research article modeling and optimization of vehicle

Research ArticleModeling and Optimization of Vehicle SuspensionEmploying a Nonlinear Fluid Inerter

Yujie Shen1 Long Chen2 Yanling Liu2 and Xiaoliang Zhang2

1School of Automotive and Traffic Engineering Jiangsu University Zhenjiang 212013 China2Automotive Engineering Research Institute Jiangsu University Zhenjiang 212013 China

Correspondence should be addressed to Long Chen chenlongujseducn

Received 10 March 2016 Revised 30 May 2016 Accepted 27 June 2016

Academic Editor Mickael Lallart

Copyright copy 2016 Yujie Shen et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

An ideal inerter has been applied to various vibration engineering fields because of its superior vibration isolation performanceThis paper proposes a new type of fluid inerter and analyzes the nonlinearities including friction and nonlinear damping forcecaused by the viscosity of fluid The nonlinear model of fluid inerter is demonstrated by the experiments analysis Furthermorethe full-car dynamic model involving the nonlinear fluid inerter is established It has been detected that the performance of thevehicle suspension may be influenced by the nonlinearities of inerter So parameters of the suspension system including the springstiffness and the damping coefficient are optimized by means of QGA (quantum genetic algorithm) which combines the geneticalgorithm and quantum computing Results indicate that compared with the original nonlinear suspension system the RMS (root-mean-square) of vertical body acceleration of optimized suspension has decreased by 90 the RMS of pitch angular accelerationhas decreased by 199 and the RMS of roll angular acceleration has decreased by 96

1 Introduction

Ride comfort is one of the most important performances ofthe vehicle and is greatly influenced by the vehicle suspensionsystem set between the vehicle body and thewheelsThere aremainly three kinds of suspension systems including activesuspension [1ndash3] semiactive suspension [4ndash6] and passivesuspension For active suspension and semiactive suspensionthey always have a changeable suspension parameter likespring stiffness or damping coefficient to preserve the twodesired aimswhich are vehicle handling and ride comfort Butfor passive suspension system it only has two parallel springand damper elements In this case the vibration isolationperformance can be improved by parametersrsquo optimization[7 8] or involving new mechanical element

In [9 10] inerter was proposed with the property that theforce in the mechanical device is proportional to the relativeaccelerations of the two terminals It was widely known asldquoJ-damperrdquo [11] in Formula One Racing since it deliveredsignificant performance gains in handling and grip As a newmechanical network element corresponding to the capacitor

inerter was widely used in passive network synthesis [12ndash14] Also the vibration isolation performance of conven-tional spring-damper isolation system can be significantlyimproved by involving the inerter which has been verified invehicle suspension [15ndash18] train suspension [19 20] buildingsuspension [21] dynamic vibration absorber [22ndash24] andvibration isolation [25 26]

Researches above are all concentrated on the ideal inerterignoring considering its nonlinearities In [27] the nonlinear-ities including friction elastic effect and backlash of a rack-and-pinion and ball-screw inerter were taken into consider-ationThe above study found that the performance of vehiclesuspension may be significantly degraded by the nonlineareffects It seems important to investigate the influences ofthe nonlinearities of inerter on vehicle suspension systemIn [28 29] a new kind of inerter called fluid inerter wasproposed and the inner force was investigated but the effectof the nonlinear model on the suspension performance wasnot discussed This paper explores a new vehicle suspensionemploying the nonlinear fluid inerter The parameters ofthe suspension are optimized by using optimal algorithm to

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 2623017 9 pageshttpdxdoiorg10115520162623017

2 Shock and Vibration

xF F

Piston rod Piston Hydraulic cylinder

Helical channel

Figure 1 Prototype of fluid inerter

reduce the effect of the nonlinearities based on the full-carmodel including the nonlinear inerter The paper is arrangedas follows

In Section 2 a new fluid inerter is designed and the non-linearities involving friction and damping force are taken intoconsideration Experiments are finished to demonstrate thenonlinear model Suspension system involving the nonlinearfluid inerter is built in Section 3 and the parameters of thesuspension are obtained by means of QGA (quantum geneticalgorithm) in Section 4 Dynamic performance of the vehiclesuspension is analyzed in Section 5 Finally some conclusionsare shown in Section 6

2 Design and Test of Fluid Inerter

There are many mechanical types of inerter such as ball-screw inerter rack-and-pinion inerter [30] hydraulic-motorinerter [31] and fluid inerter [28 29] Motion transferredmechanics are used in the device to enlarge the inertia forcewhich is the most meaningful thing to inerter The dynamicequation of inerter is

119865 = 119887 (V1minus V2) (1)

where 119865 is the force in the device V1and V2are the velocities

and 119887 is the inertance with the unit kg In this sectiona new type of fluid inerter is designed to investigate thenonlinearities

21 Ideal Fluid Inerter The fluid inerter is similar to rack-and-pinion inerter or ball-screw inerter because the fluidflowing in the helical channel can be taken as a ldquofluidflywheelrdquo to provide the inertia force The prototype of thefluid inerter with the helical channel outside the cylinder isshown in Figure 1

In Figure 1 119865 is the force applied in the fluid inerter and119909 is the displacement of the piston The hydraulic cylinderand the piston rod are the two terminals of the fluid inerterWhen the piston pushes the fluid in the left of the hydrauliccylinder into the helical channel the fluid flows through thehelical channel into the right hydraulic cylinder in order tocompensate the pressure loss

Firstly the fluid in the cylinder and the helical channelis taken as nondisclosure and noncompressed According to[29] the inertance 119887 can be gained as

119887 =119898

1 + (ℎ21205871199034)2(1198781

1198782

)

2

(2)

where119898 is themass of the fluid in the helical channel 1198781is the

effective area of the hydraulic cylinder 1198782is the section area

of the helical channel ℎ is the pitch of the helical channel and1199034is the helix radius of the helical channel1198781can be gained by the equation

1198781= 120587 (119903

2

2minus 1199032

1) (3)

where 1199031is the radius of the piston rod and 119903

2is the inside

radius of the hydraulic cylinder1198782can be gained by the equation

1198782= 1205871199032

3 (4)

where 1199033is the radius of the helical channel

119898 can be gained by the equation

119898 = 1205881198782119897 (5)

where 120588 is the oil density and 119897 is the length of helical channeland can be approximated as

