minus 119881ℎ

1015840(1199092)

times 120601119879119897(1199092)

minus 1198981+1

2119898V119897V119892120601119879119897 (1199091) minus 1198982 +

2119898V119897V

times 119892120601119879119897(1199092)

times 1198981

119899119879

119897=1

120601119879119897(1199091) 120574119897(119905) + 2119881

119899119879

119897=1

1206011015840

119879119897(1199091) 120574119897(119905)

+1198812

119899119879

119897=1

12060110158401015840

119879119897(1199091) 120574119897(119905) + 4119881ℎ

10158401015840(1199091)

times 120601119879119897(1199091)

+ 1198982

119899119879

119897=1

120601119879119897(1199092) 120574119897(119905) + 2119881

119899119879

119897=1

1206011015840

119879119896(1199092) 120574119897(119905)

+1198812

119899119879

119896=1

12060110158401015840

119879119896(1199092) 120574119897(119905) + 4119881ℎ

10158401015840(1199092)

times 120601119879119897(1199092) ]

119887+

[119872] 119903 (119905) + [119862] 119903 (119905) + [119870] 119903 (119905) = 119865 (119905) (30)

where 119903(119905) = 1205781(119905) 1205782(119905) 120578

119899V(119905) 1199111(119905) 1199112(119905) 1199021(119905)1199022(119905) 119902

119899119887(119905) 1205741(119905) 1205742(119905) 120574

(119905) + [119860] 119901 (119905) = 119875 (119905) (31)

119901 (119905) = 119903 (119905)

119903 (119905) 119875 (119905) =

[119872]minus1119865 (119905)

[119860] = [119872]minus1[119862]

minus [119868]

[119872]minus1[119870]

11199062 119906

2119899] (33)

[119880]minus1[119860] [119880] = diagonal [120572

1 1205722 1205723 120572

2119899] (34)

Let us assume119901 = [119880] V (35)

V119895(119905) + 120572

119895V119895(119905) = 119877

119895(119905) 119895 = 1 2 3 2119899 (36)

119877119895=

119899

119896=1

1199061015840

119895119904

119899

119904=1

1198981015840

119904119896119865119896(119905) (37)

1199061015840

119904119896

may be expressed as

V119895(119905) = 119883

0119895exp (minus120572

119895119905) + int

minusinfin

119867119895(120596 119905) 119889119878 [119865

119896(120596)] (38)

where 1198830119895

119867119895(120596 119905) =

minus119894120596 + 120572119895

119903119898(119905) =

2119899

119895=1

119906119898+119899119895

1198830119895exp (minus120572

119895119905)

2119899

119895=1

119906119898+119899119895

119899

119904=1

1199061015840

119895119904

119899

119896=1

1198981015840

119904119896int

minusinfin

119867119895(120596 119905) 119889119878 (119865

119896(120596))

119898 = 1 2 3 119899

119903119875119898

(119905) =

2119899

119895=1

119906119898+119899119895

119899

119904=1

1199061015840

119895119904

119899

119896=1

1198981015840

119904119896int

minusinfin

exp (minus119894120596119905)minus119894120596 + 120572

119895

119865119896(120596) 119889120596

119903119875119898

(119905)

2119899

119895=1

119906119898+119899119895

119899

119904=1

1199061015840

119895119904

119899

119896=1

1198981015840

119904119896

times int

minusinfin

119865119896(120591) [

2120587int

minusinfin

exp [119894120596 (120591 minus 119905)]minus119894120596 + 120572

119895

119889120596]119889120591

119903119875119898

(119905) =

2119899

119895=1

119906119898+119899119895

119899

119904=1

1199061015840

119895119904

119899

119896=1

1198981015840

119904119896119868119895119896 (43)

119868119895119896= int

119905

exp [120572119895(120591 minus 119905)] 119865

119896(120591) 119889120591 (44)

119868119895119896= 119868119895119908+ 119868119895119909+ 119868119895119910+ 119868119895119911 (45)

where 119908 = 1 2 3 119899V 119909 = 119899V + 1 119899V + 2 119910 = 119899V + 3119899V+4 119899V+2+119899119887 119911 = 119899V+119899119887+3 119899V+119899119887+4 119899V+119899119887+119899119879+2

119884(119905119896) and

0 05 1 15 2 25

Time (s)

50E minus 4

00E + 0

minus50E minus 4

minus10E minus 3

minus15E minus 3

minus20E minus 3

119887119868119887) 248 times 10

119887119869119887) 248 times 10

5 kgm2m 119864119887= 32GPa deck = 2512GPa

1199051 1198961199052) 197 times 10

6Nm 474 times 106Nm

119887119868119887) 37times

119887119869119887) 1695 times 10

10N-m2

063 063 063

300 300

265100

(1198961199051 1198961199052) 09 times 10

119904= 16 times

119889119911)

075 120 075

0 05 1 15 2 25 3 35 4

Time (s)10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

120583dy

50kmh60kmh80kmh

0 05 1 15 2 25 3 35

Time (s)

minus80E minus 3

40E minus 3

minus10E minus 18

minus40E minus 3

minus80E minus 3

minus12E minus 2

120583y

50kmh60kmh80kmh

0 05 1 15 2 25 3 35

Time (s)40E minus 2

00E + 0

minus40E minus 2

minus80E minus 2

minus12E minus 1

minus16E minus 1

120583ay

50kmh60kmh80kmh

0 05 1 15 2 25 3 35 4Time (s)

40E minus 5

30E minus 5

20E minus 5

10E minus 5

00E + 0

minus10E minus 5

120590dy

50kmh60kmh80kmh

119889119911) velocity (120583V119911)

119886119911) is shown in

0 05 1 15 2 25 3 35Time (s)

16E minus 3

12E minus 3

80E minus 4

40E minus 4

00E + 0

minus40E minus 4

120590y

50kmh60kmh80kmh

0 05 1 15 2 25 3 35Time (s)

16E minus 2

12E minus 2

80E minus 3

40E minus 3

00E + 0

120590ay

50kmh60kmh80kmh

0 05 1 15 2 25 3Time (s)

80E minus 3

40E minus 3

00E + 0

minus40E minus 3

minus80E minus 3

minus12E minus 2

minus16E minus 2

120583dz

50kmh60kmh80kmh

0 05 1 15 2 25 3Time (s)

12E minus 1

80E minus 2

40E minus 2

00E + 0

minus40E minus 2

minus80E minus 2

120583z

50kmh60kmh80kmh

0 1 2 3Time (s)

80E minus 1

40E minus 1

00E + 0

minus40E minus 1

minus80E minus 1

50kmh60kmh80kmh

120583az

0 05 1 15 2 25 3Time (s)

10E minus 3

80E minus 4

60E minus 4

40E minus 4

20E minus 4

00E + 0

120590dz (m

50kmh60kmh80kmh

0 1 2 3Time (s)

80E minus 3

60E minus 3

40E minus 3

20E minus 3

00E + 0

120590z

50kmh60kmh80kmh

0 05 1 15 2 25 3Time (s)

40E minus 2

30E minus 2

20E minus 2

10E minus 2

00E + 0

120590az

50kmh60kmh80kmh

A B C D

50E minus 04

40E minus 04

30E minus 04

20E minus 04

10E minus 04

00E + 00

120583dy

A B C D

40E minus 05

30E minus 05

20E minus 05

10E minus 05

00E + 00

120590dy

0 025 05 075 1 125 15 175Time (s)

30E + 5

25E + 5

20E + 5

15E + 5

10E + 5

50E + 4

00E + 0

120583F

0 025 05 075 1 125 15 175Time (s)

25E + 4

20E + 4

15E + 4

10E + 4

50E + 3

00E + 0

minus50 + 3

120590F

0 05 1 15 2 25 3 35

Time (s)

10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

minus40E minus 4

120583dy

82 times 1010 N-m2

0 05 1 15 2 25 3 35Time (s)

20E minus 5

16E minus 5

12E minus 5

80E minus 6

40E minus 6

00E + 0

120590dy

6N-m2 (B) 119864V119868V = 64 times 107N-m2

and (C) 119864V119868V = 82 times 1010N-m2

0 05 1 15 2 25 3 35

Time (s)

10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

120583dy

119909) (A) 119890

119909= 05m (B) 119890

119909= 1m and

(C) 119890119909= 15m

0 05 1 15 2 25 3 35Time (s)

30E minus 5

25E minus 5

20E minus 5

15E minus 5

10E minus 5

50E minus 6

00E + 0

120590dy

119909) (A) 119890

119909= 05m (B) 119890

119909=

1m and (C) 119890119909= 15m

119884dynamic =1003816100381610038161003816120583119884 (119909119896 119905119896) + 120590119884 (119909119896 119905119896)

1003816100381610038161003816 (46)

0 0002 0004 0006 0008 001

GbJbEbIb

119887119869119887119864119887119868119887) for different

107 N-m2 and (C) 119864V119868V = 82 times 10

10 N-m2

119884static (47)

119887119868119887)

5 Conclusions

1520 25 30 35 40

45 5055

404550556065707580146148

15152154156158

16162164

times103

8013514

Vehicle speed (kmh)

14161820222426283013

Appendix

119865119895(119905) = 0 for119895 = 1 2 3 119899V

= 119896119905119895minus119899V

ℎ (119909119895minus119899V

) + 119888119905119895minus119899V

ℎ (119909119895minus119899V

For119895 = 119899V + 1 119899V + 2

= 1198961199051ℎ (1199091) 120601119887119895minus(119899V+2) (1199091) + 1198961199052ℎ (1199092) 120601119887119895minus(119899V+2) (1199092)

+ 1198881199051ℎ (1199091) 120601119887119895minus(119899V+2) (1199091) + 1198881199052ℎ (1199092) 120601119887119895minus(119899V+2) (1199092)

For 119895 = 119899V + 3 119899V + 4 119899V + 2 + 119899119887

= 1198901199091198961199051ℎ (1199091) 120601119905119895minus(119899V+2+119899119887)

(1199091)

+ 1198961199052ℎ (1199092) 120601119905119895minus(119899V+2+119899119887)

(1199092)

+ 1198881199051ℎ (1199091) 120601119905119895minus(119899V+2+119899119887)

(1199091)

+1198881199052ℎ1015840(1199092) 120601119905119895minus(119899V+2+119899119887)

(1199092)

For 119895 = 119899V + 119899119887 + 3

119899V + 119899119887 + 4 119899V + 119899119887 + 119899119905 + 2

ℎ119898(119909) =

1198712119909 (119871 minus 119909) (A2)

ℎ119903(119909) =

119873

119904=1

119860119904cos (2120587Ω

119904119909 + 120579119904) (A3)

119868119895119896= 119868119895119908+ 119868119895119909+ 119868119895119910+ 119868119895119911 (A4)

where 119908 = 1 2 3 119899V 119909 = 119899V + 1 119899V + 2 119910 = 119899V + 2 + 1119899V + 2 + 2 119899V + 2 + 119899

119887 119911 = 119899V + 2 + 119899

119887+ 1 119899V + 2 + 119899

119887+

2 119899V + 2 + 119899119887 + 119899119879Thefirst term 119868

1198681198951= 1198681198952= sdot sdot sdot = 119868

119895119899V= 0 (A5)

119868119895119909= 1198681198951199091

+ 1198681198951199092 (A6)

1198681198951199091

= 11986811989511990911

+ 11986811989511990912

119904)minus2 and 119861

119904= 2120587Ω

119871+ (119904 minus

11986811989511990911

= 1198961199051[

119873

119904=1

119860119904

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2minus120572119895cos (120579

119904) minus 119861119904119881 sin (120579

119904)

+ exp (120572119895119905)

times [120572119895cos (119861

119904119881119905 + 120579

119904)

+119861119904119881 cos (119861

119904119881119905 + 120579

119904)]

+4ℎ0119881

12057231198951198712

120572119895119871 (120572119895119905 + exp (minus120572

119895119905) minus 1)

minus119881

12057231198951198712

119895119905) + 2 ]

11986811989511990912

= 1198881199051[

119873

119904=1

119881119860119904119861119904

exp (minus120572119895119905)

119904)

+ exp (120572119895119905)

times [120572119895sin (119861

119904119881119905 + 120579

119904)

minus119861119904119881 sin (119861

119904119881119905 + 120579

119904)]

119895119905)

12057231198951198712

minus1

120572119895119871exp (120572

119895119905) minus 2]

1198681198951199092

= 11986811989511990921

+ 11986811989511990922

11986811989511990921

= 1198961199052[

119873

119904=1

119860119904

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2

times minus 120572119895cos (119861

119904119897V minus 120579119904) minus 119861119904119881 sin (119861

119904119897V minus 120579119904)

+ exp (120572119895119905) [120572119895cos (119861

119904(119897 minus 119881119905) minus 120579

119904)

+119861119904119881 sin (119861

119904(119897V minus 119881119905) + 120579119904)]

+1198812[2 minus exp (120572

119895119905) 120572119895119905 (120572119895119905 minus 2) + 2]

+4ℎ0

12057231198951198712

minus1205722

119895119897V (119897V + 119871) (exp (120572119895119905) minus 1)

+120572119895119881 (2119897V + 119871) [exp (120572119895119905) (120572119895119905 minus 1) + 1] ]

11986811989511990922

= 1198881199052[

119873

119904=1

119860119904119861119904119881

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2120572119895sin (119861

119904119897V minus 120579119904)

+ 119861119904119881 cos (119861

119904119897V minus 120579119904)

minus exp (120572119895119905) [120572119895sin (119861

119904(119897 minus 119881119905) minus 120579

119904)

+119861119904119881 cos (119861

119904(119897V minus 119881119905) + 120579119904)]

+4ℎ0119881 exp (minus120572

119895119905)

12057221198951198712

times 2119881 [exp (120572119895119905) (120572119895119905 minus 1) + 1]

minus120572119895(2119897V + 119871) (exp (120572119895119905) minus 1)]

119868119895119910=

119899119887

119896=1

119868119895119899V+2+119896

119886=1

119868(119895119899V+2+119896)119886

(119896 = 1 2 3 119899119887) (A11)

119876119904= 119861119904119881 119877

119896=119896120587119881

119871

119878119896=119896120587

119871 119885

119904= 119861119904119897V

119868(119895119899V+2+119896)

21198961199051

119873

119904=1

119860119904 ((2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) cos (120579

119904)

+2120572119895119876119904sin (120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times cos (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119876

119904+ 119877119896) 119905 + 120579

119904

minus (119877119896+ 119876119904)

times cos (119876119904+ 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

21198961199051[

4ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198762119904)3

times 119871 (1205722

119895+ 1198762

119904) [exp (120572

119895119905) sin (119876

119904119905)

times 1198762

119904(120572119895119905 + 1)

+1205722

119895(120572119895119905 minus 1)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904119905 + 120572119895(120572119895119905 minus 2)] minus 2120572

119895119876119904

+ 119881 (21198763

119904+ 119876119904exp (120572

119895119905)) cos (119876

119904119905)

times 1198764

1199041199052+ 21198762

119904(1205722

1198951199052minus 2120572119895119905 minus 1)

+1205722

119895(1205722

1198951199052minus 4120572119895119905 + 6)

minus exp (120572119895119905) sin (119876

119904119905)

times 1198764

119904(120572119895119905 + 2) + 120572

1198951198762

119904(1205722

1198951199052minus 3)

+1205723

119895(1205722

1198951199052minus 2120572119895119905 + 2) minus 6120572

119895119876119904]

119868(119895119899V+2+119896)

21198961199052

119873

119904=1

119860119904[ exp (minus120572

119895119905)

times (((119877119896+ 119876119904) cos (119878

119904minus 119885119904minus 120579119904)

+120572119895sin (119878119896minus 119885119904minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

times (((119877119896minus 119876119904) cos (119878

119896minus 119885119904+ 120579119904)

+120572119895sin (119878119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 119878

119896minus 119885119904+ 120579119904

minus (119877119896+ 119876119904) cos (119876

119904minus 119877119896) 119905

+119878119896minus 119885119904+ 120579119904)

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

+ ((120572119895sin 119878119896+ 119885119904minus (119876119904+ 119877) 119905 minus 120579

119904

+ (119877119896+ 119876119904) cos 119878

119896+ 119885119904

+ (119876119904+ 119877119896) 119905

minus120579119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1

21198961199052[

4ℎ0119881

1198712(1205722119895+ 1198812)

times sin (119881119905 minus 119897V) [1205721198951198972

V(1205722

119895+ 1198812)2

minus 119897V (1205722

119895+ 1198812)

times minus1205723

119895119871 minus 1205721198951198711198812

+ 21198813(120572119895119905 + 1)

+21205722

119895119881(120572119895119905 minus 1)

+ 119881 [119881 21205723

119895+ 120572119895119905(1205722

119895+ 1198812)2

+2119905 (1205724

119895minus 1198814) minus 6120572

1198951198812

minus 119871 (1205722

119895+ 1198812)

times 1205722

119895(120572119895119905 minus 1)+ 119881

2(120572119895119905 +1)]

+ 119881 cos (119897V minus 119881119905) [1198814(1198972

V + 119897V119871

+21205722

1198951199052minus 4120572119895119905 minus 2)

+ 1205722

1198951198812(21198972

V + 2119897V119871

+41205722

1198951199052+ 6)

+ 1205724

119895119897V (119897V + 119871)

minus 1198815119905 (2119897V + 119871)

times (120572119895119905 minus 1)

minus 1205723

119895119881 (2119897V + 119871)

times (120572119895119905 minus 2) + 119897

V1198816]]

119868(119895119899V+2+119896)

21198881199051

119873

119904=1

119860119904119876119904( minus (2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) sin (120579

119904)

minus2120572119895119876119904cos (120579

119904))

times (1205722

119895+ (119876119904minus 119877)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times sin (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos (119876

119904+ 119877119896) 119905 + 120579

119904

+ (119877119896+ 119876119904)

times sin (119876119904+ 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199051

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) cos (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) sin (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

21198881199052

119873

119904=1

119860119904119876119904

times [ exp (minus120572119895119905 ( ( (119877

119896+ 119876119904) sin (119878

119896+ 119885119904minus 120579119904)

minus120572119895cos (119878

119896+ 119885119904minus 120579119904))

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

times ( ( (119876119904minus 119877119896) sin (119878

119896minus 119885119904+ 120579119904)

+120572119895cos (119878

119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos 119878

119896+ 119885119904minus (119876119904+ 119877119896) 119905 minus 120579

119904

minus (119877119896+ 119876119904) sin 119878

119896+ 119885119904

minus (119876119904+ 119877119896) 119905 minus 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos 119878

119896minus 119885119904+ (119876119904minus 119877119896) 119905 + 120579

119904

+ (119876119904minus 119877119896) sin 119878

119896minus 119885119904

+ (119876119904minus 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199052

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) sin (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881 + exp (120572119895119905) 119896120587119881 cos(119896120587119881119905

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881cos(119896120587119863119871

119871)

minus 119871120572119895sin(119896120587119863

119871)

119871)]

119868(119895119899V+2+119896)

= minus21198711198812(1198981199081

+ 1198981199082)

119873

119904=1

1198601199041198612

119904[(((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881 cos [(119876

119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

119904119871)119881 cos 120579

119904

+119871120572119895sin 120579119904)

times (1198612

11990411987121198812

+ 21198961205871198611199041198711198812

+ 119896212058721198812

+11987121205722

119895)minus1

+ ((119896120587 minus 119861119904119871)119881 cos 120579

119904

minus119871120572119895sin 120579119904)

times ((119896120587 minus 119861119904119871)2

1198812

+11987121205722

119895)minus1

minus8ℎ01198812(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

times [119896120587119881

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119868119895119911=

119899119905

119896=1

119868119895119899V+2+119899119887+119896

119868119895119899V+2+119899119887+119897

119886=1

119868(119895119899V+2+119899119887+119897)119886

(119896 = 1 2 3 119899119879) (A14)

119868(119895119899V+2+119899119887+119897)1

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895cos (120579

119904) + (119876

119904minus 119877119896) sin (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895cos (120579

119904) + (119876

119904+ 119877119896) sin (120579

119904)

1205722119895+ (119876119904+ 119877119896)2

+ ((120572119895cos [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119876119904minus 119877119896) sin [(119876

119904minus 119877119896) 119905 minus 120579

119904])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [(119876

119904+ 119877119896) 119905 + 120579

119904]

+ (119876119904+ 119877119896) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)2

=41198961199051119890119909ℎ0119871

119881(1205722119895+ 119877119896

times [ (1205722

119895+ 119877119896

2) exp (minus120572

119895119905)

times 1205722

119895+ exp (minus120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)] minus 119877

119896

+ 2120572119895exp (minus120572

119895119905) (1205722

119895+ 3119877119896

+ 119877119896sin (119877

119896119905) 6120572

119895+ 1199052(1205722

119895+ 119877119896

+4120572119895119905 (1205722

119895+ 119877119896

2) minus 2119877

119896

+ cos (119877119896119905) 2119905 (119877

119896

4minus 1205724

119895)

+ 2120572119895(1205722

119895minus 3119877119896

times 1205721198951199052(1205722

119895+ 119877119896

119868(119895119899V+2+119899119887+119897)3

= minus1

21198961199052119890119909

119873

119904=1

[119860119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119877119896minus 119876119904) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) sin (119861

119904119897V + 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos [(119877

119896minus 119876119904) 119905 + (119861

119904119897V minus 119878119896 minus 120579119904)]

+ (119877119896minus 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [minus (119877

119896+ 119876119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

minus (119877119896+ 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)4

=41198961199052119890119909ℎ0

1198712[((119897V (119897V + 119871) exp (minus120572119895119905)

times 120572119895cos (119878

119896minus 119877119896119905)

minus119877119896sin (119878119896minus 119877119896119905)

+119877119896sin (119878) minus 120572

119895cos (119878

119896))

times (1205722

119895+ 119877119896

2)minus1

+2119871119881 exp (minus120572

119895119905)

(1205722119895+ 119877119896

times (1205722

119895+ 119877119896

2) cos (119878

119896)

+ exp (120572119895119905) cos (119878

119896minus 119877119896119905)

times [1198772(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896[119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

119895119877119896sin (119878119896)

minus1198812

(1205722119895+ 119877119896

times119877119896sin (119878119896minus 119877119896119905)[61205722

119895+1199052(1205722

119895+ 119877119896

minus 4120572119895119905 (1205722

119895+ 119877119896

minus2119877119896

+ 2 exp (minus120572119895119905)

times [(1205723

119895minus 3120572119895119877119896

2) cos (119878

119896)

+ 119877119896(119877119896

2minus 31205722

119895)

times sin (119878119896) ]

minus [2119905 (119877119896

4minus 1205724

119895) + 2120572

119895(1205722

119895minus 3119877119896

+1205721198951199052(1205722

119895+119877119896

]cos (119878119896minus 119877119896119905)]

119868(119895119899V+2+119899119887+119897)5

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895sin (120579119904) + (119876

119904minus 119877119896) cos (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895sin (120579119904) minus (119876

119904+ 119877119896) cos (120579

119904)

1205722119895+ (119876119904+ 119877)2

+ ((120572119895sin [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119877119896minus 119876119904) cos [(119876

119904minus 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895sin [(119876

119904+ 119877119896) 119905 + 120579

119904]

minus (119876119904+ 119877) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)6

=41198881199051119890119909ℎ0119871

(1205722119895+ 119877119896

2)3(119881

119871)

times [(1205722

119895+ 119877119896

2) 120572119895cos (119877

119896119905)

+ 119877119896sin (119877

119896119905) minus 120572

119895exp (minus120572

119895119905)

+ 2 exp (minus120572119895119905) 1205722

119895+ exp (120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

minus119877119896

119868(119895119899V+2+119899119887+119897)7

=41198881199052119890119909ℎ0

119881(1205722119895+ 119877119896

times 119897V (1205722

119895+ 119877119896

2) + 119871 (120572

119895+ 119877119896

minus2119881 [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

+ cos (119878119896minus 119877119896119905)

times 119877119896

2[120572119895(119897V + 119871) minus 2119881 (120572

119895119905 + 1)]

+1205722

119895[120572119895(119897V + 119871) + 2119881 (1 minus 120572

119895119905)]

+ exp (minus120572119895119905) 119877119896sin (119878119896)

times [119897V (1205722

119895+ 119877119896

+119871 (1205722

119895+ 119877119896

2)+4120572119895119881]

+ cos (119878119896)

times [119877119896

22119881 minus 120572

119895(119897V + 119871)

minus 1205722

119895120572119895(119897V + 119871)

+2119881]]

119868(119895119899V+2+119899119887+119897)8

21198881199052119890119909

119873

119904=1

[119860119904119876119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119876119904minus 119877119896) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) cos (119861

119904119897V + 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin [(119877 minus 119876

119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)]

+ (119876119904minus 119877119896) cos [(119877

119896minus 119876119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)])

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

minus ((120572119895sin [minus (119877 + 119876

119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

+ (119877119896+ 119876119904) cos [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)9

= minus119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [exp (120572119895119905) 119896120587119881 sin(119896120587119881119905

119871)

minus119871120572119895+ exp 119871120572

119895(120572119895119905) cos(119896120587119881119905

119871)]

minus 119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881sin(119896120587119863119871

119871)

minus 119890119909119871120572119895cos(119896120587119863

119871)

minus exp (120572119895119905)

times cos(119896120587119863 minus 119881119905

119871)]

119868(119895119899V+2+119899119887+119897)10

= minus21198711198812(1198981199081

+ 1198981199082) 119890119909

119873

119904=1

1198601199041198612

119904[ (((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881

times cos [(119876119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus exp (minus120572119895119905)

times ((minus (119896120587 +119861119904119871)119881 cos 120579

119904+119871120572119895sin 120579119904)

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

+ (((119896120587 minus 119861119904119871)119881 cos 120579

119904minus 119871120572119895sin 120579119904)

times((119896120587 minus 119861119904119871)2

1198812+ 11987121205722

119895)minus1

minus8ℎ01198812119890119909(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

119871)

+119871120572119895sin(119896120587119881119905

119871)]

References

RoboticsJournal of

Journal of

RotatingMachinery

VLSI Design

Shock and Vibration

Journal of

Volume 2014

SensorsJournal of

Propagation

119903119875119898

(119905) =

2119899

119895=1

119906119898+119899119895

119899

119904=1

1199061015840

119895119904

119899

119896=1

1198981015840

119904119896int

minusinfin

exp (minus119894120596119905)minus119894120596 + 120572

119895

119865119896(120596) 119889120596

119903119875119898

(119905)

2119899

119895=1

119906119898+119899119895

119899

119904=1

1199061015840

119895119904

119899

119896=1

1198981015840

119904119896

times int

minusinfin

119865119896(120591) [

2120587int

minusinfin

exp [119894120596 (120591 minus 119905)]minus119894120596 + 120572

119895

119889120596]119889120591

119903119875119898

(119905) =

2119899

119895=1

119906119898+119899119895

119899

119904=1

1199061015840

119895119904

119899

119896=1

1198981015840

119904119896119868119895119896 (43)

119868119895119896= int

119905

exp [120572119895(120591 minus 119905)] 119865

119896(120591) 119889120591 (44)

119868119895119896= 119868119895119908+ 119868119895119909+ 119868119895119910+ 119868119895119911 (45)

where 119908 = 1 2 3 119899V 119909 = 119899V + 1 119899V + 2 119910 = 119899V + 3119899V+4 119899V+2+119899119887 119911 = 119899V+119899119887+3 119899V+119899119887+4 119899V+119899119887+119899119879+2

119884(119905119896) and

0 05 1 15 2 25

Time (s)

50E minus 4

00E + 0

minus50E minus 4

minus10E minus 3

minus15E minus 3

minus20E minus 3

119887119868119887) 248 times 10

119887119869119887) 248 times 10

5 kgm2m 119864119887= 32GPa deck = 2512GPa

1199051 1198961199052) 197 times 10

6Nm 474 times 106Nm

119887119868119887) 37times

119887119869119887) 1695 times 10

10N-m2

063 063 063

300 300

265100

(1198961199051 1198961199052) 09 times 10

119904= 16 times

119889119911)

075 120 075

0 05 1 15 2 25 3 35 4

Time (s)10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

120583dy

50kmh60kmh80kmh

0 05 1 15 2 25 3 35

Time (s)

minus80E minus 3

40E minus 3

minus10E minus 18

minus40E minus 3

minus80E minus 3

minus12E minus 2

120583y

50kmh60kmh80kmh

0 05 1 15 2 25 3 35

Time (s)40E minus 2

00E + 0

minus40E minus 2

minus80E minus 2

minus12E minus 1

minus16E minus 1

120583ay

50kmh60kmh80kmh

0 05 1 15 2 25 3 35 4Time (s)

40E minus 5

30E minus 5

20E minus 5

10E minus 5

00E + 0

minus10E minus 5

120590dy

50kmh60kmh80kmh

119889119911) velocity (120583V119911)

119886119911) is shown in

0 05 1 15 2 25 3 35Time (s)

16E minus 3

12E minus 3

80E minus 4

40E minus 4

00E + 0

minus40E minus 4

120590y

50kmh60kmh80kmh

0 05 1 15 2 25 3 35Time (s)

16E minus 2

12E minus 2

80E minus 3

40E minus 3

00E + 0

120590ay

50kmh60kmh80kmh

0 05 1 15 2 25 3Time (s)

80E minus 3

40E minus 3

00E + 0

minus40E minus 3

minus80E minus 3

minus12E minus 2

minus16E minus 2

120583dz

50kmh60kmh80kmh

0 05 1 15 2 25 3Time (s)

12E minus 1

80E minus 2

40E minus 2

00E + 0

minus40E minus 2

minus80E minus 2

120583z

50kmh60kmh80kmh

0 1 2 3Time (s)

80E minus 1

40E minus 1

00E + 0

minus40E minus 1

minus80E minus 1

50kmh60kmh80kmh

120583az

0 05 1 15 2 25 3Time (s)

10E minus 3

80E minus 4

60E minus 4

40E minus 4

20E minus 4

00E + 0

120590dz (m

50kmh60kmh80kmh

0 1 2 3Time (s)

80E minus 3

60E minus 3

40E minus 3

20E minus 3

00E + 0

120590z

50kmh60kmh80kmh

0 05 1 15 2 25 3Time (s)

40E minus 2

30E minus 2

20E minus 2

10E minus 2

00E + 0

120590az

50kmh60kmh80kmh

A B C D

50E minus 04

40E minus 04

30E minus 04

20E minus 04

10E minus 04

00E + 00

120583dy

A B C D

40E minus 05

30E minus 05

20E minus 05

10E minus 05

00E + 00

120590dy

0 025 05 075 1 125 15 175Time (s)

30E + 5

25E + 5

20E + 5

15E + 5

10E + 5

50E + 4

00E + 0

120583F

0 025 05 075 1 125 15 175Time (s)

25E + 4

20E + 4

15E + 4

10E + 4

50E + 3

00E + 0

minus50 + 3

120590F

0 05 1 15 2 25 3 35

Time (s)

10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

minus40E minus 4

120583dy

82 times 1010 N-m2

0 05 1 15 2 25 3 35Time (s)

20E minus 5

16E minus 5

12E minus 5

80E minus 6

40E minus 6

00E + 0

120590dy

6N-m2 (B) 119864V119868V = 64 times 107N-m2

and (C) 119864V119868V = 82 times 1010N-m2

0 05 1 15 2 25 3 35

Time (s)

10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

120583dy

119909) (A) 119890

119909= 05m (B) 119890

119909= 1m and

(C) 119890119909= 15m

0 05 1 15 2 25 3 35Time (s)

30E minus 5

25E minus 5

20E minus 5

15E minus 5

10E minus 5

50E minus 6

00E + 0

120590dy

119909) (A) 119890

119909= 05m (B) 119890

119909=

1m and (C) 119890119909= 15m

119884dynamic =1003816100381610038161003816120583119884 (119909119896 119905119896) + 120590119884 (119909119896 119905119896)

1003816100381610038161003816 (46)

0 0002 0004 0006 0008 001

GbJbEbIb

119887119869119887119864119887119868119887) for different

107 N-m2 and (C) 119864V119868V = 82 times 10

10 N-m2

119884static (47)

119887119868119887)

5 Conclusions

1520 25 30 35 40

45 5055

404550556065707580146148

15152154156158

16162164

times103

8013514

Vehicle speed (kmh)

