research article vertical track irregularity influence on...

Research ArticleVertical Track Irregularity Influence on the WheelHigh-Frequency Vibration in Wheel-Rail System

Yang Lei Xinyu Tian Falin Qi Dongsheng Chen and Tian Tian

China Academy of Railway Sciences Beijing 100081 China

Correspondence should be addressed to Yang Lei leiyangrailscn

Received 1 February 2016 Revised 14 July 2016 Accepted 18 July 2016

Academic Editor Muhammad N Akram

Copyright copy 2016 Yang Lei et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A three-dimensional deformable finite element model of wheel-rail system is establishedThe vertical vibration natural frequenciesare calculated by modal analysis The improved central difference method by ABAQUS explicit nonlinear dynamics module ischosen to calculate the vertical vibration of wheel axle Through the vibration measurement experiment of LMA wheel by ChinaAcademy of Railway Sciences the contact parameter is optimized and themodel ismodified According to the nonlinear simulationresults the important influence of vertical track irregularity on the wheel set high-frequency vibration is discoveredThe advantagefrequencies through spectrumare close to the vertical vibration natural frequencies of seventh eighth ninth and tenthmodeWhenthe velocity is 350 kmh systemrsquos nonlinear dynamic characteristic is higher than the results at 200 kmh The critical wavelengthof vertical track irregularity on the wheel vertical vibration is about 08m In addition a particular attention should be paid to onesituation that the main dominant frequency amplitude is more than 50 of the acceleration amplitude even if the peak accelerationis low

1 Introduction

With the development of Chinarsquos railway and urban railtransportation the speed of the train is rapidly improvingthe safety and comfort issues have become more prominentMany domestic and global scholars have carried on a lot ofresearch [1] In 1922 Timoshenko studied track dynamic byusing elastic foundation beam model which had become aclassic method [2] In 1972 Lyon and Mather established thebasic model of wheel-rail dynamic analysis by consideringthe basic parameters of track and vehicle [3 4] In themodel track was described as continuous elastic foundationldquoEuler Beamrdquo and vehicle is simplified to spring-mass systemHertz nonlinear spring model was established to simulate thewheel-rail contact

In 1987 Li gave finite element analysis formulation of theDerby basic model Xu et al made a simulation analysis onwheel-rail impact of track joints by using the Timoshenkobeam model of continuous elastic foundation [5 6] In 1992Zhai a professor in Southwest Jiaotong University combinedthe vehicle system with the track system as a coupled systemfor further study This researcher established a series of

vehicle-track interaction coupled model the major dynamicfactors of the vehicle-track structure had been taken intodetailed consideration through the whole system [7 8] In2000 Lei and Chen used finite element theory to establish afreedomvibrationmodel of track structure an analysis on thenatural frequencies and vibration characteristics of existingrailway lines in different support conditions had beenmade inthe researcherrsquos paper [9] Zhang and Dhanasekar examinedthe effect of brakingtraction torque to the longitudinal andlateral dynamics of railway wagons in 2009 [10] this paperdealt with contribution of defective tracks to vehicle safelyThen in 2013 he presented a computational method foreliminating severe stress concentration at the unsupportedrailhead ends in rail joints through innovative shape opti-mization of the contact zone With a view to minimizing thecomputational efforts hybrid genetic algorithmmethod cou-pled with parametric finite element has been developed andcompared with the traditional genetic algorithm (GA) [11]

At present there are not many researches on high-frequency nonlinear vibration characteristics in the field ofhigh-speed trains Over the past 10 years many scholars hadstudied the wheel-rail coupling dynamics whichmainly used

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 5082319 13 pageshttpdxdoiorg10115520165082319

2 Mathematical Problems in Engineering

the method of numerical calculation Numerical calculationmethod such as finite elementmethod can simulate the cou-pling situationwith a good applicability andpowerful analysisability [12ndash14] However the time step must be short whenthe characteristic was calculated at high-speed conditionTheworkload could be quite large by using the iteration methodIn addition to numerical calculation method some currentresearches are carried out by using analytical method [15ndash17]

Track irregularity is an important source of vehicle vibra-tion It is one of the core issues of high-speed railway to realizethe high track smoothness Therefore it is very importantto control the influence of track irregularity on the dynamicperformance of high speed The short vertical track irreg-ularity mainly causes the high-frequency vibration whosewavelength is less than 2m in the small range

This paper presents a solution for inherent characteristicof wheel system of LMA type Meanwhile FE dynamic modelof the wheel-rail system using the contact dynamics theoryis established The wheel systemrsquos high-frequency nonlinearvibration response is presented which is in the influence ofvertical track irregularity by utilizing ABAQUS explicitdynamics simulation module with a wheel-rail three-dimen-sional solid finite element dynamic model The importantinfluence of short wave track vertical irregularity on verticalvibration of wheelrail system is discovered The paperprovides a theoretical basis for the study of high-frequencynonlinear vibration characteristics in wheel-rail system

2 The Linear Inherent CharacteristicCalculation of the Vertical Vibration inWheel System

The structure of LMA type wheel is shown in Figure 1 thewheel tread surface is shown in Figure 2

In the vertical track irregularity excitation due to therole of spring damper high-frequency vibration componentof the bogie frame is greatly weakened which is mainly onthe wheel set Therefore in the analysis of high-frequencyvibration of wheel-rail system model above incentive pointsignores spring structure and only retains wheel setThewheeland axle section are set up on sketch and the section is rotatedin order to obtain the wheel finite element model

The ABAQUS finite element wheel model is establishedby using three-dimensional deformable elastomeric structuretype The global seeds dimension is 05 and the unit type isC3D20 The model is divided into 3552 units The model cansatisfy the requirement of calculation accuracy and speedThe finite element model is shown in Figure 3

Lanczos algorithm is chosen to do the matrix operationsthrough linear perturbation frequency method and thehigh order natural frequencies can be effectively solved byusing ABAQUS program In this paper the finite elementmodel natural frequencies are solved Natural frequenciescalculation results are presented in Table 1

Through the ABAQUS program all the natural frequen-cies and vibration modes from 50Hz to 5000Hz are solvedtotally 63 orders Vertical transverse and bending vibrations

Table 1 The wheel system vertical vibration natural frequencies

Order FrequencyHz1 8052 17323 33814 61625 78666 101487 138118 198049 2295210 2423211 2694612 3218713 3660314 3990115 4365916 47547

are included There are 16 vertical vibration natural frequen-cies and the frequency distribution is relatively dispersed

In order to verify the accuracy of the finite elementmodelnatural frequency the conventional spring-mass method canbe used to make a comparison The spring-mass method iscommonly used to solve the natural characteristic whichconverts the wheel axle into Timoshenko beam with variablecross section and then the wheel is converted into concen-trated mass

The differential equations of the motion represented bythe displacement of free vibration Timoshenko beam are

1198641198681205974119904 (119910 119905)

1205971199054+ 120588119860

1205972119904 (119910 119905)

1205971199052+ 120588119868 (

119864

1198961015840119866+ 1)

sdot1205974119904 (119910 119905)

12059711991021205971199052+ 120588119860

1205974119904 (119910 119905)

1205971199054= 0

(1)

In the equation 1198961015840 is cross-sectional shape factor 119860 is cross-sectional area 119868 is the cross-sectional moment of inertia119904(119910 119905) is vibration displacement 120588 is density 119864 is the beamrsquoselastic modulus 119866 is the beamrsquos shear modulus

The general dynamic vibration differential equation is

[119872] 11990410158401015840 + [119862] 119904

1015840 + [119870] 119904 = 119875 (119905) (2)

In the equation [119872] is mass matrix [119862] is damping matrix[119870] is stiffness matrix 119904 is inertial element displacementvector 1199041015840 is inertial element velocity vector 11990410158401015840 is inertialelement acceleration vector 119875(119905) is the matrix of systemrsquosexternal load

Spring-mass method is more suitable for solving thelow band natural frequency because the model complexityshould not be too high The natural frequency numbersolved by the spring-mass method is restricted to the modeldegrees of freedom In condition of considering the modelsimplification without affecting the accuracy of solution

Mathematical Problems in Engineering 3

120601440

120601355

120601263+20

445

plusmn01

30∘

32

R3

R15

R30

R3

R15 R30

120601314

32

115

32

32

32

23

32

135

55

155

60

5 5

5

5

285

32 32

32

32

32

32

32

85

185

185

32

120601201 plusmn 003

0292 plusmn 02

0860+20

63

32

32

R15

155plusmn1

17plusmn1

17plusmn1

63

lowastlowastR40

lowastlowastR4

09+02

09+02

0

1 15

1 40

R90

R14

R12

12

R10

R30

R3

5

R10

43+10

0 25

R15R25

70∘

25 10

070

Figure 1 The structure of wheel lowastlowastR40 needs smooth transition

28 R2512

16

70

32

R14

70∘

R90

55 155

R450

Tread surfacedatum point

Tread surfacedatum line

100

5

5

1 40 1 15

R12

R15

Figure 2 LMA wheel tread surface

Figure 3 The ABAQUS FE model of wheel

a six-degree-of-freedom wheel set model is establishedwhich can solve a total of five natural frequencies