119897 = 119899radicℎ2 + (21205871199034)2

+ 21198970 (6)

where 119899 is the circle number of the helical channel and 1198970is

the length from the inlet and outlet to the helical channel

22 Nonlinearities Analysis of Fluid Inerter When the force119865 is applied to the fluid inerter both the friction caused bythe motion of piston rod and hydraulic cylinder 119891 and thedamping force caused by the viscosity of the fluid119891

119888will block

the motion of the piston The dynamic equation is

119865 minus 119891 minus 119891119888= 119887 (7)

The friction model can be described by the ColumnFriction Model as follows

119891 = 1198910sign (V) (8)

where1198910is the amplitude of the friction V is the velocity of the

piston and sign(V) is a symbolic function it is 1 when V gt 0

and minus1 when V lt 0 The direction of friction 119891 between thepiston and hydraulic cylinder is related to the velocity of thepiston and has the constant value

For a hydraulic device the damping force caused by theviscosity of the fluid is the most important impact on thedynamic performance According to the Hagen PoiseuilleLaw the flow of the fluid in the helical channel 119876 and thepressure drop between the two terminalsΔ119901 have the relationas follows

119876 =1205871199034

3

8120583119897Δ119901 (9)

in which 120583 is viscosity

Shock and Vibration 3

z1

z2

f

b fc

Figure 2 Dynamic model of fluid inerter

The average flow velocity of fluid in the helical channel 119906is

119906 =119876

1198782

=1199032

3

8120583119897Δ119901 (10)

A fluid flow entering a new channel will also experiencean energy loss at the inlet and outlet of the hydraulic cylinderthe empirical formula for the resulting pressure drop is givenin [32] The pressure drop across the inlet and outlet is thusestimated to be

Δ1199011= 05

1205881199062

2

Δ1199012=

1205881199062

2

(11)

According to the volume conservation law

1198781 = 1198782119906 (12)

So the damping force of the inerter can be gained asfollows

119891119888= (Δ119901 + Δ119901

1+ Δ1199012) 1198781=

31205881198783

1

41198782

2

2+

81205831198971198782

1

1199032

31198782

(13)

The damping force is related not only to the velocity butalso to the square of the velocity of the pistonThe coefficientof the square of the velocity is set as 119862

1and the coefficient of

the velocity is set as 1198622

The dynamic model of the fluid inerter can be shown inFigure 2

In Figure 2 1199111and 1199112are displacements

The total force of the inerter can be gained as follows

119865 = 1198910sign (

1minus 2) + 1198621(1minus 2)2

+ 1198622(1minus 2)

+ 119887 (1minus 2)

(14)

Helicalchannel

Vibration head

Fluid inerter

Figure 3 Bench test

10 20 30 400Time (s)

minus06

minus04

minus02

0

02

04

06

Forc

e (kN

)

Figure 4 Force in 01 Hz

23 Experiment Identification In this part experiments arecarried out in the INSTRON 8800 bench to test the perfor-mance of the fluid inerter The parameters of the designedfluid inerter are shown in Table 1

So the parameters of the nonlinear fluid inerter can becalculated where 119887 is 370 kg 119862

1is 790Nsdots2sdotmminus2 and 119862

2is

4001Nsdotssdotmminus1The structure of the bench test is shown in Figure 3The displacement and the force signals can be collected

by the sensors in the device In order to gain the amplitude offriction the displacement input is set as a triangle wave andthe amplitude is 10mm Figure 4 shows the force under 01 Hzcondition

It can be noted that the force shapes like a square wavebecause both the damping force and the inertia force aresmall enough It can be inferred that the friction force is aconstant value which is 400N Furthermore the fluid inerteris tested under different frequency to verify the theoreticalmodel The amplitude of the sinusoidal displacement inputis set as 20mm under the frequency of 01 Hz 03Hz and05Hz 10mm under the frequency of 3Hz 5Hz and 8Hzand 5mm under the frequency of 10Hz 12Hz and 15Hz


0

2

4

6

8

10

12

14

16

Diff

eren

ce (

)

5 10 150Frequency (Hz)

Figure 5 Difference between theoretical model and experiment

Table 1 Parameters of the fluid inerter

Name ValueRadius of piston rod 119903

1m 0012

Inner radius of the cylinder 1199032m 0028

Radius of the helical channel 1199033m 0005

Helix radius of channel 1199034m 01

Pitch of helical channel ℎm 0012Circle number of helical channel 119899 14Oil density 120588kgsdotmminus3 800Length of transition section 119897

0m 01

Viscosity of fluid 120583Pasdots 0027

In order to test the effectiveness of the nonlinear modelthe mean value of force amplitude under the sinuous dis-placement input of simulation and experiment is taken intoconsideration Figure 5 shows the absolute difference of theforce amplitude between theoretical analysis and practicalexperiment It can be seen that the second data point isa little larger than the first data point and the third datapoint but the absolute differences are all less than 3 so itcan be inferred that the simulation results are very close tothe experiment results in low frequency Furthermore theabsolute difference becomes a little larger with the increaseof the frequency Figure 6 shows the force of the fluid inerterbetween the theoretical analysis and experiment in 05Hz and12Hz It can be inferred that the experiment results in higherfrequency coincide less well with the simulation than that inlow frequency which may be because the temperature of thefluid has increased for the vibration of the system and thefluid flow is not laminar In conclusion the experiment resultscoincide with the simulation well which demonstrates theeffectiveness of the theoretical model of fluid inerter

3 Suspension Involving Nonlinear Inerter

31 Nonlinear Model of Suspension Structure In [33] a newvehicle suspension employing two springs one damper andone inerter was proposed based on the dynamic vibration

absorber theory But the models are all linear which are notexactly in the real condition In this paper vehicle suspensionemploying a nonlinear fluid inerter is investigated Thesuspension structure involving the nonlinear fluid inerter isshown in Figure 7

In Figure 7 1198961is the stiffness of main spring 119896

2is the

stiffness of deputy spring and 119888 is the coefficient of thedamper In this paper the parameters are optimized bymeansof the method in [33] where 119896

1is 238 kNm 119896

2is 89 kNm

and 119888 is 887Nsdotssdotmminus1

32 Full-Car Model A dynamic full-car model is establishedin Figure 8

The vehicle body vertical motion equation is

119898119886119886= 11986510

+ 11986520

+ 11986530

+ 11986540 (15)