14161820222426283013

Appendix

119865119895(119905) = 0 for119895 = 1 2 3 119899V

= 119896119905119895minus119899V

ℎ (119909119895minus119899V

) + 119888119905119895minus119899V

ℎ (119909119895minus119899V

For119895 = 119899V + 1 119899V + 2

= 1198961199051ℎ (1199091) 120601119887119895minus(119899V+2) (1199091) + 1198961199052ℎ (1199092) 120601119887119895minus(119899V+2) (1199092)

+ 1198881199051ℎ (1199091) 120601119887119895minus(119899V+2) (1199091) + 1198881199052ℎ (1199092) 120601119887119895minus(119899V+2) (1199092)

For 119895 = 119899V + 3 119899V + 4 119899V + 2 + 119899119887

= 1198901199091198961199051ℎ (1199091) 120601119905119895minus(119899V+2+119899119887)

(1199091)

+ 1198961199052ℎ (1199092) 120601119905119895minus(119899V+2+119899119887)

(1199092)

+ 1198881199051ℎ (1199091) 120601119905119895minus(119899V+2+119899119887)

(1199091)

+1198881199052ℎ1015840(1199092) 120601119905119895minus(119899V+2+119899119887)

(1199092)

For 119895 = 119899V + 119899119887 + 3

119899V + 119899119887 + 4 119899V + 119899119887 + 119899119905 + 2

ℎ119898(119909) =

1198712119909 (119871 minus 119909) (A2)

ℎ119903(119909) =

119873

119904=1

119860119904cos (2120587Ω

119904119909 + 120579119904) (A3)

119868119895119896= 119868119895119908+ 119868119895119909+ 119868119895119910+ 119868119895119911 (A4)

where 119908 = 1 2 3 119899V 119909 = 119899V + 1 119899V + 2 119910 = 119899V + 2 + 1119899V + 2 + 2 119899V + 2 + 119899

119887 119911 = 119899V + 2 + 119899

119887+ 1 119899V + 2 + 119899

119887+

2 119899V + 2 + 119899119887 + 119899119879Thefirst term 119868

1198681198951= 1198681198952= sdot sdot sdot = 119868

119895119899V= 0 (A5)

119868119895119909= 1198681198951199091

+ 1198681198951199092 (A6)

1198681198951199091

= 11986811989511990911

+ 11986811989511990912

119904)minus2 and 119861

119904= 2120587Ω

119871+ (119904 minus

11986811989511990911

= 1198961199051[

119873

119904=1

119860119904

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2minus120572119895cos (120579

119904) minus 119861119904119881 sin (120579

119904)

+ exp (120572119895119905)

times [120572119895cos (119861

119904119881119905 + 120579

119904)

+119861119904119881 cos (119861

119904119881119905 + 120579

119904)]

+4ℎ0119881

12057231198951198712

120572119895119871 (120572119895119905 + exp (minus120572

119895119905) minus 1)

minus119881

12057231198951198712

119895119905) + 2 ]

11986811989511990912

= 1198881199051[

119873

119904=1

119881119860119904119861119904

exp (minus120572119895119905)

119904)

+ exp (120572119895119905)

times [120572119895sin (119861

119904119881119905 + 120579

119904)

minus119861119904119881 sin (119861

119904119881119905 + 120579

119904)]

119895119905)

12057231198951198712

minus1

120572119895119871exp (120572

119895119905) minus 2]

1198681198951199092

= 11986811989511990921

+ 11986811989511990922

11986811989511990921

= 1198961199052[

119873

119904=1

119860119904

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2

times minus 120572119895cos (119861

119904119897V minus 120579119904) minus 119861119904119881 sin (119861

119904119897V minus 120579119904)

+ exp (120572119895119905) [120572119895cos (119861

119904(119897 minus 119881119905) minus 120579

119904)

+119861119904119881 sin (119861

119904(119897V minus 119881119905) + 120579119904)]

+1198812[2 minus exp (120572

119895119905) 120572119895119905 (120572119895119905 minus 2) + 2]

+4ℎ0

12057231198951198712

minus1205722

119895119897V (119897V + 119871) (exp (120572119895119905) minus 1)

+120572119895119881 (2119897V + 119871) [exp (120572119895119905) (120572119895119905 minus 1) + 1] ]

11986811989511990922

= 1198881199052[

119873

119904=1

119860119904119861119904119881

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2120572119895sin (119861

119904119897V minus 120579119904)

+ 119861119904119881 cos (119861

119904119897V minus 120579119904)

minus exp (120572119895119905) [120572119895sin (119861

119904(119897 minus 119881119905) minus 120579

119904)

+119861119904119881 cos (119861

119904(119897V minus 119881119905) + 120579119904)]

+4ℎ0119881 exp (minus120572

119895119905)

12057221198951198712

times 2119881 [exp (120572119895119905) (120572119895119905 minus 1) + 1]

minus120572119895(2119897V + 119871) (exp (120572119895119905) minus 1)]

119868119895119910=

119899119887

119896=1

119868119895119899V+2+119896

119886=1

119868(119895119899V+2+119896)119886

(119896 = 1 2 3 119899119887) (A11)

119876119904= 119861119904119881 119877

119896=119896120587119881

119871

119878119896=119896120587

119871 119885

119904= 119861119904119897V

119868(119895119899V+2+119896)

21198961199051

119873

119904=1

119860119904 ((2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) cos (120579

119904)

+2120572119895119876119904sin (120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times cos (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119876

119904+ 119877119896) 119905 + 120579

119904

minus (119877119896+ 119876119904)

times cos (119876119904+ 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

21198961199051[

4ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198762119904)3

times 119871 (1205722

119895+ 1198762

119904) [exp (120572

119895119905) sin (119876

119904119905)

times 1198762

119904(120572119895119905 + 1)

+1205722

119895(120572119895119905 minus 1)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904119905 + 120572119895(120572119895119905 minus 2)] minus 2120572

119895119876119904

+ 119881 (21198763

119904+ 119876119904exp (120572

119895119905)) cos (119876

119904119905)

times 1198764

1199041199052+ 21198762

119904(1205722

1198951199052minus 2120572119895119905 minus 1)

+1205722

119895(1205722

1198951199052minus 4120572119895119905 + 6)

minus exp (120572119895119905) sin (119876

119904119905)

times 1198764

119904(120572119895119905 + 2) + 120572

1198951198762

119904(1205722

1198951199052minus 3)

+1205723

119895(1205722

1198951199052minus 2120572119895119905 + 2) minus 6120572

119895119876119904]

119868(119895119899V+2+119896)

21198961199052

119873

119904=1

119860119904[ exp (minus120572

119895119905)

times (((119877119896+ 119876119904) cos (119878

119904minus 119885119904minus 120579119904)

+120572119895sin (119878119896minus 119885119904minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

times (((119877119896minus 119876119904) cos (119878

119896minus 119885119904+ 120579119904)

+120572119895sin (119878119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 119878

119896minus 119885119904+ 120579119904

minus (119877119896+ 119876119904) cos (119876

119904minus 119877119896) 119905

+119878119896minus 119885119904+ 120579119904)

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

+ ((120572119895sin 119878119896+ 119885119904minus (119876119904+ 119877) 119905 minus 120579

119904

+ (119877119896+ 119876119904) cos 119878

119896+ 119885119904

+ (119876119904+ 119877119896) 119905

minus120579119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1

21198961199052[

4ℎ0119881

1198712(1205722119895+ 1198812)

times sin (119881119905 minus 119897V) [1205721198951198972

V(1205722

119895+ 1198812)2

minus 119897V (1205722

119895+ 1198812)

times minus1205723

119895119871 minus 1205721198951198711198812

+ 21198813(120572119895119905 + 1)

+21205722

119895119881(120572119895119905 minus 1)

+ 119881 [119881 21205723

119895+ 120572119895119905(1205722

119895+ 1198812)2

+2119905 (1205724

119895minus 1198814) minus 6120572

1198951198812

minus 119871 (1205722

119895+ 1198812)

times 1205722

119895(120572119895119905 minus 1)+ 119881

2(120572119895119905 +1)]

+ 119881 cos (119897V minus 119881119905) [1198814(1198972

V + 119897V119871

+21205722

1198951199052minus 4120572119895119905 minus 2)

+ 1205722

1198951198812(21198972

V + 2119897V119871

+41205722

1198951199052+ 6)

+ 1205724

119895119897V (119897V + 119871)

minus 1198815119905 (2119897V + 119871)

times (120572119895119905 minus 1)

minus 1205723

119895119881 (2119897V + 119871)

times (120572119895119905 minus 2) + 119897

V1198816]]

119868(119895119899V+2+119896)

21198881199051

119873

119904=1

119860119904119876119904( minus (2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) sin (120579

119904)

minus2120572119895119876119904cos (120579

119904))

times (1205722

119895+ (119876119904minus 119877)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times sin (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos (119876

119904+ 119877119896) 119905 + 120579

119904

+ (119877119896+ 119876119904)

times sin (119876119904+ 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199051

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) cos (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) sin (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

21198881199052

119873

119904=1

119860119904119876119904

times [ exp (minus120572119895119905 ( ( (119877

119896+ 119876119904) sin (119878

119896+ 119885119904minus 120579119904)

minus120572119895cos (119878

119896+ 119885119904minus 120579119904))

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

times ( ( (119876119904minus 119877119896) sin (119878

119896minus 119885119904+ 120579119904)

+120572119895cos (119878

119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos 119878

119896+ 119885119904minus (119876119904+ 119877119896) 119905 minus 120579

119904

minus (119877119896+ 119876119904) sin 119878

119896+ 119885119904

minus (119876119904+ 119877119896) 119905 minus 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos 119878

119896minus 119885119904+ (119876119904minus 119877119896) 119905 + 120579

119904

+ (119876119904minus 119877119896) sin 119878

119896minus 119885119904

+ (119876119904minus 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199052

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) sin (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881 + exp (120572119895119905) 119896120587119881 cos(119896120587119881119905

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881cos(119896120587119863119871

119871)

minus 119871120572119895sin(119896120587119863

119871)

119871)]

119868(119895119899V+2+119896)

= minus21198711198812(1198981199081

+ 1198981199082)

119873

119904=1

1198601199041198612

119904[(((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881 cos [(119876

119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

119904119871)119881 cos 120579

119904

+119871120572119895sin 120579119904)

times (1198612

11990411987121198812

+ 21198961205871198611199041198711198812

+ 119896212058721198812

+11987121205722

119895)minus1

+ ((119896120587 minus 119861119904119871)119881 cos 120579

119904

minus119871120572119895sin 120579119904)

times ((119896120587 minus 119861119904119871)2

1198812

+11987121205722

119895)minus1

minus8ℎ01198812(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

times [119896120587119881

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119868119895119911=

119899119905

119896=1

119868119895119899V+2+119899119887+119896

119868119895119899V+2+119899119887+119897

119886=1

119868(119895119899V+2+119899119887+119897)119886

(119896 = 1 2 3 119899119879) (A14)

119868(119895119899V+2+119899119887+119897)1

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895cos (120579

119904) + (119876

119904minus 119877119896) sin (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895cos (120579

119904) + (119876

119904+ 119877119896) sin (120579

119904)

1205722119895+ (119876119904+ 119877119896)2

+ ((120572119895cos [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119876119904minus 119877119896) sin [(119876

119904minus 119877119896) 119905 minus 120579

119904])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [(119876

119904+ 119877119896) 119905 + 120579

119904]

+ (119876119904+ 119877119896) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)2

=41198961199051119890119909ℎ0119871

119881(1205722119895+ 119877119896

times [ (1205722

119895+ 119877119896

2) exp (minus120572

119895119905)

times 1205722

119895+ exp (minus120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)] minus 119877

119896

+ 2120572119895exp (minus120572

119895119905) (1205722

119895+ 3119877119896

+ 119877119896sin (119877

119896119905) 6120572

119895+ 1199052(1205722

119895+ 119877119896

+4120572119895119905 (1205722

119895+ 119877119896

2) minus 2119877

119896

+ cos (119877119896119905) 2119905 (119877

119896

4minus 1205724

119895)

+ 2120572119895(1205722

119895minus 3119877119896

times 1205721198951199052(1205722

119895+ 119877119896

119868(119895119899V+2+119899119887+119897)3

= minus1

21198961199052119890119909

119873

119904=1

[119860119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119877119896minus 119876119904) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) sin (119861

119904119897V + 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos [(119877

119896minus 119876119904) 119905 + (119861

119904119897V minus 119878119896 minus 120579119904)]

+ (119877119896minus 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [minus (119877

119896+ 119876119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

minus (119877119896+ 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)4

=41198961199052119890119909ℎ0

1198712[((119897V (119897V + 119871) exp (minus120572119895119905)

times 120572119895cos (119878

119896minus 119877119896119905)

minus119877119896sin (119878119896minus 119877119896119905)

+119877119896sin (119878) minus 120572

119895cos (119878

119896))

times (1205722

119895+ 119877119896

2)minus1

+2119871119881 exp (minus120572

119895119905)

(1205722119895+ 119877119896

times (1205722

119895+ 119877119896

2) cos (119878

119896)

+ exp (120572119895119905) cos (119878

119896minus 119877119896119905)

times [1198772(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896[119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

119895119877119896sin (119878119896)

minus1198812

(1205722119895+ 119877119896

times119877119896sin (119878119896minus 119877119896119905)[61205722

119895+1199052(1205722

119895+ 119877119896

minus 4120572119895119905 (1205722

119895+ 119877119896

minus2119877119896

+ 2 exp (minus120572119895119905)

times [(1205723

119895minus 3120572119895119877119896

2) cos (119878

119896)

+ 119877119896(119877119896

2minus 31205722

119895)

times sin (119878119896) ]

minus [2119905 (119877119896

4minus 1205724

119895) + 2120572

119895(1205722

119895minus 3119877119896

+1205721198951199052(1205722

119895+119877119896

]cos (119878119896minus 119877119896119905)]

119868(119895119899V+2+119899119887+119897)5

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895sin (120579119904) + (119876

119904minus 119877119896) cos (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895sin (120579119904) minus (119876

119904+ 119877119896) cos (120579

119904)

1205722119895+ (119876119904+ 119877)2

+ ((120572119895sin [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119877119896minus 119876119904) cos [(119876

119904minus 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895sin [(119876

119904+ 119877119896) 119905 + 120579

119904]

minus (119876119904+ 119877) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)6

=41198881199051119890119909ℎ0119871

(1205722119895+ 119877119896

2)3(119881

119871)

times [(1205722

119895+ 119877119896

2) 120572119895cos (119877

119896119905)

+ 119877119896sin (119877

119896119905) minus 120572

119895exp (minus120572

119895119905)

+ 2 exp (minus120572119895119905) 1205722

119895+ exp (120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

minus119877119896

119868(119895119899V+2+119899119887+119897)7

=41198881199052119890119909ℎ0

119881(1205722119895+ 119877119896

times 119897V (1205722

119895+ 119877119896

2) + 119871 (120572

119895+ 119877119896

minus2119881 [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

+ cos (119878119896minus 119877119896119905)

times 119877119896

2[120572119895(119897V + 119871) minus 2119881 (120572

119895119905 + 1)]

+1205722

119895[120572119895(119897V + 119871) + 2119881 (1 minus 120572

119895119905)]

+ exp (minus120572119895119905) 119877119896sin (119878119896)

times [119897V (1205722

119895+ 119877119896

+119871 (1205722

119895+ 119877119896

2)+4120572119895119881]

+ cos (119878119896)

times [119877119896

22119881 minus 120572

119895(119897V + 119871)

minus 1205722

119895120572119895(119897V + 119871)

+2119881]]

119868(119895119899V+2+119899119887+119897)8

21198881199052119890119909

119873

119904=1

[119860119904119876119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119876119904minus 119877119896) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) cos (119861

119904119897V + 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin [(119877 minus 119876

119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)]

+ (119876119904minus 119877119896) cos [(119877

119896minus 119876119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)])

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

minus ((120572119895sin [minus (119877 + 119876

119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

+ (119877119896+ 119876119904) cos [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)9

= minus119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [exp (120572119895119905) 119896120587119881 sin(119896120587119881119905

119871)

minus119871120572119895+ exp 119871120572

119895(120572119895119905) cos(119896120587119881119905

119871)]

minus 119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881sin(119896120587119863119871

119871)

minus 119890119909119871120572119895cos(119896120587119863

119871)

minus exp (120572119895119905)

times cos(119896120587119863 minus 119881119905

119871)]

119868(119895119899V+2+119899119887+119897)10

= minus21198711198812(1198981199081

+ 1198981199082) 119890119909

119873

119904=1

1198601199041198612

119904[ (((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881

times cos [(119876119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus exp (minus120572119895119905)

times ((minus (119896120587 +119861119904119871)119881 cos 120579

119904+119871120572119895sin 120579119904)

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

+ (((119896120587 minus 119861119904119871)119881 cos 120579

119904minus 119871120572119895sin 120579119904)

times((119896120587 minus 119861119904119871)2

1198812+ 11987121205722

119895)minus1

minus8ℎ01198812119890119909(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

119871)

+119871120572119895sin(119896120587119881119905

119871)]

References

RoboticsJournal of

Journal of

RotatingMachinery

VLSI Design

Shock and Vibration

Journal of

Volume 2014

SensorsJournal of

Propagation

063 063 063

300 300

265100

(1198961199051 1198961199052) 09 times 10

119904= 16 times

119889119911)

075 120 075

0 05 1 15 2 25 3 35 4

Time (s)10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

120583dy

50kmh60kmh80kmh

0 05 1 15 2 25 3 35

Time (s)

minus80E minus 3

40E minus 3

minus10E minus 18

minus40E minus 3

minus80E minus 3

minus12E minus 2

120583y

50kmh60kmh80kmh

0 05 1 15 2 25 3 35

Time (s)40E minus 2

00E + 0

minus40E minus 2

minus80E minus 2

minus12E minus 1

minus16E minus 1

120583ay

50kmh60kmh80kmh

0 05 1 15 2 25 3 35 4Time (s)

40E minus 5

30E minus 5

20E minus 5

10E minus 5

00E + 0

minus10E minus 5

120590dy

50kmh60kmh80kmh

119889119911) velocity (120583V119911)

119886119911) is shown in

0 05 1 15 2 25 3 35Time (s)

16E minus 3

12E minus 3

80E minus 4

40E minus 4

00E + 0

minus40E minus 4

120590y

50kmh60kmh80kmh

0 05 1 15 2 25 3 35Time (s)

16E minus 2

12E minus 2

80E minus 3

40E minus 3

00E + 0

120590ay

50kmh60kmh80kmh

0 05 1 15 2 25 3Time (s)

80E minus 3

40E minus 3

00E + 0

minus40E minus 3

minus80E minus 3

minus12E minus 2

minus16E minus 2

120583dz

50kmh60kmh80kmh

0 05 1 15 2 25 3Time (s)

12E minus 1

80E minus 2

40E minus 2

00E + 0

minus40E minus 2

minus80E minus 2

120583z

50kmh60kmh80kmh

0 1 2 3Time (s)

80E minus 1

40E minus 1

00E + 0

minus40E minus 1

minus80E minus 1

50kmh60kmh80kmh

120583az

0 05 1 15 2 25 3Time (s)

10E minus 3

80E minus 4

60E minus 4

40E minus 4

20E minus 4

00E + 0

120590dz (m

50kmh60kmh80kmh

0 1 2 3Time (s)

80E minus 3

60E minus 3

40E minus 3

20E minus 3

00E + 0

120590z

50kmh60kmh80kmh

0 05 1 15 2 25 3Time (s)

40E minus 2

30E minus 2

20E minus 2

10E minus 2

00E + 0

120590az

50kmh60kmh80kmh

A B C D

50E minus 04

40E minus 04

30E minus 04

20E minus 04

10E minus 04

00E + 00

120583dy

A B C D

40E minus 05

30E minus 05

20E minus 05

10E minus 05

00E + 00

120590dy

0 025 05 075 1 125 15 175Time (s)

30E + 5

25E + 5

20E + 5

15E + 5

10E + 5

50E + 4

00E + 0

120583F

0 025 05 075 1 125 15 175Time (s)

25E + 4

20E + 4

15E + 4

10E + 4

50E + 3

00E + 0

minus50 + 3

120590F

0 05 1 15 2 25 3 35

Time (s)

10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

minus40E minus 4

120583dy

82 times 1010 N-m2

0 05 1 15 2 25 3 35Time (s)

20E minus 5

16E minus 5

12E minus 5

80E minus 6

40E minus 6

00E + 0

120590dy

6N-m2 (B) 119864V119868V = 64 times 107N-m2

and (C) 119864V119868V = 82 times 1010N-m2

0 05 1 15 2 25 3 35

Time (s)

10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

120583dy

119909) (A) 119890

119909= 05m (B) 119890

119909= 1m and

(C) 119890119909= 15m

0 05 1 15 2 25 3 35Time (s)

30E minus 5

25E minus 5

20E minus 5

15E minus 5

10E minus 5

50E minus 6

00E + 0

120590dy

119909) (A) 119890

119909= 05m (B) 119890

119909=

1m and (C) 119890119909= 15m

119884dynamic =1003816100381610038161003816120583119884 (119909119896 119905119896) + 120590119884 (119909119896 119905119896)

1003816100381610038161003816 (46)

0 0002 0004 0006 0008 001

GbJbEbIb

119887119869119887119864119887119868119887) for different

107 N-m2 and (C) 119864V119868V = 82 times 10

10 N-m2

119884static (47)

119887119868119887)

5 Conclusions

1520 25 30 35 40

45 5055

404550556065707580146148

15152154156158

16162164

times103

8013514

Vehicle speed (kmh)

14161820222426283013

Appendix

119865119895(119905) = 0 for119895 = 1 2 3 119899V

= 119896119905119895minus119899V

ℎ (119909119895minus119899V

) + 119888119905119895minus119899V

ℎ (119909119895minus119899V

For119895 = 119899V + 1 119899V + 2

= 1198961199051ℎ (1199091) 120601119887119895minus(119899V+2) (1199091) + 1198961199052ℎ (1199092) 120601119887119895minus(119899V+2) (1199092)

+ 1198881199051ℎ (1199091) 120601119887119895minus(119899V+2) (1199091) + 1198881199052ℎ (1199092) 120601119887119895minus(119899V+2) (1199092)

For 119895 = 119899V + 3 119899V + 4 119899V + 2 + 119899119887

= 1198901199091198961199051ℎ (1199091) 120601119905119895minus(119899V+2+119899119887)

(1199091)

+ 1198961199052ℎ (1199092) 120601119905119895minus(119899V+2+119899119887)

(1199092)

+ 1198881199051ℎ (1199091) 120601119905119895minus(119899V+2+119899119887)

(1199091)

+1198881199052ℎ1015840(1199092) 120601119905119895minus(119899V+2+119899119887)

(1199092)

For 119895 = 119899V + 119899119887 + 3

119899V + 119899119887 + 4 119899V + 119899119887 + 119899119905 + 2

ℎ119898(119909) =

1198712119909 (119871 minus 119909) (A2)

ℎ119903(119909) =

119873

119904=1

119860119904cos (2120587Ω

119904119909 + 120579119904) (A3)

119868119895119896= 119868119895119908+ 119868119895119909+ 119868119895119910+ 119868119895119911 (A4)

where 119908 = 1 2 3 119899V 119909 = 119899V + 1 119899V + 2 119910 = 119899V + 2 + 1119899V + 2 + 2 119899V + 2 + 119899

119887 119911 = 119899V + 2 + 119899

119887+ 1 119899V + 2 + 119899

119887+

2 119899V + 2 + 119899119887 + 119899119879Thefirst term 119868

1198681198951= 1198681198952= sdot sdot sdot = 119868

119895119899V= 0 (A5)

119868119895119909= 1198681198951199091

+ 1198681198951199092 (A6)

1198681198951199091

= 11986811989511990911

+ 11986811989511990912

119904)minus2 and 119861

119904= 2120587Ω

119871+ (119904 minus

11986811989511990911

= 1198961199051[

119873

119904=1

119860119904

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2minus120572119895cos (120579

119904) minus 119861119904119881 sin (120579

119904)

+ exp (120572119895119905)

times [120572119895cos (119861

119904119881119905 + 120579

119904)

+119861119904119881 cos (119861

119904119881119905 + 120579

119904)]

+4ℎ0119881

12057231198951198712

120572119895119871 (120572119895119905 + exp (minus120572

119895119905) minus 1)

minus119881

12057231198951198712

119895119905) + 2 ]

11986811989511990912

= 1198881199051[

119873

119904=1

119881119860119904119861119904

exp (minus120572119895119905)

119904)

+ exp (120572119895119905)

times [120572119895sin (119861

119904119881119905 + 120579

119904)

minus119861119904119881 sin (119861

119904119881119905 + 120579

119904)]

119895119905)

12057231198951198712

minus1

120572119895119871exp (120572

119895119905) minus 2]

1198681198951199092

= 11986811989511990921

+ 11986811989511990922

11986811989511990921

= 1198961199052[

119873

119904=1

119860119904

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2

times minus 120572119895cos (119861

119904119897V minus 120579119904) minus 119861119904119881 sin (119861

119904119897V minus 120579119904)

+ exp (120572119895119905) [120572119895cos (119861

119904(119897 minus 119881119905) minus 120579

119904)

+119861119904119881 sin (119861

119904(119897V minus 119881119905) + 120579119904)]

+1198812[2 minus exp (120572

119895119905) 120572119895119905 (120572119895119905 minus 2) + 2]

+4ℎ0

12057231198951198712

minus1205722

119895119897V (119897V + 119871) (exp (120572119895119905) minus 1)

+120572119895119881 (2119897V + 119871) [exp (120572119895119905) (120572119895119905 minus 1) + 1] ]

11986811989511990922

= 1198881199052[

119873

119904=1

119860119904119861119904119881

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2120572119895sin (119861

119904119897V minus 120579119904)

+ 119861119904119881 cos (119861

119904119897V minus 120579119904)

minus exp (120572119895119905) [120572119895sin (119861

119904(119897 minus 119881119905) minus 120579

119904)

+119861119904119881 cos (119861

119904(119897V minus 119881119905) + 120579119904)]

+4ℎ0119881 exp (minus120572

119895119905)

12057221198951198712

times 2119881 [exp (120572119895119905) (120572119895119905 minus 1) + 1]

minus120572119895(2119897V + 119871) (exp (120572119895119905) minus 1)]

119868119895119910=

119899119887

119896=1

119868119895119899V+2+119896

119886=1

119868(119895119899V+2+119896)119886

(119896 = 1 2 3 119899119887) (A11)

119876119904= 119861119904119881 119877

119896=119896120587119881

119871

119878119896=119896120587

119871 119885

119904= 119861119904119897V

119868(119895119899V+2+119896)

21198961199051

119873

119904=1

119860119904 ((2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) cos (120579

119904)

+2120572119895119876119904sin (120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times cos (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119876

119904+ 119877119896) 119905 + 120579

119904

minus (119877119896+ 119876119904)

times cos (119876119904+ 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

21198961199051[

4ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198762119904)3

times 119871 (1205722

119895+ 1198762

119904) [exp (120572

119895119905) sin (119876

119904119905)

times 1198762

119904(120572119895119905 + 1)

+1205722

119895(120572119895119905 minus 1)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904119905 + 120572119895(120572119895119905 minus 2)] minus 2120572

119895119876119904

+ 119881 (21198763

119904+ 119876119904exp (120572

119895119905)) cos (119876

119904119905)

times 1198764

1199041199052+ 21198762

119904(1205722

1198951199052minus 2120572119895119905 minus 1)

+1205722

119895(1205722

1198951199052minus 4120572119895119905 + 6)

minus exp (120572119895119905) sin (119876

119904119905)

times 1198764

119904(120572119895119905 + 2) + 120572

1198951198762

119904(1205722

1198951199052minus 3)

+1205723

119895(1205722

1198951199052minus 2120572119895119905 + 2) minus 6120572

119895119876119904]

119868(119895119899V+2+119896)

21198961199052

119873

119904=1

119860119904[ exp (minus120572

119895119905)

times (((119877119896+ 119876119904) cos (119878

119904minus 119885119904minus 120579119904)

+120572119895sin (119878119896minus 119885119904minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

times (((119877119896minus 119876119904) cos (119878

119896minus 119885119904+ 120579119904)

+120572119895sin (119878119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 119878

119896minus 119885119904+ 120579119904

minus (119877119896+ 119876119904) cos (119876

119904minus 119877119896) 119905

+119878119896minus 119885119904+ 120579119904)

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

+ ((120572119895sin 119878119896+ 119885119904minus (119876119904+ 119877) 119905 minus 120579

119904

+ (119877119896+ 119876119904) cos 119878

119896+ 119885119904

+ (119876119904+ 119877119896) 119905

minus120579119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1

21198961199052[

4ℎ0119881

1198712(1205722119895+ 1198812)

times sin (119881119905 minus 119897V) [1205721198951198972

V(1205722

119895+ 1198812)2

minus 119897V (1205722

119895+ 1198812)

times minus1205723

119895119871 minus 1205721198951198711198812

+ 21198813(120572119895119905 + 1)

+21205722

119895119881(120572119895119905 minus 1)

+ 119881 [119881 21205723

119895+ 120572119895119905(1205722

119895+ 1198812)2

+2119905 (1205724

119895minus 1198814) minus 6120572

1198951198812

minus 119871 (1205722

119895+ 1198812)

times 1205722

119895(120572119895119905 minus 1)+ 119881

2(120572119895119905 +1)]

+ 119881 cos (119897V minus 119881119905) [1198814(1198972

V + 119897V119871

+21205722

1198951199052minus 4120572119895119905 minus 2)

+ 1205722

1198951198812(21198972

V + 2119897V119871

+41205722

1198951199052+ 6)

+ 1205724

119895119897V (119897V + 119871)

minus 1198815119905 (2119897V + 119871)

times (120572119895119905 minus 1)

minus 1205723

119895119881 (2119897V + 119871)

times (120572119895119905 minus 2) + 119897

V1198816]]

119868(119895119899V+2+119896)

21198881199051

119873

119904=1

119860119904119876119904( minus (2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) sin (120579

119904)

minus2120572119895119876119904cos (120579

119904))

times (1205722

119895+ (119876119904minus 119877)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times sin (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos (119876

119904+ 119877119896) 119905 + 120579

119904

+ (119877119896+ 119876119904)

times sin (119876119904+ 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199051

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) cos (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) sin (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

21198881199052

119873

119904=1

119860119904119876119904

times [ exp (minus120572119895119905 ( ( (119877

119896+ 119876119904) sin (119878

119896+ 119885119904minus 120579119904)

minus120572119895cos (119878

119896+ 119885119904minus 120579119904))

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

times ( ( (119876119904minus 119877119896) sin (119878

119896minus 119885119904+ 120579119904)

+120572119895cos (119878

119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos 119878

119896+ 119885119904minus (119876119904+ 119877119896) 119905 minus 120579

119904

minus (119877119896+ 119876119904) sin 119878

119896+ 119885119904

minus (119876119904+ 119877119896) 119905 minus 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos 119878

119896minus 119885119904+ (119876119904minus 119877119896) 119905 + 120579

119904

+ (119876119904minus 119877119896) sin 119878

119896minus 119885119904

+ (119876119904minus 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199052

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) sin (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881 + exp (120572119895119905) 119896120587119881 cos(119896120587119881119905

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881cos(119896120587119863119871

119871)

minus 119871120572119895sin(119896120587119863

119871)

119871)]

119868(119895119899V+2+119896)

= minus21198711198812(1198981199081

+ 1198981199082)

119873

119904=1

1198601199041198612

119904[(((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881 cos [(119876

119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

119904119871)119881 cos 120579

119904

+119871120572119895sin 120579119904)

times (1198612

11990411987121198812

+ 21198961205871198611199041198711198812

+ 119896212058721198812

+11987121205722

119895)minus1

+ ((119896120587 minus 119861119904119871)119881 cos 120579

119904

minus119871120572119895sin 120579119904)

times ((119896120587 minus 119861119904119871)2

1198812

+11987121205722

119895)minus1

minus8ℎ01198812(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

times [119896120587119881

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119868119895119911=

119899119905

119896=1

119868119895119899V+2+119899119887+119896

119868119895119899V+2+119899119887+119897

119886=1

119868(119895119899V+2+119899119887+119897)119886

(119896 = 1 2 3 119899119879) (A14)