[119862] = 0 119875(119905) = 0 are defined Then the differentialequations of system for the free vibration without dampingare obtained

[119872] 11990410158401015840 + [119870] 119904 = 0 (3)

According to the conventional theory of simple harmonicvibration method the equation solution is set up as

120601119894= 120579119894cos (120596 sdot 119905) (119894 = 1 2 119899) (4)

In the equation 120596 is natural circular frequency 120579119894is ampli-

tude of inertia 119868


Substitute (4) into (3)

([119870] minus 1205962[119872]) 119904 = 0

Or [119860] 119904 = 0

(5)

In the equation [119860] is the characteristic matrix calculatingnatural frequency is converted into computing the algebraicEigen values for [119860]Meanwhile the corresponding eigenvec-tors of the Eigen values are the vibrationmodes of natural fre-quencies the five vertical vibration frequencies are 825Hz1668Hz 3442Hz 6275Hz and 8201 Hz

Compared with the result of conventional spring-massmethod the result difference of ABAQUS FEmodel is within5 Furthermore the high order natural frequencies can beeffectively solved by using ABAQUS program It is presentedthat the ABAQUS FE model is reasonable

3 Nonlinear Contact Dynamic Finite ElementModel of Wheel-Rail System

Usually in multibody dynamics and conventional spring-massmethod the systemrsquos double nonlinear vertical vibrationequation containing nonlinear damping and nonlinear stiff-ness is established [18]

For example a two-degree-of-freedom system Duffingoscillator is used to define nonlinear stiffness and then cubicnonlinear elastic force is obtained Vanderpol oscillator isused to define nonlinear damping and then cubic nonlinearfriction force is obtained

119872111990410158401015840

1+ 119862 (119904

1015840

1minus 1199041015840

2) + 1198621015840(1199041015840

1minus 1199041015840

2)3

+ 119870 (1199041minus 1199042)

+ 1198701015840(1199041minus 1199042)3

= 1198751

119872211990410158401015840

2minus 119862 (119904

1015840

1minus 1199041015840

2) minus 1198621015840(1199041015840

1minus 1199041015840

2)3

minus 1198701015840(1199041minus 1199042)3

minus 1198701015840(1199041minus 1199042)3

= minus1198752

(6)

In the equation1198701015840 is nonlinear stiffness value1198621015840 is nonlineardamping coefficient

However multibody dynamics method is mainly suitablefor solving the low-frequency dynamics of integral structureproblems such as the whole vehicle model on high-speedrailway The Runge-Kutta algorithm commonly used forsolving ordinary differential equations in the multibodydynamicsmethod is time consuming and it is prone to diver-gence instability when solving the problem of high-frequencyvibration

The ABAQUS finite element analysis is quite valid tosolve contact problems such as high-frequency vibrationand systemrsquos nonlinear characteristics [19] Based on theformer analysis of systemrsquos linear natural characteristicsthe ABAQUS deformable contact nonlinear finite elementdynamic model is used

The model is the symmetrical complete one which takesactual complexity of boundary condition at the symmetryaxis into account without distorted rather than one-sidedwheel model In addition the boundary constraint is rel-atively simple without taking other regulation of couple

Figure 4 The region applied the kinematics parameter

moment into consideration The incorrect result caused byoverconstraint is avoided

When defining load and boundary condition the load ofgravitational field is taking vertical downward applied to theentire model Commonly the load of high-speed EMUwheelaxle is from 14 t to 16 t Spring damper is set to the wheel andtrack system furthermore spring damper of track system isconnected to the groundThen the wheel velocity parameteris applied as in Figure 4

The type and quality of grid unit closely influencethe accuracy the smoothness and spectrum of simulationresults In order to acquire the accurate solution of contactdynamics the fine linear hexahedral hourglass-stiffness-control-type reducing-integral element C3D8R and hexahe-dral nonconforming element C3D8I are divided into wheeland track respectively in this paper [11] The grid of wheelis separated quite densely divided into 113500 units Gridseed of the track is laid out relatively sparsely but fine inthe region of irregularity ensures the accuracy of wave curvetotally divided into 1860 units The grid of model is shown inFigure 5 In this model the same irregularity curves are setup in two tracks

4 Nonlinear Dynamic Finite Element ModelAlgorithm and the Optimization of ContactStiffness Factor Based on Experiment

41 Solving Nonlinear Dynamics of Wheel-Rail System Basedon ABAQUSExplicit by Improved Central Difference MethodFor nonlinear dynamics explicit dynamics analysis andimplicit dynamics analysis are usually used For implicit andexplicit integral procedures

11987211990610158401015840= 119875 minus 119868 (7)

where119872 is the node mass matrix 119875 is the applied force 119868 isthe internal force 11990610158401015840 is the node acceleration

Implicit dynamics analysis method is universally used tosolve the general nonlinear problems The method has noinherent limit on the size of time step incrementThe solving


Figure 5 The grid of model

time is typically dependent on the solution accuracy andconvergence [20]

Thismethod itself has the characteristics of unconditionalstability However the stiffness matrix is very complicatedParticularly when the mesh is meticulous it may lead to hugecost of iteration the accuracy has no significant increase andit has no obvious advantages in solving the high-frequencycharacteristics

Explicit analysis method is quite effective on the non-linear model with an incentive impact in short time whichhas the advantage in the fast speed for each incremental stepwithout iterative calculation The fidelity and accuracy couldbe higher than the implicit analysis with any time integralparameter which can solve the nonlinear high-frequencyvibration system quite accurately The explicit module ofABAQUS uses central differencemethod to give explicit solu-tion of the motionrsquos equation with parametric improvementThe dynamic condition of incremental step is used to calcu-late the next one [21]

The dynamic explicit method is used in this paperExplicit method requires minor step increment which onlydepends on the model highest natural frequency and isunrelated to the type and duration of load At the beginningof increment (119905) the acceleration 119906

10158401015840|(119905)

is

1199061015840101584010038161003816100381610038161003816(119905)

= (119872)minus1

(119875 minus 119868)|(119905) (8)

It is simple to calculate the acceleration since explicit methodalways uses diagonal or concentrate mass matrix

Constant acceleration for the change in calculating veloc-ity is assumed To determine the midpoint velocity of incre-mental step |

(119905+Δ1199052) changes in the velocity value are added

to the former midpoint velocity |(119905minusΔ1199052)

|(119905+Δ1199052)

= |(119905minusΔ1199052)

+Δ119905|(119905+Δ119905)

+ Δ119905|(119905)

2|(119905) (9)

The beginning increment displacement 119906|(119905)

plus the integralof velocity to determine the displacement at the end ofincrement 119906|

(119905+Δ119905)is as follows

119906|(119905+Δ119905)

= 119906|(119905)

+ Δ119905|(119905+Δ119905)

|(119905+Δ1199052)

(10)

The acceleration with the condition of dynamics balance isprovided at the beginning of incremental step When theacceleration is obtained the velocity and displacement arecalculated ldquoexplicitlyrdquo on time [19]

In order to ensure the analysis correctness for the elasticbody of large displacement the geometric nonlinear must beconsidered in the nonlinear contact

42 Contact Parameter Optimization Because the contactdynamic calculation is required in nonlinear analysis of thefinite element model contact parameter has great influenceon the simulation results of the amplitude and frequencyspectrum Therefore contact theory is the key to nonlineardynamics Surface to surface contact is chosen to be theinteraction mode Track surface is set as the first surfacewhile wheel surface is set as the second surface The mesh offirst surface is divided more sparsely than the second one Arational weighting coefficient of contact could prevent wheelnodes from inserting into the track surface which leads amore accurate result in the important region Due to thedistinct time-varying characteristic of contact surface it isnecessary to define the slip formula as the finite slip

Then the contact property should be defined accuratelyand scientifically The contact stiffness is quite important todynamic calculation The larger the contact stiffness is thesmaller the penetration is However excessive contact stiff-ness may lead the overall stiffness matrix into morbid statethe convergence difficulty of solution is aroused [22] Basedon Hertz contact theory the contact stiffness factor is usuallyset from 01 to 1

The optimal value not only makes the results morereasonable but also makes the displacement and accelerationachieve the best convergence effectThe specific value shouldbe obtained through experiment and simulation of severalattempts

The peak accelerations and the convergence of variousoperating conditions are obtained by different stiffness factorvalues It is shown in Figure 6 and Table 2

In principle the appropriate contact stiffness factor is easyto converge Factors 01 03 04 and 07 are appropriateFor further verification of the stiffness factor it is necessaryto select the best one according to the actual experimentalresults

43 Vibration Measurement Experiment of LMA WheelWhen the load applied in different track irregularities anddifferent high velocities it is difficult to measure the wheelaxle vertical acceleration especially difficult to acquire thehigh-frequency characteristics within a very short periodBecause of the elastic contact in wheel-rail system the actualwheel sets on irregularity excitation is not a simple harmonicexcitationHowever in general only the equivalent technique


050

100150200250300350

01 02 03 04 05 06 07 08 09 1Contact stiffness factor

Peak

acce

lera

tion

(ms

2)