The vehicle body pitching motion equation is

119868119910 = 119897119903(11986530

+ 11986540) minus 119897119891(11986510

+ 11986520) (16)

The vehicle body roll motion equation is

119868119909 = (119865

20+ 11986540

minus 11986510

minus 11986530)119889

2 (17)

where119898119886is vehicle bodymass 119911

119886is the vertical displacement

of the vehicle body 119897119891is the length from the front wheel to the

center of vehicle body 119897119903is the length from the rear wheel to

the center of vehicle body 119889 is the tread between left wheelsand right wheels 119868

119909is the body roll moment of inertia 119868

119910is

the body pitch moment of inertia 120579 is the roll angle of thevehicle body 120593 is the pitch angle of the vehicle body and 119865

1198940

is the suspension forceThe motions of four corners of vehicle body are

11988510

= 119911119886minus 119897119891120593 minus

119889

2120579

11988520

= 119911119886minus 119897119891120593 +

119889

2120579

11988530

= 119911119886+ 119897119903120593 minus

119889

2120579

11988540

= 119911119886+ 119897119903120593 +

119889

2120579

(18)

where 1198851198940

is the displacement of the four corners of thevehicle body mass

For the unsprung mass the equations are shown asfollows

11989811= 119896119905(1198761minus 1198851) minus 11986510

11989822= 119896119905(1198762minus 1198852) minus 11986520

11989833= 119896119905(1198763minus 1198853) minus 11986530

11989844= 119896119905(1198764minus 1198854) minus 11986540

(19)

where 119898119894is unsprung mass 119896

119905is the tire equivalent stiffness

119876119894is the displacements of road and 119885

119894is vertical displace-

ment of the four unsprung mass


TheoreticalExperiment

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Forc

e (kN

)

1 2 3 4 5 6 7 8 9 100Time (s)

(a) 05Hz


minus15

minus10

minus5

0

5

10

15

Forc

e (kN

)

005 01 015 02 025 03 035 040Time (s)

(b) 12Hz

Figure 6 Force in time domain

k1

f

bfc

k2c

Figure 7 Nonlinear model of suspension structure

The equations of the suspension force are shown in thefollowing

11986510

= 1198961(1198851minus 11988510) + 1198962(1198851minus 1198851198871) + 119888 (

1minus 1198871)

1198962(1198851minus 1198851198871) + 119888 (

1minus 1198871)

= 1198910sign (

1198871minus 10) + 119887 (

1198871minus 10)

+ 1198621(1198871

minus 10)2

+ 1198622(1198871

minus 10)

11986520

= 1198961(1198852minus 11988520) + 1198962(1198852minus 1198851198872) + 119888 (

2minus 1198872)

1198962(1198852minus 1198851198872) + 119888 (

2minus 1198872)

= 1198910sign (

1198872minus 20) + 119887 (

1198872minus 20)

+ 1198621(1198872

minus 20)2

+ 1198622(1198872

minus 20)

11986530

= 1198961(1198853minus 11988530) + 1198962(1198853minus 1198851198873) + 119888 (

1minus 1198873)

1198962(1198853minus 1198851198873) + 119888 (

3minus 1198873)

= 1198910sign (

1198873minus 30) + 119887 (

1198873minus 30)

+ 1198621(1198873

minus 30)2

+ 1198622(1198873

minus 30)

11986540

= 1198961(1198854minus 11988540) + 1198962(1198854minus 1198851198874) + 119888 (

4minus 1198874)

1198962(1198854minus 1198851198874) + 119888 (

4minus 1198874)

= 1198910sign (

1198874minus 40) + 119887 (

1198874minus 40)

+ 1198621(1198874

minus 40)2

+ 1198622(1198874

minus 40)

(20)

where 119885119887119894is the displacement of inerter

4 Optimization of the Suspension Parameters

According to [27] the nonlinearities of inerter may havea bad effect on the vehicle suspension system so it seemsimportant to optimize the parameters of the suspension ele-ments Genetic optimal algorithm originates from the naturalselection of biological evolution It begins from a potentialpopulation and then selects the individual according to thefitness By means of natural genetics operators combiningcrossover and mutation new solution stocks will be createdThe QGA (quantum genetic algorithm) is the combination


Suspension 1

Suspension 2

Suspension 3

Suspension 4

X

Y

Z

d

z10 F10

z20F20

120579za

ma

120593z30

F30

z40F40

m1

m2

m3

m4

z1

z2

z3

z4

kt

kt

kt

kt

Q1

Q2

Q3

Q4

lf lr

Figure 8 Full-car model

of genetic algorithm and quantum computing The quantumstate vector is applied to the genetic encoding and theevolution of chromosome is realized by quantum logic gate

The updated rules of the quantum bits are

[

1205721015840

119894

1205731015840

119894

] = 119880 (120579119894) [

120572119894

120573119894

] (21)

where [1205721015840

1198941205731015840

119894]119879 is the updated quantum bit [120572119894 120573

119894]119879 is the

original quantum bit and 119880(120579119894) is the updated rotate matrix

shown in the following

119880(120579119894) = [

cos (120579119894) minus sin (120579

119894)

sin (120579119894) cos (120579

119894)] (22)

where 120579119894is the rotate angle and kept the default values in this

paperIn order to gain the satisfied parameters of suspension

the QGA is used to optimize the parameters of suspensionThe population is set as 100 the max number of the popula-tion is set as 200 119896

1 1198962 and 119888 are chosen as the optimized

variables and their ranges are limited as

10000N sdotmminus1 le 1198961le 30000N sdotmminus1


100N sdot s sdotmminus1 le 119888 le 3000N sdot s sdotmminus1

(23)

The fitness is set as

min 119869 = 1199081

119860

1198600

+ 1199082

119861

1198610

+ 1199083

119862

1198620

(24)

where 119869 is the fitness function of the genetic algorithm1199081 1199082 and 119908

3are the weighting coefficients and set as

033 119860 is the RMS (root-mean-square) of the vertical bodyacceleration 119861 is the RMS of the pitch angular acceleration119862 is the RMS of the roll angular acceleration of the optimizedsuspension119860