119868(119895119899V+2+119899119887+119897)1

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895cos (120579

119904) + (119876

119904minus 119877119896) sin (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895cos (120579

119904) + (119876

119904+ 119877119896) sin (120579

119904)

1205722119895+ (119876119904+ 119877119896)2

+ ((120572119895cos [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119876119904minus 119877119896) sin [(119876

119904minus 119877119896) 119905 minus 120579

119904])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [(119876

119904+ 119877119896) 119905 + 120579

119904]

+ (119876119904+ 119877119896) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)2

=41198961199051119890119909ℎ0119871

119881(1205722119895+ 119877119896

times [ (1205722

119895+ 119877119896

2) exp (minus120572

119895119905)

times 1205722

119895+ exp (minus120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)] minus 119877

119896

+ 2120572119895exp (minus120572

119895119905) (1205722

119895+ 3119877119896

+ 119877119896sin (119877

119896119905) 6120572

119895+ 1199052(1205722

119895+ 119877119896

+4120572119895119905 (1205722

119895+ 119877119896

2) minus 2119877

119896

+ cos (119877119896119905) 2119905 (119877

119896

4minus 1205724

119895)

+ 2120572119895(1205722

119895minus 3119877119896

times 1205721198951199052(1205722

119895+ 119877119896

119868(119895119899V+2+119899119887+119897)3

= minus1

21198961199052119890119909

119873

119904=1

[119860119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119877119896minus 119876119904) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) sin (119861

119904119897V + 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos [(119877

119896minus 119876119904) 119905 + (119861

119904119897V minus 119878119896 minus 120579119904)]

+ (119877119896minus 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [minus (119877

119896+ 119876119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

minus (119877119896+ 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)4

=41198961199052119890119909ℎ0

1198712[((119897V (119897V + 119871) exp (minus120572119895119905)

times 120572119895cos (119878

119896minus 119877119896119905)

minus119877119896sin (119878119896minus 119877119896119905)

+119877119896sin (119878) minus 120572

119895cos (119878

119896))

times (1205722

119895+ 119877119896

2)minus1

+2119871119881 exp (minus120572

119895119905)

(1205722119895+ 119877119896

times (1205722

119895+ 119877119896

2) cos (119878

119896)

+ exp (120572119895119905) cos (119878

119896minus 119877119896119905)

times [1198772(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896[119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

119895119877119896sin (119878119896)

minus1198812

(1205722119895+ 119877119896

times119877119896sin (119878119896minus 119877119896119905)[61205722

119895+1199052(1205722

119895+ 119877119896

minus 4120572119895119905 (1205722

119895+ 119877119896

minus2119877119896

+ 2 exp (minus120572119895119905)

times [(1205723

119895minus 3120572119895119877119896

2) cos (119878

119896)

+ 119877119896(119877119896

2minus 31205722

119895)

times sin (119878119896) ]

minus [2119905 (119877119896

4minus 1205724

119895) + 2120572

119895(1205722

119895minus 3119877119896

+1205721198951199052(1205722

119895+119877119896

]cos (119878119896minus 119877119896119905)]

119868(119895119899V+2+119899119887+119897)5

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895sin (120579119904) + (119876

119904minus 119877119896) cos (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895sin (120579119904) minus (119876

119904+ 119877119896) cos (120579

119904)

1205722119895+ (119876119904+ 119877)2

+ ((120572119895sin [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119877119896minus 119876119904) cos [(119876

119904minus 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895sin [(119876

119904+ 119877119896) 119905 + 120579

119904]

minus (119876119904+ 119877) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)6

=41198881199051119890119909ℎ0119871

(1205722119895+ 119877119896

2)3(119881

119871)

times [(1205722

119895+ 119877119896

2) 120572119895cos (119877

119896119905)

+ 119877119896sin (119877

119896119905) minus 120572

119895exp (minus120572

119895119905)

+ 2 exp (minus120572119895119905) 1205722

119895+ exp (120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

minus119877119896

119868(119895119899V+2+119899119887+119897)7

=41198881199052119890119909ℎ0

119881(1205722119895+ 119877119896

times 119897V (1205722

119895+ 119877119896

2) + 119871 (120572

119895+ 119877119896

minus2119881 [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

+ cos (119878119896minus 119877119896119905)

times 119877119896

2[120572119895(119897V + 119871) minus 2119881 (120572

119895119905 + 1)]

+1205722

119895[120572119895(119897V + 119871) + 2119881 (1 minus 120572

119895119905)]

+ exp (minus120572119895119905) 119877119896sin (119878119896)

times [119897V (1205722

119895+ 119877119896

+119871 (1205722

119895+ 119877119896

2)+4120572119895119881]

+ cos (119878119896)

times [119877119896

22119881 minus 120572

119895(119897V + 119871)

minus 1205722

119895120572119895(119897V + 119871)

+2119881]]

119868(119895119899V+2+119899119887+119897)8

21198881199052119890119909

119873

119904=1

[119860119904119876119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119876119904minus 119877119896) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) cos (119861

119904119897V + 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin [(119877 minus 119876

119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)]

+ (119876119904minus 119877119896) cos [(119877

119896minus 119876119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)])

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

minus ((120572119895sin [minus (119877 + 119876

119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

+ (119877119896+ 119876119904) cos [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)9

= minus119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [exp (120572119895119905) 119896120587119881 sin(119896120587119881119905

119871)

minus119871120572119895+ exp 119871120572

119895(120572119895119905) cos(119896120587119881119905

119871)]

minus 119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881sin(119896120587119863119871

119871)

minus 119890119909119871120572119895cos(119896120587119863

119871)

minus exp (120572119895119905)

times cos(119896120587119863 minus 119881119905

119871)]

119868(119895119899V+2+119899119887+119897)10

= minus21198711198812(1198981199081

+ 1198981199082) 119890119909

119873

119904=1

1198601199041198612

119904[ (((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881

times cos [(119876119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus exp (minus120572119895119905)

times ((minus (119896120587 +119861119904119871)119881 cos 120579

119904+119871120572119895sin 120579119904)

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

+ (((119896120587 minus 119861119904119871)119881 cos 120579

119904minus 119871120572119895sin 120579119904)

times((119896120587 minus 119861119904119871)2

1198812+ 11987121205722

119895)minus1

minus8ℎ01198812119890119909(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

119871)

+119871120572119895sin(119896120587119881119905

119871)]

References

RoboticsJournal of

Journal of

RotatingMachinery

VLSI Design

Shock and Vibration

Journal of

Volume 2014

SensorsJournal of

Propagation

075 120 075

0 05 1 15 2 25 3 35 4

Time (s)10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

120583dy

50kmh60kmh80kmh

0 05 1 15 2 25 3 35

Time (s)

minus80E minus 3

40E minus 3

minus10E minus 18

minus40E minus 3

minus80E minus 3

minus12E minus 2

120583y

50kmh60kmh80kmh

0 05 1 15 2 25 3 35

Time (s)40E minus 2

00E + 0

minus40E minus 2

minus80E minus 2

minus12E minus 1

minus16E minus 1

120583ay

50kmh60kmh80kmh

0 05 1 15 2 25 3 35 4Time (s)

40E minus 5

30E minus 5

20E minus 5

10E minus 5

00E + 0

minus10E minus 5

120590dy

50kmh60kmh80kmh

119889119911) velocity (120583V119911)

119886119911) is shown in

0 05 1 15 2 25 3 35Time (s)

16E minus 3

12E minus 3

80E minus 4

40E minus 4

00E + 0

minus40E minus 4

120590y

50kmh60kmh80kmh

0 05 1 15 2 25 3 35Time (s)

16E minus 2

12E minus 2

80E minus 3

40E minus 3

00E + 0

120590ay

50kmh60kmh80kmh

0 05 1 15 2 25 3Time (s)

80E minus 3

40E minus 3

00E + 0

minus40E minus 3

minus80E minus 3

minus12E minus 2

minus16E minus 2

120583dz

50kmh60kmh80kmh

0 05 1 15 2 25 3Time (s)

12E minus 1

80E minus 2

40E minus 2

00E + 0

minus40E minus 2

minus80E minus 2

120583z

50kmh60kmh80kmh

0 1 2 3Time (s)

80E minus 1

40E minus 1

00E + 0

minus40E minus 1

minus80E minus 1

50kmh60kmh80kmh

120583az

0 05 1 15 2 25 3Time (s)

10E minus 3

80E minus 4

60E minus 4

40E minus 4

20E minus 4

00E + 0

120590dz (m

50kmh60kmh80kmh

0 1 2 3Time (s)

80E minus 3

60E minus 3

40E minus 3

20E minus 3

00E + 0

120590z

50kmh60kmh80kmh

0 05 1 15 2 25 3Time (s)

40E minus 2

30E minus 2

20E minus 2

10E minus 2

00E + 0

120590az

50kmh60kmh80kmh

A B C D

50E minus 04

40E minus 04

30E minus 04

20E minus 04

10E minus 04

00E + 00

120583dy

A B C D

40E minus 05

30E minus 05

20E minus 05

10E minus 05

00E + 00

120590dy

0 025 05 075 1 125 15 175Time (s)

30E + 5

25E + 5

20E + 5

15E + 5

10E + 5

50E + 4

00E + 0

120583F

0 025 05 075 1 125 15 175Time (s)

25E + 4

20E + 4

15E + 4

10E + 4

50E + 3

00E + 0

minus50 + 3

120590F

0 05 1 15 2 25 3 35

Time (s)

10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

minus40E minus 4

120583dy

82 times 1010 N-m2

0 05 1 15 2 25 3 35Time (s)

20E minus 5

16E minus 5

12E minus 5

80E minus 6

40E minus 6

00E + 0

120590dy

6N-m2 (B) 119864V119868V = 64 times 107N-m2

and (C) 119864V119868V = 82 times 1010N-m2

0 05 1 15 2 25 3 35

Time (s)

10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

120583dy

119909) (A) 119890

119909= 05m (B) 119890

119909= 1m and

(C) 119890119909= 15m

0 05 1 15 2 25 3 35Time (s)

30E minus 5

25E minus 5

20E minus 5

15E minus 5

10E minus 5

50E minus 6

00E + 0

120590dy

119909) (A) 119890

119909= 05m (B) 119890

119909=

1m and (C) 119890119909= 15m

119884dynamic =1003816100381610038161003816120583119884 (119909119896 119905119896) + 120590119884 (119909119896 119905119896)

1003816100381610038161003816 (46)

0 0002 0004 0006 0008 001

GbJbEbIb

119887119869119887119864119887119868119887) for different

107 N-m2 and (C) 119864V119868V = 82 times 10

10 N-m2

119884static (47)

119887119868119887)

5 Conclusions

1520 25 30 35 40

45 5055

404550556065707580146148

15152154156158

16162164

times103

8013514

Vehicle speed (kmh)

14161820222426283013

Appendix

119865119895(119905) = 0 for119895 = 1 2 3 119899V

= 119896119905119895minus119899V

ℎ (119909119895minus119899V

) + 119888119905119895minus119899V

ℎ (119909119895minus119899V

For119895 = 119899V + 1 119899V + 2

= 1198961199051ℎ (1199091) 120601119887119895minus(119899V+2) (1199091) + 1198961199052ℎ (1199092) 120601119887119895minus(119899V+2) (1199092)

+ 1198881199051ℎ (1199091) 120601119887119895minus(119899V+2) (1199091) + 1198881199052ℎ (1199092) 120601119887119895minus(119899V+2) (1199092)

For 119895 = 119899V + 3 119899V + 4 119899V + 2 + 119899119887

= 1198901199091198961199051ℎ (1199091) 120601119905119895minus(119899V+2+119899119887)

(1199091)

+ 1198961199052ℎ (1199092) 120601119905119895minus(119899V+2+119899119887)

(1199092)

+ 1198881199051ℎ (1199091) 120601119905119895minus(119899V+2+119899119887)

(1199091)

+1198881199052ℎ1015840(1199092) 120601119905119895minus(119899V+2+119899119887)

(1199092)

For 119895 = 119899V + 119899119887 + 3

119899V + 119899119887 + 4 119899V + 119899119887 + 119899119905 + 2

ℎ119898(119909) =

1198712119909 (119871 minus 119909) (A2)

ℎ119903(119909) =

119873

119904=1

119860119904cos (2120587Ω

119904119909 + 120579119904) (A3)

119868119895119896= 119868119895119908+ 119868119895119909+ 119868119895119910+ 119868119895119911 (A4)

where 119908 = 1 2 3 119899V 119909 = 119899V + 1 119899V + 2 119910 = 119899V + 2 + 1119899V + 2 + 2 119899V + 2 + 119899

119887 119911 = 119899V + 2 + 119899

119887+ 1 119899V + 2 + 119899

119887+

2 119899V + 2 + 119899119887 + 119899119879Thefirst term 119868

1198681198951= 1198681198952= sdot sdot sdot = 119868

119895119899V= 0 (A5)

119868119895119909= 1198681198951199091

+ 1198681198951199092 (A6)

1198681198951199091

= 11986811989511990911

+ 11986811989511990912

119904)minus2 and 119861

119904= 2120587Ω

119871+ (119904 minus

11986811989511990911

= 1198961199051[

119873

119904=1

119860119904

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2minus120572119895cos (120579

119904) minus 119861119904119881 sin (120579

119904)

+ exp (120572119895119905)

times [120572119895cos (119861

119904119881119905 + 120579

119904)

+119861119904119881 cos (119861

119904119881119905 + 120579

119904)]

+4ℎ0119881

12057231198951198712

120572119895119871 (120572119895119905 + exp (minus120572

119895119905) minus 1)

minus119881

12057231198951198712

119895119905) + 2 ]

11986811989511990912

= 1198881199051[

119873

119904=1

119881119860119904119861119904

exp (minus120572119895119905)

119904)

+ exp (120572119895119905)

times [120572119895sin (119861

119904119881119905 + 120579

119904)

minus119861119904119881 sin (119861

119904119881119905 + 120579

119904)]

119895119905)

12057231198951198712

minus1

120572119895119871exp (120572

119895119905) minus 2]

1198681198951199092

= 11986811989511990921

+ 11986811989511990922

11986811989511990921

= 1198961199052[

119873

119904=1

119860119904

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2

times minus 120572119895cos (119861

119904119897V minus 120579119904) minus 119861119904119881 sin (119861

119904119897V minus 120579119904)

+ exp (120572119895119905) [120572119895cos (119861

119904(119897 minus 119881119905) minus 120579

119904)

+119861119904119881 sin (119861

119904(119897V minus 119881119905) + 120579119904)]

+1198812[2 minus exp (120572

119895119905) 120572119895119905 (120572119895119905 minus 2) + 2]

+4ℎ0

12057231198951198712

minus1205722

119895119897V (119897V + 119871) (exp (120572119895119905) minus 1)

+120572119895119881 (2119897V + 119871) [exp (120572119895119905) (120572119895119905 minus 1) + 1] ]

11986811989511990922

= 1198881199052[

119873

119904=1

119860119904119861119904119881

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2120572119895sin (119861

119904119897V minus 120579119904)

+ 119861119904119881 cos (119861

119904119897V minus 120579119904)

minus exp (120572119895119905) [120572119895sin (119861

119904(119897 minus 119881119905) minus 120579

119904)

+119861119904119881 cos (119861

119904(119897V minus 119881119905) + 120579119904)]

+4ℎ0119881 exp (minus120572

119895119905)

12057221198951198712

times 2119881 [exp (120572119895119905) (120572119895119905 minus 1) + 1]

minus120572119895(2119897V + 119871) (exp (120572119895119905) minus 1)]

119868119895119910=

119899119887

119896=1

119868119895119899V+2+119896

119886=1

119868(119895119899V+2+119896)119886

(119896 = 1 2 3 119899119887) (A11)

119876119904= 119861119904119881 119877

119896=119896120587119881

119871

119878119896=119896120587

119871 119885

119904= 119861119904119897V

119868(119895119899V+2+119896)

21198961199051

119873

119904=1

119860119904 ((2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) cos (120579

119904)

+2120572119895119876119904sin (120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times cos (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119876

119904+ 119877119896) 119905 + 120579

119904

minus (119877119896+ 119876119904)

times cos (119876119904+ 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

21198961199051[

4ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198762119904)3

times 119871 (1205722

119895+ 1198762

119904) [exp (120572

119895119905) sin (119876

119904119905)

times 1198762

119904(120572119895119905 + 1)

+1205722

119895(120572119895119905 minus 1)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904119905 + 120572119895(120572119895119905 minus 2)] minus 2120572

119895119876119904

+ 119881 (21198763

119904+ 119876119904exp (120572

119895119905)) cos (119876

119904119905)

times 1198764

1199041199052+ 21198762

119904(1205722

1198951199052minus 2120572119895119905 minus 1)

+1205722

119895(1205722

1198951199052minus 4120572119895119905 + 6)

minus exp (120572119895119905) sin (119876

119904119905)

times 1198764

119904(120572119895119905 + 2) + 120572

1198951198762

119904(1205722

1198951199052minus 3)

+1205723

119895(1205722

1198951199052minus 2120572119895119905 + 2) minus 6120572

119895119876119904]

119868(119895119899V+2+119896)

21198961199052

119873

119904=1

119860119904[ exp (minus120572

119895119905)

times (((119877119896+ 119876119904) cos (119878

119904minus 119885119904minus 120579119904)

+120572119895sin (119878119896minus 119885119904minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

times (((119877119896minus 119876119904) cos (119878

119896minus 119885119904+ 120579119904)

+120572119895sin (119878119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 119878

119896minus 119885119904+ 120579119904

minus (119877119896+ 119876119904) cos (119876

119904minus 119877119896) 119905

+119878119896minus 119885119904+ 120579119904)

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

+ ((120572119895sin 119878119896+ 119885119904minus (119876119904+ 119877) 119905 minus 120579

119904

+ (119877119896+ 119876119904) cos 119878

119896+ 119885119904

+ (119876119904+ 119877119896) 119905

minus120579119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1

21198961199052[

4ℎ0119881

1198712(1205722119895+ 1198812)

times sin (119881119905 minus 119897V) [1205721198951198972

V(1205722

119895+ 1198812)2

minus 119897V (1205722

119895+ 1198812)

times minus1205723

119895119871 minus 1205721198951198711198812

+ 21198813(120572119895119905 + 1)

+21205722

119895119881(120572119895119905 minus 1)

+ 119881 [119881 21205723

119895+ 120572119895119905(1205722

119895+ 1198812)2

+2119905 (1205724

119895minus 1198814) minus 6120572

1198951198812

minus 119871 (1205722

119895+ 1198812)

times 1205722

119895(120572119895119905 minus 1)+ 119881

2(120572119895119905 +1)]

+ 119881 cos (119897V minus 119881119905) [1198814(1198972

V + 119897V119871

+21205722

1198951199052minus 4120572119895119905 minus 2)

+ 1205722

1198951198812(21198972

V + 2119897V119871

+41205722

1198951199052+ 6)

+ 1205724

119895119897V (119897V + 119871)

minus 1198815119905 (2119897V + 119871)

times (120572119895119905 minus 1)

minus 1205723

119895119881 (2119897V + 119871)

times (120572119895119905 minus 2) + 119897

V1198816]]

119868(119895119899V+2+119896)

21198881199051

119873

119904=1

119860119904119876119904( minus (2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) sin (120579

119904)

minus2120572119895119876119904cos (120579

119904))

times (1205722

119895+ (119876119904minus 119877)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times sin (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos (119876

119904+ 119877119896) 119905 + 120579

119904

+ (119877119896+ 119876119904)

times sin (119876119904+ 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199051

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) cos (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) sin (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

21198881199052

119873

119904=1

119860119904119876119904

times [ exp (minus120572119895119905 ( ( (119877

119896+ 119876119904) sin (119878

119896+ 119885119904minus 120579119904)

minus120572119895cos (119878

119896+ 119885119904minus 120579119904))

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

times ( ( (119876119904minus 119877119896) sin (119878

119896minus 119885119904+ 120579119904)

+120572119895cos (119878

119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos 119878

119896+ 119885119904minus (119876119904+ 119877119896) 119905 minus 120579

119904

minus (119877119896+ 119876119904) sin 119878

119896+ 119885119904

minus (119876119904+ 119877119896) 119905 minus 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos 119878

119896minus 119885119904+ (119876119904minus 119877119896) 119905 + 120579

119904

+ (119876119904minus 119877119896) sin 119878

119896minus 119885119904

+ (119876119904minus 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199052

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) sin (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881 + exp (120572119895119905) 119896120587119881 cos(119896120587119881119905

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881cos(119896120587119863119871

119871)

minus 119871120572119895sin(119896120587119863

119871)

119871)]

119868(119895119899V+2+119896)

= minus21198711198812(1198981199081

+ 1198981199082)

119873

119904=1

1198601199041198612

119904[(((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881 cos [(119876

119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

119904119871)119881 cos 120579

119904

+119871120572119895sin 120579119904)

times (1198612

11990411987121198812

+ 21198961205871198611199041198711198812

+ 119896212058721198812

+11987121205722

119895)minus1

+ ((119896120587 minus 119861119904119871)119881 cos 120579

119904

minus119871120572119895sin 120579119904)

times ((119896120587 minus 119861119904119871)2

1198812

+11987121205722

119895)minus1

minus8ℎ01198812(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

times [119896120587119881

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119868119895119911=

119899119905

119896=1

119868119895119899V+2+119899119887+119896

119868119895119899V+2+119899119887+119897

119886=1

119868(119895119899V+2+119899119887+119897)119886

(119896 = 1 2 3 119899119879) (A14)

119868(119895119899V+2+119899119887+119897)1

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895cos (120579

119904) + (119876

119904minus 119877119896) sin (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895cos (120579

119904) + (119876

119904+ 119877119896) sin (120579

119904)

1205722119895+ (119876119904+ 119877119896)2

+ ((120572119895cos [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119876119904minus 119877119896) sin [(119876

119904minus 119877119896) 119905 minus 120579

119904])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [(119876

119904+ 119877119896) 119905 + 120579

119904]

+ (119876119904+ 119877119896) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)2

=41198961199051119890119909ℎ0119871

119881(1205722119895+ 119877119896

times [ (1205722

119895+ 119877119896

2) exp (minus120572

119895119905)

times 1205722

119895+ exp (minus120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)] minus 119877

119896

+ 2120572119895exp (minus120572

119895119905) (1205722

119895+ 3119877119896

+ 119877119896sin (119877

119896119905) 6120572

119895+ 1199052(1205722

119895+ 119877119896

+4120572119895119905 (1205722

119895+ 119877119896

2) minus 2119877

119896

+ cos (119877119896119905) 2119905 (119877

119896

4minus 1205724

119895)

+ 2120572119895(1205722

119895minus 3119877119896

times 1205721198951199052(1205722

119895+ 119877119896

119868(119895119899V+2+119899119887+119897)3

= minus1

21198961199052119890119909

119873

119904=1

[119860119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119877119896minus 119876119904) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) sin (119861

119904119897V + 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos [(119877

119896minus 119876119904) 119905 + (119861

119904119897V minus 119878119896 minus 120579119904)]

+ (119877119896minus 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [minus (119877

119896+ 119876119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

minus (119877119896+ 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)4

=41198961199052119890119909ℎ0

1198712[((119897V (119897V + 119871) exp (minus120572119895119905)

times 120572119895cos (119878

119896minus 119877119896119905)

minus119877119896sin (119878119896minus 119877119896119905)

+119877119896sin (119878) minus 120572

119895cos (119878

119896))

times (1205722

119895+ 119877119896

2)minus1

+2119871119881 exp (minus120572

119895119905)

(1205722119895+ 119877119896

times (1205722

119895+ 119877119896

2) cos (119878

119896)

+ exp (120572119895119905) cos (119878

119896minus 119877119896119905)

times [1198772(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896[119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

119895119877119896sin (119878119896)

minus1198812

(1205722119895+ 119877119896

times119877119896sin (119878119896minus 119877119896119905)[61205722

119895+1199052(1205722

119895+ 119877119896

minus 4120572119895119905 (1205722

119895+ 119877119896

minus2119877119896

+ 2 exp (minus120572119895119905)

times [(1205723

119895minus 3120572119895119877119896

2) cos (119878

119896)

+ 119877119896(119877119896

2minus 31205722

119895)

times sin (119878119896) ]

minus [2119905 (119877119896

4minus 1205724

119895) + 2120572

119895(1205722

119895minus 3119877119896

+1205721198951199052(1205722

119895+119877119896

]cos (119878119896minus 119877119896119905)]

119868(119895119899V+2+119899119887+119897)5

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895sin (120579119904) + (119876

119904minus 119877119896) cos (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895sin (120579119904) minus (119876

119904+ 119877119896) cos (120579

119904)

1205722119895+ (119876119904+ 119877)2

+ ((120572119895sin [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119877119896minus 119876119904) cos [(119876

119904minus 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895sin [(119876

119904+ 119877119896) 119905 + 120579

119904]

minus (119876119904+ 119877) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)6

=41198881199051119890119909ℎ0119871

(1205722119895+ 119877119896

2)3(119881

119871)

times [(1205722

119895+ 119877119896

2) 120572119895cos (119877

119896119905)

+ 119877119896sin (119877

119896119905) minus 120572

119895exp (minus120572

119895119905)

+ 2 exp (minus120572119895119905) 1205722

119895+ exp (120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

minus119877119896

119868(119895119899V+2+119899119887+119897)7

=41198881199052119890119909ℎ0

119881(1205722119895+ 119877119896

times 119897V (1205722

119895+ 119877119896

2) + 119871 (120572

119895+ 119877119896

minus2119881 [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

+ cos (119878119896minus 119877119896119905)

times 119877119896

2[120572119895(119897V + 119871) minus 2119881 (120572

119895119905 + 1)]

+1205722

119895[120572119895(119897V + 119871) + 2119881 (1 minus 120572

119895119905)]

+ exp (minus120572119895119905) 119877119896sin (119878119896)

times [119897V (1205722

119895+ 119877119896

+119871 (1205722

119895+ 119877119896

2)+4120572119895119881]

+ cos (119878119896)

times [119877119896

22119881 minus 120572

119895(119897V + 119871)

minus 1205722

119895120572119895(119897V + 119871)

+2119881]]

119868(119895119899V+2+119899119887+119897)8

21198881199052119890119909

119873

119904=1

[119860119904119876119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119876119904minus 119877119896) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) cos (119861

119904119897V + 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin [(119877 minus 119876

119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)]

+ (119876119904minus 119877119896) cos [(119877

119896minus 119876119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)])

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

minus ((120572119895sin [minus (119877 + 119876

119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

+ (119877119896+ 119876119904) cos [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)9

= minus119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [exp (120572119895119905) 119896120587119881 sin(119896120587119881119905

119871)

minus119871120572119895+ exp 119871120572

119895(120572119895119905) cos(119896120587119881119905

119871)]

minus 119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881sin(119896120587119863119871

119871)

minus 119890119909119871120572119895cos(119896120587119863

119871)

minus exp (120572119895119905)

times cos(119896120587119863 minus 119881119905

119871)]

119868(119895119899V+2+119899119887+119897)10

= minus21198711198812(1198981199081

+ 1198981199082) 119890119909

119873

119904=1

1198601199041198612

119904[ (((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881

times cos [(119876119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus exp (minus120572119895119905)

times ((minus (119896120587 +119861119904119871)119881 cos 120579

119904+119871120572119895sin 120579119904)

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

+ (((119896120587 minus 119861119904119871)119881 cos 120579

119904minus 119871120572119895sin 120579119904)

times((119896120587 minus 119861119904119871)2

1198812+ 11987121205722

119895)minus1

minus8ℎ01198812119890119909(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

119871)

+119871120572119895sin(119896120587119881119905

119871)]

References

RoboticsJournal of

Journal of

RotatingMachinery

VLSI Design

Shock and Vibration

Journal of

Volume 2014

SensorsJournal of

Propagation

0 05 1 15 2 25 3 35Time (s)

16E minus 3

12E minus 3

80E minus 4

40E minus 4

00E + 0

minus40E minus 4

120590y

50kmh60kmh80kmh

0 05 1 15 2 25 3 35Time (s)

16E minus 2

12E minus 2

80E minus 3

40E minus 3

00E + 0

120590ay

50kmh60kmh80kmh

0 05 1 15 2 25 3Time (s)

80E minus 3

40E minus 3

00E + 0

minus40E minus 3

minus80E minus 3

minus12E minus 2

minus16E minus 2

120583dz

50kmh60kmh80kmh

0 05 1 15 2 25 3Time (s)

12E minus 1

80E minus 2

40E minus 2

00E + 0

minus40E minus 2

minus80E minus 2

120583z

50kmh60kmh80kmh

0 1 2 3Time (s)

80E minus 1

40E minus 1

00E + 0

minus40E minus 1

minus80E minus 1

50kmh60kmh80kmh

120583az

0 05 1 15 2 25 3Time (s)

10E minus 3

80E minus 4

60E minus 4

40E minus 4

20E minus 4

00E + 0

120590dz (m

50kmh60kmh80kmh

0 1 2 3Time (s)

80E minus 3

60E minus 3

40E minus 3

20E minus 3

00E + 0

120590z

50kmh60kmh80kmh

0 05 1 15 2 25 3Time (s)

40E minus 2

30E minus 2

20E minus 2

10E minus 2

00E + 0

120590az

50kmh60kmh80kmh

A B C D

50E minus 04

40E minus 04

30E minus 04

20E minus 04

10E minus 04

00E + 00

120583dy

A B C D

40E minus 05

30E minus 05

20E minus 05

10E minus 05

00E + 00

120590dy

0 025 05 075 1 125 15 175Time (s)

30E + 5

25E + 5

20E + 5

15E + 5

10E + 5

50E + 4

00E + 0

120583F

0 025 05 075 1 125 15 175Time (s)

25E + 4

20E + 4

15E + 4

10E + 4

50E + 3

00E + 0

minus50 + 3

120590F

0 05 1 15 2 25 3 35

Time (s)

10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

minus40E minus 4

120583dy

82 times 1010 N-m2

0 05 1 15 2 25 3 35Time (s)

20E minus 5

16E minus 5

12E minus 5

80E minus 6

40E minus 6

00E + 0

120590dy

6N-m2 (B) 119864V119868V = 64 times 107N-m2

and (C) 119864V119868V = 82 times 1010N-m2

0 05 1 15 2 25 3 35

Time (s)

10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

120583dy

119909) (A) 119890

119909= 05m (B) 119890

119909= 1m and

(C) 119890119909= 15m

0 05 1 15 2 25 3 35Time (s)

30E minus 5

25E minus 5

20E minus 5

15E minus 5

10E minus 5

50E minus 6

00E + 0

120590dy

119909) (A) 119890

119909= 05m (B) 119890

119909=

1m and (C) 119890119909= 15m

119884dynamic =1003816100381610038161003816120583119884 (119909119896 119905119896) + 120590119884 (119909119896 119905119896)

1003816100381610038161003816 (46)

0 0002 0004 0006 0008 001

GbJbEbIb

119887119869119887119864119887119868119887) for different

107 N-m2 and (C) 119864V119868V = 82 times 10

10 N-m2

119884static (47)

119887119868119887)

5 Conclusions

1520 25 30 35 40

45 5055

404550556065707580146148

15152154156158

16162164

times103

8013514

Vehicle speed (kmh)

14161820222426283013

Appendix

119865119895(119905) = 0 for119895 = 1 2 3 119899V

= 119896119905119895minus119899V

ℎ (119909119895minus119899V

) + 119888119905119895minus119899V

ℎ (119909119895minus119899V

For119895 = 119899V + 1 119899V + 2

= 1198961199051ℎ (1199091) 120601119887119895minus(119899V+2) (1199091) + 1198961199052ℎ (1199092) 120601119887119895minus(119899V+2) (1199092)

+ 1198881199051ℎ (1199091) 120601119887119895minus(119899V+2) (1199091) + 1198881199052ℎ (1199092) 120601119887119895minus(119899V+2) (1199092)

For 119895 = 119899V + 3 119899V + 4 119899V + 2 + 119899119887

= 1198901199091198961199051ℎ (1199091) 120601119905119895minus(119899V+2+119899119887)

(1199091)