Wavelength 05mmWavelength 15mm

(a) Velocity is 160 kmh

050

100150200250300350400450500


Peak

acce

lera

tion

(ms

2)


(b) Velocity is 200 kmh

0100200300400500600700800900


Peak

acce

lera

tion

(ms

2)


(c) Velocity is 300 kmh

0200400600800

100012001400


Peak

acce

lera

tion

(ms

2)


(d) Velocity is 350 kmh

Figure 6 Acceleration simulated with different stiffness factor in different conditions

Table 2 The convergence of different contact stiffness factor

Stiffness factor Convergence01 Easy02 Normal03 Easy04 Easy05 Normal06 Normal07 Easy08 Normal09 Hard10 Hard

is used to verify the amplitude At present the vertical dis-placement excitation of harmonic wave is usually equivalentto the track irregularity in relevant test

According to the vibration measurement experiment ofLMA wheel by China Academy of Railway Sciences differenttypes of harmonic waves are taken as the track irregularityThe corresponding relationship is

119910 =119860

2[1 minus cos (2120587119891119905 + 120601)] (11)

where 119891 = V times 1000(3600 times 119871) 119891 is the frequency oftrack irregularity 119910 is waveform 119860 is the amplitude of track

irregularity V is the velocity of wheel 120601 is the phase of trackirregularity two tracks with the same phase 119905 is time 119871 is thewavelength of track irregularity

The axle vertical accelerations are tested in the conditionof 16 t axial loaded and then the vertical peak accelerationsare presented in Table 3

Combined with the test results and simulation resultsunder different stiffness factors mentioned above it can beconcluded that the average difference of factor 01 is 589the average difference of factor 03 is 871 the averagedifference of factor 05 is 825 and the average differenceof factor 07 is 273

It is presented that the contact stiffness factor 07 is theoptimal value The results analyzed are close to the measuredvalue of these experiments with a fine convergence

5 The Vertical Nonlinear VibrationSimulation Results and Analysis

Predecessors have done some related research includingwheel-rail system joint spectrum such as Sato formula toinput stimulated spectrum of wheel-rail random high-frequency vibration and noise radiation model The equationis [23]

119878 (119896) =119860

1198963 (12)


Table 3 The vertical accelerations tested by the vibration measurement experiment

Wheel velocity (kmh) Irregularity excitation Peak acceleration (ms2)Wavelength (m) Wave amplitude (mm)

160 05 10 272160 15 10 72200 05 10 382200 15 10 97300 05 10 798300 15 10 342350 05 10 946350 15 10 407

0 0005 001 0015 002 0025 003 0035 004 0045minus60

minus40

minus20

0

20

40

60

t (s)

a(m

s2)

(a) Acceleration curve

0 1000 2000 3000 4000 50000123456789

X 2382Y 5662

Am

plitu

de

X 7793Y 4043

X 1403Y 881

f (Hz)

(b) Fourier spectrum

Figure 7 Acceleration and spectrum at 200 kmh

where 119878(119896) is spectral density of wheel-rail surface roughness119896 is roughness wave number 119860 is wheel-rail surface rough-ness coefficient

Wheel-rail under load action has elastic deformation andthe formation of an elliptical contact zone occurs The influ-ence of wheel-rail surface roughness on the wheel-rail sys-temrsquos vibration is affected by the size of contact areaWhen theroughness wavelength is less than the contact patch the exci-tation effect will be weakened Considering the effect filterfunction is used

|119867 (119896)|2=4

119886

1

(119896119887)2int

arctan(119886)

0

[119869119896119887 sec (119909)]2 119889119909 (13)

where 119886 is the roughness correlation coefficient 119887 is thecontact zone circle radius 119869 is Bessel function 119909 is thelongitudinal distance along the track

In this paper the authors set the wavelengths from 02mto 20m The irregularity amplitudes are generally from02mm to 1mm based on the actual tests 03mm 06mmand 09mm are chosen as examples for simulation Wheelaxle is the focus According to the velocity criteria of Chinahigh-speed railway 200 kmh and 350 kmh are selected to bemain velocities

Figures 7ndash14 present the axle vertical acceleration in sometypical conditions and the results of Fourier spectrum Thespectrum results of low frequency below 400Hz are ignored

51 The effect of 2m Wavelength-09mm Amplitude Irregu-larity When the velocity is 200 kmh the peak accelerationis 60ms2 acceleration amplitude is 30ms2 It is shown onFourier spectrum that the main dominant frequency is1403Hz while the secondary dominant frequencies are2382Hz and 7703Hz Because of the wave filter and restric-tion on the sampling number the highest frequency isgenerally lower than 5000Hz

When the train velocity is 350 kmh the peak accelerationis 150ms2 acceleration amplitude is 97ms2 It is shownon Fourier spectrum that the main dominant frequency is1938Hz and the secondary dominant frequencies are2372Hz and 1403Hz

52 The Effect of 1m Wavelength-09mm Amplitude Irregu-larity When the velocity is 200 kmh the peak accelerationis 130ms2 acceleration amplitude is 72ms2 The maindominant frequency is 1403Hz the secondary dominantfrequency is 2306Hz

When the velocity is 350 kmh the peak accelerationis 720ms2 acceleration amplitude is 340ms2 It is shownon Fourier spectrum that the main dominant frequency is1914Hz and the secondary dominant frequencies are 2324Hzand 3600Hz

53 The Effect of 04mWavelength-09mm Amplitude Irregu-larity When the train velocity is 200 kmh the peak accel-eration is 400ms2 acceleration amplitude is 160ms2 It is


0 0005 001 0015 002 0025 003minus150

minus100

minus50

0

50

100

150

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

5

10

15

20

25

30

X 2372Y 1611

Am

plitu

de X 1403Y 1578

X 1938Y 2768

f (Hz)



0 0005 001 0015 002 0025 003minus100

minus50

0

50

100

150

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

5

10

15

20

25

X 2306Y 1308

Am

plitu

de

X 1403Y 2312

f (Hz)



0

000

2

000

4

000

6

000

8

001

001

2

001

4

001

6

001

8

002

minus400

minus200

0

200

400

600

800

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

102030405060708090

X 1914Y 8087

Am

plitu

de

X 2324Y 3404

X 3600Y 2469

f (Hz)



shown on Fourier spectrum that the main dominant fre-quency is 2372Hz and the secondary dominant frequenciesare 1403Hz and 4344Hz

When the train velocity is 350 kmh the peak accelerationis 950ms2 acceleration amplitude is 490ms2 It is shownon Fourier spectrum that the main dominant frequencies are

1914Hz and 2324Hz and the secondary dominant frequencyis 4283Hz

54The Effect of 1mWavelength-06mmAmplitude Irregular-ity When the train velocity is 200 kmh the peak accelera-tion is 91ms2 acceleration amplitude is 49ms2 It is shown


0 0005 001 0015 002 0025 003minus400

minus300

minus200

minus100

0

100

200

300

400

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

102030405060708090

X 1403Y 3225A

mpl

itude

X 2372Y 8395

X 4344Y 159

f (Hz)



0

000

2

000

4

000

6

000

8

001

001

2

001

4

001

6

001

8

002

minus800minus600minus400minus200

0200400600800

1000

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

102030405060708090

100X 1914Y 9623

Am

plitu

de

X 2324Y 9094

X 3600Y 4002

X 4283Y 6066

f (Hz)



0 0005 001 0015 002 0025 003minus100


020406080

100

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

5

10

15

20

25X 2372Y 2491

Am

plitu

de X 1403Y 1401

f (Hz)



on Fourier spectrum that the main dominant frequency is2372Hz and the secondary dominant frequency is 1403Hz


2324Hz and 1914Hz and the secondary dominant frequen-cies are 3554Hz and 4283Hz

As can be seen from the results list above in differentworking conditions changes of track irregularity wave-length will affect Fourier spectrum of the corresponding


0

000

2

000

4

000

6

000

8

001

001

2

001

4

001

6

001

8

002

minus400

minus300

minus200

minus100

0

100

200

300

400

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

10

20

30

40

50

60

X 1914Y 5186

Am

plitu

de

X 2324Y 5187 X 3554

Y 4166

X 4283Y 2967

f (Hz)



0

100

200

300

400

500

600

700

02 04 06 08 1 13 17 2Wavelength (m)

Peak

acce

lera

tion

(ms

2)

Wave amplitudeWave amplitudeWave amplitude

03mm06mm09mm


Wavelength (m)

0100200300400500600700800900

1000

02 04 06 08 1 13 17 2

Peak

acce

lera

tion

(ms

2)


03mm06mm09mm


Figure 15 Peak accelerations simulated in each condition

acceleration curve The main dominant frequency variessignificantly when the velocity is 350 kmh

Finally in order to make the analysis comprehensiveand persuasive the analysis includes the conditions of eightirregularity-wavelengths from 02m to 2m Combine withthe simulation above Figure 15 and Table 4 are obtained

It is concluded through the analysis of the simulationresults in various working conditions that