0is the RMS of vertical body acceleration 119861

0is

the RMS of pitch angular acceleration and 1198620is the RMS of

roll angular acceleration of the original suspensionThere are limitations of the suspension deflection and the

dynamic tire load For suspension deflection there is 3120590119891119889

le

[119891119889] where 120590

119891119889is the standard deviation of suspension

deflection and [119891119889] is the limitation of the tire beat distance

and set as 8 cm For dynamic tire load there is 3120590119865119889

le 119866where 120590

119865119889is the RMS of the dynamic tire load and 119866 is the

vehicle static load and set as 2500NThe model parameters are listed in Table 2The optimal parameters of suspension are listed in

Table 3

5 Performance Analysis

The random road model can be shown in equation

119894(119905) = minus2120587119891cut119876119894 (119905) + 2120587radic119866

0119881119908 (119905) (25)

where 119876119894(119905) is the displacement 119866

0is the roughness coef-

ficient 119891cut is cut-off frequency and 119881 is the speed In thisstudy 119866

0is 5 times 10minus6m3 cycleminus1 and 119891cut is 001Hz

In order to show the influences of the nonlinearities ofthe fluid inerter on the vehicle suspension system when thespeed is set as 20ms Table 4 shows the performance indexesamong the linear suspension the nonlinear suspensionand the optimized suspension The linear suspension is thesuspension with a linear inerter the nonlinear suspensionis the suspension with a nonlinear fluid inerter and theoptimized suspension is the suspension with a nonlinearinerter whose parameters have been optimized


Table 2 Model parameters

Parameter name ValueVehicle body mass119898

119886kg 770

Unsprung mass (front) 1198981 1198982kg 37

Unsprung mass (rear) 1198983 1198984kg 33

Tread 119889m 136Length from front axle to centroid 119897

119891m 0955

Length from rear axle to centroid 119897119903m 1380

Roll moment inertia 119868119909kgsdotm2 293

Pitch moment inertia 119868119910kgsdotm2 1074

Stiffness of tire 119896119905kNsdotmminus1 192

Table 3 Optimal parameters

Parameters ValueMain spring stiffness 119896

1kNsdotmminus1 201

Deputy spring stiffness 1198962kNsdotmminus1 102

Damper coefficient 119888Nsdotssdotmminus1 607

It is noted that the suspension performances are alldegraded by the nonlinearities of the fluid inerter the verticalbody acceleration increases by 83 the pitch angular accel-eration increases by 134 and the roll angular accelerationincreases by 13 In [27] the nonlinear ball-screw inerterinvolving friction and elastic effect was discussed Resultsshowed that the performance benefits were slightly degradedby ball-screw inerter nonlinearities which is the same asthe fluid inerter So we can conclude that the nonlinearitiesof the inerter will have a slight effect on the suspensionperformance

Furthermore the performance indexes of the optimizedsuspension are all decreased compared with the nonlinearsuspension The RMS of pitch angular acceleration hasdramatically decreased by 199 The RMS of vertical bodyacceleration has decreased by 90 and the RMS of rollangular acceleration decreased by 96 Figures 9ndash11 show theperformance indexes in the time domain

It can be inferred that the vertical body acceleration thepitch angular acceleration and the roll angular accelerationare all decreased which means the vibration isolation perfor-mance of optimized suspension is significantly superior to thenonlinear one

6 Conclusion

In this paper a new fluid inerter is designed and thenonlinearities factors including friction and damping forceare analyzed Experiments have been finished in the benchto test the fluid inerter and to verify the effectivenessof the nonlinear model of fluid inerter Furthermore anew vehicle suspension system employing the nonlinearfluid inerter is built and the parameters are obtained bymeans of quantum genetic algorithm Results show that thedynamic performance indexes of the optimized suspensionall decreased compared with the nonlinear suspensionThereis a dramatic decrease which is at most 199 in the RMS

NonlinearOptimized

Vert

ical

bod

y ac

cele

ratio

n (m

middotsminus2)

minus3

minus2

minus1

0

1

2

3

1 2 3 4 5 6 7 8 9 100Time (s)

Figure 9 Vertical body acceleration

NonlinearOptimized

Pitc

h an

gula

r acc

eler

atio

n (r

admiddotsminus

2)

minus3

minus2

minus1

0

1

2

3

1 2 3 4 5 6 7 8 9 100Time (s)

Figure 10 Pitch angular acceleration

NonlinearOptimized

Roll

angu

lar a

ccel

erat

ion

(rad

middotsminus2)

minus5

0

5

1 2 3 4 5 6 7 8 9 100Time (s)

Figure 11 Roll angular acceleration


Table 4 Performance indexes

Name Linear Nonlinear Optimized ImprovementVertical body acceleration(msdotsminus2) 05230 05668 05156 90Pitch angular acceleration(radsdotsminus2) 04201 04764 03815 199Roll angular acceleration(radsdotsminus2) 09359 09478 08566 96

of pitch angular acceleration the decrease of the RMS ofvertical body acceleration is 90 and the decrease of theRMSof roll angular acceleration is 96 The vibration isolationperformance is significantly improved

Competing Interests

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

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant no 51405202) the ScientificResearch Innovation Projects of Jiangsu Province (Grant noKYLX15 1081) the China Postdoctoral Science Foundation(Grant no 2015M570408) and the ldquoSix Talent Peaksrdquo High-Level project of JiangsuProvince (Grant no 2013-JNHB-001)

References

[1] H Du and N Zhang ldquoFuzzy control for nonlinear uncertainelectrohydraulic active suspensionswith input constraintrdquo IEEETransactions on Fuzzy Systems vol 17 no 2 pp 343ndash356 2009

[2] H H Pan W C Sun H J Gao and J Y Yu ldquoFinite-timestabilization for vehicle active suspension systems with hardconstraintsrdquo IEEE Transactions on Intelligent TransportationSystems vol 16 no 5 pp 2663ndash2672 2015

[3] I Youn and A Hac ldquoPreview control of active suspension withintegral actionrdquo International Journal of Automotive Technologyvol 7 no 5 pp 547ndash554 2006

[4] M A Eltantawie ldquoDecentralized neuro-fuzzy control for halfcar with semi-active suspension systemrdquo International Journalof Automotive Technology vol 13 no 3 pp 423ndash431 2012

[5] J C Tudon-Martinez S Fergani S Varrier et al ldquoRoadadaptive semi-active suspension in an automotive vehicle usingan LPV controllerrdquo in Proceedings of the 7th IFAC Symposium onAdvances in Automotive Control (AAC rsquo13) vol 7 pp 231ndash236Tokyo Japan September 2013