+ 1198961199052ℎ (1199092) 120601119905119895minus(119899V+2+119899119887)

(1199092)

+ 1198881199051ℎ (1199091) 120601119905119895minus(119899V+2+119899119887)

(1199091)

+1198881199052ℎ1015840(1199092) 120601119905119895minus(119899V+2+119899119887)

(1199092)

For 119895 = 119899V + 119899119887 + 3

119899V + 119899119887 + 4 119899V + 119899119887 + 119899119905 + 2

ℎ119898(119909) =

1198712119909 (119871 minus 119909) (A2)

ℎ119903(119909) =

119873

119904=1

119860119904cos (2120587Ω

119904119909 + 120579119904) (A3)

119868119895119896= 119868119895119908+ 119868119895119909+ 119868119895119910+ 119868119895119911 (A4)

where 119908 = 1 2 3 119899V 119909 = 119899V + 1 119899V + 2 119910 = 119899V + 2 + 1119899V + 2 + 2 119899V + 2 + 119899

119887 119911 = 119899V + 2 + 119899

119887+ 1 119899V + 2 + 119899

119887+

2 119899V + 2 + 119899119887 + 119899119879Thefirst term 119868

1198681198951= 1198681198952= sdot sdot sdot = 119868

119895119899V= 0 (A5)

119868119895119909= 1198681198951199091

+ 1198681198951199092 (A6)

1198681198951199091

= 11986811989511990911

+ 11986811989511990912

119904)minus2 and 119861

119904= 2120587Ω

119871+ (119904 minus

11986811989511990911

= 1198961199051[

119873

119904=1

119860119904

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2minus120572119895cos (120579

119904) minus 119861119904119881 sin (120579

119904)

+ exp (120572119895119905)

times [120572119895cos (119861

119904119881119905 + 120579

119904)

+119861119904119881 cos (119861

119904119881119905 + 120579

119904)]

+4ℎ0119881

12057231198951198712

120572119895119871 (120572119895119905 + exp (minus120572

119895119905) minus 1)

minus119881

12057231198951198712

119895119905) + 2 ]

11986811989511990912

= 1198881199051[

119873

119904=1

119881119860119904119861119904

exp (minus120572119895119905)

119904)

+ exp (120572119895119905)

times [120572119895sin (119861

119904119881119905 + 120579

119904)

minus119861119904119881 sin (119861

119904119881119905 + 120579

119904)]

119895119905)

12057231198951198712

minus1

120572119895119871exp (120572

119895119905) minus 2]

1198681198951199092

= 11986811989511990921

+ 11986811989511990922

11986811989511990921

= 1198961199052[

119873

119904=1

119860119904

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2

times minus 120572119895cos (119861

119904119897V minus 120579119904) minus 119861119904119881 sin (119861

119904119897V minus 120579119904)

+ exp (120572119895119905) [120572119895cos (119861

119904(119897 minus 119881119905) minus 120579

119904)

+119861119904119881 sin (119861

119904(119897V minus 119881119905) + 120579119904)]

+1198812[2 minus exp (120572

119895119905) 120572119895119905 (120572119895119905 minus 2) + 2]

+4ℎ0

12057231198951198712

minus1205722

119895119897V (119897V + 119871) (exp (120572119895119905) minus 1)

+120572119895119881 (2119897V + 119871) [exp (120572119895119905) (120572119895119905 minus 1) + 1] ]

11986811989511990922

= 1198881199052[

119873

119904=1

119860119904119861119904119881

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2120572119895sin (119861

119904119897V minus 120579119904)

+ 119861119904119881 cos (119861

119904119897V minus 120579119904)

minus exp (120572119895119905) [120572119895sin (119861

119904(119897 minus 119881119905) minus 120579

119904)

+119861119904119881 cos (119861

119904(119897V minus 119881119905) + 120579119904)]

+4ℎ0119881 exp (minus120572

119895119905)

12057221198951198712

times 2119881 [exp (120572119895119905) (120572119895119905 minus 1) + 1]

minus120572119895(2119897V + 119871) (exp (120572119895119905) minus 1)]

119868119895119910=

119899119887

119896=1

119868119895119899V+2+119896

119886=1

119868(119895119899V+2+119896)119886

(119896 = 1 2 3 119899119887) (A11)

119876119904= 119861119904119881 119877

119896=119896120587119881

119871

119878119896=119896120587

119871 119885

119904= 119861119904119897V

119868(119895119899V+2+119896)

21198961199051

119873

119904=1

119860119904 ((2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) cos (120579

119904)

+2120572119895119876119904sin (120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times cos (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119876

119904+ 119877119896) 119905 + 120579

119904

minus (119877119896+ 119876119904)

times cos (119876119904+ 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

21198961199051[

4ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198762119904)3

times 119871 (1205722

119895+ 1198762

119904) [exp (120572

119895119905) sin (119876

119904119905)

times 1198762

119904(120572119895119905 + 1)

+1205722

119895(120572119895119905 minus 1)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904119905 + 120572119895(120572119895119905 minus 2)] minus 2120572

119895119876119904

+ 119881 (21198763

119904+ 119876119904exp (120572

119895119905)) cos (119876

119904119905)

times 1198764

1199041199052+ 21198762

119904(1205722

1198951199052minus 2120572119895119905 minus 1)

+1205722

119895(1205722

1198951199052minus 4120572119895119905 + 6)

minus exp (120572119895119905) sin (119876

119904119905)

times 1198764

119904(120572119895119905 + 2) + 120572

1198951198762

119904(1205722

1198951199052minus 3)

+1205723

119895(1205722

1198951199052minus 2120572119895119905 + 2) minus 6120572

119895119876119904]

119868(119895119899V+2+119896)

21198961199052

119873

119904=1

119860119904[ exp (minus120572

119895119905)

times (((119877119896+ 119876119904) cos (119878

119904minus 119885119904minus 120579119904)

+120572119895sin (119878119896minus 119885119904minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

times (((119877119896minus 119876119904) cos (119878

119896minus 119885119904+ 120579119904)

+120572119895sin (119878119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 119878

119896minus 119885119904+ 120579119904

minus (119877119896+ 119876119904) cos (119876

119904minus 119877119896) 119905

+119878119896minus 119885119904+ 120579119904)

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

+ ((120572119895sin 119878119896+ 119885119904minus (119876119904+ 119877) 119905 minus 120579

119904

+ (119877119896+ 119876119904) cos 119878

119896+ 119885119904

+ (119876119904+ 119877119896) 119905

minus120579119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1

21198961199052[

4ℎ0119881

1198712(1205722119895+ 1198812)

times sin (119881119905 minus 119897V) [1205721198951198972

V(1205722

119895+ 1198812)2

minus 119897V (1205722

119895+ 1198812)

times minus1205723

119895119871 minus 1205721198951198711198812

+ 21198813(120572119895119905 + 1)

+21205722

119895119881(120572119895119905 minus 1)

+ 119881 [119881 21205723

119895+ 120572119895119905(1205722

119895+ 1198812)2

+2119905 (1205724

119895minus 1198814) minus 6120572

1198951198812

minus 119871 (1205722

119895+ 1198812)

times 1205722

119895(120572119895119905 minus 1)+ 119881

2(120572119895119905 +1)]

+ 119881 cos (119897V minus 119881119905) [1198814(1198972

V + 119897V119871

+21205722

1198951199052minus 4120572119895119905 minus 2)

+ 1205722

1198951198812(21198972

V + 2119897V119871

+41205722

1198951199052+ 6)

+ 1205724

119895119897V (119897V + 119871)

minus 1198815119905 (2119897V + 119871)

times (120572119895119905 minus 1)

minus 1205723

119895119881 (2119897V + 119871)

times (120572119895119905 minus 2) + 119897

V1198816]]

119868(119895119899V+2+119896)

21198881199051

119873

119904=1

119860119904119876119904( minus (2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) sin (120579

119904)

minus2120572119895119876119904cos (120579

119904))

times (1205722

119895+ (119876119904minus 119877)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times sin (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos (119876

119904+ 119877119896) 119905 + 120579

119904

+ (119877119896+ 119876119904)

times sin (119876119904+ 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199051

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) cos (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) sin (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

21198881199052

119873

119904=1

119860119904119876119904

times [ exp (minus120572119895119905 ( ( (119877

119896+ 119876119904) sin (119878

119896+ 119885119904minus 120579119904)

minus120572119895cos (119878

119896+ 119885119904minus 120579119904))

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

times ( ( (119876119904minus 119877119896) sin (119878

119896minus 119885119904+ 120579119904)

+120572119895cos (119878

119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos 119878

119896+ 119885119904minus (119876119904+ 119877119896) 119905 minus 120579

119904

minus (119877119896+ 119876119904) sin 119878

119896+ 119885119904

minus (119876119904+ 119877119896) 119905 minus 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos 119878

119896minus 119885119904+ (119876119904minus 119877119896) 119905 + 120579

119904

+ (119876119904minus 119877119896) sin 119878

119896minus 119885119904

+ (119876119904minus 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199052

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) sin (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881 + exp (120572119895119905) 119896120587119881 cos(119896120587119881119905

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881cos(119896120587119863119871

119871)

minus 119871120572119895sin(119896120587119863

119871)

119871)]

119868(119895119899V+2+119896)

= minus21198711198812(1198981199081

+ 1198981199082)

119873

119904=1

1198601199041198612

119904[(((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881 cos [(119876

119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

119904119871)119881 cos 120579

119904

+119871120572119895sin 120579119904)

times (1198612

11990411987121198812

+ 21198961205871198611199041198711198812

+ 119896212058721198812

+11987121205722

119895)minus1

+ ((119896120587 minus 119861119904119871)119881 cos 120579

119904

minus119871120572119895sin 120579119904)

times ((119896120587 minus 119861119904119871)2

1198812

+11987121205722

119895)minus1

minus8ℎ01198812(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

times [119896120587119881

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119868119895119911=

119899119905

119896=1

119868119895119899V+2+119899119887+119896

119868119895119899V+2+119899119887+119897

119886=1

119868(119895119899V+2+119899119887+119897)119886

(119896 = 1 2 3 119899119879) (A14)

119868(119895119899V+2+119899119887+119897)1

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895cos (120579

119904) + (119876

119904minus 119877119896) sin (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895cos (120579

119904) + (119876

119904+ 119877119896) sin (120579

119904)

1205722119895+ (119876119904+ 119877119896)2

+ ((120572119895cos [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119876119904minus 119877119896) sin [(119876

119904minus 119877119896) 119905 minus 120579

119904])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [(119876

119904+ 119877119896) 119905 + 120579

119904]

+ (119876119904+ 119877119896) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)2

=41198961199051119890119909ℎ0119871

119881(1205722119895+ 119877119896

times [ (1205722

119895+ 119877119896

2) exp (minus120572

119895119905)

times 1205722

119895+ exp (minus120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)] minus 119877

119896

+ 2120572119895exp (minus120572

119895119905) (1205722

119895+ 3119877119896

+ 119877119896sin (119877

119896119905) 6120572

119895+ 1199052(1205722

119895+ 119877119896

+4120572119895119905 (1205722

119895+ 119877119896

2) minus 2119877

119896

+ cos (119877119896119905) 2119905 (119877

119896

4minus 1205724

119895)

+ 2120572119895(1205722

119895minus 3119877119896

times 1205721198951199052(1205722

119895+ 119877119896

119868(119895119899V+2+119899119887+119897)3

= minus1

21198961199052119890119909

119873

119904=1

[119860119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119877119896minus 119876119904) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) sin (119861

119904119897V + 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos [(119877

119896minus 119876119904) 119905 + (119861

119904119897V minus 119878119896 minus 120579119904)]

+ (119877119896minus 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [minus (119877

119896+ 119876119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

minus (119877119896+ 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)4

=41198961199052119890119909ℎ0

1198712[((119897V (119897V + 119871) exp (minus120572119895119905)

times 120572119895cos (119878

119896minus 119877119896119905)

minus119877119896sin (119878119896minus 119877119896119905)

+119877119896sin (119878) minus 120572

119895cos (119878

119896))

times (1205722

119895+ 119877119896

2)minus1

+2119871119881 exp (minus120572

119895119905)

(1205722119895+ 119877119896

times (1205722

119895+ 119877119896

2) cos (119878

119896)

+ exp (120572119895119905) cos (119878

119896minus 119877119896119905)

times [1198772(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896[119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

119895119877119896sin (119878119896)

minus1198812

(1205722119895+ 119877119896

times119877119896sin (119878119896minus 119877119896119905)[61205722

119895+1199052(1205722

119895+ 119877119896

minus 4120572119895119905 (1205722

119895+ 119877119896

minus2119877119896

+ 2 exp (minus120572119895119905)

times [(1205723

119895minus 3120572119895119877119896

2) cos (119878

119896)

+ 119877119896(119877119896

2minus 31205722

119895)

times sin (119878119896) ]

minus [2119905 (119877119896

4minus 1205724

119895) + 2120572

119895(1205722

119895minus 3119877119896

+1205721198951199052(1205722

119895+119877119896

]cos (119878119896minus 119877119896119905)]

119868(119895119899V+2+119899119887+119897)5

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895sin (120579119904) + (119876

119904minus 119877119896) cos (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895sin (120579119904) minus (119876

119904+ 119877119896) cos (120579

119904)

1205722119895+ (119876119904+ 119877)2

+ ((120572119895sin [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119877119896minus 119876119904) cos [(119876

119904minus 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895sin [(119876

119904+ 119877119896) 119905 + 120579

119904]

minus (119876119904+ 119877) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)6

=41198881199051119890119909ℎ0119871

(1205722119895+ 119877119896

2)3(119881

119871)

times [(1205722

119895+ 119877119896

2) 120572119895cos (119877

119896119905)

+ 119877119896sin (119877

119896119905) minus 120572

119895exp (minus120572

119895119905)

+ 2 exp (minus120572119895119905) 1205722

119895+ exp (120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

minus119877119896

119868(119895119899V+2+119899119887+119897)7

=41198881199052119890119909ℎ0

119881(1205722119895+ 119877119896

times 119897V (1205722

119895+ 119877119896

2) + 119871 (120572

119895+ 119877119896

minus2119881 [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

+ cos (119878119896minus 119877119896119905)

times 119877119896

2[120572119895(119897V + 119871) minus 2119881 (120572

119895119905 + 1)]

+1205722

119895[120572119895(119897V + 119871) + 2119881 (1 minus 120572

119895119905)]

+ exp (minus120572119895119905) 119877119896sin (119878119896)

times [119897V (1205722

119895+ 119877119896

+119871 (1205722

119895+ 119877119896

2)+4120572119895119881]

+ cos (119878119896)

times [119877119896

22119881 minus 120572

119895(119897V + 119871)

minus 1205722

119895120572119895(119897V + 119871)

+2119881]]

119868(119895119899V+2+119899119887+119897)8

21198881199052119890119909

119873

119904=1

[119860119904119876119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119876119904minus 119877119896) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) cos (119861

119904119897V + 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin [(119877 minus 119876

119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)]

+ (119876119904minus 119877119896) cos [(119877

119896minus 119876119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)])

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

minus ((120572119895sin [minus (119877 + 119876

119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

+ (119877119896+ 119876119904) cos [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)9

= minus119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [exp (120572119895119905) 119896120587119881 sin(119896120587119881119905

119871)

minus119871120572119895+ exp 119871120572

119895(120572119895119905) cos(119896120587119881119905

119871)]

minus 119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881sin(119896120587119863119871

119871)

minus 119890119909119871120572119895cos(119896120587119863

119871)

minus exp (120572119895119905)

times cos(119896120587119863 minus 119881119905

119871)]

119868(119895119899V+2+119899119887+119897)10

= minus21198711198812(1198981199081

+ 1198981199082) 119890119909

119873

119904=1

1198601199041198612

119904[ (((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881

times cos [(119876119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus exp (minus120572119895119905)

times ((minus (119896120587 +119861119904119871)119881 cos 120579

119904+119871120572119895sin 120579119904)

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

+ (((119896120587 minus 119861119904119871)119881 cos 120579

119904minus 119871120572119895sin 120579119904)

times((119896120587 minus 119861119904119871)2

1198812+ 11987121205722

119895)minus1

minus8ℎ01198812119890119909(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

119871)

+119871120572119895sin(119896120587119881119905

119871)]

References

RoboticsJournal of

Journal of

RotatingMachinery

VLSI Design

Shock and Vibration

Journal of

Volume 2014

SensorsJournal of

Propagation

0 1 2 3Time (s)

80E minus 3

60E minus 3

40E minus 3

20E minus 3

00E + 0

120590z

50kmh60kmh80kmh

0 05 1 15 2 25 3Time (s)

40E minus 2

30E minus 2

20E minus 2

10E minus 2

00E + 0

120590az

50kmh60kmh80kmh

A B C D

50E minus 04

40E minus 04

30E minus 04

20E minus 04

10E minus 04

00E + 00

120583dy

A B C D

40E minus 05

30E minus 05

20E minus 05

10E minus 05

00E + 00

120590dy

0 025 05 075 1 125 15 175Time (s)

30E + 5

25E + 5

20E + 5

15E + 5

10E + 5

50E + 4

00E + 0

120583F

0 025 05 075 1 125 15 175Time (s)

25E + 4

20E + 4

15E + 4

10E + 4

50E + 3

00E + 0

minus50 + 3

120590F

0 05 1 15 2 25 3 35

Time (s)

10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

minus40E minus 4

120583dy

82 times 1010 N-m2

0 05 1 15 2 25 3 35Time (s)

20E minus 5

16E minus 5

12E minus 5

80E minus 6

40E minus 6

00E + 0

120590dy

6N-m2 (B) 119864V119868V = 64 times 107N-m2

and (C) 119864V119868V = 82 times 1010N-m2

0 05 1 15 2 25 3 35

Time (s)

10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

120583dy

119909) (A) 119890

119909= 05m (B) 119890

119909= 1m and

(C) 119890119909= 15m

0 05 1 15 2 25 3 35Time (s)

30E minus 5

25E minus 5

20E minus 5

15E minus 5

10E minus 5

50E minus 6

00E + 0

120590dy

119909) (A) 119890

119909= 05m (B) 119890

119909=

1m and (C) 119890119909= 15m

119884dynamic =1003816100381610038161003816120583119884 (119909119896 119905119896) + 120590119884 (119909119896 119905119896)

1003816100381610038161003816 (46)

0 0002 0004 0006 0008 001

GbJbEbIb

119887119869119887119864119887119868119887) for different

107 N-m2 and (C) 119864V119868V = 82 times 10

10 N-m2

119884static (47)

119887119868119887)

5 Conclusions

1520 25 30 35 40

45 5055

404550556065707580146148

15152154156158

16162164

times103

8013514

Vehicle speed (kmh)

14161820222426283013

Appendix

119865119895(119905) = 0 for119895 = 1 2 3 119899V

= 119896119905119895minus119899V

ℎ (119909119895minus119899V

) + 119888119905119895minus119899V

ℎ (119909119895minus119899V

For119895 = 119899V + 1 119899V + 2

= 1198961199051ℎ (1199091) 120601119887119895minus(119899V+2) (1199091) + 1198961199052ℎ (1199092) 120601119887119895minus(119899V+2) (1199092)

+ 1198881199051ℎ (1199091) 120601119887119895minus(119899V+2) (1199091) + 1198881199052ℎ (1199092) 120601119887119895minus(119899V+2) (1199092)

For 119895 = 119899V + 3 119899V + 4 119899V + 2 + 119899119887

= 1198901199091198961199051ℎ (1199091) 120601119905119895minus(119899V+2+119899119887)

(1199091)

+ 1198961199052ℎ (1199092) 120601119905119895minus(119899V+2+119899119887)

(1199092)

+ 1198881199051ℎ (1199091) 120601119905119895minus(119899V+2+119899119887)

(1199091)

+1198881199052ℎ1015840(1199092) 120601119905119895minus(119899V+2+119899119887)

(1199092)

For 119895 = 119899V + 119899119887 + 3

119899V + 119899119887 + 4 119899V + 119899119887 + 119899119905 + 2

ℎ119898(119909) =

1198712119909 (119871 minus 119909) (A2)

ℎ119903(119909) =

119873

119904=1

119860119904cos (2120587Ω

119904119909 + 120579119904) (A3)

119868119895119896= 119868119895119908+ 119868119895119909+ 119868119895119910+ 119868119895119911 (A4)

where 119908 = 1 2 3 119899V 119909 = 119899V + 1 119899V + 2 119910 = 119899V + 2 + 1119899V + 2 + 2 119899V + 2 + 119899

119887 119911 = 119899V + 2 + 119899

119887+ 1 119899V + 2 + 119899

119887+

2 119899V + 2 + 119899119887 + 119899119879Thefirst term 119868

1198681198951= 1198681198952= sdot sdot sdot = 119868

119895119899V= 0 (A5)

119868119895119909= 1198681198951199091

+ 1198681198951199092 (A6)

1198681198951199091

= 11986811989511990911

+ 11986811989511990912

119904)minus2 and 119861

119904= 2120587Ω

119871+ (119904 minus

11986811989511990911

= 1198961199051[

119873

119904=1

119860119904

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2minus120572119895cos (120579

119904) minus 119861119904119881 sin (120579

119904)

+ exp (120572119895119905)

times [120572119895cos (119861

119904119881119905 + 120579

119904)

+119861119904119881 cos (119861

119904119881119905 + 120579

119904)]

+4ℎ0119881

12057231198951198712

120572119895119871 (120572119895119905 + exp (minus120572

119895119905) minus 1)

minus119881

12057231198951198712

119895119905) + 2 ]

11986811989511990912

= 1198881199051[

119873

119904=1

119881119860119904119861119904

exp (minus120572119895119905)

119904)

+ exp (120572119895119905)

times [120572119895sin (119861

119904119881119905 + 120579

119904)

minus119861119904119881 sin (119861

119904119881119905 + 120579

119904)]

119895119905)

12057231198951198712

minus1

120572119895119871exp (120572

119895119905) minus 2]

1198681198951199092

= 11986811989511990921

+ 11986811989511990922

11986811989511990921

= 1198961199052[

119873

119904=1

119860119904

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2

times minus 120572119895cos (119861

119904119897V minus 120579119904) minus 119861119904119881 sin (119861

119904119897V minus 120579119904)

+ exp (120572119895119905) [120572119895cos (119861

119904(119897 minus 119881119905) minus 120579

119904)

+119861119904119881 sin (119861

119904(119897V minus 119881119905) + 120579119904)]

+1198812[2 minus exp (120572

119895119905) 120572119895119905 (120572119895119905 minus 2) + 2]

+4ℎ0

12057231198951198712

minus1205722

119895119897V (119897V + 119871) (exp (120572119895119905) minus 1)

+120572119895119881 (2119897V + 119871) [exp (120572119895119905) (120572119895119905 minus 1) + 1] ]

11986811989511990922

= 1198881199052[

119873

119904=1

119860119904119861119904119881

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2120572119895sin (119861

119904119897V minus 120579119904)

+ 119861119904119881 cos (119861

119904119897V minus 120579119904)

minus exp (120572119895119905) [120572119895sin (119861

119904(119897 minus 119881119905) minus 120579

119904)

+119861119904119881 cos (119861

119904(119897V minus 119881119905) + 120579119904)]

+4ℎ0119881 exp (minus120572

119895119905)

12057221198951198712

times 2119881 [exp (120572119895119905) (120572119895119905 minus 1) + 1]

minus120572119895(2119897V + 119871) (exp (120572119895119905) minus 1)]

119868119895119910=

119899119887

119896=1

119868119895119899V+2+119896

119886=1

119868(119895119899V+2+119896)119886

(119896 = 1 2 3 119899119887) (A11)

119876119904= 119861119904119881 119877

119896=119896120587119881

119871

119878119896=119896120587

119871 119885

119904= 119861119904119897V

119868(119895119899V+2+119896)

21198961199051

119873

119904=1

119860119904 ((2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) cos (120579

119904)

+2120572119895119876119904sin (120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times cos (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119876

119904+ 119877119896) 119905 + 120579

119904

minus (119877119896+ 119876119904)

times cos (119876119904+ 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

21198961199051[

4ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198762119904)3

times 119871 (1205722

119895+ 1198762

119904) [exp (120572

119895119905) sin (119876

119904119905)

times 1198762

119904(120572119895119905 + 1)

+1205722

119895(120572119895119905 minus 1)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904119905 + 120572119895(120572119895119905 minus 2)] minus 2120572

119895119876119904

+ 119881 (21198763

119904+ 119876119904exp (120572

119895119905)) cos (119876

119904119905)

times 1198764

1199041199052+ 21198762

119904(1205722

1198951199052minus 2120572119895119905 minus 1)

+1205722

119895(1205722

1198951199052minus 4120572119895119905 + 6)

minus exp (120572119895119905) sin (119876

119904119905)

times 1198764

119904(120572119895119905 + 2) + 120572

1198951198762

119904(1205722

1198951199052minus 3)

+1205723

119895(1205722

1198951199052minus 2120572119895119905 + 2) minus 6120572

119895119876119904]

119868(119895119899V+2+119896)

21198961199052

119873

119904=1

119860119904[ exp (minus120572

119895119905)

times (((119877119896+ 119876119904) cos (119878

119904minus 119885119904minus 120579119904)

+120572119895sin (119878119896minus 119885119904minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

times (((119877119896minus 119876119904) cos (119878

119896minus 119885119904+ 120579119904)

+120572119895sin (119878119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 119878

119896minus 119885119904+ 120579119904

minus (119877119896+ 119876119904) cos (119876

119904minus 119877119896) 119905

+119878119896minus 119885119904+ 120579119904)

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

+ ((120572119895sin 119878119896+ 119885119904minus (119876119904+ 119877) 119905 minus 120579

119904

+ (119877119896+ 119876119904) cos 119878

119896+ 119885119904

+ (119876119904+ 119877119896) 119905

minus120579119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1

21198961199052[

4ℎ0119881

1198712(1205722119895+ 1198812)

times sin (119881119905 minus 119897V) [1205721198951198972

V(1205722

119895+ 1198812)2

minus 119897V (1205722

119895+ 1198812)

times minus1205723

119895119871 minus 1205721198951198711198812

+ 21198813(120572119895119905 + 1)

+21205722

119895119881(120572119895119905 minus 1)

+ 119881 [119881 21205723

119895+ 120572119895119905(1205722

119895+ 1198812)2

+2119905 (1205724

119895minus 1198814) minus 6120572

1198951198812

minus 119871 (1205722

119895+ 1198812)

times 1205722

119895(120572119895119905 minus 1)+ 119881

2(120572119895119905 +1)]

+ 119881 cos (119897V minus 119881119905) [1198814(1198972

V + 119897V119871

+21205722

1198951199052minus 4120572119895119905 minus 2)

+ 1205722

1198951198812(21198972

V + 2119897V119871

+41205722

1198951199052+ 6)

+ 1205724

119895119897V (119897V + 119871)

minus 1198815119905 (2119897V + 119871)

times (120572119895119905 minus 1)

minus 1205723

119895119881 (2119897V + 119871)

times (120572119895119905 minus 2) + 119897

V1198816]]

119868(119895119899V+2+119896)

21198881199051

119873

119904=1

119860119904119876119904( minus (2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) sin (120579

119904)

minus2120572119895119876119904cos (120579

119904))

times (1205722

119895+ (119876119904minus 119877)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times sin (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos (119876

119904+ 119877119896) 119905 + 120579

119904

+ (119877119896+ 119876119904)

times sin (119876119904+ 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199051

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) cos (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) sin (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

21198881199052

119873

119904=1

119860119904119876119904

times [ exp (minus120572119895119905 ( ( (119877

119896+ 119876119904) sin (119878

119896+ 119885119904minus 120579119904)

minus120572119895cos (119878

119896+ 119885119904minus 120579119904))

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

times ( ( (119876119904minus 119877119896) sin (119878

119896minus 119885119904+ 120579119904)

+120572119895cos (119878

119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos 119878

119896+ 119885119904minus (119876119904+ 119877119896) 119905 minus 120579

119904

minus (119877119896+ 119876119904) sin 119878

119896+ 119885119904

minus (119876119904+ 119877119896) 119905 minus 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos 119878

119896minus 119885119904+ (119876119904minus 119877119896) 119905 + 120579

119904

+ (119876119904minus 119877119896) sin 119878

119896minus 119885119904

+ (119876119904minus 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199052

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) sin (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881 + exp (120572119895119905) 119896120587119881 cos(119896120587119881119905

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881cos(119896120587119863119871

119871)

minus 119871120572119895sin(119896120587119863

119871)

119871)]

119868(119895119899V+2+119896)

= minus21198711198812(1198981199081

+ 1198981199082)

119873

119904=1

1198601199041198612

119904[(((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881 cos [(119876

119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

119904119871)119881 cos 120579

119904

+119871120572119895sin 120579119904)

times (1198612

11990411987121198812

+ 21198961205871198611199041198711198812

+ 119896212058721198812

+11987121205722

119895)minus1

+ ((119896120587 minus 119861119904119871)119881 cos 120579

119904

minus119871120572119895sin 120579119904)

times ((119896120587 minus 119861119904119871)2

1198812

+11987121205722

119895)minus1

minus8ℎ01198812(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

times [119896120587119881

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119868119895119911=

119899119905

119896=1

119868119895119899V+2+119899119887+119896

119868119895119899V+2+119899119887+119897

119886=1

119868(119895119899V+2+119899119887+119897)119886

(119896 = 1 2 3 119899119879) (A14)

119868(119895119899V+2+119899119887+119897)1

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895cos (120579

119904) + (119876

119904minus 119877119896) sin (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895cos (120579

119904) + (119876

119904+ 119877119896) sin (120579

119904)

1205722119895+ (119876119904+ 119877119896)2

+ ((120572119895cos [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119876119904minus 119877119896) sin [(119876

119904minus 119877119896) 119905 minus 120579

119904])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [(119876

119904+ 119877119896) 119905 + 120579

119904]

+ (119876119904+ 119877119896) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)2

=41198961199051119890119909ℎ0119871

119881(1205722119895+ 119877119896

times [ (1205722

119895+ 119877119896

2) exp (minus120572

119895119905)

times 1205722

119895+ exp (minus120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)] minus 119877

119896

+ 2120572119895exp (minus120572

119895119905) (1205722

119895+ 3119877119896

+ 119877119896sin (119877

119896119905) 6120572

119895+ 1199052(1205722

119895+ 119877119896

+4120572119895119905 (1205722

119895+ 119877119896

2) minus 2119877

119896

+ cos (119877119896119905) 2119905 (119877

119896

4minus 1205724

119895)

+ 2120572119895(1205722

119895minus 3119877119896

times 1205721198951199052(1205722

119895+ 119877119896

119868(119895119899V+2+119899119887+119897)3

= minus1

21198961199052119890119909

119873

119904=1

[119860119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119877119896minus 119876119904) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) sin (119861

119904119897V + 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos [(119877

119896minus 119876119904) 119905 + (119861

119904119897V minus 119878119896 minus 120579119904)]

+ (119877119896minus 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [minus (119877

119896+ 119876119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

minus (119877119896+ 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)4

=41198961199052119890119909ℎ0

1198712[((119897V (119897V + 119871) exp (minus120572119895119905)

times 120572119895cos (119878

119896minus 119877119896119905)

minus119877119896sin (119878119896minus 119877119896119905)

+119877119896sin (119878) minus 120572

119895cos (119878

119896))

times (1205722

119895+ 119877119896

2)minus1

+2119871119881 exp (minus120572

119895119905)

(1205722119895+ 119877119896

times (1205722

119895+ 119877119896

2) cos (119878

119896)

+ exp (120572119895119905) cos (119878

119896minus 119877119896119905)

times [1198772(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896[119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

119895119877119896sin (119878119896)

minus1198812

(1205722119895+ 119877119896

times119877119896sin (119878119896minus 119877119896119905)[61205722

119895+1199052(1205722

119895+ 119877119896

minus 4120572119895119905 (1205722

119895+ 119877119896

minus2119877119896

+ 2 exp (minus120572119895119905)

times [(1205723

119895minus 3120572119895119877119896

2) cos (119878

119896)

+ 119877119896(119877119896

2minus 31205722

119895)

times sin (119878119896) ]

minus [2119905 (119877119896

4minus 1205724

119895) + 2120572

119895(1205722

119895minus 3119877119896

+1205721198951199052(1205722

119895+119877119896

]cos (119878119896minus 119877119896119905)]