(1) the main dominant frequency includes 1400Hz1900Hz and 2350Hz

(2) in the majority of operating conditions the peakacceleration at velocity of 350 kmh is higher than thatat 200 kmh significantly and themain frequency alsoshifts or changes Meanwhile the nonlinear strengthof system is enhanced obviously

In the past few years many experts have made a conclusionabout the influence of track irregularity on wheel set Whenthe speed is constant the increase of irregularity wavelengthwill lead to the peak acceleration and nonlinearity decreasingthe increase of irregularity-amplitude will lead to the peak

acceleration increasing However the decrease of wavelengthor amplitude has an opposite result When the amplitude andwavelength are constant the decrease of velocity will weakenthe vibration

However it is not the case In vibration it does not havea single dynamic characteristic for a system Gao et al haddiscovered the sensitive wavelength of the long-wavelengthtrack irregularity on the vehicle vibration [24]

In this paper it is presented from the simulation resultsthat vertical track irregularity (wavelength less than 20m)has a critical wavelength on the wheel vertical vibrationbecause of its inherent characteristics and nonlinear factorsWhen irregularity wavelength is close to the critical rangethe dynamic characteristics of vibration change obviously aresummarized as follows

For the irregularity whose wavelength is more than 08mwhether the velocity is 200 kmh or 350 kmh the accel-eration is reduced by decreasing the irregularity-amplitudeor increasing wavelength The increase of irregularity wave-length will significantly lead to the main frequency ampli-tude decreasing when the velocity is 350 kmh The main


Table 4 The simulation results of the acceleration and the main dominant frequency in each working condition

Wl (m) Wa (mm) 200 kmh 350 kmh119886 (ms2)Amp (ms2) 119891 (Hz)Amp (ms2) 119886 (ms2)Amp (ms2) 119891 (Hz)Amp (ms2)

02 03 580191 23721124 220100 231221602 06 600270 23721627 400220 231850802 09 640236 237212492 870445 190782504 03 350144 23728105 240126 23242804 06 360180 23721028 440280 233658804 09 400160 23728395 950490 1914962306 03 24091 23725133 250108 233422106 06 255127 23726478 424205 234159106 09 280116 2372467 920410 191495508 03 11541 23722225 260110 233423808 06 12952 23722944 460284 233460208 09 15081 23723065 900410 1914951 03 6036 14031965 18095 233818471 06 9149 23722491 310255 232441871 09 13072 14032312 720340 1914808713 03 4822 1403911 9041 235519213 06 6739 23721211 21096 2355302113 09 9964 14031923 490221 191871317 03 3712 1403692 8650 2367227517 06 6429 14031039 11077 2355324717 09 8741 14032233 290180 19265952 03 4311 1403439 8933 239211962 06 3622 1403919 7851 237717212 09 6030 1403881 15097 19382768Note Wl = wavelength Wa = wave amplitude Amp = amplitude 119891 = main dominant frequency 119886 = peak acceleration simulated

frequency amplitude changes little when the velocity is200 kmh

However for the irregularity whose wavelength is lessthan 08m one has the following

The increase of irregularity wavelength will significantlylead to the main frequency amplitude decreasing when thevelocity is 200 kmh However themain frequency amplitudechanges little when the velocity is 350 kmh

The peak acceleration is not sensitive to the verticaltrack irregularity-amplitude when the velocity is 200 kmhHowever it is sensitive to the wavelength The decrease ofwavelength will significantly increase the acceleration whilethe increase of irregularity-amplitude affects little increasingthe acceleration

The peak acceleration is not sensitive to the verticaltrack irregularity wavelength when the velocity is 350 kmhHowever it is sensitive to the amplitude The increase ofamplitude will significantly increase the acceleration whilethe decrease of irregularity wavelength affects little increasingthe acceleration

It is shown inTable 4 that for example under the action of02m wavelength-03mm amplitude irregularity the verticalpeak acceleration at 200 kmh is obviously 25 times the valueat 350 kmh Under the action of 06m wavelength-03mm

amplitude irregularity the vertical peak acceleration at200 kmh is almost equal to the value at 350 kmh Howeverunder the action of 06m wavelength-09mm amplitudeirregularity the peak acceleration at 350 kmh is 33 timesthe value at 200 kmh Under the action of 08m wavelength-09mm amplitude irregularity the peak acceleration at350 kmh is 6 times the value at 200 kmh It is explained thatthe acceleration at 300 kmh is not always absolutely higherthan the value at 200 kmh Sometimes under the same con-ditions of vertical irregularities the vibration is less when thespeed is high

It can be determined that the critical wavelength is about08m

In addition special attention should be paid to some con-ditions that the main dominant high-frequency amplitude ismore than 50 of the acceleration amplitude From the view-point of energy the abnormal state of spectrum amplitude isdefinitely not being overlooked Even if the peak accelerationis low the destruction will be huge to the internal ofwheel-rail system Furthermore theseworking conditions aremostly the ldquosoftrdquo vibration conditions at 200 kmh It is pre-sented that the wheel vibration response under the action of avertical track irregularity is not always positively related to thetrain speed


6 Conclusions

In this paper Based on the theory of nonlinear contactdynamics of wheel-rail system a symmetrical completethree-dimensional finite element model of wheel-rail systemis established The authors discover the important influenceof vertical track irregularity on the wheel set high-frequencyvibration

(1) The inherent linear characteristic of wheel set issolved Compared with the result of conventionalspring-mass method the result difference ofABAQUS FE model is within 5 Furthermore thehigh order natural frequencies can be effectivelysolved by using ABAQUS program It is presentedthat the ABAQUS FE model is reasonable

(2) The improved central differencemethod by ABAQUSexplicit nonlinear dynamics module is suitable forcalculating the high-frequency vertical vibration ofwheel axle Through the vibration measurementexperiment of LMA wheel by China Academy of Rail-way Sciences the contact stiffness factor is optimizeddefined as 07

(3) The nonlinear vibrations and frequencies responsecharacteristics of wheel axle are simulated It is shownthat themain frequencies of vertical vibration acceler-ation onwheel axle in different conditions are concen-trated in the range of 1400Hz to 4500Hz very closeto the vertical vibration natural frequency of 7th 8th9th 10th 13th and 15th mode with the obvious char-acteristics of high-frequency vibration In additionthe frequencies offset in a certain range is affected bynonlinear characteristics

(4) The critical wavelength of vertical track irregularityon the vertical vibration at wheel axle of wheel-railsystem is about 08mWhen the velocity is 350 kmhsystemrsquos nonlinear characteristics are higher than theresults at 200 kmh Furthermore a particular atten-tion should be paid to the Fourier spectrum whoseldquoenergyrdquo is quite strong even if the peak acceleration isquite ldquoweakrdquo especially whose main dominant fre-quency amplitude is more than 50 of the accelera-tion amplitude

The authors believe that the conclusion of this paper isof relatively high value Referring to the amplitude-frequencydynamics characteristics the appropriate measures are for-mulated which have special guiding significance to the on-site tests and have already been applied to the operationalmaintenance

Additional Points

Research area includes dynamics and infrastructure inspec-tion

Disclosure

Yang Lei is the first author

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

First it is an honor for the authors that the paper is basedon the project named ldquoInvestigation of Long-Term Vibra-tion Response and Deterioration Mechanism of a HS Rail-way Tunnel Subjected to Train Loading and AerodynamicLoading under the Cumulative Effect of the Typical InitialDamagesrdquoThe ldquoJoint Fund for Basic Research of High-speedRailwayrdquo whose Award no is U1434211 supports the projectldquoThe Joint Fund for Basic Research of High-speed Railwayrdquofocuses on the basic theoretical research of the high-speedtrain and railway engineering technology which is designedto play the guiding and coordinating role of the NationalNatural Science Foundation of China (NSFC) The recipientis researcher Qianli ZhangThe authors would like to expressthe heartfelt thanks to themThis works is supported by JointFund for Basic Research of High-speed Railway of NationalNatural Science Foundation of China China Railway Corpo-ration (U1434211)

References

[1] T-LWang V K Garg and K-H Chu ldquoRailway bridgevehicleinteraction studies with new vehicle modelrdquo Journal of Struc-tural Engineering vol 117 no 7 pp 2099ndash2116 1991

[2] S P Timoshenko ldquoOn the forced vibrations of bridgesrdquo Philo-sophical Magazine vol 43 no 257 pp 1018ndash1019 1922

[3] D Lyon ldquoThe calculation of track forces due to dipped railjoints wheel flat and rail weldsrdquo The Second ORE Colloquiumon Technical Computer Program 1972

[4] D Lyon and J G Mather ldquoTrack stiffness and damping mea-surements and theoretical considerationrdquo BR Tech Note TNTS22 British Railways Research amp Development Division DerbyUK 1974

[5] S R Xu W J Xu and Y X Zhong ldquoA simulation analysis ofwheelrail impact force at the rail jointrdquo Journal of the ChinaRailway Society vol 1 pp 99ndash109 1989

[6] D Q Li ldquoWheeltrack vertical dynamic action and responsesrdquoJournal of the China Railway Society vol 9 no 1 pp 1ndash8 1987