[6] H Chen C Long C-C Yuan and H-B Jiang ldquoNon-linearmodelling and control of semi-active suspensions with variabledampingrdquo Vehicle System Dynamics vol 51 no 10 pp 1568ndash1587 2013

[7] M F Aly A O Nassef and K Hamza ldquoMulti-objective designof vehicle suspension systems via a local diffusion genetic algo-rithm for disjoint Pareto frontiersrdquo Engineering Optimizationvol 47 no 5 pp 706ndash717 2015

[8] L R C Drehmer W J P Casas and H M Gomes ldquoParametersoptimisation of a vehicle suspension system using a particleswarm optimisation algorithmrdquo Vehicle System Dynamics vol53 no 4 pp 449ndash474 2015

[9] M C Smith ldquoSynthesis of mechanical networks the inerterrdquoIEEE Transactions on Automatic Control vol 47 no 10 pp1648ndash1662 2002

[10] M C Smith ldquoForce-controlling mechanical devicerdquo PatentPending International Application no PCTGB0203056 2001

[11] M Z Q Chen C Papageorgiou F Scheibe F-CWang andMSmith ldquoThe missing mechanical circuit elementrdquo IEEE Circuitsand Systems Magazine vol 9 no 1 pp 10ndash26 2009

[12] M Z Chen K Wang Y Zou and G Chen ldquoRealization ofthree-port spring networks with inerter for effectivemechanicalcontrolrdquo IEEETransactions onAutomatic Control vol 60 no 10pp 2722ndash2727 2015

[13] KWang andM ZQ Chen ldquoMinimal realizations of three-portresistive networksrdquo IEEE Transactions on Circuits and Systems IRegular Papers vol 62 no 4 pp 986ndash994 2015

[14] M Z Q Chen K Wang M H Yin C Y Li Z Q Zuo andG R Chen ldquoSynthesis of n-port resistive networks containing2n terminalsrdquo International Journal of Circuit Theory andApplications vol 43 no 4 pp 427ndash437 2015

[15] X-J Zhang M Ahmadian and K-H Guo ldquoOn the benefits ofsemi-active suspensionswith inertersrdquo Shock andVibration vol19 no 3 pp 257ndash272 2012

[16] Y LHuMZQChen andZ Shu ldquoPassive vehicle suspensionsemploying inerters with multiple performance requirementsrdquoJournal of Sound and Vibration vol 333 no 8 pp 2212ndash22252014

[17] P Li J Lam and K C Cheung ldquoControl of vehicle suspensionusing an adaptive inerterrdquo Proceedings of the Institution ofMechanical Engineers Part D Journal of Automobile Engineer-ing vol 229 no 14 pp 1934ndash1943 2015

[18] M Z Q Chen Y L Hu C Y Li and G R Chen ldquoPerformancebenefits of using inerter in semiactive suspensionsrdquo IEEETransactions on Control Systems Technology vol 23 no 4 pp1571ndash1577 2015

[19] F-C Wang M-K Liao B-H Liao W-J Su and H-A ChanldquoThe performance improvements of train suspension systemswith mechanical networks employing inertersrdquo Vehicle SystemDynamics vol 47 no 7 pp 805ndash830 2009

[20] F-C Wang and M-K Liao ldquoThe lateral stability of train sus-pension systems employing inertersrdquo Vehicle System Dynamicsvol 48 no 5 pp 619ndash643 2010

[21] F C Wang M F Hong and C W Chen ldquoBuilding suspensionwith inertersrdquo Proceedings of the Institution of MechanicalEngineers Part C Journal of Mechanical Engineering Sciencevol 224 no 8 pp 1605ndash1616 2010

[22] H L Sun P Q Zhang H B Chen K Zhang and X LGong ldquoApplication of dynamic vibration absorbers in structuralvibration control undermulti-frequency harmonic excitationsrdquoApplied Acoustics vol 69 no 12 pp 1361ndash1367 2008

[23] L Marian and A Giaralis ldquoOptimal design of a novel tunedmass-damper-inerter (TMDI) passive vibration control config-uration for stochastically support-excited structural systemsrdquoProbabilistic Engineering Mechanics vol 38 pp 156ndash164 2014


[24] Y L Hu and M Z Q Chen ldquoPerformance evaluation forinerter-based dynamic vibration absorbersrdquo International Jour-nal of Mechanical Sciences vol 99 pp 297ndash307 2015

[25] M Z Q Chen Y Hu L Huang and G Chen ldquoInfluence ofinerter on natural frequencies of vibration systemsrdquo Journal ofSound and Vibration vol 333 no 7 pp 1874ndash1887 2014

[26] Y L Hu M Z Q Chen Z Shu and L X Huang ldquoAnalysis andoptimisation for inerter-based isolators via fixed-point theoryand algebraic solutionrdquo Journal of Sound andVibration vol 346no 1 pp 17ndash36 2015

[27] F-C Wang and W-J Su ldquoImpact of inerter nonlinearities onvehicle suspension controlrdquo Vehicle System Dynamics vol 46no 7 pp 575ndash595 2008

[28] S J Swift M C Smith A R Glover C Papageorgiou BGartner and N E Houghton ldquoDesign and modelling of a fluidinerterrdquo International Journal of Control vol 86 no 11 pp2035ndash2051 2013

[29] A R Glover M C Smith and N E Houghton ldquoForce-controlling hydraulic devicerdquo International Patent ApplicationNo PCTGB2010001491 2009

[30] C Papageorgiou and M C Smith ldquoLaboratory experimentaltesting of inertersrdquo in Proceedings of the 44th IEEE Conferenceon Decision and Control and the European Control Conferencepp 3351ndash3356 Seville Spain December 2005

[31] F-C Wang M-F Hong and T-C Lin ldquoDesigning and testinga hydraulic inerterrdquo Proceedings of the Institution of MechanicalEngineers Part C Journal of Mechanical Engineering Sciencevol 225 no 1 pp 66ndash72 2011

[32] B S Massey Mechanics of Fluids Chapman amp Hall LondonUK 6th edition 1997

[33] Y Shen L Chen X Yang D Shi and J Yang ldquoImproveddesign of dynamic vibration absorber by using the inerter andits application in vehicle suspensionrdquo Journal of Sound andVibration vol 361 pp 148ndash158 2016

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

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



xF F

Piston rod Piston Hydraulic cylinder

Helical channel

Figure 1 Prototype of fluid inerter

reduce the effect of the nonlinearities based on the full-carmodel including the nonlinear inerter The paper is arrangedas follows