119868(119895119899V+2+119899119887+119897)5

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895sin (120579119904) + (119876

119904minus 119877119896) cos (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895sin (120579119904) minus (119876

119904+ 119877119896) cos (120579

119904)

1205722119895+ (119876119904+ 119877)2

+ ((120572119895sin [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119877119896minus 119876119904) cos [(119876

119904minus 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895sin [(119876

119904+ 119877119896) 119905 + 120579

119904]

minus (119876119904+ 119877) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)6

=41198881199051119890119909ℎ0119871

(1205722119895+ 119877119896

2)3(119881

119871)

times [(1205722

119895+ 119877119896

2) 120572119895cos (119877

119896119905)

+ 119877119896sin (119877

119896119905) minus 120572

119895exp (minus120572

119895119905)

+ 2 exp (minus120572119895119905) 1205722

119895+ exp (120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

minus119877119896

119868(119895119899V+2+119899119887+119897)7

=41198881199052119890119909ℎ0

119881(1205722119895+ 119877119896

times 119897V (1205722

119895+ 119877119896

2) + 119871 (120572

119895+ 119877119896

minus2119881 [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

+ cos (119878119896minus 119877119896119905)

times 119877119896

2[120572119895(119897V + 119871) minus 2119881 (120572

119895119905 + 1)]

+1205722

119895[120572119895(119897V + 119871) + 2119881 (1 minus 120572

119895119905)]

+ exp (minus120572119895119905) 119877119896sin (119878119896)

times [119897V (1205722

119895+ 119877119896

+119871 (1205722

119895+ 119877119896

2)+4120572119895119881]

+ cos (119878119896)

times [119877119896

22119881 minus 120572

119895(119897V + 119871)

minus 1205722

119895120572119895(119897V + 119871)

+2119881]]

119868(119895119899V+2+119899119887+119897)8

21198881199052119890119909

119873

119904=1

[119860119904119876119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119876119904minus 119877119896) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) cos (119861

119904119897V + 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin [(119877 minus 119876

119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)]

+ (119876119904minus 119877119896) cos [(119877

119896minus 119876119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)])

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

minus ((120572119895sin [minus (119877 + 119876

119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

+ (119877119896+ 119876119904) cos [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)9

= minus119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [exp (120572119895119905) 119896120587119881 sin(119896120587119881119905

119871)

minus119871120572119895+ exp 119871120572

119895(120572119895119905) cos(119896120587119881119905

119871)]

minus 119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881sin(119896120587119863119871

119871)

minus 119890119909119871120572119895cos(119896120587119863

119871)

minus exp (120572119895119905)

times cos(119896120587119863 minus 119881119905

119871)]

119868(119895119899V+2+119899119887+119897)10

= minus21198711198812(1198981199081

+ 1198981199082) 119890119909

119873

119904=1

1198601199041198612

119904[ (((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881

times cos [(119876119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus exp (minus120572119895119905)

times ((minus (119896120587 +119861119904119871)119881 cos 120579

119904+119871120572119895sin 120579119904)

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

+ (((119896120587 minus 119861119904119871)119881 cos 120579

119904minus 119871120572119895sin 120579119904)

times((119896120587 minus 119861119904119871)2

1198812+ 11987121205722

119895)minus1

minus8ℎ01198812119890119909(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

119871)

+119871120572119895sin(119896120587119881119905

119871)]

References

RoboticsJournal of

Journal of

RotatingMachinery

VLSI Design

Shock and Vibration

Journal of

Volume 2014

SensorsJournal of

Propagation

0 05 1 15 2 25 3 35

Time (s)

10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

minus40E minus 4

120583dy

82 times 1010 N-m2

0 05 1 15 2 25 3 35Time (s)

20E minus 5

16E minus 5

12E minus 5

80E minus 6

40E minus 6

00E + 0

120590dy

6N-m2 (B) 119864V119868V = 64 times 107N-m2

and (C) 119864V119868V = 82 times 1010N-m2

0 05 1 15 2 25 3 35

Time (s)

10E minus 4

00E + 0

minus10E minus 4

minus20E minus 4

minus30E minus 4

120583dy

119909) (A) 119890

119909= 05m (B) 119890

119909= 1m and

(C) 119890119909= 15m

0 05 1 15 2 25 3 35Time (s)

30E minus 5

25E minus 5

20E minus 5

15E minus 5

10E minus 5

50E minus 6

00E + 0

120590dy

119909) (A) 119890

119909= 05m (B) 119890

119909=

1m and (C) 119890119909= 15m

119884dynamic =1003816100381610038161003816120583119884 (119909119896 119905119896) + 120590119884 (119909119896 119905119896)

1003816100381610038161003816 (46)

0 0002 0004 0006 0008 001

GbJbEbIb

119887119869119887119864119887119868119887) for different

107 N-m2 and (C) 119864V119868V = 82 times 10

10 N-m2

119884static (47)

119887119868119887)

5 Conclusions

1520 25 30 35 40

45 5055

404550556065707580146148

15152154156158

16162164

times103

8013514

Vehicle speed (kmh)

14161820222426283013

Appendix

119865119895(119905) = 0 for119895 = 1 2 3 119899V

= 119896119905119895minus119899V

ℎ (119909119895minus119899V

) + 119888119905119895minus119899V

ℎ (119909119895minus119899V

For119895 = 119899V + 1 119899V + 2

= 1198961199051ℎ (1199091) 120601119887119895minus(119899V+2) (1199091) + 1198961199052ℎ (1199092) 120601119887119895minus(119899V+2) (1199092)

+ 1198881199051ℎ (1199091) 120601119887119895minus(119899V+2) (1199091) + 1198881199052ℎ (1199092) 120601119887119895minus(119899V+2) (1199092)

For 119895 = 119899V + 3 119899V + 4 119899V + 2 + 119899119887

= 1198901199091198961199051ℎ (1199091) 120601119905119895minus(119899V+2+119899119887)

(1199091)

+ 1198961199052ℎ (1199092) 120601119905119895minus(119899V+2+119899119887)

(1199092)

+ 1198881199051ℎ (1199091) 120601119905119895minus(119899V+2+119899119887)

(1199091)

+1198881199052ℎ1015840(1199092) 120601119905119895minus(119899V+2+119899119887)

(1199092)

For 119895 = 119899V + 119899119887 + 3

119899V + 119899119887 + 4 119899V + 119899119887 + 119899119905 + 2

ℎ119898(119909) =

1198712119909 (119871 minus 119909) (A2)

ℎ119903(119909) =

119873

119904=1

119860119904cos (2120587Ω

119904119909 + 120579119904) (A3)

119868119895119896= 119868119895119908+ 119868119895119909+ 119868119895119910+ 119868119895119911 (A4)

where 119908 = 1 2 3 119899V 119909 = 119899V + 1 119899V + 2 119910 = 119899V + 2 + 1119899V + 2 + 2 119899V + 2 + 119899

119887 119911 = 119899V + 2 + 119899

119887+ 1 119899V + 2 + 119899

119887+

2 119899V + 2 + 119899119887 + 119899119879Thefirst term 119868

1198681198951= 1198681198952= sdot sdot sdot = 119868

119895119899V= 0 (A5)

119868119895119909= 1198681198951199091

+ 1198681198951199092 (A6)

1198681198951199091

= 11986811989511990911

+ 11986811989511990912

119904)minus2 and 119861

119904= 2120587Ω

119871+ (119904 minus

11986811989511990911

= 1198961199051[

119873

119904=1

119860119904

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2minus120572119895cos (120579

119904) minus 119861119904119881 sin (120579

119904)

+ exp (120572119895119905)

times [120572119895cos (119861

119904119881119905 + 120579

119904)

+119861119904119881 cos (119861

119904119881119905 + 120579

119904)]

+4ℎ0119881

12057231198951198712

120572119895119871 (120572119895119905 + exp (minus120572

119895119905) minus 1)

minus119881

12057231198951198712

119895119905) + 2 ]

11986811989511990912

= 1198881199051[

119873

119904=1

119881119860119904119861119904

exp (minus120572119895119905)

119904)

+ exp (120572119895119905)

times [120572119895sin (119861

119904119881119905 + 120579

119904)

minus119861119904119881 sin (119861

119904119881119905 + 120579

119904)]

119895119905)

12057231198951198712

minus1

120572119895119871exp (120572

119895119905) minus 2]

1198681198951199092

= 11986811989511990921

+ 11986811989511990922

11986811989511990921

= 1198961199052[

119873

119904=1

119860119904

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2

times minus 120572119895cos (119861

119904119897V minus 120579119904) minus 119861119904119881 sin (119861

119904119897V minus 120579119904)

+ exp (120572119895119905) [120572119895cos (119861

119904(119897 minus 119881119905) minus 120579

119904)

+119861119904119881 sin (119861

119904(119897V minus 119881119905) + 120579119904)]

+1198812[2 minus exp (120572

119895119905) 120572119895119905 (120572119895119905 minus 2) + 2]

+4ℎ0

12057231198951198712

minus1205722

119895119897V (119897V + 119871) (exp (120572119895119905) minus 1)

+120572119895119881 (2119897V + 119871) [exp (120572119895119905) (120572119895119905 minus 1) + 1] ]

11986811989511990922

= 1198881199052[

119873

119904=1

119860119904119861119904119881

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2120572119895sin (119861

119904119897V minus 120579119904)

+ 119861119904119881 cos (119861

119904119897V minus 120579119904)

minus exp (120572119895119905) [120572119895sin (119861

119904(119897 minus 119881119905) minus 120579

119904)

+119861119904119881 cos (119861

119904(119897V minus 119881119905) + 120579119904)]

+4ℎ0119881 exp (minus120572

119895119905)

12057221198951198712

times 2119881 [exp (120572119895119905) (120572119895119905 minus 1) + 1]

minus120572119895(2119897V + 119871) (exp (120572119895119905) minus 1)]

119868119895119910=

119899119887

119896=1

119868119895119899V+2+119896

119886=1

119868(119895119899V+2+119896)119886

(119896 = 1 2 3 119899119887) (A11)

119876119904= 119861119904119881 119877

119896=119896120587119881

119871

119878119896=119896120587

119871 119885

119904= 119861119904119897V

119868(119895119899V+2+119896)

21198961199051

119873

119904=1

119860119904 ((2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) cos (120579

119904)

+2120572119895119876119904sin (120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times cos (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119876

119904+ 119877119896) 119905 + 120579

119904

minus (119877119896+ 119876119904)

times cos (119876119904+ 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

21198961199051[

4ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198762119904)3

times 119871 (1205722

119895+ 1198762

119904) [exp (120572

119895119905) sin (119876

119904119905)

times 1198762

119904(120572119895119905 + 1)

+1205722

119895(120572119895119905 minus 1)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904119905 + 120572119895(120572119895119905 minus 2)] minus 2120572

119895119876119904

+ 119881 (21198763

119904+ 119876119904exp (120572

119895119905)) cos (119876

119904119905)

times 1198764

1199041199052+ 21198762

119904(1205722

1198951199052minus 2120572119895119905 minus 1)

+1205722

119895(1205722

1198951199052minus 4120572119895119905 + 6)

minus exp (120572119895119905) sin (119876

119904119905)

times 1198764

119904(120572119895119905 + 2) + 120572

1198951198762

119904(1205722

1198951199052minus 3)

+1205723

119895(1205722

1198951199052minus 2120572119895119905 + 2) minus 6120572

119895119876119904]

119868(119895119899V+2+119896)

21198961199052

119873

119904=1

119860119904[ exp (minus120572

119895119905)

times (((119877119896+ 119876119904) cos (119878

119904minus 119885119904minus 120579119904)

+120572119895sin (119878119896minus 119885119904minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

times (((119877119896minus 119876119904) cos (119878

119896minus 119885119904+ 120579119904)

+120572119895sin (119878119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 119878

119896minus 119885119904+ 120579119904

minus (119877119896+ 119876119904) cos (119876

119904minus 119877119896) 119905

+119878119896minus 119885119904+ 120579119904)

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

+ ((120572119895sin 119878119896+ 119885119904minus (119876119904+ 119877) 119905 minus 120579

119904

+ (119877119896+ 119876119904) cos 119878

119896+ 119885119904

+ (119876119904+ 119877119896) 119905

minus120579119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1

21198961199052[

4ℎ0119881

1198712(1205722119895+ 1198812)

times sin (119881119905 minus 119897V) [1205721198951198972

V(1205722

119895+ 1198812)2

minus 119897V (1205722

119895+ 1198812)

times minus1205723

119895119871 minus 1205721198951198711198812

+ 21198813(120572119895119905 + 1)

+21205722

119895119881(120572119895119905 minus 1)

+ 119881 [119881 21205723

119895+ 120572119895119905(1205722

119895+ 1198812)2

+2119905 (1205724

119895minus 1198814) minus 6120572

1198951198812

minus 119871 (1205722

119895+ 1198812)

times 1205722

119895(120572119895119905 minus 1)+ 119881

2(120572119895119905 +1)]

+ 119881 cos (119897V minus 119881119905) [1198814(1198972

V + 119897V119871

+21205722

1198951199052minus 4120572119895119905 minus 2)

+ 1205722

1198951198812(21198972

V + 2119897V119871

+41205722

1198951199052+ 6)

+ 1205724

119895119897V (119897V + 119871)

minus 1198815119905 (2119897V + 119871)

times (120572119895119905 minus 1)

minus 1205723

119895119881 (2119897V + 119871)

times (120572119895119905 minus 2) + 119897

V1198816]]

119868(119895119899V+2+119896)

21198881199051

119873

119904=1

119860119904119876119904( minus (2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) sin (120579

119904)

minus2120572119895119876119904cos (120579

119904))

times (1205722

119895+ (119876119904minus 119877)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times sin (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos (119876

119904+ 119877119896) 119905 + 120579

119904

+ (119877119896+ 119876119904)

times sin (119876119904+ 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199051

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) cos (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) sin (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

21198881199052

119873

119904=1

119860119904119876119904

times [ exp (minus120572119895119905 ( ( (119877

119896+ 119876119904) sin (119878

119896+ 119885119904minus 120579119904)

minus120572119895cos (119878

119896+ 119885119904minus 120579119904))

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

times ( ( (119876119904minus 119877119896) sin (119878

119896minus 119885119904+ 120579119904)

+120572119895cos (119878

119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos 119878

119896+ 119885119904minus (119876119904+ 119877119896) 119905 minus 120579

119904

minus (119877119896+ 119876119904) sin 119878

119896+ 119885119904

minus (119876119904+ 119877119896) 119905 minus 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos 119878

119896minus 119885119904+ (119876119904minus 119877119896) 119905 + 120579

119904

+ (119876119904minus 119877119896) sin 119878

119896minus 119885119904

+ (119876119904minus 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199052

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) sin (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881 + exp (120572119895119905) 119896120587119881 cos(119896120587119881119905

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881cos(119896120587119863119871

119871)

minus 119871120572119895sin(119896120587119863

119871)

119871)]

119868(119895119899V+2+119896)

= minus21198711198812(1198981199081

+ 1198981199082)

119873

119904=1

1198601199041198612

119904[(((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881 cos [(119876

119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

119904119871)119881 cos 120579

119904

+119871120572119895sin 120579119904)

times (1198612

11990411987121198812

+ 21198961205871198611199041198711198812

+ 119896212058721198812

+11987121205722

119895)minus1

+ ((119896120587 minus 119861119904119871)119881 cos 120579

119904

minus119871120572119895sin 120579119904)

times ((119896120587 minus 119861119904119871)2

1198812

+11987121205722

119895)minus1

minus8ℎ01198812(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

times [119896120587119881

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119868119895119911=

119899119905

119896=1

119868119895119899V+2+119899119887+119896

119868119895119899V+2+119899119887+119897

119886=1

119868(119895119899V+2+119899119887+119897)119886

(119896 = 1 2 3 119899119879) (A14)

119868(119895119899V+2+119899119887+119897)1

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895cos (120579

119904) + (119876

119904minus 119877119896) sin (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895cos (120579

119904) + (119876

119904+ 119877119896) sin (120579

119904)

1205722119895+ (119876119904+ 119877119896)2

+ ((120572119895cos [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119876119904minus 119877119896) sin [(119876

119904minus 119877119896) 119905 minus 120579

119904])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [(119876

119904+ 119877119896) 119905 + 120579

119904]

+ (119876119904+ 119877119896) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)2

=41198961199051119890119909ℎ0119871

119881(1205722119895+ 119877119896

times [ (1205722

119895+ 119877119896

2) exp (minus120572

119895119905)

times 1205722

119895+ exp (minus120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)] minus 119877

119896

+ 2120572119895exp (minus120572

119895119905) (1205722

119895+ 3119877119896

+ 119877119896sin (119877

119896119905) 6120572

119895+ 1199052(1205722

119895+ 119877119896

+4120572119895119905 (1205722

119895+ 119877119896

2) minus 2119877

119896

+ cos (119877119896119905) 2119905 (119877

119896

4minus 1205724

119895)

+ 2120572119895(1205722

119895minus 3119877119896

times 1205721198951199052(1205722

119895+ 119877119896

119868(119895119899V+2+119899119887+119897)3

= minus1

21198961199052119890119909

119873

119904=1

[119860119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119877119896minus 119876119904) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) sin (119861

119904119897V + 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos [(119877

119896minus 119876119904) 119905 + (119861

119904119897V minus 119878119896 minus 120579119904)]

+ (119877119896minus 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [minus (119877

119896+ 119876119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

minus (119877119896+ 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)4

=41198961199052119890119909ℎ0

1198712[((119897V (119897V + 119871) exp (minus120572119895119905)

times 120572119895cos (119878

119896minus 119877119896119905)

minus119877119896sin (119878119896minus 119877119896119905)

+119877119896sin (119878) minus 120572

119895cos (119878

119896))

times (1205722

119895+ 119877119896

2)minus1

+2119871119881 exp (minus120572

119895119905)

(1205722119895+ 119877119896

times (1205722

119895+ 119877119896

2) cos (119878

119896)

+ exp (120572119895119905) cos (119878

119896minus 119877119896119905)

times [1198772(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896[119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

119895119877119896sin (119878119896)

minus1198812

(1205722119895+ 119877119896

times119877119896sin (119878119896minus 119877119896119905)[61205722

119895+1199052(1205722

119895+ 119877119896

minus 4120572119895119905 (1205722

119895+ 119877119896

minus2119877119896

+ 2 exp (minus120572119895119905)

times [(1205723

119895minus 3120572119895119877119896

2) cos (119878

119896)

+ 119877119896(119877119896

2minus 31205722

119895)

times sin (119878119896) ]

minus [2119905 (119877119896

4minus 1205724

119895) + 2120572

119895(1205722

119895minus 3119877119896

+1205721198951199052(1205722

119895+119877119896

]cos (119878119896minus 119877119896119905)]

119868(119895119899V+2+119899119887+119897)5

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895sin (120579119904) + (119876

119904minus 119877119896) cos (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895sin (120579119904) minus (119876

119904+ 119877119896) cos (120579

119904)

1205722119895+ (119876119904+ 119877)2

+ ((120572119895sin [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119877119896minus 119876119904) cos [(119876

119904minus 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895sin [(119876

119904+ 119877119896) 119905 + 120579

119904]

minus (119876119904+ 119877) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)6

=41198881199051119890119909ℎ0119871

(1205722119895+ 119877119896

2)3(119881

119871)

times [(1205722

119895+ 119877119896

2) 120572119895cos (119877

119896119905)

+ 119877119896sin (119877

119896119905) minus 120572

119895exp (minus120572

119895119905)

+ 2 exp (minus120572119895119905) 1205722

119895+ exp (120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

minus119877119896

119868(119895119899V+2+119899119887+119897)7

=41198881199052119890119909ℎ0

119881(1205722119895+ 119877119896

times 119897V (1205722

119895+ 119877119896

2) + 119871 (120572

119895+ 119877119896

minus2119881 [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

+ cos (119878119896minus 119877119896119905)

times 119877119896

2[120572119895(119897V + 119871) minus 2119881 (120572

119895119905 + 1)]

+1205722

119895[120572119895(119897V + 119871) + 2119881 (1 minus 120572

119895119905)]

+ exp (minus120572119895119905) 119877119896sin (119878119896)

times [119897V (1205722

119895+ 119877119896

+119871 (1205722

119895+ 119877119896

2)+4120572119895119881]

+ cos (119878119896)

times [119877119896

22119881 minus 120572

119895(119897V + 119871)

minus 1205722

119895120572119895(119897V + 119871)

+2119881]]

119868(119895119899V+2+119899119887+119897)8

21198881199052119890119909

119873

119904=1

[119860119904119876119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119876119904minus 119877119896) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) cos (119861

119904119897V + 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin [(119877 minus 119876

119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)]

+ (119876119904minus 119877119896) cos [(119877

119896minus 119876119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)])

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

minus ((120572119895sin [minus (119877 + 119876

119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

+ (119877119896+ 119876119904) cos [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)9

= minus119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [exp (120572119895119905) 119896120587119881 sin(119896120587119881119905

119871)

minus119871120572119895+ exp 119871120572

119895(120572119895119905) cos(119896120587119881119905

119871)]

minus 119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881sin(119896120587119863119871

119871)

minus 119890119909119871120572119895cos(119896120587119863

119871)

minus exp (120572119895119905)

times cos(119896120587119863 minus 119881119905

119871)]

119868(119895119899V+2+119899119887+119897)10

= minus21198711198812(1198981199081

+ 1198981199082) 119890119909

119873

119904=1

1198601199041198612

119904[ (((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881

times cos [(119876119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus exp (minus120572119895119905)

times ((minus (119896120587 +119861119904119871)119881 cos 120579

119904+119871120572119895sin 120579119904)

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

+ (((119896120587 minus 119861119904119871)119881 cos 120579

119904minus 119871120572119895sin 120579119904)

times((119896120587 minus 119861119904119871)2

1198812+ 11987121205722

119895)minus1

minus8ℎ01198812119890119909(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

119871)

+119871120572119895sin(119896120587119881119905

119871)]

References

RoboticsJournal of

Journal of

RotatingMachinery

VLSI Design

Shock and Vibration

Journal of

Volume 2014

SensorsJournal of

Propagation

0 0002 0004 0006 0008 001

GbJbEbIb

119887119869119887119864119887119868119887) for different

107 N-m2 and (C) 119864V119868V = 82 times 10

10 N-m2

119884static (47)

119887119868119887)

5 Conclusions

1520 25 30 35 40

45 5055

404550556065707580146148

15152154156158

16162164

times103

8013514

Vehicle speed (kmh)

14161820222426283013

Appendix

119865119895(119905) = 0 for119895 = 1 2 3 119899V

= 119896119905119895minus119899V

ℎ (119909119895minus119899V

) + 119888119905119895minus119899V

ℎ (119909119895minus119899V

For119895 = 119899V + 1 119899V + 2

= 1198961199051ℎ (1199091) 120601119887119895minus(119899V+2) (1199091) + 1198961199052ℎ (1199092) 120601119887119895minus(119899V+2) (1199092)

+ 1198881199051ℎ (1199091) 120601119887119895minus(119899V+2) (1199091) + 1198881199052ℎ (1199092) 120601119887119895minus(119899V+2) (1199092)

For 119895 = 119899V + 3 119899V + 4 119899V + 2 + 119899119887

= 1198901199091198961199051ℎ (1199091) 120601119905119895minus(119899V+2+119899119887)

(1199091)

+ 1198961199052ℎ (1199092) 120601119905119895minus(119899V+2+119899119887)

(1199092)

+ 1198881199051ℎ (1199091) 120601119905119895minus(119899V+2+119899119887)

(1199091)

+1198881199052ℎ1015840(1199092) 120601119905119895minus(119899V+2+119899119887)

(1199092)

For 119895 = 119899V + 119899119887 + 3

119899V + 119899119887 + 4 119899V + 119899119887 + 119899119905 + 2

ℎ119898(119909) =

1198712119909 (119871 minus 119909) (A2)

ℎ119903(119909) =

119873

119904=1

119860119904cos (2120587Ω

119904119909 + 120579119904) (A3)

119868119895119896= 119868119895119908+ 119868119895119909+ 119868119895119910+ 119868119895119911 (A4)

where 119908 = 1 2 3 119899V 119909 = 119899V + 1 119899V + 2 119910 = 119899V + 2 + 1119899V + 2 + 2 119899V + 2 + 119899

119887 119911 = 119899V + 2 + 119899

119887+ 1 119899V + 2 + 119899

119887+

2 119899V + 2 + 119899119887 + 119899119879Thefirst term 119868

1198681198951= 1198681198952= sdot sdot sdot = 119868

119895119899V= 0 (A5)

119868119895119909= 1198681198951199091

+ 1198681198951199092 (A6)

1198681198951199091

= 11986811989511990911

+ 11986811989511990912

119904)minus2 and 119861

119904= 2120587Ω

119871+ (119904 minus

11986811989511990911

= 1198961199051[

119873

119904=1

119860119904

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2minus120572119895cos (120579

119904) minus 119861119904119881 sin (120579

119904)

+ exp (120572119895119905)

times [120572119895cos (119861

119904119881119905 + 120579

119904)

+119861119904119881 cos (119861

119904119881119905 + 120579

119904)]

+4ℎ0119881

12057231198951198712

120572119895119871 (120572119895119905 + exp (minus120572

119895119905) minus 1)

minus119881

12057231198951198712

119895119905) + 2 ]

11986811989511990912

= 1198881199051[

119873

119904=1

119881119860119904119861119904

exp (minus120572119895119905)

119904)

+ exp (120572119895119905)

times [120572119895sin (119861

119904119881119905 + 120579

119904)

minus119861119904119881 sin (119861

119904119881119905 + 120579

119904)]

119895119905)

12057231198951198712

minus1

120572119895119871exp (120572

119895119905) minus 2]

1198681198951199092

= 11986811989511990921

+ 11986811989511990922

11986811989511990921

= 1198961199052[

119873

119904=1

119860119904

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2

times minus 120572119895cos (119861

119904119897V minus 120579119904) minus 119861119904119881 sin (119861

119904119897V minus 120579119904)

+ exp (120572119895119905) [120572119895cos (119861

119904(119897 minus 119881119905) minus 120579

119904)

+119861119904119881 sin (119861

119904(119897V minus 119881119905) + 120579119904)]

+1198812[2 minus exp (120572

119895119905) 120572119895119905 (120572119895119905 minus 2) + 2]

+4ℎ0

12057231198951198712

minus1205722

119895119897V (119897V + 119871) (exp (120572119895119905) minus 1)

+120572119895119881 (2119897V + 119871) [exp (120572119895119905) (120572119895119905 minus 1) + 1] ]

11986811989511990922

= 1198881199052[

119873

119904=1

119860119904119861119904119881

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2120572119895sin (119861

119904119897V minus 120579119904)

+ 119861119904119881 cos (119861

119904119897V minus 120579119904)

minus exp (120572119895119905) [120572119895sin (119861

119904(119897 minus 119881119905) minus 120579

119904)

+119861119904119881 cos (119861

119904(119897V minus 119881119905) + 120579119904)]

+4ℎ0119881 exp (minus120572

119895119905)

12057221198951198712

times 2119881 [exp (120572119895119905) (120572119895119905 minus 1) + 1]

minus120572119895(2119897V + 119871) (exp (120572119895119905) minus 1)]

119868119895119910=

119899119887

119896=1

119868119895119899V+2+119896

119886=1

119868(119895119899V+2+119896)119886

(119896 = 1 2 3 119899119887) (A11)

119876119904= 119861119904119881 119877

119896=119896120587119881

119871

119878119896=119896120587

119871 119885

119904= 119861119904119897V

119868(119895119899V+2+119896)

21198961199051

119873

119904=1

119860119904 ((2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) cos (120579

119904)

+2120572119895119876119904sin (120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times cos (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119876

119904+ 119877119896) 119905 + 120579

119904

minus (119877119896+ 119876119904)

times cos (119876119904+ 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

21198961199051[

4ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198762119904)3

times 119871 (1205722

119895+ 1198762

119904) [exp (120572

119895119905) sin (119876

119904119905)

times 1198762

119904(120572119895119905 + 1)

+1205722

119895(120572119895119905 minus 1)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904119905 + 120572119895(120572119895119905 minus 2)] minus 2120572

119895119876119904

+ 119881 (21198763

119904+ 119876119904exp (120572

119895119905)) cos (119876

119904119905)

times 1198764

1199041199052+ 21198762

119904(1205722

1198951199052minus 2120572119895119905 minus 1)

+1205722

119895(1205722

1198951199052minus 4120572119895119905 + 6)

minus exp (120572119895119905) sin (119876

119904119905)

times 1198764

119904(120572119895119905 + 2) + 120572

1198951198762

119904(1205722

1198951199052minus 3)

+1205723

119895(1205722

1198951199052minus 2120572119895119905 + 2) minus 6120572

119895119876119904]

119868(119895119899V+2+119896)

21198961199052

119873

119904=1

119860119904[ exp (minus120572

119895119905)

times (((119877119896+ 119876119904) cos (119878

119904minus 119885119904minus 120579119904)

+120572119895sin (119878119896minus 119885119904minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

times (((119877119896minus 119876119904) cos (119878

119896minus 119885119904+ 120579119904)

+120572119895sin (119878119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 119878

119896minus 119885119904+ 120579119904

minus (119877119896+ 119876119904) cos (119876

119904minus 119877119896) 119905

+119878119896minus 119885119904+ 120579119904)

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

+ ((120572119895sin 119878119896+ 119885119904minus (119876119904+ 119877) 119905 minus 120579

119904

+ (119877119896+ 119876119904) cos 119878

119896+ 119885119904

+ (119876119904+ 119877119896) 119905

minus120579119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1

21198961199052[

4ℎ0119881

1198712(1205722119895+ 1198812)

times sin (119881119905 minus 119897V) [1205721198951198972

V(1205722

119895+ 1198812)2

minus 119897V (1205722

119895+ 1198812)

times minus1205723

119895119871 minus 1205721198951198711198812

+ 21198813(120572119895119905 + 1)

+21205722

119895119881(120572119895119905 minus 1)

+ 119881 [119881 21205723

119895+ 120572119895119905(1205722

119895+ 1198812)2

+2119905 (1205724

119895minus 1198814) minus 6120572

1198951198812

minus 119871 (1205722

119895+ 1198812)

times 1205722

119895(120572119895119905 minus 1)+ 119881

2(120572119895119905 +1)]

+ 119881 cos (119897V minus 119881119905) [1198814(1198972

V + 119897V119871

+21205722

1198951199052minus 4120572119895119905 minus 2)

+ 1205722

1198951198812(21198972

V + 2119897V119871

+41205722

1198951199052+ 6)

+ 1205724

119895119897V (119897V + 119871)

minus 1198815119905 (2119897V + 119871)

times (120572119895119905 minus 1)

minus 1205723

119895119881 (2119897V + 119871)

times (120572119895119905 minus 2) + 119897

V1198816]]

119868(119895119899V+2+119896)

21198881199051

119873

119904=1

119860119904119876119904( minus (2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) sin (120579

119904)

minus2120572119895119876119904cos (120579

119904))

times (1205722

119895+ (119876119904minus 119877)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times sin (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos (119876

119904+ 119877119896) 119905 + 120579

119904

+ (119877119896+ 119876119904)

times sin (119876119904+ 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199051

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) cos (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) sin (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

21198881199052

119873

119904=1

119860119904119876119904

times [ exp (minus120572119895119905 ( ( (119877

119896+ 119876119904) sin (119878

119896+ 119885119904minus 120579119904)

minus120572119895cos (119878

119896+ 119885119904minus 120579119904))

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

times ( ( (119876119904minus 119877119896) sin (119878

119896minus 119885119904+ 120579119904)

+120572119895cos (119878

119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos 119878

119896+ 119885119904minus (119876119904+ 119877119896) 119905 minus 120579

119904

minus (119877119896+ 119876119904) sin 119878

119896+ 119885119904

minus (119876119904+ 119877119896) 119905 minus 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos 119878

119896minus 119885119904+ (119876119904minus 119877119896) 119905 + 120579

119904

+ (119876119904minus 119877119896) sin 119878

119896minus 119885119904

+ (119876119904minus 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199052

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) sin (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881 + exp (120572119895119905) 119896120587119881 cos(119896120587119881119905