[7] W M Zhai ldquoThe vertical model of vehicle-track system and itscoupling dynamicsrdquo Journal of the China Railway Society vol14 no 3 pp 10ndash21 1992

[8] G Chen and W M Zhai ldquoNumerical simulation of thestochastic process of Railway track irregularitiesrdquo Journal ofSouthwest Jiaotong University vol 34 no 2 pp 138ndash142 1999

[9] X Y Lei and S S Chen ldquoSpace dynamic analyses for track struc-ture of high speed railwayrdquo Journal of the China Railway Societyvol 5 pp 76ndash80 2000

[10] Z Zhang and M Dhanasekar ldquoDynamics of railway wagonssubjected to brakingtraction torquerdquoVehicle SystemDynamicsvol 47 no 3 pp 285ndash307 2009

[11] N Zong and M Dhanasekar ldquoHybrid genetic algorithm forelimination of severe stress concentration in railhead endsrdquoJournal of Computing in Civil Engineering vol 29 no 5 ArticleID 4014075 2015

[12] Y Q Sun andMDhanasekar ldquoA dynamicmodel for the verticalinteraction of the rail track and wagon systemrdquo International


Journal of Solids and Structures vol 39 no 5 pp 1337ndash13592002

[13] J Oscarson and T Dahlberg ldquoDynamic traintrackballastinteractionmdashcomputer models and full-scale experimentsrdquoVehicle System Dynamics vol 29 pp 73ndash84 1998

[14] L Y Liu H L Liu and X Y Lei ldquoHigh-frequency vibrationcharacteristics of wheels in wheel-rail systemrdquo Journal of Trafficand Transportation Engineering vol 11 no 6 pp 44ndash67 2011

[15] X Sheng C J C Jones and D J Thompson ldquoA theoreticalmodel for ground vibration from trains generated by verticaltrack irregularitiesrdquo Journal of Sound and Vibration vol 272no 3ndash5 pp 937ndash965 2004

[16] G Lombaert G Degrande J Kogut and S Francois ldquoTheexperimental validation of a numericalmodel for the predictionof railway induced vibrationsrdquo Journal of Sound and Vibrationvol 297 no 3ndash5 pp 512ndash535 2006

[17] Quarterly Report of Railway Technical Research InstituteldquoMultiple-wheel induced vibration of rail with surface irregu-larityrdquoManabe Katsushi vol 45 no 3 pp 136ndash141 2004

[18] J Dai ldquoStochastic dynamic analysis of vehicles with randomparameters based on a quarter-car modelrdquo Journal of Vibrationand Shock vol 29 no 6 pp 211ndash215 2010

[19] ldquoMathematical collection of paper of the academy of sciencesrdquoMonument to Ivan Krylov St Petersburg Russia 61 1995

[20] M H Bhatti Vertical and lateral dynamic response of railwaybridges due to nonlinear vehicle and track irregularities [PhDthesis] Illinois Institute of Technology Chicago Ill USA 1982

[21] Z Zhuang ABAQUS Nonlinear Finite Element Analysis andExamples Science Press Beijing China 2005

[22] X F Ma ABAQUS 611 Finite Element Analysis of ChineseVersion from the Entry to the Proficient Tsinghua UniversityPress Beijing China 2013

[23] Y Sato ldquoStudy on high-frequency vibration in track operationwith high-speed trainsrdquoQuarterly Report of RTRI vol 18 no 3pp 109ndash114 1977

[24] JMGaoWM Zhai andK YWang ldquoStudy on sensitive wave-lengths of irregularities in high-speed operationrdquo Journal of theChina Railway Society vol 34 no 7 pp 83ndash88 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


the method of numerical calculation Numerical calculationmethod such as finite elementmethod can simulate the cou-pling situationwith a good applicability andpowerful analysisability [12ndash14] However the time step must be short whenthe characteristic was calculated at high-speed conditionTheworkload could be quite large by using the iteration methodIn addition to numerical calculation method some currentresearches are carried out by using analytical method [15ndash17]

Track irregularity is an important source of vehicle vibra-tion It is one of the core issues of high-speed railway to realizethe high track smoothness Therefore it is very importantto control the influence of track irregularity on the dynamicperformance of high speed The short vertical track irreg-ularity mainly causes the high-frequency vibration whosewavelength is less than 2m in the small range

This paper presents a solution for inherent characteristicof wheel system of LMA type Meanwhile FE dynamic modelof the wheel-rail system using the contact dynamics theoryis established The wheel systemrsquos high-frequency nonlinearvibration response is presented which is in the influence ofvertical track irregularity by utilizing ABAQUS explicitdynamics simulation module with a wheel-rail three-dimen-sional solid finite element dynamic model The importantinfluence of short wave track vertical irregularity on verticalvibration of wheelrail system is discovered The paperprovides a theoretical basis for the study of high-frequencynonlinear vibration characteristics in wheel-rail system

2 The Linear Inherent CharacteristicCalculation of the Vertical Vibration inWheel System

The structure of LMA type wheel is shown in Figure 1 thewheel tread surface is shown in Figure 2

In the vertical track irregularity excitation due to therole of spring damper high-frequency vibration componentof the bogie frame is greatly weakened which is mainly onthe wheel set Therefore in the analysis of high-frequencyvibration of wheel-rail system model above incentive pointsignores spring structure and only retains wheel setThewheeland axle section are set up on sketch and the section is rotatedin order to obtain the wheel finite element model

The ABAQUS finite element wheel model is establishedby using three-dimensional deformable elastomeric structuretype The global seeds dimension is 05 and the unit type isC3D20 The model is divided into 3552 units The model cansatisfy the requirement of calculation accuracy and speedThe finite element model is shown in Figure 3

Lanczos algorithm is chosen to do the matrix operationsthrough linear perturbation frequency method and thehigh order natural frequencies can be effectively solved byusing ABAQUS program In this paper the finite elementmodel natural frequencies are solved Natural frequenciescalculation results are presented in Table 1

Through the ABAQUS program all the natural frequen-cies and vibration modes from 50Hz to 5000Hz are solvedtotally 63 orders Vertical transverse and bending vibrations

Table 1 The wheel system vertical vibration natural frequencies

Order FrequencyHz1 8052 17323 33814 61625 78666 101487 138118 198049 2295210 2423211 2694612 3218713 3660314 3990115 4365916 47547

are included There are 16 vertical vibration natural frequen-cies and the frequency distribution is relatively dispersed

In order to verify the accuracy of the finite elementmodelnatural frequency the conventional spring-mass method canbe used to make a comparison The spring-mass method iscommonly used to solve the natural characteristic whichconverts the wheel axle into Timoshenko beam with variablecross section and then the wheel is converted into concen-trated mass

The differential equations of the motion represented bythe displacement of free vibration Timoshenko beam are

1198641198681205974119904 (119910 119905)

1205971199054+ 120588119860

1205972119904 (119910 119905)

1205971199052+ 120588119868 (

119864

1198961015840119866+ 1)

sdot1205974119904 (119910 119905)

12059711991021205971199052+ 120588119860

1205974119904 (119910 119905)

1205971199054= 0

(1)

In the equation 1198961015840 is cross-sectional shape factor 119860 is cross-sectional area 119868 is the cross-sectional moment of inertia119904(119910 119905) is vibration displacement 120588 is density 119864 is the beamrsquoselastic modulus 119866 is the beamrsquos shear modulus

The general dynamic vibration differential equation is

[119872] 11990410158401015840 + [119862] 119904

1015840 + [119870] 119904 = 119875 (119905) (2)

In the equation [119872] is mass matrix [119862] is damping matrix[119870] is stiffness matrix 119904 is inertial element displacementvector 1199041015840 is inertial element velocity vector 11990410158401015840 is inertialelement acceleration vector 119875(119905) is the matrix of systemrsquosexternal load

Spring-mass method is more suitable for solving thelow band natural frequency because the model complexityshould not be too high The natural frequency numbersolved by the spring-mass method is restricted to the modeldegrees of freedom In condition of considering the modelsimplification without affecting the accuracy of solution


120601440

120601355

120601263+20

445

plusmn01

30∘

32

R3

R15

R30

R3

R15 R30

120601314

32

115

32

32

32

23

32

135

55

155

60

5 5

5

5

285

32 32

32

32

32

32

32

85

185

185

32

120601201 plusmn 003

0292 plusmn 02

0860+20

63

32

32

R15

155plusmn1

17plusmn1

17plusmn1

63

lowastlowastR40

lowastlowastR4

09+02

09+02

0

1 15

1 40

R90

R14

R12

12

R10

R30

R3

5

R10

43+10

0 25

R15R25

70∘

25 10

070


28 R2512

16

70

32

R14

70∘

R90

55 155

R450



100

5

5

1 40 1 15

R12

R15





[119872] 11990410158401015840 + [119870] 119904 = 0 (3)


120601119894= 120579119894cos (120596 sdot 119905) (119894 = 1 2 119899) (4)





([119870] minus 1205962[119872]) 119904 = 0

Or [119860] 119904 = 0

(5)






119872111990410158401015840

1+ 119862 (119904

1015840

1minus 1199041015840

2) + 1198621015840(1199041015840

1minus 1199041015840

2)3

+ 119870 (1199041minus 1199042)