In Section 2 a new fluid inerter is designed and the non-linearities involving friction and damping force are taken intoconsideration Experiments are finished to demonstrate thenonlinear model Suspension system involving the nonlinearfluid inerter is built in Section 3 and the parameters of thesuspension are obtained by means of QGA (quantum geneticalgorithm) in Section 4 Dynamic performance of the vehiclesuspension is analyzed in Section 5 Finally some conclusionsare shown in Section 6

2 Design and Test of Fluid Inerter

There are many mechanical types of inerter such as ball-screw inerter rack-and-pinion inerter [30] hydraulic-motorinerter [31] and fluid inerter [28 29] Motion transferredmechanics are used in the device to enlarge the inertia forcewhich is the most meaningful thing to inerter The dynamicequation of inerter is

119865 = 119887 (V1minus V2) (1)

where 119865 is the force in the device V1and V2are the velocities

and 119887 is the inertance with the unit kg In this sectiona new type of fluid inerter is designed to investigate thenonlinearities

21 Ideal Fluid Inerter The fluid inerter is similar to rack-and-pinion inerter or ball-screw inerter because the fluidflowing in the helical channel can be taken as a ldquofluidflywheelrdquo to provide the inertia force The prototype of thefluid inerter with the helical channel outside the cylinder isshown in Figure 1

In Figure 1 119865 is the force applied in the fluid inerter and119909 is the displacement of the piston The hydraulic cylinderand the piston rod are the two terminals of the fluid inerterWhen the piston pushes the fluid in the left of the hydrauliccylinder into the helical channel the fluid flows through thehelical channel into the right hydraulic cylinder in order tocompensate the pressure loss

Firstly the fluid in the cylinder and the helical channelis taken as nondisclosure and noncompressed According to[29] the inertance 119887 can be gained as

119887 =119898

1 + (ℎ21205871199034)2(1198781

1198782

)

2

(2)

where119898 is themass of the fluid in the helical channel 1198781is the

effective area of the hydraulic cylinder 1198782is the section area

of the helical channel ℎ is the pitch of the helical channel and1199034is the helix radius of the helical channel1198781can be gained by the equation

1198781= 120587 (119903

2

2minus 1199032

1) (3)

where 1199031is the radius of the piston rod and 119903

2is the inside

radius of the hydraulic cylinder1198782can be gained by the equation

1198782= 1205871199032

3 (4)

where 1199033is the radius of the helical channel

119898 can be gained by the equation

119898 = 1205881198782119897 (5)

where 120588 is the oil density and 119897 is the length of helical channeland can be approximated as

119897 = 119899radicℎ2 + (21205871199034)2

+ 21198970 (6)

where 119899 is the circle number of the helical channel and 1198970is

the length from the inlet and outlet to the helical channel

22 Nonlinearities Analysis of Fluid Inerter When the force119865 is applied to the fluid inerter both the friction caused bythe motion of piston rod and hydraulic cylinder 119891 and thedamping force caused by the viscosity of the fluid119891

119888will block

the motion of the piston The dynamic equation is

119865 minus 119891 minus 119891119888= 119887 (7)

The friction model can be described by the ColumnFriction Model as follows

119891 = 1198910sign (V) (8)

where1198910is the amplitude of the friction V is the velocity of the

piston and sign(V) is a symbolic function it is 1 when V gt 0

and minus1 when V lt 0 The direction of friction 119891 between thepiston and hydraulic cylinder is related to the velocity of thepiston and has the constant value

For a hydraulic device the damping force caused by theviscosity of the fluid is the most important impact on thedynamic performance According to the Hagen PoiseuilleLaw the flow of the fluid in the helical channel 119876 and thepressure drop between the two terminalsΔ119901 have the relationas follows

119876 =1205871199034

3

8120583119897Δ119901 (9)

in which 120583 is viscosity


z1

z2

f

b fc



119906 =119876

1198782

=1199032

3

8120583119897Δ119901 (10)


Δ1199011= 05

1205881199062

2

Δ1199012=

1205881199062

2

(11)


1198781 = 1198782119906 (12)


119891119888= (Δ119901 + Δ119901

1+ Δ1199012) 1198781=

31205881198783

1

41198782

2

2+

81205831198971198782

1

1199032

31198782

(13)







119865 = 1198910sign (

1minus 2) + 1198621(1minus 2)2

+ 1198622(1minus 2)

+ 119887 (1minus 2)

(14)

Helicalchannel

Vibration head

Fluid inerter

Figure 3 Bench test

10 20 30 400Time (s)

minus06

minus04

minus02

0

02

04

06

Forc

e (kN

)





2is





0

2

4

6

8

10

12

14

16

Diff

eren

ce (

)





1m 0012





0m 01







2is the


1is 238 kNm 119896

2is 89 kNm




119898119886119886= 11986510

+ 11986520

+ 11986530

+ 11986540 (15)


119868119910 = 119897119903(11986530

+ 11986540) minus 119897119891(11986510

+ 11986520) (16)


119868119909 = (119865

20+ 11986540

minus 11986510

minus 11986530)119889

2 (17)







119910is


1198940


11988510

= 119911119886minus 119897119891120593 minus

119889

2120579

11988520

= 119911119886minus 119897119891120593 +

119889

2120579

11988530

= 119911119886+ 119897119903120593 minus

119889

2120579

11988540

= 119911119886+ 119897119903120593 +

119889

2120579

(18)

where 1198851198940



11989811= 119896119905(1198761minus 1198851) minus 11986510

11989822= 119896119905(1198762minus 1198852) minus 11986520

11989833= 119896119905(1198763minus 1198853) minus 11986530

11989844= 119896119905(1198764minus 1198854) minus 11986540

(19)








minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Forc

e (kN

)

1 2 3 4 5 6 7 8 9 100Time (s)

(a) 05Hz


minus15

minus10

minus5

0

5

10

15

Forc

e (kN

)

005 01 015 02 025 03 035 040Time (s)

(b) 12Hz


k1

f

bfc

k2c



11986510

= 1198961(1198851minus 11988510) + 1198962(1198851minus 1198851198871) + 119888 (

1minus 1198871)

1198962(1198851minus 1198851198871) + 119888 (

1minus 1198871)

= 1198910sign (

1198871minus 10) + 119887 (

1198871minus 10)