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881cos(119896120587119863119871

119871)

minus 119871120572119895sin(119896120587119863

119871)

119871)]

119868(119895119899V+2+119896)

= minus21198711198812(1198981199081

+ 1198981199082)

119873

119904=1

1198601199041198612

119904[(((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881 cos [(119876

119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

119904119871)119881 cos 120579

119904

+119871120572119895sin 120579119904)

times (1198612

11990411987121198812

+ 21198961205871198611199041198711198812

+ 119896212058721198812

+11987121205722

119895)minus1

+ ((119896120587 minus 119861119904119871)119881 cos 120579

119904

minus119871120572119895sin 120579119904)

times ((119896120587 minus 119861119904119871)2

1198812

+11987121205722

119895)minus1

minus8ℎ01198812(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

times [119896120587119881

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119868119895119911=

119899119905

119896=1

119868119895119899V+2+119899119887+119896

119868119895119899V+2+119899119887+119897

119886=1

119868(119895119899V+2+119899119887+119897)119886

(119896 = 1 2 3 119899119879) (A14)

119868(119895119899V+2+119899119887+119897)1

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895cos (120579

119904) + (119876

119904minus 119877119896) sin (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895cos (120579

119904) + (119876

119904+ 119877119896) sin (120579

119904)

1205722119895+ (119876119904+ 119877119896)2

+ ((120572119895cos [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119876119904minus 119877119896) sin [(119876

119904minus 119877119896) 119905 minus 120579

119904])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [(119876

119904+ 119877119896) 119905 + 120579

119904]

+ (119876119904+ 119877119896) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)2

=41198961199051119890119909ℎ0119871

119881(1205722119895+ 119877119896

times [ (1205722

119895+ 119877119896

2) exp (minus120572

119895119905)

times 1205722

119895+ exp (minus120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)] minus 119877

119896

+ 2120572119895exp (minus120572

119895119905) (1205722

119895+ 3119877119896

+ 119877119896sin (119877

119896119905) 6120572

119895+ 1199052(1205722

119895+ 119877119896

+4120572119895119905 (1205722

119895+ 119877119896

2) minus 2119877

119896

+ cos (119877119896119905) 2119905 (119877

119896

4minus 1205724

119895)

+ 2120572119895(1205722

119895minus 3119877119896

times 1205721198951199052(1205722

119895+ 119877119896

119868(119895119899V+2+119899119887+119897)3

= minus1

21198961199052119890119909

119873

119904=1

[119860119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119877119896minus 119876119904) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) sin (119861

119904119897V + 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos [(119877

119896minus 119876119904) 119905 + (119861

119904119897V minus 119878119896 minus 120579119904)]

+ (119877119896minus 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [minus (119877

119896+ 119876119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

minus (119877119896+ 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)4

=41198961199052119890119909ℎ0

1198712[((119897V (119897V + 119871) exp (minus120572119895119905)

times 120572119895cos (119878

119896minus 119877119896119905)

minus119877119896sin (119878119896minus 119877119896119905)

+119877119896sin (119878) minus 120572

119895cos (119878

119896))

times (1205722

119895+ 119877119896

2)minus1

+2119871119881 exp (minus120572

119895119905)

(1205722119895+ 119877119896

times (1205722

119895+ 119877119896

2) cos (119878

119896)

+ exp (120572119895119905) cos (119878

119896minus 119877119896119905)

times [1198772(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896[119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

119895119877119896sin (119878119896)

minus1198812

(1205722119895+ 119877119896

times119877119896sin (119878119896minus 119877119896119905)[61205722

119895+1199052(1205722

119895+ 119877119896

minus 4120572119895119905 (1205722

119895+ 119877119896

minus2119877119896

+ 2 exp (minus120572119895119905)

times [(1205723

119895minus 3120572119895119877119896

2) cos (119878

119896)

+ 119877119896(119877119896

2minus 31205722

119895)

times sin (119878119896) ]

minus [2119905 (119877119896

4minus 1205724

119895) + 2120572

119895(1205722

119895minus 3119877119896

+1205721198951199052(1205722

119895+119877119896

]cos (119878119896minus 119877119896119905)]

119868(119895119899V+2+119899119887+119897)5

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895sin (120579119904) + (119876

119904minus 119877119896) cos (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895sin (120579119904) minus (119876

119904+ 119877119896) cos (120579

119904)

1205722119895+ (119876119904+ 119877)2

+ ((120572119895sin [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119877119896minus 119876119904) cos [(119876

119904minus 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895sin [(119876

119904+ 119877119896) 119905 + 120579

119904]

minus (119876119904+ 119877) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)6

=41198881199051119890119909ℎ0119871

(1205722119895+ 119877119896

2)3(119881

119871)

times [(1205722

119895+ 119877119896

2) 120572119895cos (119877

119896119905)

+ 119877119896sin (119877

119896119905) minus 120572

119895exp (minus120572

119895119905)

+ 2 exp (minus120572119895119905) 1205722

119895+ exp (120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

minus119877119896

119868(119895119899V+2+119899119887+119897)7

=41198881199052119890119909ℎ0

119881(1205722119895+ 119877119896

times 119897V (1205722

119895+ 119877119896

2) + 119871 (120572

119895+ 119877119896

minus2119881 [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

+ cos (119878119896minus 119877119896119905)

times 119877119896

2[120572119895(119897V + 119871) minus 2119881 (120572

119895119905 + 1)]

+1205722

119895[120572119895(119897V + 119871) + 2119881 (1 minus 120572

119895119905)]

+ exp (minus120572119895119905) 119877119896sin (119878119896)

times [119897V (1205722

119895+ 119877119896

+119871 (1205722

119895+ 119877119896

2)+4120572119895119881]

+ cos (119878119896)

times [119877119896

22119881 minus 120572

119895(119897V + 119871)

minus 1205722

119895120572119895(119897V + 119871)

+2119881]]

119868(119895119899V+2+119899119887+119897)8

21198881199052119890119909

119873

119904=1

[119860119904119876119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119876119904minus 119877119896) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) cos (119861

119904119897V + 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin [(119877 minus 119876

119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)]

+ (119876119904minus 119877119896) cos [(119877

119896minus 119876119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)])

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

minus ((120572119895sin [minus (119877 + 119876

119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

+ (119877119896+ 119876119904) cos [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)9

= minus119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [exp (120572119895119905) 119896120587119881 sin(119896120587119881119905

119871)

minus119871120572119895+ exp 119871120572

119895(120572119895119905) cos(119896120587119881119905

119871)]

minus 119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881sin(119896120587119863119871

119871)

minus 119890119909119871120572119895cos(119896120587119863

119871)

minus exp (120572119895119905)

times cos(119896120587119863 minus 119881119905

119871)]

119868(119895119899V+2+119899119887+119897)10

= minus21198711198812(1198981199081

+ 1198981199082) 119890119909

119873

119904=1

1198601199041198612

119904[ (((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881

times cos [(119876119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus exp (minus120572119895119905)

times ((minus (119896120587 +119861119904119871)119881 cos 120579

119904+119871120572119895sin 120579119904)

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

+ (((119896120587 minus 119861119904119871)119881 cos 120579

119904minus 119871120572119895sin 120579119904)

times((119896120587 minus 119861119904119871)2

1198812+ 11987121205722

119895)minus1

minus8ℎ01198812119890119909(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

119871)

+119871120572119895sin(119896120587119881119905

119871)]

References

RoboticsJournal of

Journal of

RotatingMachinery

VLSI Design

Shock and Vibration

Journal of

Volume 2014

SensorsJournal of

Propagation

1520 25 30 35 40

45 5055

404550556065707580146148

15152154156158

16162164

times103

8013514

Vehicle speed (kmh)

14161820222426283013

Appendix

119865119895(119905) = 0 for119895 = 1 2 3 119899V

= 119896119905119895minus119899V

ℎ (119909119895minus119899V

) + 119888119905119895minus119899V

ℎ (119909119895minus119899V

For119895 = 119899V + 1 119899V + 2

= 1198961199051ℎ (1199091) 120601119887119895minus(119899V+2) (1199091) + 1198961199052ℎ (1199092) 120601119887119895minus(119899V+2) (1199092)

+ 1198881199051ℎ (1199091) 120601119887119895minus(119899V+2) (1199091) + 1198881199052ℎ (1199092) 120601119887119895minus(119899V+2) (1199092)

For 119895 = 119899V + 3 119899V + 4 119899V + 2 + 119899119887

= 1198901199091198961199051ℎ (1199091) 120601119905119895minus(119899V+2+119899119887)

(1199091)

+ 1198961199052ℎ (1199092) 120601119905119895minus(119899V+2+119899119887)

(1199092)

+ 1198881199051ℎ (1199091) 120601119905119895minus(119899V+2+119899119887)

(1199091)

+1198881199052ℎ1015840(1199092) 120601119905119895minus(119899V+2+119899119887)

(1199092)

For 119895 = 119899V + 119899119887 + 3

119899V + 119899119887 + 4 119899V + 119899119887 + 119899119905 + 2

ℎ119898(119909) =

1198712119909 (119871 minus 119909) (A2)

ℎ119903(119909) =

119873

119904=1

119860119904cos (2120587Ω

119904119909 + 120579119904) (A3)

119868119895119896= 119868119895119908+ 119868119895119909+ 119868119895119910+ 119868119895119911 (A4)

where 119908 = 1 2 3 119899V 119909 = 119899V + 1 119899V + 2 119910 = 119899V + 2 + 1119899V + 2 + 2 119899V + 2 + 119899

119887 119911 = 119899V + 2 + 119899

119887+ 1 119899V + 2 + 119899

119887+

2 119899V + 2 + 119899119887 + 119899119879Thefirst term 119868

1198681198951= 1198681198952= sdot sdot sdot = 119868

119895119899V= 0 (A5)

119868119895119909= 1198681198951199091

+ 1198681198951199092 (A6)

1198681198951199091

= 11986811989511990911

+ 11986811989511990912

119904)minus2 and 119861

119904= 2120587Ω

119871+ (119904 minus

11986811989511990911

= 1198961199051[

119873

119904=1

119860119904

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2minus120572119895cos (120579

119904) minus 119861119904119881 sin (120579

119904)

+ exp (120572119895119905)

times [120572119895cos (119861

119904119881119905 + 120579

119904)

+119861119904119881 cos (119861

119904119881119905 + 120579

119904)]

+4ℎ0119881

12057231198951198712

120572119895119871 (120572119895119905 + exp (minus120572

119895119905) minus 1)

minus119881

12057231198951198712

119895119905) + 2 ]

11986811989511990912

= 1198881199051[

119873

119904=1

119881119860119904119861119904

exp (minus120572119895119905)

119904)

+ exp (120572119895119905)

times [120572119895sin (119861

119904119881119905 + 120579

119904)

minus119861119904119881 sin (119861

119904119881119905 + 120579

119904)]

119895119905)

12057231198951198712

minus1

120572119895119871exp (120572

119895119905) minus 2]

1198681198951199092

= 11986811989511990921

+ 11986811989511990922

11986811989511990921

= 1198961199052[

119873

119904=1

119860119904

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2

times minus 120572119895cos (119861

119904119897V minus 120579119904) minus 119861119904119881 sin (119861

119904119897V minus 120579119904)

+ exp (120572119895119905) [120572119895cos (119861

119904(119897 minus 119881119905) minus 120579

119904)

+119861119904119881 sin (119861

119904(119897V minus 119881119905) + 120579119904)]

+1198812[2 minus exp (120572

119895119905) 120572119895119905 (120572119895119905 minus 2) + 2]

+4ℎ0

12057231198951198712

minus1205722

119895119897V (119897V + 119871) (exp (120572119895119905) minus 1)

+120572119895119881 (2119897V + 119871) [exp (120572119895119905) (120572119895119905 minus 1) + 1] ]

11986811989511990922

= 1198881199052[

119873

119904=1

119860119904119861119904119881

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2120572119895sin (119861

119904119897V minus 120579119904)

+ 119861119904119881 cos (119861

119904119897V minus 120579119904)

minus exp (120572119895119905) [120572119895sin (119861

119904(119897 minus 119881119905) minus 120579

119904)

+119861119904119881 cos (119861

119904(119897V minus 119881119905) + 120579119904)]

+4ℎ0119881 exp (minus120572

119895119905)

12057221198951198712

times 2119881 [exp (120572119895119905) (120572119895119905 minus 1) + 1]

minus120572119895(2119897V + 119871) (exp (120572119895119905) minus 1)]

119868119895119910=

119899119887

119896=1

119868119895119899V+2+119896

119886=1

119868(119895119899V+2+119896)119886

(119896 = 1 2 3 119899119887) (A11)

119876119904= 119861119904119881 119877

119896=119896120587119881

119871

119878119896=119896120587

119871 119885

119904= 119861119904119897V

119868(119895119899V+2+119896)

21198961199051

119873

119904=1

119860119904 ((2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) cos (120579

119904)

+2120572119895119876119904sin (120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times cos (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119876

119904+ 119877119896) 119905 + 120579

119904

minus (119877119896+ 119876119904)

times cos (119876119904+ 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

21198961199051[

4ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198762119904)3

times 119871 (1205722

119895+ 1198762

119904) [exp (120572

119895119905) sin (119876

119904119905)

times 1198762

119904(120572119895119905 + 1)

+1205722

119895(120572119895119905 minus 1)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904119905 + 120572119895(120572119895119905 minus 2)] minus 2120572

119895119876119904

+ 119881 (21198763

119904+ 119876119904exp (120572

119895119905)) cos (119876

119904119905)

times 1198764

1199041199052+ 21198762

119904(1205722

1198951199052minus 2120572119895119905 minus 1)

+1205722

119895(1205722

1198951199052minus 4120572119895119905 + 6)

minus exp (120572119895119905) sin (119876

119904119905)

times 1198764

119904(120572119895119905 + 2) + 120572

1198951198762

119904(1205722

1198951199052minus 3)

+1205723

119895(1205722

1198951199052minus 2120572119895119905 + 2) minus 6120572

119895119876119904]

119868(119895119899V+2+119896)

21198961199052

119873

119904=1

119860119904[ exp (minus120572

119895119905)

times (((119877119896+ 119876119904) cos (119878

119904minus 119885119904minus 120579119904)

+120572119895sin (119878119896minus 119885119904minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

times (((119877119896minus 119876119904) cos (119878

119896minus 119885119904+ 120579119904)

+120572119895sin (119878119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 119878

119896minus 119885119904+ 120579119904

minus (119877119896+ 119876119904) cos (119876

119904minus 119877119896) 119905

+119878119896minus 119885119904+ 120579119904)

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

+ ((120572119895sin 119878119896+ 119885119904minus (119876119904+ 119877) 119905 minus 120579

119904

+ (119877119896+ 119876119904) cos 119878

119896+ 119885119904

+ (119876119904+ 119877119896) 119905

minus120579119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1

21198961199052[

4ℎ0119881

1198712(1205722119895+ 1198812)

times sin (119881119905 minus 119897V) [1205721198951198972

V(1205722

119895+ 1198812)2

minus 119897V (1205722

119895+ 1198812)

times minus1205723

119895119871 minus 1205721198951198711198812

+ 21198813(120572119895119905 + 1)

+21205722

119895119881(120572119895119905 minus 1)

+ 119881 [119881 21205723

119895+ 120572119895119905(1205722

119895+ 1198812)2

+2119905 (1205724

119895minus 1198814) minus 6120572

1198951198812

minus 119871 (1205722

119895+ 1198812)

times 1205722

119895(120572119895119905 minus 1)+ 119881

2(120572119895119905 +1)]

+ 119881 cos (119897V minus 119881119905) [1198814(1198972

V + 119897V119871

+21205722

1198951199052minus 4120572119895119905 minus 2)

+ 1205722

1198951198812(21198972

V + 2119897V119871

+41205722

1198951199052+ 6)

+ 1205724

119895119897V (119897V + 119871)

minus 1198815119905 (2119897V + 119871)

times (120572119895119905 minus 1)

minus 1205723

119895119881 (2119897V + 119871)

times (120572119895119905 minus 2) + 119897

V1198816]]

119868(119895119899V+2+119896)

21198881199051

119873

119904=1

119860119904119876119904( minus (2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) sin (120579

119904)

minus2120572119895119876119904cos (120579

119904))

times (1205722

119895+ (119876119904minus 119877)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times sin (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos (119876

119904+ 119877119896) 119905 + 120579

119904

+ (119877119896+ 119876119904)

times sin (119876119904+ 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199051

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) cos (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) sin (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

21198881199052

119873

119904=1

119860119904119876119904

times [ exp (minus120572119895119905 ( ( (119877

119896+ 119876119904) sin (119878

119896+ 119885119904minus 120579119904)

minus120572119895cos (119878

119896+ 119885119904minus 120579119904))

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

times ( ( (119876119904minus 119877119896) sin (119878

119896minus 119885119904+ 120579119904)

+120572119895cos (119878

119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos 119878

119896+ 119885119904minus (119876119904+ 119877119896) 119905 minus 120579

119904

minus (119877119896+ 119876119904) sin 119878

119896+ 119885119904

minus (119876119904+ 119877119896) 119905 minus 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos 119878

119896minus 119885119904+ (119876119904minus 119877119896) 119905 + 120579

119904

+ (119876119904minus 119877119896) sin 119878

119896minus 119885119904

+ (119876119904minus 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199052

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) sin (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881 + exp (120572119895119905) 119896120587119881 cos(119896120587119881119905

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881cos(119896120587119863119871

119871)

minus 119871120572119895sin(119896120587119863

119871)

119871)]

119868(119895119899V+2+119896)

= minus21198711198812(1198981199081

+ 1198981199082)

119873

119904=1

1198601199041198612

119904[(((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881 cos [(119876

119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

119904119871)119881 cos 120579

119904

+119871120572119895sin 120579119904)

times (1198612

11990411987121198812

+ 21198961205871198611199041198711198812

+ 119896212058721198812

+11987121205722

119895)minus1

+ ((119896120587 minus 119861119904119871)119881 cos 120579

119904

minus119871120572119895sin 120579119904)

times ((119896120587 minus 119861119904119871)2

1198812

+11987121205722

119895)minus1

minus8ℎ01198812(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

times [119896120587119881

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119868119895119911=

119899119905

119896=1

119868119895119899V+2+119899119887+119896

119868119895119899V+2+119899119887+119897

119886=1

119868(119895119899V+2+119899119887+119897)119886

(119896 = 1 2 3 119899119879) (A14)

119868(119895119899V+2+119899119887+119897)1

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895cos (120579

119904) + (119876

119904minus 119877119896) sin (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895cos (120579

119904) + (119876

119904+ 119877119896) sin (120579

119904)

1205722119895+ (119876119904+ 119877119896)2

+ ((120572119895cos [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119876119904minus 119877119896) sin [(119876

119904minus 119877119896) 119905 minus 120579

119904])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [(119876

119904+ 119877119896) 119905 + 120579

119904]

+ (119876119904+ 119877119896) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)2

=41198961199051119890119909ℎ0119871

119881(1205722119895+ 119877119896

times [ (1205722

119895+ 119877119896

2) exp (minus120572

119895119905)

times 1205722

119895+ exp (minus120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)] minus 119877

119896

+ 2120572119895exp (minus120572

119895119905) (1205722

119895+ 3119877119896

+ 119877119896sin (119877

119896119905) 6120572

119895+ 1199052(1205722

119895+ 119877119896

+4120572119895119905 (1205722

119895+ 119877119896

2) minus 2119877

119896

+ cos (119877119896119905) 2119905 (119877

119896

4minus 1205724

119895)

+ 2120572119895(1205722

119895minus 3119877119896

times 1205721198951199052(1205722

119895+ 119877119896

119868(119895119899V+2+119899119887+119897)3

= minus1

21198961199052119890119909

119873

119904=1

[119860119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119877119896minus 119876119904) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) sin (119861

119904119897V + 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos [(119877

119896minus 119876119904) 119905 + (119861

119904119897V minus 119878119896 minus 120579119904)]

+ (119877119896minus 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [minus (119877

119896+ 119876119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

minus (119877119896+ 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)4

=41198961199052119890119909ℎ0

1198712[((119897V (119897V + 119871) exp (minus120572119895119905)

times 120572119895cos (119878

119896minus 119877119896119905)

minus119877119896sin (119878119896minus 119877119896119905)

+119877119896sin (119878) minus 120572

119895cos (119878

119896))

times (1205722

119895+ 119877119896

2)minus1

+2119871119881 exp (minus120572

119895119905)

(1205722119895+ 119877119896

times (1205722

119895+ 119877119896

2) cos (119878

119896)

+ exp (120572119895119905) cos (119878

119896minus 119877119896119905)

times [1198772(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896[119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

119895119877119896sin (119878119896)

minus1198812

(1205722119895+ 119877119896

times119877119896sin (119878119896minus 119877119896119905)[61205722

119895+1199052(1205722

119895+ 119877119896

minus 4120572119895119905 (1205722

119895+ 119877119896

minus2119877119896

+ 2 exp (minus120572119895119905)

times [(1205723

119895minus 3120572119895119877119896

2) cos (119878

119896)

+ 119877119896(119877119896

2minus 31205722

119895)

times sin (119878119896) ]

minus [2119905 (119877119896

4minus 1205724

119895) + 2120572

119895(1205722

119895minus 3119877119896

+1205721198951199052(1205722

119895+119877119896

]cos (119878119896minus 119877119896119905)]

119868(119895119899V+2+119899119887+119897)5

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895sin (120579119904) + (119876

119904minus 119877119896) cos (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895sin (120579119904) minus (119876

119904+ 119877119896) cos (120579

119904)

1205722119895+ (119876119904+ 119877)2

+ ((120572119895sin [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119877119896minus 119876119904) cos [(119876

119904minus 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895sin [(119876

119904+ 119877119896) 119905 + 120579

119904]

minus (119876119904+ 119877) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)6

=41198881199051119890119909ℎ0119871

(1205722119895+ 119877119896

2)3(119881

119871)

times [(1205722

119895+ 119877119896

2) 120572119895cos (119877

119896119905)

+ 119877119896sin (119877

119896119905) minus 120572

119895exp (minus120572

119895119905)

+ 2 exp (minus120572119895119905) 1205722

119895+ exp (120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

minus119877119896

119868(119895119899V+2+119899119887+119897)7

=41198881199052119890119909ℎ0

119881(1205722119895+ 119877119896

times 119897V (1205722

119895+ 119877119896

2) + 119871 (120572

119895+ 119877119896

minus2119881 [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

+ cos (119878119896minus 119877119896119905)

times 119877119896

2[120572119895(119897V + 119871) minus 2119881 (120572

119895119905 + 1)]

+1205722

119895[120572119895(119897V + 119871) + 2119881 (1 minus 120572

119895119905)]

+ exp (minus120572119895119905) 119877119896sin (119878119896)

times [119897V (1205722

119895+ 119877119896

+119871 (1205722

119895+ 119877119896

2)+4120572119895119881]

+ cos (119878119896)

times [119877119896

22119881 minus 120572

119895(119897V + 119871)

minus 1205722

119895120572119895(119897V + 119871)

+2119881]]

119868(119895119899V+2+119899119887+119897)8

21198881199052119890119909

119873

119904=1

[119860119904119876119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119876119904minus 119877119896) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) cos (119861

119904119897V + 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin [(119877 minus 119876

119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)]

+ (119876119904minus 119877119896) cos [(119877

119896minus 119876119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)])

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

minus ((120572119895sin [minus (119877 + 119876

119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

+ (119877119896+ 119876119904) cos [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)9

= minus119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [exp (120572119895119905) 119896120587119881 sin(119896120587119881119905

119871)

minus119871120572119895+ exp 119871120572

119895(120572119895119905) cos(119896120587119881119905

119871)]

minus 119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881sin(119896120587119863119871

119871)

minus 119890119909119871120572119895cos(119896120587119863

119871)

minus exp (120572119895119905)

times cos(119896120587119863 minus 119881119905

119871)]

119868(119895119899V+2+119899119887+119897)10

= minus21198711198812(1198981199081

+ 1198981199082) 119890119909

119873

119904=1

1198601199041198612

119904[ (((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881

times cos [(119876119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus exp (minus120572119895119905)

times ((minus (119896120587 +119861119904119871)119881 cos 120579

119904+119871120572119895sin 120579119904)

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

+ (((119896120587 minus 119861119904119871)119881 cos 120579

119904minus 119871120572119895sin 120579119904)

times((119896120587 minus 119861119904119871)2

1198812+ 11987121205722

119895)minus1

minus8ℎ01198812119890119909(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

119871)

+119871120572119895sin(119896120587119881119905

119871)]

References

RoboticsJournal of

Journal of

RotatingMachinery

VLSI Design

Shock and Vibration

Journal of

Volume 2014

SensorsJournal of

Propagation

Appendix

119865119895(119905) = 0 for119895 = 1 2 3 119899V

= 119896119905119895minus119899V

ℎ (119909119895minus119899V

) + 119888119905119895minus119899V

ℎ (119909119895minus119899V

For119895 = 119899V + 1 119899V + 2

= 1198961199051ℎ (1199091) 120601119887119895minus(119899V+2) (1199091) + 1198961199052ℎ (1199092) 120601119887119895minus(119899V+2) (1199092)

+ 1198881199051ℎ (1199091) 120601119887119895minus(119899V+2) (1199091) + 1198881199052ℎ (1199092) 120601119887119895minus(119899V+2) (1199092)

For 119895 = 119899V + 3 119899V + 4 119899V + 2 + 119899119887

= 1198901199091198961199051ℎ (1199091) 120601119905119895minus(119899V+2+119899119887)

(1199091)

+ 1198961199052ℎ (1199092) 120601119905119895minus(119899V+2+119899119887)

(1199092)

+ 1198881199051ℎ (1199091) 120601119905119895minus(119899V+2+119899119887)

(1199091)

+1198881199052ℎ1015840(1199092) 120601119905119895minus(119899V+2+119899119887)

(1199092)

For 119895 = 119899V + 119899119887 + 3

119899V + 119899119887 + 4 119899V + 119899119887 + 119899119905 + 2

ℎ119898(119909) =

1198712119909 (119871 minus 119909) (A2)

ℎ119903(119909) =

119873

119904=1

119860119904cos (2120587Ω

119904119909 + 120579119904) (A3)

119868119895119896= 119868119895119908+ 119868119895119909+ 119868119895119910+ 119868119895119911 (A4)

where 119908 = 1 2 3 119899V 119909 = 119899V + 1 119899V + 2 119910 = 119899V + 2 + 1119899V + 2 + 2 119899V + 2 + 119899

119887 119911 = 119899V + 2 + 119899

119887+ 1 119899V + 2 + 119899

119887+

2 119899V + 2 + 119899119887 + 119899119879Thefirst term 119868

1198681198951= 1198681198952= sdot sdot sdot = 119868

119895119899V= 0 (A5)

119868119895119909= 1198681198951199091

+ 1198681198951199092 (A6)

1198681198951199091

= 11986811989511990911

+ 11986811989511990912

119904)minus2 and 119861

119904= 2120587Ω

119871+ (119904 minus

11986811989511990911

= 1198961199051[

119873

119904=1

119860119904

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2minus120572119895cos (120579

119904) minus 119861119904119881 sin (120579

119904)

+ exp (120572119895119905)

times [120572119895cos (119861

119904119881119905 + 120579

119904)

+119861119904119881 cos (119861

119904119881119905 + 120579

119904)]

+4ℎ0119881

12057231198951198712

120572119895119871 (120572119895119905 + exp (minus120572

119895119905) minus 1)

minus119881

12057231198951198712

119895119905) + 2 ]

11986811989511990912

= 1198881199051[

119873

119904=1

119881119860119904119861119904

exp (minus120572119895119905)

119904)

+ exp (120572119895119905)

times [120572119895sin (119861

119904119881119905 + 120579

119904)

minus119861119904119881 sin (119861

119904119881119905 + 120579

119904)]

119895119905)

12057231198951198712

minus1

120572119895119871exp (120572

119895119905) minus 2]

1198681198951199092

= 11986811989511990921

+ 11986811989511990922

11986811989511990921

= 1198961199052[

119873

119904=1

119860119904

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2

times minus 120572119895cos (119861

119904119897V minus 120579119904) minus 119861119904119881 sin (119861

119904119897V minus 120579119904)

+ exp (120572119895119905) [120572119895cos (119861

119904(119897 minus 119881119905) minus 120579

119904)

+119861119904119881 sin (119861

119904(119897V minus 119881119905) + 120579119904)]

+1198812[2 minus exp (120572

119895119905) 120572119895119905 (120572119895119905 minus 2) + 2]

+4ℎ0

12057231198951198712

minus1205722

119895119897V (119897V + 119871) (exp (120572119895119905) minus 1)

+120572119895119881 (2119897V + 119871) [exp (120572119895119905) (120572119895119905 minus 1) + 1] ]

11986811989511990922

= 1198881199052[

119873

119904=1

119860119904119861119904119881

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2120572119895sin (119861

119904119897V minus 120579119904)

+ 119861119904119881 cos (119861

119904119897V minus 120579119904)

minus exp (120572119895119905) [120572119895sin (119861

119904(119897 minus 119881119905) minus 120579

119904)

+119861119904119881 cos (119861

119904(119897V minus 119881119905) + 120579119904)]

+4ℎ0119881 exp (minus120572

119895119905)

12057221198951198712

times 2119881 [exp (120572119895119905) (120572119895119905 minus 1) + 1]

minus120572119895(2119897V + 119871) (exp (120572119895119905) minus 1)]

119868119895119910=

119899119887

119896=1

119868119895119899V+2+119896

119886=1

119868(119895119899V+2+119896)119886

(119896 = 1 2 3 119899119887) (A11)

119876119904= 119861119904119881 119877

119896=119896120587119881

119871

119878119896=119896120587

119871 119885

119904= 119861119904119897V

119868(119895119899V+2+119896)

21198961199051

119873

119904=1

119860119904 ((2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) cos (120579

119904)

+2120572119895119876119904sin (120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times cos (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119876

119904+ 119877119896) 119905 + 120579

119904

minus (119877119896+ 119876119904)

times cos (119876119904+ 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

21198961199051[

4ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198762119904)3

times 119871 (1205722

119895+ 1198762

119904) [exp (120572

119895119905) sin (119876

119904119905)

times 1198762

119904(120572119895119905 + 1)

+1205722

119895(120572119895119905 minus 1)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904119905 + 120572119895(120572119895119905 minus 2)] minus 2120572

119895119876119904

+ 119881 (21198763

119904+ 119876119904exp (120572

119895119905)) cos (119876

119904119905)

times 1198764

1199041199052+ 21198762

119904(1205722

1198951199052minus 2120572119895119905 minus 1)

+1205722

119895(1205722

1198951199052minus 4120572119895119905 + 6)

minus exp (120572119895119905) sin (119876

119904119905)

times 1198764

119904(120572119895119905 + 2) + 120572

1198951198762

119904(1205722

1198951199052minus 3)

+1205723

119895(1205722

1198951199052minus 2120572119895119905 + 2) minus 6120572

119895119876119904]

119868(119895119899V+2+119896)

21198961199052

119873

119904=1

119860119904[ exp (minus120572

119895119905)

times (((119877119896+ 119876119904) cos (119878

119904minus 119885119904minus 120579119904)

+120572119895sin (119878119896minus 119885119904minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

times (((119877119896minus 119876119904) cos (119878

119896minus 119885119904+ 120579119904)

+120572119895sin (119878119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 119878

119896minus 119885119904+ 120579119904

minus (119877119896+ 119876119904) cos (119876

119904minus 119877119896) 119905

+119878119896minus 119885119904+ 120579119904)

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

+ ((120572119895sin 119878119896+ 119885119904minus (119876119904+ 119877) 119905 minus 120579

119904

+ (119877119896+ 119876119904) cos 119878

119896+ 119885119904

+ (119876119904+ 119877119896) 119905

minus120579119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1

21198961199052[

4ℎ0119881

1198712(1205722119895+ 1198812)

times sin (119881119905 minus 119897V) [1205721198951198972

V(1205722

119895+ 1198812)2

minus 119897V (1205722

119895+ 1198812)

times minus1205723

119895119871 minus 1205721198951198711198812

+ 21198813(120572119895119905 + 1)