+ 1198701015840(1199041minus 1199042)3

= 1198751

119872211990410158401015840

2minus 119862 (119904

1015840

1minus 1199041015840

2) minus 1198621015840(1199041015840

1minus 1199041015840

2)3

minus 1198701015840(1199041minus 1199042)3

minus 1198701015840(1199041minus 1199042)3

= minus1198752

(6)











11987211990610158401015840= 119875 minus 119868 (7)









10158401015840|(119905)

is

1199061015840101584010038161003816100381610038161003816(119905)

= (119872)minus1

(119875 minus 119868)|(119905) (8)





|(119905+Δ1199052)

= |(119905minusΔ1199052)

+Δ119905|(119905+Δ119905)

+ Δ119905|(119905)

2|(119905) (9)




119906|(119905+Δ119905)

= 119906|(119905)

+ Δ119905|(119905+Δ119905)

|(119905+Δ1199052)

(10)










050

100150200250300350


Peak

acce

lera

tion

(ms

2)



050

100150200250300350400450500


Peak

acce

lera

tion

(ms

2)



0100200300400500600700800900


Peak

acce

lera

tion

(ms

2)



0200400600800

100012001400


Peak

acce

lera

tion

(ms

2)








119910 =119860

2[1 minus cos (2120587119891119905 + 120601)] (11)








119878 (119896) =119860

1198963 (12)




160 05 10 272160 15 10 72200 05 10 382200 15 10 97300 05 10 798300 15 10 342350 05 10 946350 15 10 407

0 0005 001 0015 002 0025 003 0035 004 0045minus60

minus40

minus20

0

20

40

60

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000123456789

X 2382Y 5662

Am

plitu

de

X 7793Y 4043

X 1403Y 881

f (Hz)





|119867 (119896)|2=4

119886

1

(119896119887)2int

arctan(119886)

0

[119869119896119887 sec (119909)]2 119889119909 (13)










0 0005 001 0015 002 0025 003minus150

minus100

minus50

0

50

100

150

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

5

10

15

20

25

30

X 2372Y 1611

Am

plitu

de X 1403Y 1578

X 1938Y 2768

f (Hz)



0 0005 001 0015 002 0025 003minus100

minus50

0

50

100

150

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

5

10

15

20

25

X 2306Y 1308

Am

plitu

de

X 1403Y 2312

f (Hz)



0

000

2

000

4

000

6

000

8

001

001

2

001

4

001

6

001

8

002

minus400

minus200

0

200

400

600

800

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

102030405060708090

X 1914Y 8087

Am

plitu

de

X 2324Y 3404

X 3600Y 2469

f (Hz)








0 0005 001 0015 002 0025 003minus400

minus300

minus200

minus100

0

100

200

300

400

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

102030405060708090

X 1403Y 3225A

mpl

itude

X 2372Y 8395

X 4344Y 159

f (Hz)



0

000

2

000

4

000

6

000

8

001

001

2

001

4

001

6

001

8

002


0200400600800

1000

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

102030405060708090

100X 1914Y 9623

Am

plitu

de

X 2324Y 9094

X 3600Y 4002

X 4283Y 6066

f (Hz)



0 0005 001 0015 002 0025 003minus100


020406080

100

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

5

10

15

20

25X 2372Y 2491

Am

plitu

de X 1403Y 1401

f (Hz)








0

000

2

000

4

000

6

000

8

001

001

2

001

4

001

6

001

8

002

minus400

minus300

minus200

minus100

0

100

200

300

400

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

10

20

30

40

50

60

X 1914Y 5186

Am

plitu

de

X 2324Y 5187 X 3554

Y 4166

X 4283Y 2967

f (Hz)



0

100

200

300

400

500

600

700

02 04 06 08 1 13 17 2Wavelength (m)

Peak

acce

lera

tion

(ms

2)


03mm06mm09mm


Wavelength (m)

0100200300400500600700800900

1000

02 04 06 08 1 13 17 2

Peak

acce

lera

tion

(ms

2)


03mm06mm09mm



























6 Conclusions







Additional Points


Disclosure


Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120601440

120601355

120601263+20

445

plusmn01

30∘

32

R3

R15

R30

R3

R15 R30

120601314

32

115

32

32

32

23

32

135

55

155

60

5 5

5

5

285

32 32

32

32

32

32

32

85

185

185

32

120601201 plusmn 003

0292 plusmn 02

0860+20

63

32

32

R15

155plusmn1

17plusmn1

17plusmn1

63

lowastlowastR40

lowastlowastR4

09+02

09+02

0

1 15

1 40

R90

R14

R12

12

R10

R30

R3

5

R10

43+10

0 25

R15R25

70∘

25 10

070


28 R2512

16

70

32

R14

70∘

R90

55 155

R450



100

5

5

1 40 1 15

R12

R15





[119872] 11990410158401015840 + [119870] 119904 = 0 (3)


120601119894= 120579119894cos (120596 sdot 119905) (119894 = 1 2 119899) (4)





([119870] minus 1205962[119872]) 119904 = 0

Or [119860] 119904 = 0

(5)






119872111990410158401015840

1+ 119862 (119904

1015840

1minus 1199041015840

2) + 1198621015840(1199041015840

1minus 1199041015840

2)3

+ 119870 (1199041minus 1199042)

+ 1198701015840(1199041minus 1199042)3

= 1198751

119872211990410158401015840

2minus 119862 (119904

1015840

1minus 1199041015840

2) minus 1198621015840(1199041015840

1minus 1199041015840

2)3

minus 1198701015840(1199041minus 1199042)3

minus 1198701015840(1199041minus 1199042)3

= minus1198752

(6)











11987211990610158401015840= 119875 minus 119868 (7)









10158401015840|(119905)

is

1199061015840101584010038161003816100381610038161003816(119905)

= (119872)minus1

(119875 minus 119868)|(119905) (8)





|(119905+Δ1199052)

= |(119905minusΔ1199052)

+Δ119905|(119905+Δ119905)

+ Δ119905|(119905)

2|(119905) (9)




119906|(119905+Δ119905)

= 119906|(119905)

+ Δ119905|(119905+Δ119905)

|(119905+Δ1199052)

(10)










050

100150200250300350


Peak

acce

lera

tion

(ms

2)



050

100150200250300350400450500


Peak

acce

lera

tion

(ms

2)



0100200300400500600700800900


Peak

acce

lera

tion

(ms

2)



0200400600800

100012001400


Peak

acce

lera

tion

(ms

2)








119910 =119860

2[1 minus cos (2120587119891119905 + 120601)] (11)








119878 (119896) =119860

1198963 (12)




160 05 10 272160 15 10 72200 05 10 382200 15 10 97300 05 10 798300 15 10 342350 05 10 946350 15 10 407

0 0005 001 0015 002 0025 003 0035 004 0045minus60

minus40

minus20

0

20

40

60

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000123456789

X 2382Y 5662

Am

plitu

de

X 7793Y 4043

X 1403Y 881

f (Hz)





|119867 (119896)|2=4

119886

1

(119896119887)2int

arctan(119886)

0

[119869119896119887 sec (119909)]2 119889119909 (13)










0 0005 001 0015 002 0025 003minus150

minus100

minus50

0

50

100

150

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

5

10

15

20

25

30

X 2372Y 1611

Am

plitu

de X 1403Y 1578

X 1938Y 2768

f (Hz)



0 0005 001 0015 002 0025 003minus100

minus50

0

50

100

150

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

5

10

15

20

25

X 2306Y 1308

Am

plitu

de

X 1403Y 2312

f (Hz)



0

000

2

000

4

000

6

000

8

001

001

2

001

4

001

6

001

8

002

minus400

minus200

0

200

400

600

800

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

102030405060708090

X 1914Y 8087

Am

plitu

de

X 2324Y 3404

X 3600Y 2469

f (Hz)








0 0005 001 0015 002 0025 003minus400

minus300

minus200

minus100

0

100

200

300

400

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

102030405060708090

X 1403Y 3225A

mpl

itude

X 2372Y 8395

X 4344Y 159

f (Hz)



0

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4

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000

8

001

001

2

001

4

001

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001

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002


0200400600800

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t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

102030405060708090

100X 1914Y 9623

Am

plitu

de

X 2324Y 9094

X 3600Y 4002

X 4283Y 6066

f (Hz)



0 0005 001 0015 002 0025 003minus100


020406080

100

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

5

10

15

20

25X 2372Y 2491

Am

plitu

de X 1403Y 1401

f (Hz)








0

000

2

000

4

000

6

000

8

001

001

2

001

4

001

6

001

8

002

minus400

minus300

minus200

minus100

0

100

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400

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

10

20

30

40

50

60

X 1914Y 5186

Am

plitu

de

X 2324Y 5187 X 3554

Y 4166

X 4283Y 2967

f (Hz)



0

100

200

300

400

500

600

700

02 04 06 08 1 13 17 2Wavelength (m)

Peak

acce

lera

tion

(ms

2)


03mm06mm09mm


Wavelength (m)