+ 1198621(1198871

minus 10)2

+ 1198622(1198871

minus 10)

11986520

= 1198961(1198852minus 11988520) + 1198962(1198852minus 1198851198872) + 119888 (

2minus 1198872)

1198962(1198852minus 1198851198872) + 119888 (

2minus 1198872)

= 1198910sign (

1198872minus 20) + 119887 (

1198872minus 20)

+ 1198621(1198872

minus 20)2

+ 1198622(1198872

minus 20)

11986530

= 1198961(1198853minus 11988530) + 1198962(1198853minus 1198851198873) + 119888 (

1minus 1198873)

1198962(1198853minus 1198851198873) + 119888 (

3minus 1198873)

= 1198910sign (

1198873minus 30) + 119887 (

1198873minus 30)

+ 1198621(1198873

minus 30)2

+ 1198622(1198873

minus 30)

11986540

= 1198961(1198854minus 11988540) + 1198962(1198854minus 1198851198874) + 119888 (

4minus 1198874)

1198962(1198854minus 1198851198874) + 119888 (

4minus 1198874)

= 1198910sign (

1198874minus 40) + 119887 (

1198874minus 40)

+ 1198621(1198874

minus 40)2

+ 1198622(1198874

minus 40)

(20)





Suspension 1

Suspension 2

Suspension 3

Suspension 4

X

Y

Z

d

z10 F10

z20F20

120579za

ma

120593z30

F30

z40F40

m1

m2

m3

m4

z1

z2

z3

z4

kt

kt

kt

kt

Q1

Q2

Q3

Q4

lf lr




[

1205721015840

119894

1205731015840

119894

] = 119880 (120579119894) [

120572119894

120573119894

] (21)

where [1205721015840

1198941205731015840


119894]119879 is the



119880(120579119894) = [

cos (120579119894) minus sin (120579

119894)

sin (120579119894) cos (120579

119894)] (22)









(23)


min 119869 = 1199081

119860

1198600

+ 1199082

119861

1198610

+ 1199083

119862

1198620

(24)





0is




le

[119891119889] where 120590




le 119866where 120590



Table 3



119894(119905) = minus2120587119891cut119876119894 (119905) + 2120587radic119866

0119881119908 (119905) (25)









119886kg 770




119891m 0955







1kNsdotmminus1 201






6 Conclusion


NonlinearOptimized

Vert

ical

bod

y ac

cele

ratio

n (m

middotsminus2)

minus3

minus2

minus1

0

1

2

3

1 2 3 4 5 6 7 8 9 100Time (s)


NonlinearOptimized

Pitc

h an

gula

r acc

eler

atio

n (r

admiddotsminus

2)

minus3

minus2

minus1

0

1

2

3

1 2 3 4 5 6 7 8 9 100Time (s)


NonlinearOptimized

Roll

angu

lar a

ccel

erat

ion

(rad

middotsminus2)

minus5

0

5

1 2 3 4 5 6 7 8 9 100Time (s)






Competing Interests


Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










z1

z2

f

b fc



119906 =119876

1198782

=1199032

3

8120583119897Δ119901 (10)


Δ1199011= 05

1205881199062

2

Δ1199012=

1205881199062

2

(11)


1198781 = 1198782119906 (12)


119891119888= (Δ119901 + Δ119901

1+ Δ1199012) 1198781=

31205881198783

1

41198782

2

2+

81205831198971198782

1

1199032

31198782

(13)







119865 = 1198910sign (

1minus 2) + 1198621(1minus 2)2

+ 1198622(1minus 2)

+ 119887 (1minus 2)

(14)

Helicalchannel

Vibration head

Fluid inerter

Figure 3 Bench test

10 20 30 400Time (s)

minus06

minus04

minus02

0

02

04

06

Forc

e (kN

)





2is





0

2

4

6

8

10

12

14

16

Diff

eren

ce (

)





1m 0012





0m 01







2is the


1is 238 kNm 119896

2is 89 kNm




119898119886119886= 11986510

+ 11986520

+ 11986530

+ 11986540 (15)


119868119910 = 119897119903(11986530

+ 11986540) minus 119897119891(11986510

+ 11986520) (16)


119868119909 = (119865

20+ 11986540

minus 11986510

minus 11986530)119889

2 (17)







119910is


1198940


11988510

= 119911119886minus 119897119891120593 minus

119889

2120579

11988520

= 119911119886minus 119897119891120593 +

119889

2120579

11988530

= 119911119886+ 119897119903120593 minus

119889

2120579

11988540

= 119911119886+ 119897119903120593 +

119889

2120579

(18)

where 1198851198940



11989811= 119896119905(1198761minus 1198851) minus 11986510

11989822= 119896119905(1198762minus 1198852) minus 11986520

11989833= 119896119905(1198763minus 1198853) minus 11986530

11989844= 119896119905(1198764minus 1198854) minus 11986540

(19)








minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Forc

e (kN

)

1 2 3 4 5 6 7 8 9 100Time (s)

(a) 05Hz


minus15

minus10

minus5

0

5

10

15

Forc

e (kN

)

005 01 015 02 025 03 035 040Time (s)

(b) 12Hz


k1

f

bfc

k2c



11986510

= 1198961(1198851minus 11988510) + 1198962(1198851minus 1198851198871) + 119888 (

1minus 1198871)

1198962(1198851minus 1198851198871) + 119888 (

1minus 1198871)

= 1198910sign (

1198871minus 10) + 119887 (

1198871minus 10)

+ 1198621(1198871

minus 10)2

+ 1198622(1198871

minus 10)

11986520

= 1198961(1198852minus 11988520) + 1198962(1198852minus 1198851198872) + 119888 (

2minus 1198872)

1198962(1198852minus 1198851198872) + 119888 (

2minus 1198872)

= 1198910sign (

1198872minus 20) + 119887 (

1198872minus 20)

+ 1198621(1198872

minus 20)2

+ 1198622(1198872

minus 20)

11986530

= 1198961(1198853minus 11988530) + 1198962(1198853minus 1198851198873) + 119888 (

1minus 1198873)

1198962(1198853minus 1198851198873) + 119888 (

3minus 1198873)

= 1198910sign (

1198873minus 30) + 119887 (

1198873minus 30)

+ 1198621(1198873

minus 30)2

+ 1198622(1198873

minus 30)