+21205722

119895119881(120572119895119905 minus 1)

+ 119881 [119881 21205723

119895+ 120572119895119905(1205722

119895+ 1198812)2

+2119905 (1205724

119895minus 1198814) minus 6120572

1198951198812

minus 119871 (1205722

119895+ 1198812)

times 1205722

119895(120572119895119905 minus 1)+ 119881

2(120572119895119905 +1)]

+ 119881 cos (119897V minus 119881119905) [1198814(1198972

V + 119897V119871

+21205722

1198951199052minus 4120572119895119905 minus 2)

+ 1205722

1198951198812(21198972

V + 2119897V119871

+41205722

1198951199052+ 6)

+ 1205724

119895119897V (119897V + 119871)

minus 1198815119905 (2119897V + 119871)

times (120572119895119905 minus 1)

minus 1205723

119895119881 (2119897V + 119871)

times (120572119895119905 minus 2) + 119897

V1198816]]

119868(119895119899V+2+119896)

21198881199051

119873

119904=1

119860119904119876119904( minus (2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) sin (120579

119904)

minus2120572119895119876119904cos (120579

119904))

times (1205722

119895+ (119876119904minus 119877)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times sin (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos (119876

119904+ 119877119896) 119905 + 120579

119904

+ (119877119896+ 119876119904)

times sin (119876119904+ 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199051

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) cos (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) sin (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

21198881199052

119873

119904=1

119860119904119876119904

times [ exp (minus120572119895119905 ( ( (119877

119896+ 119876119904) sin (119878

119896+ 119885119904minus 120579119904)

minus120572119895cos (119878

119896+ 119885119904minus 120579119904))

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

times ( ( (119876119904minus 119877119896) sin (119878

119896minus 119885119904+ 120579119904)

+120572119895cos (119878

119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos 119878

119896+ 119885119904minus (119876119904+ 119877119896) 119905 minus 120579

119904

minus (119877119896+ 119876119904) sin 119878

119896+ 119885119904

minus (119876119904+ 119877119896) 119905 minus 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos 119878

119896minus 119885119904+ (119876119904minus 119877119896) 119905 + 120579

119904

+ (119876119904minus 119877119896) sin 119878

119896minus 119885119904

+ (119876119904minus 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199052

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) sin (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881 + exp (120572119895119905) 119896120587119881 cos(119896120587119881119905

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881cos(119896120587119863119871

119871)

minus 119871120572119895sin(119896120587119863

119871)

119871)]

119868(119895119899V+2+119896)

= minus21198711198812(1198981199081

+ 1198981199082)

119873

119904=1

1198601199041198612

119904[(((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881 cos [(119876

119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

119904119871)119881 cos 120579

119904

+119871120572119895sin 120579119904)

times (1198612

11990411987121198812

+ 21198961205871198611199041198711198812

+ 119896212058721198812

+11987121205722

119895)minus1

+ ((119896120587 minus 119861119904119871)119881 cos 120579

119904

minus119871120572119895sin 120579119904)

times ((119896120587 minus 119861119904119871)2

1198812

+11987121205722

119895)minus1

minus8ℎ01198812(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

times [119896120587119881

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119868119895119911=

119899119905

119896=1

119868119895119899V+2+119899119887+119896

119868119895119899V+2+119899119887+119897

119886=1

119868(119895119899V+2+119899119887+119897)119886

(119896 = 1 2 3 119899119879) (A14)

119868(119895119899V+2+119899119887+119897)1

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895cos (120579

119904) + (119876

119904minus 119877119896) sin (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895cos (120579

119904) + (119876

119904+ 119877119896) sin (120579

119904)

1205722119895+ (119876119904+ 119877119896)2

+ ((120572119895cos [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119876119904minus 119877119896) sin [(119876

119904minus 119877119896) 119905 minus 120579

119904])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [(119876

119904+ 119877119896) 119905 + 120579

119904]

+ (119876119904+ 119877119896) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)2

=41198961199051119890119909ℎ0119871

119881(1205722119895+ 119877119896

times [ (1205722

119895+ 119877119896

2) exp (minus120572

119895119905)

times 1205722

119895+ exp (minus120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)] minus 119877

119896

+ 2120572119895exp (minus120572

119895119905) (1205722

119895+ 3119877119896

+ 119877119896sin (119877

119896119905) 6120572

119895+ 1199052(1205722

119895+ 119877119896

+4120572119895119905 (1205722

119895+ 119877119896

2) minus 2119877

119896

+ cos (119877119896119905) 2119905 (119877

119896

4minus 1205724

119895)

+ 2120572119895(1205722

119895minus 3119877119896

times 1205721198951199052(1205722

119895+ 119877119896

119868(119895119899V+2+119899119887+119897)3

= minus1

21198961199052119890119909

119873

119904=1

[119860119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119877119896minus 119876119904) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) sin (119861

119904119897V + 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos [(119877

119896minus 119876119904) 119905 + (119861

119904119897V minus 119878119896 minus 120579119904)]

+ (119877119896minus 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [minus (119877

119896+ 119876119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

minus (119877119896+ 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)4

=41198961199052119890119909ℎ0

1198712[((119897V (119897V + 119871) exp (minus120572119895119905)

times 120572119895cos (119878

119896minus 119877119896119905)

minus119877119896sin (119878119896minus 119877119896119905)

+119877119896sin (119878) minus 120572

119895cos (119878

119896))

times (1205722

119895+ 119877119896

2)minus1

+2119871119881 exp (minus120572

119895119905)

(1205722119895+ 119877119896

times (1205722

119895+ 119877119896

2) cos (119878

119896)

+ exp (120572119895119905) cos (119878

119896minus 119877119896119905)

times [1198772(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896[119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

119895119877119896sin (119878119896)

minus1198812

(1205722119895+ 119877119896

times119877119896sin (119878119896minus 119877119896119905)[61205722

119895+1199052(1205722

119895+ 119877119896

minus 4120572119895119905 (1205722

119895+ 119877119896

minus2119877119896

+ 2 exp (minus120572119895119905)

times [(1205723

119895minus 3120572119895119877119896

2) cos (119878

119896)

+ 119877119896(119877119896

2minus 31205722

119895)

times sin (119878119896) ]

minus [2119905 (119877119896

4minus 1205724

119895) + 2120572

119895(1205722

119895minus 3119877119896

+1205721198951199052(1205722

119895+119877119896

]cos (119878119896minus 119877119896119905)]

119868(119895119899V+2+119899119887+119897)5

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895sin (120579119904) + (119876

119904minus 119877119896) cos (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895sin (120579119904) minus (119876

119904+ 119877119896) cos (120579

119904)

1205722119895+ (119876119904+ 119877)2

+ ((120572119895sin [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119877119896minus 119876119904) cos [(119876

119904minus 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895sin [(119876

119904+ 119877119896) 119905 + 120579

119904]

minus (119876119904+ 119877) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)6

=41198881199051119890119909ℎ0119871

(1205722119895+ 119877119896

2)3(119881

119871)

times [(1205722

119895+ 119877119896

2) 120572119895cos (119877

119896119905)

+ 119877119896sin (119877

119896119905) minus 120572

119895exp (minus120572

119895119905)

+ 2 exp (minus120572119895119905) 1205722

119895+ exp (120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

minus119877119896

119868(119895119899V+2+119899119887+119897)7

=41198881199052119890119909ℎ0

119881(1205722119895+ 119877119896

times 119897V (1205722

119895+ 119877119896

2) + 119871 (120572

119895+ 119877119896

minus2119881 [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

+ cos (119878119896minus 119877119896119905)

times 119877119896

2[120572119895(119897V + 119871) minus 2119881 (120572

119895119905 + 1)]

+1205722

119895[120572119895(119897V + 119871) + 2119881 (1 minus 120572

119895119905)]

+ exp (minus120572119895119905) 119877119896sin (119878119896)

times [119897V (1205722

119895+ 119877119896

+119871 (1205722

119895+ 119877119896

2)+4120572119895119881]

+ cos (119878119896)

times [119877119896

22119881 minus 120572

119895(119897V + 119871)

minus 1205722

119895120572119895(119897V + 119871)

+2119881]]

119868(119895119899V+2+119899119887+119897)8

21198881199052119890119909

119873

119904=1

[119860119904119876119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119876119904minus 119877119896) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) cos (119861

119904119897V + 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin [(119877 minus 119876

119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)]

+ (119876119904minus 119877119896) cos [(119877

119896minus 119876119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)])

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

minus ((120572119895sin [minus (119877 + 119876

119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

+ (119877119896+ 119876119904) cos [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)9

= minus119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [exp (120572119895119905) 119896120587119881 sin(119896120587119881119905

119871)

minus119871120572119895+ exp 119871120572

119895(120572119895119905) cos(119896120587119881119905

119871)]

minus 119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881sin(119896120587119863119871

119871)

minus 119890119909119871120572119895cos(119896120587119863

119871)

minus exp (120572119895119905)

times cos(119896120587119863 minus 119881119905

119871)]

119868(119895119899V+2+119899119887+119897)10

= minus21198711198812(1198981199081

+ 1198981199082) 119890119909

119873

119904=1

1198601199041198612

119904[ (((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881

times cos [(119876119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus exp (minus120572119895119905)

times ((minus (119896120587 +119861119904119871)119881 cos 120579

119904+119871120572119895sin 120579119904)

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

+ (((119896120587 minus 119861119904119871)119881 cos 120579

119904minus 119871120572119895sin 120579119904)

times((119896120587 minus 119861119904119871)2

1198812+ 11987121205722

119895)minus1

minus8ℎ01198812119890119909(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

119871)

+119871120572119895sin(119896120587119881119905

119871)]

References

RoboticsJournal of

Journal of

RotatingMachinery

VLSI Design

Shock and Vibration

Journal of

Volume 2014

SensorsJournal of

Propagation

1198681198951199092

= 11986811989511990921

+ 11986811989511990922

11986811989511990921

= 1198961199052[

119873

119904=1

119860119904

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2

times minus 120572119895cos (119861

119904119897V minus 120579119904) minus 119861119904119881 sin (119861

119904119897V minus 120579119904)

+ exp (120572119895119905) [120572119895cos (119861

119904(119897 minus 119881119905) minus 120579

119904)

+119861119904119881 sin (119861

119904(119897V minus 119881119905) + 120579119904)]

+1198812[2 minus exp (120572

119895119905) 120572119895119905 (120572119895119905 minus 2) + 2]

+4ℎ0

12057231198951198712

minus1205722

119895119897V (119897V + 119871) (exp (120572119895119905) minus 1)

+120572119895119881 (2119897V + 119871) [exp (120572119895119905) (120572119895119905 minus 1) + 1] ]

11986811989511990922

= 1198881199052[

119873

119904=1

119860119904119861119904119881

exp (minus120572119895119905)

1205722119895+ (119861119904119881)2120572119895sin (119861

119904119897V minus 120579119904)

+ 119861119904119881 cos (119861

119904119897V minus 120579119904)

minus exp (120572119895119905) [120572119895sin (119861

119904(119897 minus 119881119905) minus 120579

119904)

+119861119904119881 cos (119861

119904(119897V minus 119881119905) + 120579119904)]

+4ℎ0119881 exp (minus120572

119895119905)

12057221198951198712

times 2119881 [exp (120572119895119905) (120572119895119905 minus 1) + 1]

minus120572119895(2119897V + 119871) (exp (120572119895119905) minus 1)]

119868119895119910=

119899119887

119896=1

119868119895119899V+2+119896

119886=1

119868(119895119899V+2+119896)119886

(119896 = 1 2 3 119899119887) (A11)

119876119904= 119861119904119881 119877

119896=119896120587119881

119871

119878119896=119896120587

119871 119885

119904= 119861119904119897V

119868(119895119899V+2+119896)

21198961199051

119873

119904=1

119860119904 ((2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) cos (120579

119904)

+2120572119895119876119904sin (120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times cos (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119876

119904+ 119877119896) 119905 + 120579

119904

minus (119877119896+ 119876119904)

times cos (119876119904+ 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

21198961199051[

4ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198762119904)3

times 119871 (1205722

119895+ 1198762

119904) [exp (120572

119895119905) sin (119876

119904119905)

times 1198762

119904(120572119895119905 + 1)

+1205722

119895(120572119895119905 minus 1)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904119905 + 120572119895(120572119895119905 minus 2)] minus 2120572

119895119876119904

+ 119881 (21198763

119904+ 119876119904exp (120572

119895119905)) cos (119876

119904119905)

times 1198764

1199041199052+ 21198762

119904(1205722

1198951199052minus 2120572119895119905 minus 1)

+1205722

119895(1205722

1198951199052minus 4120572119895119905 + 6)

minus exp (120572119895119905) sin (119876

119904119905)

times 1198764

119904(120572119895119905 + 2) + 120572

1198951198762

119904(1205722

1198951199052minus 3)

+1205723

119895(1205722

1198951199052minus 2120572119895119905 + 2) minus 6120572

119895119876119904]

119868(119895119899V+2+119896)

21198961199052

119873

119904=1

119860119904[ exp (minus120572

119895119905)

times (((119877119896+ 119876119904) cos (119878

119904minus 119885119904minus 120579119904)

+120572119895sin (119878119896minus 119885119904minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

times (((119877119896minus 119876119904) cos (119878

119896minus 119885119904+ 120579119904)

+120572119895sin (119878119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 119878

119896minus 119885119904+ 120579119904

minus (119877119896+ 119876119904) cos (119876

119904minus 119877119896) 119905

+119878119896minus 119885119904+ 120579119904)

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

+ ((120572119895sin 119878119896+ 119885119904minus (119876119904+ 119877) 119905 minus 120579

119904

+ (119877119896+ 119876119904) cos 119878

119896+ 119885119904

+ (119876119904+ 119877119896) 119905

minus120579119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1

21198961199052[

4ℎ0119881

1198712(1205722119895+ 1198812)

times sin (119881119905 minus 119897V) [1205721198951198972

V(1205722

119895+ 1198812)2

minus 119897V (1205722

119895+ 1198812)

times minus1205723

119895119871 minus 1205721198951198711198812

+ 21198813(120572119895119905 + 1)

+21205722

119895119881(120572119895119905 minus 1)

+ 119881 [119881 21205723

119895+ 120572119895119905(1205722

119895+ 1198812)2

+2119905 (1205724

119895minus 1198814) minus 6120572

1198951198812

minus 119871 (1205722

119895+ 1198812)

times 1205722

119895(120572119895119905 minus 1)+ 119881

2(120572119895119905 +1)]

+ 119881 cos (119897V minus 119881119905) [1198814(1198972

V + 119897V119871

+21205722

1198951199052minus 4120572119895119905 minus 2)

+ 1205722

1198951198812(21198972

V + 2119897V119871

+41205722

1198951199052+ 6)

+ 1205724

119895119897V (119897V + 119871)

minus 1198815119905 (2119897V + 119871)

times (120572119895119905 minus 1)

minus 1205723

119895119881 (2119897V + 119871)

times (120572119895119905 minus 2) + 119897

V1198816]]

119868(119895119899V+2+119896)

21198881199051

119873

119904=1

119860119904119876119904( minus (2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) sin (120579

119904)

minus2120572119895119876119904cos (120579

119904))

times (1205722

119895+ (119876119904minus 119877)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times sin (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos (119876

119904+ 119877119896) 119905 + 120579

119904

+ (119877119896+ 119876119904)

times sin (119876119904+ 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199051

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) cos (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) sin (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

21198881199052

119873

119904=1

119860119904119876119904

times [ exp (minus120572119895119905 ( ( (119877

119896+ 119876119904) sin (119878

119896+ 119885119904minus 120579119904)

minus120572119895cos (119878

119896+ 119885119904minus 120579119904))

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

times ( ( (119876119904minus 119877119896) sin (119878

119896minus 119885119904+ 120579119904)

+120572119895cos (119878

119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos 119878

119896+ 119885119904minus (119876119904+ 119877119896) 119905 minus 120579

119904

minus (119877119896+ 119876119904) sin 119878

119896+ 119885119904

minus (119876119904+ 119877119896) 119905 minus 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos 119878

119896minus 119885119904+ (119876119904minus 119877119896) 119905 + 120579

119904

+ (119876119904minus 119877119896) sin 119878

119896minus 119885119904

+ (119876119904minus 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199052

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) sin (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881 + exp (120572119895119905) 119896120587119881 cos(119896120587119881119905

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881cos(119896120587119863119871

119871)

minus 119871120572119895sin(119896120587119863

119871)

119871)]

119868(119895119899V+2+119896)

= minus21198711198812(1198981199081

+ 1198981199082)

119873

119904=1

1198601199041198612

119904[(((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881 cos [(119876

119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

119904119871)119881 cos 120579

119904

+119871120572119895sin 120579119904)

times (1198612

11990411987121198812

+ 21198961205871198611199041198711198812

+ 119896212058721198812

+11987121205722

119895)minus1

+ ((119896120587 minus 119861119904119871)119881 cos 120579

119904

minus119871120572119895sin 120579119904)

times ((119896120587 minus 119861119904119871)2

1198812

+11987121205722

119895)minus1

minus8ℎ01198812(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

times [119896120587119881

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119868119895119911=

119899119905

119896=1

119868119895119899V+2+119899119887+119896

119868119895119899V+2+119899119887+119897

119886=1

119868(119895119899V+2+119899119887+119897)119886

(119896 = 1 2 3 119899119879) (A14)

119868(119895119899V+2+119899119887+119897)1

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895cos (120579

119904) + (119876

119904minus 119877119896) sin (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895cos (120579

119904) + (119876

119904+ 119877119896) sin (120579

119904)

1205722119895+ (119876119904+ 119877119896)2

+ ((120572119895cos [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119876119904minus 119877119896) sin [(119876

119904minus 119877119896) 119905 minus 120579

119904])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [(119876

119904+ 119877119896) 119905 + 120579

119904]

+ (119876119904+ 119877119896) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)2

=41198961199051119890119909ℎ0119871

119881(1205722119895+ 119877119896

times [ (1205722

119895+ 119877119896

2) exp (minus120572

119895119905)

times 1205722

119895+ exp (minus120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)] minus 119877

119896

+ 2120572119895exp (minus120572

119895119905) (1205722

119895+ 3119877119896

+ 119877119896sin (119877

119896119905) 6120572

119895+ 1199052(1205722

119895+ 119877119896

+4120572119895119905 (1205722

119895+ 119877119896

2) minus 2119877

119896

+ cos (119877119896119905) 2119905 (119877

119896

4minus 1205724

119895)

+ 2120572119895(1205722

119895minus 3119877119896

times 1205721198951199052(1205722

119895+ 119877119896

119868(119895119899V+2+119899119887+119897)3

= minus1

21198961199052119890119909

119873

119904=1

[119860119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119877119896minus 119876119904) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) sin (119861

119904119897V + 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos [(119877

119896minus 119876119904) 119905 + (119861

119904119897V minus 119878119896 minus 120579119904)]

+ (119877119896minus 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [minus (119877

119896+ 119876119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

minus (119877119896+ 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)4

=41198961199052119890119909ℎ0

1198712[((119897V (119897V + 119871) exp (minus120572119895119905)

times 120572119895cos (119878

119896minus 119877119896119905)

minus119877119896sin (119878119896minus 119877119896119905)

+119877119896sin (119878) minus 120572

119895cos (119878

119896))

times (1205722

119895+ 119877119896

2)minus1

+2119871119881 exp (minus120572

119895119905)

(1205722119895+ 119877119896

times (1205722

119895+ 119877119896

2) cos (119878

119896)

+ exp (120572119895119905) cos (119878

119896minus 119877119896119905)

times [1198772(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896[119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

119895119877119896sin (119878119896)

minus1198812

(1205722119895+ 119877119896

times119877119896sin (119878119896minus 119877119896119905)[61205722

119895+1199052(1205722

119895+ 119877119896

minus 4120572119895119905 (1205722

119895+ 119877119896

minus2119877119896

+ 2 exp (minus120572119895119905)

times [(1205723

119895minus 3120572119895119877119896

2) cos (119878

119896)

+ 119877119896(119877119896

2minus 31205722

119895)

times sin (119878119896) ]

minus [2119905 (119877119896

4minus 1205724

119895) + 2120572

119895(1205722

119895minus 3119877119896

+1205721198951199052(1205722

119895+119877119896

]cos (119878119896minus 119877119896119905)]

119868(119895119899V+2+119899119887+119897)5

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895sin (120579119904) + (119876

119904minus 119877119896) cos (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895sin (120579119904) minus (119876

119904+ 119877119896) cos (120579

119904)

1205722119895+ (119876119904+ 119877)2

+ ((120572119895sin [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119877119896minus 119876119904) cos [(119876

119904minus 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895sin [(119876

119904+ 119877119896) 119905 + 120579

119904]

minus (119876119904+ 119877) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)6

=41198881199051119890119909ℎ0119871

(1205722119895+ 119877119896

2)3(119881

119871)

times [(1205722

119895+ 119877119896

2) 120572119895cos (119877

119896119905)

+ 119877119896sin (119877

119896119905) minus 120572

119895exp (minus120572

119895119905)

+ 2 exp (minus120572119895119905) 1205722

119895+ exp (120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

minus119877119896

119868(119895119899V+2+119899119887+119897)7

=41198881199052119890119909ℎ0

119881(1205722119895+ 119877119896

times 119897V (1205722

119895+ 119877119896

2) + 119871 (120572

119895+ 119877119896

minus2119881 [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

+ cos (119878119896minus 119877119896119905)

times 119877119896

2[120572119895(119897V + 119871) minus 2119881 (120572

119895119905 + 1)]

+1205722

119895[120572119895(119897V + 119871) + 2119881 (1 minus 120572

119895119905)]

+ exp (minus120572119895119905) 119877119896sin (119878119896)

times [119897V (1205722

119895+ 119877119896

+119871 (1205722

119895+ 119877119896

2)+4120572119895119881]

+ cos (119878119896)

times [119877119896

22119881 minus 120572

119895(119897V + 119871)

minus 1205722

119895120572119895(119897V + 119871)

+2119881]]

119868(119895119899V+2+119899119887+119897)8

21198881199052119890119909

119873

119904=1

[119860119904119876119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119876119904minus 119877119896) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) cos (119861

119904119897V + 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin [(119877 minus 119876

119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)]

+ (119876119904minus 119877119896) cos [(119877

119896minus 119876119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)])

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

minus ((120572119895sin [minus (119877 + 119876

119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

+ (119877119896+ 119876119904) cos [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)9

= minus119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [exp (120572119895119905) 119896120587119881 sin(119896120587119881119905

119871)

minus119871120572119895+ exp 119871120572

119895(120572119895119905) cos(119896120587119881119905

119871)]

minus 119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881sin(119896120587119863119871

119871)

minus 119890119909119871120572119895cos(119896120587119863

119871)

minus exp (120572119895119905)

times cos(119896120587119863 minus 119881119905

119871)]

119868(119895119899V+2+119899119887+119897)10

= minus21198711198812(1198981199081

+ 1198981199082) 119890119909

119873

119904=1

1198601199041198612

119904[ (((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881

times cos [(119876119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus exp (minus120572119895119905)

times ((minus (119896120587 +119861119904119871)119881 cos 120579

119904+119871120572119895sin 120579119904)

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

+ (((119896120587 minus 119861119904119871)119881 cos 120579

119904minus 119871120572119895sin 120579119904)

times((119896120587 minus 119861119904119871)2

1198812+ 11987121205722

119895)minus1

minus8ℎ01198812119890119909(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

119871)

+119871120572119895sin(119896120587119881119905

119871)]

References

RoboticsJournal of

Journal of

RotatingMachinery

VLSI Design

Shock and Vibration

Journal of

Volume 2014

SensorsJournal of

Propagation

+1205722

119895(1205722

1198951199052minus 4120572119895119905 + 6)

minus exp (120572119895119905) sin (119876

119904119905)

times 1198764

119904(120572119895119905 + 2) + 120572

1198951198762

119904(1205722

1198951199052minus 3)

+1205723

119895(1205722

1198951199052minus 2120572119895119905 + 2) minus 6120572

119895119876119904]

119868(119895119899V+2+119896)

21198961199052

119873

119904=1

119860119904[ exp (minus120572

119895119905)

times (((119877119896+ 119876119904) cos (119878

119904minus 119885119904minus 120579119904)

+120572119895sin (119878119896minus 119885119904minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

times (((119877119896minus 119876119904) cos (119878

119896minus 119885119904+ 120579119904)

+120572119895sin (119878119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

minus ((120572119895sin (119876

119904minus 119877119896) 119905 + 119878

119896minus 119885119904+ 120579119904

minus (119877119896+ 119876119904) cos (119876

119904minus 119877119896) 119905

+119878119896minus 119885119904+ 120579119904)

times(1205722

119895+ (119876119904minus 119877119904)2

)minus1

+ ((120572119895sin 119878119896+ 119885119904minus (119876119904+ 119877) 119905 minus 120579

119904

+ (119877119896+ 119876119904) cos 119878

119896+ 119885119904

+ (119876119904+ 119877119896) 119905

minus120579119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1

21198961199052[

4ℎ0119881

1198712(1205722119895+ 1198812)

times sin (119881119905 minus 119897V) [1205721198951198972

V(1205722

119895+ 1198812)2

minus 119897V (1205722

119895+ 1198812)

times minus1205723

119895119871 minus 1205721198951198711198812

+ 21198813(120572119895119905 + 1)

+21205722

119895119881(120572119895119905 minus 1)

+ 119881 [119881 21205723

119895+ 120572119895119905(1205722

119895+ 1198812)2

+2119905 (1205724

119895minus 1198814) minus 6120572

1198951198812

minus 119871 (1205722

119895+ 1198812)

times 1205722

119895(120572119895119905 minus 1)+ 119881

2(120572119895119905 +1)]

+ 119881 cos (119897V minus 119881119905) [1198814(1198972

V + 119897V119871

+21205722

1198951199052minus 4120572119895119905 minus 2)

+ 1205722

1198951198812(21198972

V + 2119897V119871

+41205722

1198951199052+ 6)

+ 1205724

119895119897V (119897V + 119871)

minus 1198815119905 (2119897V + 119871)

times (120572119895119905 minus 1)

minus 1205723

119895119881 (2119897V + 119871)

times (120572119895119905 minus 2) + 119897

V1198816]]

119868(119895119899V+2+119896)

21198881199051

119873

119904=1

119860119904119876119904( minus (2119877

119896exp (minus120572

119895119905)

times (1205722

119895minus 1198762

119904+ 119877119896

2) sin (120579

119904)

minus2120572119895119876119904cos (120579

119904))

times (1205722

119895+ (119876119904minus 119877)2

times 1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos (119876

119904minus 119877119896) 119905 + 120579

119904

+ (119877119896minus 119876119904)

times sin (119876119904minus 119877119896) 119905 + 120579

119904)

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos (119876

119904+ 119877119896) 119905 + 120579

119904

+ (119877119896+ 119876119904)

times sin (119876119904+ 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199051

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) cos (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) sin (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

21198881199052

119873

119904=1

119860119904119876119904

times [ exp (minus120572119895119905 ( ( (119877

119896+ 119876119904) sin (119878

119896+ 119885119904minus 120579119904)

minus120572119895cos (119878

119896+ 119885119904minus 120579119904))

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

times ( ( (119876119904minus 119877119896) sin (119878

119896minus 119885119904+ 120579119904)

+120572119895cos (119878

119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos 119878

119896+ 119885119904minus (119876119904+ 119877119896) 119905 minus 120579

119904

minus (119877119896+ 119876119904) sin 119878

119896+ 119885119904

minus (119876119904+ 119877119896) 119905 minus 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos 119878

119896minus 119885119904+ (119876119904minus 119877119896) 119905 + 120579

119904

+ (119876119904minus 119877119896) sin 119878

119896minus 119885119904

+ (119876119904minus 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199052

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) sin (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881 + exp (120572119895119905) 119896120587119881 cos(119896120587119881119905

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881cos(119896120587119863119871

119871)

minus 119871120572119895sin(119896120587119863

119871)

119871)]

119868(119895119899V+2+119896)

= minus21198711198812(1198981199081

+ 1198981199082)

119873

119904=1

1198601199041198612

119904[(((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881 cos [(119876

119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

119904119871)119881 cos 120579

119904

+119871120572119895sin 120579119904)

times (1198612

11990411987121198812

+ 21198961205871198611199041198711198812

+ 119896212058721198812

+11987121205722

119895)minus1

+ ((119896120587 minus 119861119904119871)119881 cos 120579

119904

minus119871120572119895sin 120579119904)

times ((119896120587 minus 119861119904119871)2

1198812

+11987121205722

119895)minus1

minus8ℎ01198812(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

times [119896120587119881

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119868119895119911=

119899119905

119896=1

119868119895119899V+2+119899119887+119896

119868119895119899V+2+119899119887+119897

119886=1

119868(119895119899V+2+119899119887+119897)119886

(119896 = 1 2 3 119899119879) (A14)

119868(119895119899V+2+119899119887+119897)1

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895cos (120579

119904) + (119876

119904minus 119877119896) sin (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895cos (120579

119904) + (119876

119904+ 119877119896) sin (120579

119904)

1205722119895+ (119876119904+ 119877119896)2

+ ((120572119895cos [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119876119904minus 119877119896) sin [(119876

119904minus 119877119896) 119905 minus 120579

119904])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [(119876

119904+ 119877119896) 119905 + 120579

119904]

+ (119876119904+ 119877119896) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)2

=41198961199051119890119909ℎ0119871

119881(1205722119895+ 119877119896

times [ (1205722

119895+ 119877119896

2) exp (minus120572

119895119905)

times 1205722

119895+ exp (minus120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)] minus 119877

119896

+ 2120572119895exp (minus120572

119895119905) (1205722

119895+ 3119877119896

+ 119877119896sin (119877

119896119905) 6120572

119895+ 1199052(1205722

119895+ 119877119896

+4120572119895119905 (1205722

119895+ 119877119896

2) minus 2119877

119896

+ cos (119877119896119905) 2119905 (119877

119896

4minus 1205724

119895)

+ 2120572119895(1205722

119895minus 3119877119896

times 1205721198951199052(1205722

119895+ 119877119896

119868(119895119899V+2+119899119887+119897)3

= minus1

21198961199052119890119909

119873

119904=1

[119860119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119877119896minus 119876119904) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) sin (119861

119904119897V + 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos [(119877

119896minus 119876119904) 119905 + (119861

119904119897V minus 119878119896 minus 120579119904)]

+ (119877119896minus 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [minus (119877

119896+ 119876119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

minus (119877119896+ 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)4

=41198961199052119890119909ℎ0

1198712[((119897V (119897V + 119871) exp (minus120572119895119905)

times 120572119895cos (119878

119896minus 119877119896119905)

minus119877119896sin (119878119896minus 119877119896119905)

+119877119896sin (119878) minus 120572

119895cos (119878

119896))

times (1205722

119895+ 119877119896

2)minus1

+2119871119881 exp (minus120572

119895119905)

(1205722119895+ 119877119896

times (1205722

119895+ 119877119896

2) cos (119878

119896)

+ exp (120572119895119905) cos (119878

119896minus 119877119896119905)

times [1198772(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896[119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

119895119877119896sin (119878119896)

minus1198812

(1205722119895+ 119877119896

times119877119896sin (119878119896minus 119877119896119905)[61205722

119895+1199052(1205722

119895+ 119877119896

minus 4120572119895119905 (1205722

119895+ 119877119896

minus2119877119896

+ 2 exp (minus120572119895119905)

times [(1205723

119895minus 3120572119895119877119896

2) cos (119878

119896)

+ 119877119896(119877119896

2minus 31205722

119895)

times sin (119878119896) ]

minus [2119905 (119877119896

4minus 1205724

119895) + 2120572

119895(1205722

119895minus 3119877119896

+1205721198951199052(1205722

119895+119877119896

]cos (119878119896minus 119877119896119905)]

119868(119895119899V+2+119899119887+119897)5

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895sin (120579119904) + (119876

119904minus 119877119896) cos (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895sin (120579119904) minus (119876

119904+ 119877119896) cos (120579

119904)

1205722119895+ (119876119904+ 119877)2

+ ((120572119895sin [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119877119896minus 119876119904) cos [(119876

119904minus 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895sin [(119876

119904+ 119877119896) 119905 + 120579

119904]

minus (119876119904+ 119877) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)6