0100200300400500600700800900

1000

02 04 06 08 1 13 17 2

Peak

acce

lera

tion

(ms

2)


03mm06mm09mm



























6 Conclusions







Additional Points


Disclosure


Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











([119870] minus 1205962[119872]) 119904 = 0

Or [119860] 119904 = 0

(5)






119872111990410158401015840

1+ 119862 (119904

1015840

1minus 1199041015840

2) + 1198621015840(1199041015840

1minus 1199041015840

2)3

+ 119870 (1199041minus 1199042)

+ 1198701015840(1199041minus 1199042)3

= 1198751

119872211990410158401015840

2minus 119862 (119904

1015840

1minus 1199041015840

2) minus 1198621015840(1199041015840

1minus 1199041015840

2)3

minus 1198701015840(1199041minus 1199042)3

minus 1198701015840(1199041minus 1199042)3

= minus1198752

(6)











11987211990610158401015840= 119875 minus 119868 (7)









10158401015840|(119905)

is

1199061015840101584010038161003816100381610038161003816(119905)

= (119872)minus1

(119875 minus 119868)|(119905) (8)





|(119905+Δ1199052)

= |(119905minusΔ1199052)

+Δ119905|(119905+Δ119905)

+ Δ119905|(119905)

2|(119905) (9)




119906|(119905+Δ119905)

= 119906|(119905)

+ Δ119905|(119905+Δ119905)

|(119905+Δ1199052)

(10)










050

100150200250300350


Peak

acce

lera

tion

(ms

2)



050

100150200250300350400450500


Peak

acce

lera

tion

(ms

2)



0100200300400500600700800900


Peak

acce

lera

tion

(ms

2)



0200400600800

100012001400


Peak

acce

lera

tion

(ms

2)








119910 =119860

2[1 minus cos (2120587119891119905 + 120601)] (11)








119878 (119896) =119860

1198963 (12)




160 05 10 272160 15 10 72200 05 10 382200 15 10 97300 05 10 798300 15 10 342350 05 10 946350 15 10 407

0 0005 001 0015 002 0025 003 0035 004 0045minus60

minus40

minus20

0

20

40

60

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000123456789

X 2382Y 5662

Am

plitu

de

X 7793Y 4043

X 1403Y 881

f (Hz)





|119867 (119896)|2=4

119886

1

(119896119887)2int

arctan(119886)

0

[119869119896119887 sec (119909)]2 119889119909 (13)










0 0005 001 0015 002 0025 003minus150

minus100

minus50

0

50

100

150

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

5

10

15

20

25

30

X 2372Y 1611

Am

plitu

de X 1403Y 1578

X 1938Y 2768

f (Hz)



0 0005 001 0015 002 0025 003minus100

minus50

0

50

100

150

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

5

10

15

20

25

X 2306Y 1308

Am

plitu

de

X 1403Y 2312

f (Hz)



0

000

2

000

4

000

6

000

8

001

001

2

001

4

001

6

001

8

002

minus400

minus200

0

200

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600

800

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

102030405060708090

X 1914Y 8087

Am

plitu

de

X 2324Y 3404

X 3600Y 2469

f (Hz)








0 0005 001 0015 002 0025 003minus400

minus300

minus200

minus100

0

100

200

300

400

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

102030405060708090

X 1403Y 3225A

mpl

itude

X 2372Y 8395

X 4344Y 159

f (Hz)



0

000

2

000

4

000

6

000

8

001

001

2

001

4

001

6

001

8

002


0200400600800

1000

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

102030405060708090

100X 1914Y 9623

Am

plitu

de

X 2324Y 9094

X 3600Y 4002

X 4283Y 6066

f (Hz)



0 0005 001 0015 002 0025 003minus100


020406080

100

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

5

10

15

20

25X 2372Y 2491

Am

plitu

de X 1403Y 1401

f (Hz)








0

000

2

000

4

000

6

000

8

001

001

2

001

4

001

6

001

8

002

minus400

minus300

minus200

minus100

0

100

200

300

400

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

10

20

30

40

50

60

X 1914Y 5186

Am

plitu

de

X 2324Y 5187 X 3554

Y 4166

X 4283Y 2967

f (Hz)



0

100

200

300

400

500

600

700

02 04 06 08 1 13 17 2Wavelength (m)

Peak

acce

lera

tion

(ms

2)


03mm06mm09mm


Wavelength (m)

0100200300400500600700800900

1000

02 04 06 08 1 13 17 2

Peak

acce

lera

tion

(ms

2)


03mm06mm09mm



























6 Conclusions







Additional Points


Disclosure


Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















10158401015840|(119905)

is

1199061015840101584010038161003816100381610038161003816(119905)

= (119872)minus1

(119875 minus 119868)|(119905) (8)





|(119905+Δ1199052)

= |(119905minusΔ1199052)

+Δ119905|(119905+Δ119905)

+ Δ119905|(119905)

2|(119905) (9)




119906|(119905+Δ119905)

= 119906|(119905)

+ Δ119905|(119905+Δ119905)

|(119905+Δ1199052)

(10)










050

100150200250300350


Peak

acce

lera

tion

(ms

2)



050

100150200250300350400450500


Peak

acce

lera

tion

(ms

2)



0100200300400500600700800900


Peak

acce

lera

tion

(ms

2)



0200400600800

100012001400


Peak

acce

lera

tion

(ms

2)








119910 =119860

2[1 minus cos (2120587119891119905 + 120601)] (11)








119878 (119896) =119860

1198963 (12)




160 05 10 272160 15 10 72200 05 10 382200 15 10 97300 05 10 798300 15 10 342350 05 10 946350 15 10 407

0 0005 001 0015 002 0025 003 0035 004 0045minus60

minus40

minus20

0

20

40

60

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000123456789

X 2382Y 5662

Am

plitu

de

X 7793Y 4043

X 1403Y 881

f (Hz)





|119867 (119896)|2=4

119886

1

(119896119887)2int

arctan(119886)

0

[119869119896119887 sec (119909)]2 119889119909 (13)










0 0005 001 0015 002 0025 003minus150

minus100

minus50

0

50

100

150

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

5

10

15

20

25

30

X 2372Y 1611

Am

plitu

de X 1403Y 1578

X 1938Y 2768

f (Hz)



0 0005 001 0015 002 0025 003minus100

minus50

0

50

100

150

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

5

10

15

20

25

X 2306Y 1308

Am

plitu

de

X 1403Y 2312

f (Hz)



0

000

2

000

4

000

6

000

8

001

001

2

001

4

001

6

001

8

002

minus400

minus200

0

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600

800

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

102030405060708090

X 1914Y 8087

Am

plitu

de

X 2324Y 3404

X 3600Y 2469

f (Hz)








0 0005 001 0015 002 0025 003minus400

minus300

minus200

minus100

0

100

200

300

400

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

102030405060708090

X 1403Y 3225A

mpl

itude

X 2372Y 8395

X 4344Y 159

f (Hz)



0

000

2

000

4

000

6

000

8

001

001

2

001

4

001

6

001

8

002


0200400600800

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t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

102030405060708090

100X 1914Y 9623

Am

plitu

de

X 2324Y 9094

X 3600Y 4002

X 4283Y 6066

f (Hz)



0 0005 001 0015 002 0025 003minus100


020406080

100

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

5

10

15

20

25X 2372Y 2491

Am

plitu

de X 1403Y 1401

f (Hz)








0

000

2

000

4

000

6

000

8

001

001

2

001

4

001

6

001

8

002

minus400

minus300

minus200

minus100

0

100

200

300

400

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

10

20

30

40

50

60

X 1914Y 5186

Am

plitu

de

X 2324Y 5187 X 3554

Y 4166

X 4283Y 2967

f (Hz)



0

100

200

300

400

500

600

700

02 04 06 08 1 13 17 2Wavelength (m)

Peak

acce

lera

tion

(ms

2)


03mm06mm09mm


Wavelength (m)

0100200300400500600700800900

1000

02 04 06 08 1 13 17 2

Peak

acce

lera

tion

(ms

2)


03mm06mm09mm



























6 Conclusions







Additional Points


Disclosure


Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










050

100150200250300350


Peak

acce

lera

tion

(ms

2)



050

100150200250300350400450500


Peak

acce

lera

tion

(ms

2)



0100200300400500600700800900


Peak

acce

lera

tion

(ms

2)



0200400600800

100012001400


Peak

acce

lera

tion

(ms

2)








119910 =119860

2[1 minus cos (2120587119891119905 + 120601)] (11)








119878 (119896) =119860

1198963 (12)




160 05 10 272160 15 10 72200 05 10 382200 15 10 97300 05 10 798300 15 10 342350 05 10 946350 15 10 407

0 0005 001 0015 002 0025 003 0035 004 0045minus60

minus40

minus20

0

20

40

60

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000123456789

X 2382Y 5662

Am

plitu

de

X 7793Y 4043

X 1403Y 881

f (Hz)





|119867 (119896)|2=4

119886

1

(119896119887)2int

arctan(119886)

0

[119869119896119887 sec (119909)]2 119889119909 (13)