=41198881199051119890119909ℎ0119871

(1205722119895+ 119877119896

2)3(119881

119871)

times [(1205722

119895+ 119877119896

2) 120572119895cos (119877

119896119905)

+ 119877119896sin (119877

119896119905) minus 120572

119895exp (minus120572

119895119905)

+ 2 exp (minus120572119895119905) 1205722

119895+ exp (120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

minus119877119896

119868(119895119899V+2+119899119887+119897)7

=41198881199052119890119909ℎ0

119881(1205722119895+ 119877119896

times 119897V (1205722

119895+ 119877119896

2) + 119871 (120572

119895+ 119877119896

minus2119881 [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

+ cos (119878119896minus 119877119896119905)

times 119877119896

2[120572119895(119897V + 119871) minus 2119881 (120572

119895119905 + 1)]

+1205722

119895[120572119895(119897V + 119871) + 2119881 (1 minus 120572

119895119905)]

+ exp (minus120572119895119905) 119877119896sin (119878119896)

times [119897V (1205722

119895+ 119877119896

+119871 (1205722

119895+ 119877119896

2)+4120572119895119881]

+ cos (119878119896)

times [119877119896

22119881 minus 120572

119895(119897V + 119871)

minus 1205722

119895120572119895(119897V + 119871)

+2119881]]

119868(119895119899V+2+119899119887+119897)8

21198881199052119890119909

119873

119904=1

[119860119904119876119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119876119904minus 119877119896) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) cos (119861

119904119897V + 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin [(119877 minus 119876

119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)]

+ (119876119904minus 119877119896) cos [(119877

119896minus 119876119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)])

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

minus ((120572119895sin [minus (119877 + 119876

119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

+ (119877119896+ 119876119904) cos [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)9

= minus119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [exp (120572119895119905) 119896120587119881 sin(119896120587119881119905

119871)

minus119871120572119895+ exp 119871120572

119895(120572119895119905) cos(119896120587119881119905

119871)]

minus 119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881sin(119896120587119863119871

119871)

minus 119890119909119871120572119895cos(119896120587119863

119871)

minus exp (120572119895119905)

times cos(119896120587119863 minus 119881119905

119871)]

119868(119895119899V+2+119899119887+119897)10

= minus21198711198812(1198981199081

+ 1198981199082) 119890119909

119873

119904=1

1198601199041198612

119904[ (((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881

times cos [(119876119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus exp (minus120572119895119905)

times ((minus (119896120587 +119861119904119871)119881 cos 120579

119904+119871120572119895sin 120579119904)

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

+ (((119896120587 minus 119861119904119871)119881 cos 120579

119904minus 119871120572119895sin 120579119904)

times((119896120587 minus 119861119904119871)2

1198812+ 11987121205722

119895)minus1

minus8ℎ01198812119890119909(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

119871)

+119871120572119895sin(119896120587119881119905

119871)]

References

RoboticsJournal of

Journal of

RotatingMachinery

VLSI Design

Shock and Vibration

Journal of

Volume 2014

SensorsJournal of

Propagation

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos (119876

119904+ 119877119896) 119905 + 120579

119904

+ (119877119896+ 119876119904)

times sin (119876119904+ 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199051

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) cos (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) sin (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

21198881199052

119873

119904=1

119860119904119876119904

times [ exp (minus120572119895119905 ( ( (119877

119896+ 119876119904) sin (119878

119896+ 119885119904minus 120579119904)

minus120572119895cos (119878

119896+ 119885119904minus 120579119904))

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

times ( ( (119876119904minus 119877119896) sin (119878

119896minus 119885119904+ 120579119904)

+120572119895cos (119878

119896minus 119885119904+ 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos 119878

119896+ 119885119904minus (119876119904+ 119877119896) 119905 minus 120579

119904

minus (119877119896+ 119876119904) sin 119878

119896+ 119885119904

minus (119876119904+ 119877119896) 119905 minus 120579

119904)

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895cos 119878

119896minus 119885119904+ (119876119904minus 119877119896) 119905 + 120579

119904

+ (119876119904minus 119877119896) sin 119878

119896minus 119885119904

+ (119876119904minus 119877119896) 119905 + 120579

119904)

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

119868(119895119899V+2+119896)

= minus1198881199052

2ℎ0119881 exp (minus120572

119895119905)

1198712(1205722119895+ 1198812)

times exp (120572119895119905) sin (119876

119904119905)

times [1198762

119904(2120572119895119905119881 minus 120572

119895119871 + 2119881)

+1205722

119895(2120572119895119905119881 minus 120572

119895119871 minus 2119881)]

minus 119876119904exp (120572

119895119905) cos (119876

119904119905)

times [1198762

119904(119871 minus 119881119905)

+120572119895(120572119895119871 minus 2120572

119895119905119881 + 4119881)]

119868(119895119899V+2+119896)

119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881 + exp (120572119895119905) 119896120587119881 cos(119896120587119881119905

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881cos(119896120587119863119871

119871)

minus 119871120572119895sin(119896120587119863

119871)

119871)]

119868(119895119899V+2+119896)

= minus21198711198812(1198981199081

+ 1198981199082)

119873

119904=1

1198601199041198612

119904[(((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881 cos [(119876

119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

119904119871)119881 cos 120579

119904

+119871120572119895sin 120579119904)

times (1198612

11990411987121198812

+ 21198961205871198611199041198711198812

+ 119896212058721198812

+11987121205722

119895)minus1

+ ((119896120587 minus 119861119904119871)119881 cos 120579

119904

minus119871120572119895sin 120579119904)

times ((119896120587 minus 119861119904119871)2

1198812

+11987121205722

119895)minus1

minus8ℎ01198812(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

times [119896120587119881

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119868119895119911=

119899119905

119896=1

119868119895119899V+2+119899119887+119896

119868119895119899V+2+119899119887+119897

119886=1

119868(119895119899V+2+119899119887+119897)119886

(119896 = 1 2 3 119899119879) (A14)

119868(119895119899V+2+119899119887+119897)1

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895cos (120579

119904) + (119876

119904minus 119877119896) sin (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895cos (120579

119904) + (119876

119904+ 119877119896) sin (120579

119904)

1205722119895+ (119876119904+ 119877119896)2

+ ((120572119895cos [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119876119904minus 119877119896) sin [(119876

119904minus 119877119896) 119905 minus 120579

119904])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [(119876

119904+ 119877119896) 119905 + 120579

119904]

+ (119876119904+ 119877119896) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)2

=41198961199051119890119909ℎ0119871

119881(1205722119895+ 119877119896

times [ (1205722

119895+ 119877119896

2) exp (minus120572

119895119905)

times 1205722

119895+ exp (minus120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)] minus 119877

119896

+ 2120572119895exp (minus120572

119895119905) (1205722

119895+ 3119877119896

+ 119877119896sin (119877

119896119905) 6120572

119895+ 1199052(1205722

119895+ 119877119896

+4120572119895119905 (1205722

119895+ 119877119896

2) minus 2119877

119896

+ cos (119877119896119905) 2119905 (119877

119896

4minus 1205724

119895)

+ 2120572119895(1205722

119895minus 3119877119896

times 1205721198951199052(1205722

119895+ 119877119896

119868(119895119899V+2+119899119887+119897)3

= minus1

21198961199052119890119909

119873

119904=1

[119860119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119877119896minus 119876119904) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) sin (119861

119904119897V + 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos [(119877

119896minus 119876119904) 119905 + (119861

119904119897V minus 119878119896 minus 120579119904)]

+ (119877119896minus 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [minus (119877

119896+ 119876119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

minus (119877119896+ 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)4

=41198961199052119890119909ℎ0

1198712[((119897V (119897V + 119871) exp (minus120572119895119905)

times 120572119895cos (119878

119896minus 119877119896119905)

minus119877119896sin (119878119896minus 119877119896119905)

+119877119896sin (119878) minus 120572

119895cos (119878

119896))

times (1205722

119895+ 119877119896

2)minus1

+2119871119881 exp (minus120572

119895119905)

(1205722119895+ 119877119896

times (1205722

119895+ 119877119896

2) cos (119878

119896)

+ exp (120572119895119905) cos (119878

119896minus 119877119896119905)

times [1198772(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896[119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

119895119877119896sin (119878119896)

minus1198812

(1205722119895+ 119877119896

times119877119896sin (119878119896minus 119877119896119905)[61205722

119895+1199052(1205722

119895+ 119877119896

minus 4120572119895119905 (1205722

119895+ 119877119896

minus2119877119896

+ 2 exp (minus120572119895119905)

times [(1205723

119895minus 3120572119895119877119896

2) cos (119878

119896)

+ 119877119896(119877119896

2minus 31205722

119895)

times sin (119878119896) ]

minus [2119905 (119877119896

4minus 1205724

119895) + 2120572

119895(1205722

119895minus 3119877119896

+1205721198951199052(1205722

119895+119877119896

]cos (119878119896minus 119877119896119905)]

119868(119895119899V+2+119899119887+119897)5

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895sin (120579119904) + (119876

119904minus 119877119896) cos (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895sin (120579119904) minus (119876

119904+ 119877119896) cos (120579

119904)

1205722119895+ (119876119904+ 119877)2

+ ((120572119895sin [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119877119896minus 119876119904) cos [(119876

119904minus 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895sin [(119876

119904+ 119877119896) 119905 + 120579

119904]

minus (119876119904+ 119877) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)6

=41198881199051119890119909ℎ0119871

(1205722119895+ 119877119896

2)3(119881

119871)

times [(1205722

119895+ 119877119896

2) 120572119895cos (119877

119896119905)

+ 119877119896sin (119877

119896119905) minus 120572

119895exp (minus120572

119895119905)

+ 2 exp (minus120572119895119905) 1205722

119895+ exp (120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

minus119877119896

119868(119895119899V+2+119899119887+119897)7

=41198881199052119890119909ℎ0

119881(1205722119895+ 119877119896

times 119897V (1205722

119895+ 119877119896

2) + 119871 (120572

119895+ 119877119896

minus2119881 [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

+ cos (119878119896minus 119877119896119905)

times 119877119896

2[120572119895(119897V + 119871) minus 2119881 (120572

119895119905 + 1)]

+1205722

119895[120572119895(119897V + 119871) + 2119881 (1 minus 120572

119895119905)]

+ exp (minus120572119895119905) 119877119896sin (119878119896)

times [119897V (1205722

119895+ 119877119896

+119871 (1205722

119895+ 119877119896

2)+4120572119895119881]

+ cos (119878119896)

times [119877119896

22119881 minus 120572

119895(119897V + 119871)

minus 1205722

119895120572119895(119897V + 119871)

+2119881]]

119868(119895119899V+2+119899119887+119897)8

21198881199052119890119909

119873

119904=1

[119860119904119876119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119876119904minus 119877119896) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) cos (119861

119904119897V + 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin [(119877 minus 119876

119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)]

+ (119876119904minus 119877119896) cos [(119877

119896minus 119876119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)])

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

minus ((120572119895sin [minus (119877 + 119876

119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

+ (119877119896+ 119876119904) cos [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)9

= minus119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [exp (120572119895119905) 119896120587119881 sin(119896120587119881119905

119871)

minus119871120572119895+ exp 119871120572

119895(120572119895119905) cos(119896120587119881119905

119871)]

minus 119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881sin(119896120587119863119871

119871)

minus 119890119909119871120572119895cos(119896120587119863

119871)

minus exp (120572119895119905)

times cos(119896120587119863 minus 119881119905

119871)]

119868(119895119899V+2+119899119887+119897)10

= minus21198711198812(1198981199081

+ 1198981199082) 119890119909

119873

119904=1

1198601199041198612

119904[ (((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881

times cos [(119876119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus exp (minus120572119895119905)

times ((minus (119896120587 +119861119904119871)119881 cos 120579

119904+119871120572119895sin 120579119904)

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

+ (((119896120587 minus 119861119904119871)119881 cos 120579

119904minus 119871120572119895sin 120579119904)

times((119896120587 minus 119861119904119871)2

1198812+ 11987121205722

119895)minus1

minus8ℎ01198812119890119909(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

119871)

+119871120572119895sin(119896120587119881119905

119871)]

References

RoboticsJournal of

Journal of

RotatingMachinery

VLSI Design

Shock and Vibration

Journal of

Volume 2014

SensorsJournal of

Propagation

119873

119904=1

1198601199041198612

119904[(((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881 cos [(119876

119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

119904119871)119881 cos 120579

119904

+119871120572119895sin 120579119904)

times (1198612

11990411987121198812

+ 21198961205871198611199041198711198812

+ 119896212058721198812

+11987121205722

119895)minus1

+ ((119896120587 minus 119861119904119871)119881 cos 120579

119904

minus119871120572119895sin 120579119904)

times ((119896120587 minus 119861119904119871)2

1198812

+11987121205722

119895)minus1

minus8ℎ01198812(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

times [119896120587119881

119871)

+119871120572119895sin(119896120587119881119905

119871)]

119868119895119911=

119899119905

119896=1

119868119895119899V+2+119899119887+119896

119868119895119899V+2+119899119887+119897

119886=1

119868(119895119899V+2+119899119887+119897)119886

(119896 = 1 2 3 119899119879) (A14)

119868(119895119899V+2+119899119887+119897)1

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895cos (120579

119904) + (119876

119904minus 119877119896) sin (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895cos (120579

119904) + (119876

119904+ 119877119896) sin (120579

119904)

1205722119895+ (119876119904+ 119877119896)2

+ ((120572119895cos [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119876119904minus 119877119896) sin [(119876

119904minus 119877119896) 119905 minus 120579

119904])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [(119876

119904+ 119877119896) 119905 + 120579

119904]

+ (119876119904+ 119877119896) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)2

=41198961199051119890119909ℎ0119871

119881(1205722119895+ 119877119896

times [ (1205722

119895+ 119877119896

2) exp (minus120572

119895119905)

times 1205722

119895+ exp (minus120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)] minus 119877

119896

+ 2120572119895exp (minus120572

119895119905) (1205722

119895+ 3119877119896

+ 119877119896sin (119877

119896119905) 6120572

119895+ 1199052(1205722

119895+ 119877119896

+4120572119895119905 (1205722

119895+ 119877119896

2) minus 2119877

119896

+ cos (119877119896119905) 2119905 (119877

119896

4minus 1205724

119895)

+ 2120572119895(1205722

119895minus 3119877119896

times 1205721198951199052(1205722

119895+ 119877119896

119868(119895119899V+2+119899119887+119897)3

= minus1

21198961199052119890119909

119873

119904=1

[119860119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119877119896minus 119876119904) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) sin (119861

119904119897V + 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos [(119877

119896minus 119876119904) 119905 + (119861

119904119897V minus 119878119896 minus 120579119904)]

+ (119877119896minus 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [minus (119877

119896+ 119876119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

minus (119877119896+ 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)4

=41198961199052119890119909ℎ0

1198712[((119897V (119897V + 119871) exp (minus120572119895119905)

times 120572119895cos (119878

119896minus 119877119896119905)

minus119877119896sin (119878119896minus 119877119896119905)

+119877119896sin (119878) minus 120572

119895cos (119878

119896))

times (1205722

119895+ 119877119896

2)minus1

+2119871119881 exp (minus120572

119895119905)

(1205722119895+ 119877119896

times (1205722

119895+ 119877119896

2) cos (119878

119896)

+ exp (120572119895119905) cos (119878

119896minus 119877119896119905)

times [1198772(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896[119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

119895119877119896sin (119878119896)

minus1198812

(1205722119895+ 119877119896

times119877119896sin (119878119896minus 119877119896119905)[61205722

119895+1199052(1205722

119895+ 119877119896

minus 4120572119895119905 (1205722

119895+ 119877119896

minus2119877119896

+ 2 exp (minus120572119895119905)

times [(1205723

119895minus 3120572119895119877119896

2) cos (119878

119896)

+ 119877119896(119877119896

2minus 31205722

119895)

times sin (119878119896) ]

minus [2119905 (119877119896

4minus 1205724

119895) + 2120572

119895(1205722

119895minus 3119877119896

+1205721198951199052(1205722

119895+119877119896

]cos (119878119896minus 119877119896119905)]

119868(119895119899V+2+119899119887+119897)5

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895sin (120579119904) + (119876

119904minus 119877119896) cos (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895sin (120579119904) minus (119876

119904+ 119877119896) cos (120579

119904)

1205722119895+ (119876119904+ 119877)2

+ ((120572119895sin [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119877119896minus 119876119904) cos [(119876

119904minus 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895sin [(119876

119904+ 119877119896) 119905 + 120579

119904]

minus (119876119904+ 119877) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)6

=41198881199051119890119909ℎ0119871

(1205722119895+ 119877119896

2)3(119881

119871)

times [(1205722

119895+ 119877119896

2) 120572119895cos (119877

119896119905)

+ 119877119896sin (119877

119896119905) minus 120572

119895exp (minus120572

119895119905)

+ 2 exp (minus120572119895119905) 1205722

119895+ exp (120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

minus119877119896

119868(119895119899V+2+119899119887+119897)7

=41198881199052119890119909ℎ0

119881(1205722119895+ 119877119896

times 119897V (1205722

119895+ 119877119896

2) + 119871 (120572

119895+ 119877119896

minus2119881 [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

+ cos (119878119896minus 119877119896119905)

times 119877119896

2[120572119895(119897V + 119871) minus 2119881 (120572

119895119905 + 1)]

+1205722

119895[120572119895(119897V + 119871) + 2119881 (1 minus 120572

119895119905)]

+ exp (minus120572119895119905) 119877119896sin (119878119896)

times [119897V (1205722

119895+ 119877119896

+119871 (1205722

119895+ 119877119896

2)+4120572119895119881]

+ cos (119878119896)

times [119877119896

22119881 minus 120572

119895(119897V + 119871)

minus 1205722

119895120572119895(119897V + 119871)

+2119881]]

119868(119895119899V+2+119899119887+119897)8

21198881199052119890119909

119873

119904=1

[119860119904119876119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119876119904minus 119877119896) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) cos (119861

119904119897V + 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin [(119877 minus 119876

119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)]

+ (119876119904minus 119877119896) cos [(119877

119896minus 119876119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)])

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

minus ((120572119895sin [minus (119877 + 119876

119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

+ (119877119896+ 119876119904) cos [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)9

= minus119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [exp (120572119895119905) 119896120587119881 sin(119896120587119881119905

119871)

minus119871120572119895+ exp 119871120572

119895(120572119895119905) cos(119896120587119881119905

119871)]

minus 119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881sin(119896120587119863119871

119871)

minus 119890119909119871120572119895cos(119896120587119863

119871)

minus exp (120572119895119905)

times cos(119896120587119863 minus 119881119905

119871)]

119868(119895119899V+2+119899119887+119897)10

= minus21198711198812(1198981199081

+ 1198981199082) 119890119909

119873

119904=1

1198601199041198612

119904[ (((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881

times cos [(119876119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus exp (minus120572119895119905)

times ((minus (119896120587 +119861119904119871)119881 cos 120579

119904+119871120572119895sin 120579119904)

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

+ (((119896120587 minus 119861119904119871)119881 cos 120579

119904minus 119871120572119895sin 120579119904)

times((119896120587 minus 119861119904119871)2

1198812+ 11987121205722

119895)minus1

minus8ℎ01198812119890119909(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

119871)

+119871120572119895sin(119896120587119881119905

119871)]

References

RoboticsJournal of

Journal of

RotatingMachinery

VLSI Design

Shock and Vibration

Journal of

Volume 2014

SensorsJournal of

Propagation

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)] minus 119877

119896

+ 2120572119895exp (minus120572

119895119905) (1205722

119895+ 3119877119896

+ 119877119896sin (119877

119896119905) 6120572

119895+ 1199052(1205722

119895+ 119877119896

+4120572119895119905 (1205722

119895+ 119877119896

2) minus 2119877

119896

+ cos (119877119896119905) 2119905 (119877

119896

4minus 1205724

119895)

+ 2120572119895(1205722

119895minus 3119877119896

times 1205721198951199052(1205722

119895+ 119877119896

119868(119895119899V+2+119899119887+119897)3

= minus1

21198961199052119890119909

119873

119904=1

[119860119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119877119896minus 119876119904) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895cos (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) sin (119861

119904119897V + 119878119896 minus 120579119904))

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

+ ((120572119895cos [(119877

119896minus 119876119904) 119905 + (119861

119904119897V minus 119878119896 minus 120579119904)]

+ (119877119896minus 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895cos [minus (119877

119896+ 119876119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

minus (119877119896+ 119876119904) sin [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)4

=41198961199052119890119909ℎ0

1198712[((119897V (119897V + 119871) exp (minus120572119895119905)

times 120572119895cos (119878

119896minus 119877119896119905)

minus119877119896sin (119878119896minus 119877119896119905)

+119877119896sin (119878) minus 120572

119895cos (119878

119896))

times (1205722

119895+ 119877119896

2)minus1

+2119871119881 exp (minus120572

119895119905)

(1205722119895+ 119877119896

times (1205722

119895+ 119877119896

2) cos (119878

119896)

+ exp (120572119895119905) cos (119878

119896minus 119877119896119905)

times [1198772(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896[119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

119895119877119896sin (119878119896)

minus1198812

(1205722119895+ 119877119896

times119877119896sin (119878119896minus 119877119896119905)[61205722

119895+1199052(1205722

119895+ 119877119896

minus 4120572119895119905 (1205722

119895+ 119877119896

minus2119877119896

+ 2 exp (minus120572119895119905)

times [(1205723

119895minus 3120572119895119877119896

2) cos (119878

119896)

+ 119877119896(119877119896

2minus 31205722

119895)

times sin (119878119896) ]

minus [2119905 (119877119896

4minus 1205724

119895) + 2120572

119895(1205722

119895minus 3119877119896

+1205721198951199052(1205722

119895+119877119896

]cos (119878119896minus 119877119896119905)]

119868(119895119899V+2+119899119887+119897)5

= minus1

21198961199051119890119909

119873

119904=1

119860119904exp (minus120572

119895119905)

120572119895sin (120579119904) + (119876

119904minus 119877119896) cos (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895sin (120579119904) minus (119876

119904+ 119877119896) cos (120579

119904)

1205722119895+ (119876119904+ 119877)2

+ ((120572119895sin [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119877119896minus 119876119904) cos [(119876

119904minus 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895sin [(119876

119904+ 119877119896) 119905 + 120579

119904]

minus (119876119904+ 119877) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)6

=41198881199051119890119909ℎ0119871

(1205722119895+ 119877119896

2)3(119881

119871)

times [(1205722

119895+ 119877119896

2) 120572119895cos (119877

119896119905)

+ 119877119896sin (119877

119896119905) minus 120572

119895exp (minus120572

119895119905)

+ 2 exp (minus120572119895119905) 1205722

119895+ exp (120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

minus119877119896

119868(119895119899V+2+119899119887+119897)7

=41198881199052119890119909ℎ0

119881(1205722119895+ 119877119896

times 119897V (1205722

119895+ 119877119896

2) + 119871 (120572

119895+ 119877119896

minus2119881 [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

+ cos (119878119896minus 119877119896119905)

times 119877119896

2[120572119895(119897V + 119871) minus 2119881 (120572

119895119905 + 1)]

+1205722

119895[120572119895(119897V + 119871) + 2119881 (1 minus 120572

119895119905)]

+ exp (minus120572119895119905) 119877119896sin (119878119896)

times [119897V (1205722

119895+ 119877119896

+119871 (1205722

119895+ 119877119896

2)+4120572119895119881]

+ cos (119878119896)

times [119877119896

22119881 minus 120572

119895(119897V + 119871)

minus 1205722

119895120572119895(119897V + 119871)

+2119881]]

119868(119895119899V+2+119899119887+119897)8

21198881199052119890119909

119873

119904=1

[119860119904119876119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119876119904minus 119877119896) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) cos (119861

119904119897V + 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin [(119877 minus 119876

119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)]

+ (119876119904minus 119877119896) cos [(119877

119896minus 119876119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)])

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

minus ((120572119895sin [minus (119877 + 119876

119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

+ (119877119896+ 119876119904) cos [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)9

= minus119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [exp (120572119895119905) 119896120587119881 sin(119896120587119881119905

119871)

minus119871120572119895+ exp 119871120572

119895(120572119895119905) cos(119896120587119881119905

119871)]

minus 119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881sin(119896120587119863119871

119871)

minus 119890119909119871120572119895cos(119896120587119863

119871)

minus exp (120572119895119905)

times cos(119896120587119863 minus 119881119905

119871)]

119868(119895119899V+2+119899119887+119897)10

= minus21198711198812(1198981199081

+ 1198981199082) 119890119909

119873

119904=1

1198601199041198612

119904[ (((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881

times cos [(119876119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus exp (minus120572119895119905)

times ((minus (119896120587 +119861119904119871)119881 cos 120579

119904+119871120572119895sin 120579119904)

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

+ (((119896120587 minus 119861119904119871)119881 cos 120579

119904minus 119871120572119895sin 120579119904)

times((119896120587 minus 119861119904119871)2

1198812+ 11987121205722

119895)minus1

minus8ℎ01198812119890119909(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

119871)

+119871120572119895sin(119896120587119881119905

119871)]

References

RoboticsJournal of

Journal of

RotatingMachinery

VLSI Design

Shock and Vibration

Journal of

Volume 2014

SensorsJournal of

Propagation

120572119895sin (120579119904) + (119876

119904minus 119877119896) cos (120579

119904)

1205722119895+ (119876119904minus 119877119896)2

+120572119895sin (120579119904) minus (119876

119904+ 119877119896) cos (120579

119904)

1205722119895+ (119876119904+ 119877)2

+ ((120572119895sin [(119876

119904minus 119877119896) 119905 + 120579

119904]

+ (119877119896minus 119876119904) cos [(119876

119904minus 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904minus 119877)2

)minus1

+ ((120572119895sin [(119876

119904+ 119877119896) 119905 + 120579

119904]

minus (119876119904+ 119877) sin [(119876

119904+ 119877119896) 119905 + 120579

119904])

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)6

=41198881199051119890119909ℎ0119871

(1205722119895+ 119877119896

2)3(119881

119871)

times [(1205722

119895+ 119877119896

2) 120572119895cos (119877

119896119905)

+ 119877119896sin (119877

119896119905) minus 120572

119895exp (minus120572

119895119905)

+ 2 exp (minus120572119895119905) 1205722

119895+ exp (120572

119895119905) cos (119877

119896119905)

times [119877119896

2(120572119895119905 + 1) + 120572

119895(120572119895119905 minus 1)]

+ 119877119896exp (120572

119895119905) sin (119877

119896119905)

times [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

minus119877119896

119868(119895119899V+2+119899119887+119897)7

=41198881199052119890119909ℎ0

119881(1205722119895+ 119877119896

times 119897V (1205722

119895+ 119877119896

2) + 119871 (120572

119895+ 119877119896

minus2119881 [119877119896

2119905 + 120572119895(120572119895119905 minus 2)]

+ cos (119878119896minus 119877119896119905)

times 119877119896

2[120572119895(119897V + 119871) minus 2119881 (120572

119895119905 + 1)]

+1205722

119895[120572119895(119897V + 119871) + 2119881 (1 minus 120572

119895119905)]

+ exp (minus120572119895119905) 119877119896sin (119878119896)

times [119897V (1205722

119895+ 119877119896

+119871 (1205722

119895+ 119877119896

2)+4120572119895119881]

+ cos (119878119896)

times [119877119896

22119881 minus 120572

119895(119897V + 119871)

minus 1205722

119895120572119895(119897V + 119871)

+2119881]]

119868(119895119899V+2+119899119887+119897)8

21198881199052119890119909

119873

119904=1

[119860119904119876119904exp (minus120572

119895119905)

times ((120572119895cos (119861

119904119897V minus 119878119896 minus 120579119904)

+ (119876119904minus 119877119896) sin (119861

119904119897V minus 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

+ ((120572119895sin (119861

119904119897V + 119878119896 minus 120579119904)

+ (119877119896+ 119876119904) cos (119861

119904119897V + 119878119896 minus 120579119904))

times (1205722

119895+ (119876119904+ 119877119896)2

)minus1

minus ((120572119895sin [(119877 minus 119876

119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)]

+ (119876119904minus 119877119896) cos [(119877

119896minus 119876119904) 119905

+ (119861119904119897V minus 119878119896 minus 120579119904)])

times (1205722

119895+ (119876119904minus 119877119896)2

)minus1

minus ((120572119895sin [minus (119877 + 119876

119904) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)]

+ (119877119896+ 119876119904) cos [minus (119876

119904+ 119877119896) 119905

+ (119861119904119897V + 119878119896 minus 120579119904)])

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)9

= minus119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [exp (120572119895119905) 119896120587119881 sin(119896120587119881119905

119871)

minus119871120572119895+ exp 119871120572

119895(120572119895119905) cos(119896120587119881119905

119871)]

minus 119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881sin(119896120587119863119871

119871)

minus 119890119909119871120572119895cos(119896120587119863

119871)

minus exp (120572119895119905)

times cos(119896120587119863 minus 119881119905

119871)]

119868(119895119899V+2+119899119887+119897)10

= minus21198711198812(1198981199081

+ 1198981199082) 119890119909

119873

119904=1

1198601199041198612

119904[ (((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881

times cos [(119876119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus exp (minus120572119895119905)

times ((minus (119896120587 +119861119904119871)119881 cos 120579

119904+119871120572119895sin 120579119904)

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

+ (((119896120587 minus 119861119904119871)119881 cos 120579

119904minus 119871120572119895sin 120579119904)

times((119896120587 minus 119861119904119871)2

1198812+ 11987121205722

119895)minus1

minus8ℎ01198812119890119909(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

119871)

+119871120572119895sin(119896120587119881119905

119871)]

References

RoboticsJournal of

Journal of

RotatingMachinery

VLSI Design

Shock and Vibration

Journal of

Volume 2014

SensorsJournal of

Propagation

times(1205722

119895+ (119876119904+ 119877119896)2

)minus1

119868(119895119899V+2+119899119887+119897)9

= minus119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199081)

2 (11987121205722119895+ 119899212058721198812)

times [exp (120572119895119905) 119896120587119881 sin(119896120587119881119905

119871)

minus119871120572119895+ exp 119871120572

119895(120572119895119905) cos(119896120587119881119905

119871)]

minus 119890119909

exp (minus120572119895119905) 119892119871 (119897V119898V + 21198981199082)

2 (11987121205722119895+ 119899212058721198812)

times [119896120587119881sin(119896120587119863119871

119871)

minus 119890119909119871120572119895cos(119896120587119863

119871)

minus exp (120572119895119905)

times cos(119896120587119863 minus 119881119905

119871)]

119868(119895119899V+2+119899119887+119897)10

= minus21198711198812(1198981199081

+ 1198981199082) 119890119909

119873

119904=1

1198601199041198612

119904[ (((119896120587 minus 119861

119904119871)119881 cos [(119876

119904minus 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904minus 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812minus 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus (((119896120587 + 119861119904119871)119881

times cos [(119876119904+ 119877119904) 119905 + 120579

119904]

+119871120572119895sin [(119876

119904+ 119877119904) 119905 + 120579

119904])

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

minus exp (minus120572119895119905)

times ((minus (119896120587 +119861119904119871)119881 cos 120579

119904+119871120572119895sin 120579119904)

times (1198612

11990411987121198812+ 2119896120587119861

1199041198711198812

+119896212058721198812+ 11987121205722

119895)minus1

+ (((119896120587 minus 119861119904119871)119881 cos 120579

119904minus 119871120572119895sin 120579119904)

times((119896120587 minus 119861119904119871)2

1198812+ 11987121205722

119895)minus1

minus8ℎ01198812119890119909(1198981199081

+ 1198981199082)

119896212058721198812 + 11987121205722119895

119871)

+119871120572119895sin(119896120587119881119905

119871)]

References

RoboticsJournal of

Journal of

RotatingMachinery

VLSI Design

Shock and Vibration

Journal of

Volume 2014

SensorsJournal of

Propagation

RoboticsJournal of

Journal of

RotatingMachinery

VLSI Design

Shock and Vibration

Journal of

Volume 2014

SensorsJournal of

Propagation

RoboticsJournal of

Journal of

RotatingMachinery

VLSI Design

Shock and Vibration

Journal of

Volume 2014

SensorsJournal of

Propagation

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