0 0005 001 0015 002 0025 003minus150

minus100

minus50

0

50

100

150

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

5

10

15

20

25

30

X 2372Y 1611

Am

plitu

de X 1403Y 1578

X 1938Y 2768

f (Hz)



0 0005 001 0015 002 0025 003minus100

minus50

0

50

100

150

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

5

10

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20

25

X 2306Y 1308

Am

plitu

de

X 1403Y 2312

f (Hz)



0

000

2

000

4

000

6

000

8

001

001

2

001

4

001

6

001

8

002

minus400

minus200

0

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600

800

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

102030405060708090

X 1914Y 8087

Am

plitu

de

X 2324Y 3404

X 3600Y 2469

f (Hz)








0 0005 001 0015 002 0025 003minus400

minus300

minus200

minus100

0

100

200

300

400

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

102030405060708090

X 1403Y 3225A

mpl

itude

X 2372Y 8395

X 4344Y 159

f (Hz)



0

000

2

000

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000

6

000

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001

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2

001

4

001

6

001

8

002


0200400600800

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t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

102030405060708090

100X 1914Y 9623

Am

plitu

de

X 2324Y 9094

X 3600Y 4002

X 4283Y 6066

f (Hz)



0 0005 001 0015 002 0025 003minus100


020406080

100

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

5

10

15

20

25X 2372Y 2491

Am

plitu

de X 1403Y 1401

f (Hz)








0

000

2

000

4

000

6

000

8

001

001

2

001

4

001

6

001

8

002

minus400

minus300

minus200

minus100

0

100

200

300

400

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

10

20

30

40

50

60

X 1914Y 5186

Am

plitu

de

X 2324Y 5187 X 3554

Y 4166

X 4283Y 2967

f (Hz)



0

100

200

300

400

500

600

700

02 04 06 08 1 13 17 2Wavelength (m)

Peak

acce

lera

tion

(ms

2)


03mm06mm09mm


Wavelength (m)

0100200300400500600700800900

1000

02 04 06 08 1 13 17 2

Peak

acce

lera

tion

(ms

2)


03mm06mm09mm



























6 Conclusions







Additional Points


Disclosure


Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












160 05 10 272160 15 10 72200 05 10 382200 15 10 97300 05 10 798300 15 10 342350 05 10 946350 15 10 407

0 0005 001 0015 002 0025 003 0035 004 0045minus60

minus40

minus20

0

20

40

60

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000123456789

X 2382Y 5662

Am

plitu

de

X 7793Y 4043

X 1403Y 881

f (Hz)





|119867 (119896)|2=4

119886

1

(119896119887)2int

arctan(119886)

0

[119869119896119887 sec (119909)]2 119889119909 (13)










0 0005 001 0015 002 0025 003minus150

minus100

minus50

0

50

100

150

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

5

10

15

20

25

30

X 2372Y 1611

Am

plitu

de X 1403Y 1578

X 1938Y 2768

f (Hz)



0 0005 001 0015 002 0025 003minus100

minus50

0

50

100

150

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

5

10

15

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25

X 2306Y 1308

Am

plitu

de

X 1403Y 2312

f (Hz)



0

000

2

000

4

000

6

000

8

001

001

2

001

4

001

6

001

8

002

minus400

minus200

0

200

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800

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

102030405060708090

X 1914Y 8087

Am

plitu

de

X 2324Y 3404

X 3600Y 2469

f (Hz)








0 0005 001 0015 002 0025 003minus400

minus300

minus200

minus100

0

100

200

300

400

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

102030405060708090

X 1403Y 3225A

mpl

itude

X 2372Y 8395

X 4344Y 159

f (Hz)



0

000

2

000

4

000

6

000

8

001

001

2

001

4

001

6

001

8

002


0200400600800

1000

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

102030405060708090

100X 1914Y 9623

Am

plitu

de

X 2324Y 9094

X 3600Y 4002

X 4283Y 6066

f (Hz)



0 0005 001 0015 002 0025 003minus100


020406080

100

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

5

10

15

20

25X 2372Y 2491

Am

plitu

de X 1403Y 1401

f (Hz)








0

000

2

000

4

000

6

000

8

001

001

2

001

4

001

6

001

8

002

minus400

minus300

minus200

minus100

0

100

200

300

400

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

10

20

30

40

50

60

X 1914Y 5186

Am

plitu

de

X 2324Y 5187 X 3554

Y 4166

X 4283Y 2967

f (Hz)



0

100

200

300

400

500

600

700

02 04 06 08 1 13 17 2Wavelength (m)

Peak

acce

lera

tion

(ms

2)


03mm06mm09mm


Wavelength (m)

0100200300400500600700800900

1000

02 04 06 08 1 13 17 2

Peak

acce

lera

tion

(ms

2)


03mm06mm09mm



























6 Conclusions







Additional Points


Disclosure


Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 0005 001 0015 002 0025 003minus150

minus100

minus50

0

50

100

150

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

5

10

15

20

25

30

X 2372Y 1611

Am

plitu

de X 1403Y 1578

X 1938Y 2768

f (Hz)



0 0005 001 0015 002 0025 003minus100

minus50

0

50

100

150

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

5

10

15

20

25

X 2306Y 1308

Am

plitu

de

X 1403Y 2312

f (Hz)



0

000

2

000

4

000

6

000

8

001

001

2

001

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001

6

001

8

002

minus400

minus200

0

200

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t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

102030405060708090

X 1914Y 8087

Am

plitu

de

X 2324Y 3404

X 3600Y 2469

f (Hz)








0 0005 001 0015 002 0025 003minus400

minus300

minus200

minus100

0

100

200

300

400

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

102030405060708090

X 1403Y 3225A

mpl

itude

X 2372Y 8395

X 4344Y 159

f (Hz)



0

000

2

000

4

000

6

000

8

001

001

2

001

4

001

6

001

8

002


0200400600800

1000

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

102030405060708090

100X 1914Y 9623

Am

plitu

de

X 2324Y 9094

X 3600Y 4002

X 4283Y 6066

f (Hz)



0 0005 001 0015 002 0025 003minus100


020406080

100

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

5

10

15

20

25X 2372Y 2491

Am

plitu

de X 1403Y 1401

f (Hz)








0

000

2

000

4

000

6

000

8

001

001

2

001

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001

8

002

minus400

minus300

minus200

minus100

0

100

200

300

400

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

10

20

30

40

50

60

X 1914Y 5186

Am

plitu

de

X 2324Y 5187 X 3554

Y 4166

X 4283Y 2967

f (Hz)



0

100

200

300

400

500

600

700

02 04 06 08 1 13 17 2Wavelength (m)

Peak

acce

lera

tion

(ms

2)


03mm06mm09mm


Wavelength (m)

0100200300400500600700800900

1000

02 04 06 08 1 13 17 2

Peak

acce

lera

tion

(ms

2)


03mm06mm09mm



























6 Conclusions







Additional Points


Disclosure


Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 0005 001 0015 002 0025 003minus400

minus300

minus200

minus100

0

100

200

300

400

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

102030405060708090

X 1403Y 3225A

mpl

itude

X 2372Y 8395

X 4344Y 159

f (Hz)



0

000

2

000

4

000

6

000

8

001

001

2

001

4

001

6

001

8

002


0200400600800

1000

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

102030405060708090

100X 1914Y 9623

Am

plitu

de

X 2324Y 9094

X 3600Y 4002

X 4283Y 6066

f (Hz)



0 0005 001 0015 002 0025 003minus100


020406080

100

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

5

10

15

20

25X 2372Y 2491

Am

plitu

de X 1403Y 1401

f (Hz)








0

000

2

000

4

000

6

000

8

001

001

2

001

4

001

6

001

8

002

minus400

minus300

minus200

minus100

0

100

200

300

400

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

10

20

30

40

50

60

X 1914Y 5186

Am

plitu

de

X 2324Y 5187 X 3554

Y 4166

X 4283Y 2967

f (Hz)



0

100

200

300

400

500

600

700

02 04 06 08 1 13 17 2Wavelength (m)

Peak

acce

lera

tion

(ms

2)


03mm06mm09mm


Wavelength (m)

0100200300400500600700800900

1000

02 04 06 08 1 13 17 2

Peak

acce

lera

tion

(ms

2)


03mm06mm09mm



























6 Conclusions







Additional Points


Disclosure


Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

000

2

000

4

000

6

000

8

001

001

2

001

4

001

6

001

8

002

minus400

minus300

minus200

minus100

0

100

200

300

400

t (s)

a(m

s2)


0 1000 2000 3000 4000 50000

10

20

30

40

50

60

X 1914Y 5186

Am

plitu

de

X 2324Y 5187 X 3554

Y 4166

X 4283Y 2967

f (Hz)



0

100

200

300

400

500

600

700

02 04 06 08 1 13 17 2Wavelength (m)

Peak

acce

lera

tion

(ms

2)


03mm06mm09mm


Wavelength (m)

0100200300400500600700800900

1000

02 04 06 08 1 13 17 2

Peak

acce

lera

tion

(ms

2)


03mm06mm09mm



























6 Conclusions







Additional Points


Disclosure


Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra























6 Conclusions







Additional Points


Disclosure


Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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