11986540

= 1198961(1198854minus 11988540) + 1198962(1198854minus 1198851198874) + 119888 (

4minus 1198874)

1198962(1198854minus 1198851198874) + 119888 (

4minus 1198874)

= 1198910sign (

1198874minus 40) + 119887 (

1198874minus 40)

+ 1198621(1198874

minus 40)2

+ 1198622(1198874

minus 40)

(20)





Suspension 1

Suspension 2

Suspension 3

Suspension 4

X

Y

Z

d

z10 F10

z20F20

120579za

ma

120593z30

F30

z40F40

m1

m2

m3

m4

z1

z2

z3

z4

kt

kt

kt

kt

Q1

Q2

Q3

Q4

lf lr




[

1205721015840

119894

1205731015840

119894

] = 119880 (120579119894) [

120572119894

120573119894

] (21)

where [1205721015840

1198941205731015840


119894]119879 is the



119880(120579119894) = [

cos (120579119894) minus sin (120579

119894)

sin (120579119894) cos (120579

119894)] (22)









(23)


min 119869 = 1199081

119860

1198600

+ 1199082

119861

1198610

+ 1199083

119862

1198620

(24)





0is




le

[119891119889] where 120590




le 119866where 120590



Table 3



119894(119905) = minus2120587119891cut119876119894 (119905) + 2120587radic119866

0119881119908 (119905) (25)









119886kg 770




119891m 0955







1kNsdotmminus1 201






6 Conclusion


NonlinearOptimized

Vert

ical

bod

y ac

cele

ratio

n (m

middotsminus2)

minus3

minus2

minus1

0

1

2

3

1 2 3 4 5 6 7 8 9 100Time (s)


NonlinearOptimized

Pitc

h an

gula

r acc

eler

atio

n (r

admiddotsminus

2)

minus3

minus2

minus1

0

1

2

3

1 2 3 4 5 6 7 8 9 100Time (s)


NonlinearOptimized

Roll

angu

lar a

ccel

erat

ion

(rad

middotsminus2)

minus5

0

5

1 2 3 4 5 6 7 8 9 100Time (s)






Competing Interests


Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

2

4

6

8

10

12

14

16

Diff

eren

ce (

)





1m 0012





0m 01







2is the


1is 238 kNm 119896

2is 89 kNm




119898119886119886= 11986510

+ 11986520

+ 11986530

+ 11986540 (15)


119868119910 = 119897119903(11986530

+ 11986540) minus 119897119891(11986510

+ 11986520) (16)


119868119909 = (119865

20+ 11986540

minus 11986510

minus 11986530)119889

2 (17)







119910is


1198940


11988510

= 119911119886minus 119897119891120593 minus

119889

2120579

11988520

= 119911119886minus 119897119891120593 +

119889

2120579

11988530

= 119911119886+ 119897119903120593 minus

119889

2120579

11988540

= 119911119886+ 119897119903120593 +

119889

2120579

(18)

where 1198851198940



11989811= 119896119905(1198761minus 1198851) minus 11986510

11989822= 119896119905(1198762minus 1198852) minus 11986520

11989833= 119896119905(1198763minus 1198853) minus 11986530

11989844= 119896119905(1198764minus 1198854) minus 11986540

(19)








minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Forc

e (kN

)

1 2 3 4 5 6 7 8 9 100Time (s)

(a) 05Hz


minus15

minus10

minus5

0

5

10

15

Forc

e (kN

)

005 01 015 02 025 03 035 040Time (s)

(b) 12Hz


k1

f

bfc

k2c



11986510

= 1198961(1198851minus 11988510) + 1198962(1198851minus 1198851198871) + 119888 (

1minus 1198871)

1198962(1198851minus 1198851198871) + 119888 (

1minus 1198871)

= 1198910sign (

1198871minus 10) + 119887 (

1198871minus 10)

+ 1198621(1198871

minus 10)2

+ 1198622(1198871

minus 10)

11986520

= 1198961(1198852minus 11988520) + 1198962(1198852minus 1198851198872) + 119888 (

2minus 1198872)

1198962(1198852minus 1198851198872) + 119888 (

2minus 1198872)

= 1198910sign (

1198872minus 20) + 119887 (

1198872minus 20)

+ 1198621(1198872

minus 20)2

+ 1198622(1198872

minus 20)

11986530

= 1198961(1198853minus 11988530) + 1198962(1198853minus 1198851198873) + 119888 (

1minus 1198873)

1198962(1198853minus 1198851198873) + 119888 (

3minus 1198873)

= 1198910sign (

1198873minus 30) + 119887 (

1198873minus 30)

+ 1198621(1198873

minus 30)2

+ 1198622(1198873

minus 30)

11986540

= 1198961(1198854minus 11988540) + 1198962(1198854minus 1198851198874) + 119888 (

4minus 1198874)

1198962(1198854minus 1198851198874) + 119888 (

4minus 1198874)

= 1198910sign (

1198874minus 40) + 119887 (

1198874minus 40)

+ 1198621(1198874

minus 40)2

+ 1198622(1198874

minus 40)

(20)





Suspension 1

Suspension 2

Suspension 3

Suspension 4

X

Y

Z

d

z10 F10

z20F20

120579za

ma

120593z30

F30

z40F40

m1

m2

m3

m4

z1

z2

z3

z4

kt

kt

kt

kt

Q1

Q2

Q3

Q4

lf lr




[

1205721015840

119894

1205731015840

119894

] = 119880 (120579119894) [

120572119894

120573119894

] (21)

where [1205721015840

1198941205731015840


119894]119879 is the



119880(120579119894) = [

cos (120579119894) minus sin (120579

119894)

sin (120579119894) cos (120579

119894)] (22)









(23)


min 119869 = 1199081

119860

1198600

+ 1199082

119861

1198610

+ 1199083

119862

1198620

(24)





0is




le

[119891119889] where 120590




le 119866where 120590



Table 3



119894(119905) = minus2120587119891cut119876119894 (119905) + 2120587radic119866

0119881119908 (119905) (25)









119886kg 770




119891m 0955







1kNsdotmminus1 201






6 Conclusion


NonlinearOptimized

Vert

ical

bod

y ac

cele

ratio

n (m

middotsminus2)

minus3

minus2

minus1

0

1

2

3

1 2 3 4 5 6 7 8 9 100Time (s)


NonlinearOptimized

Pitc

h an

gula

r acc

eler

atio

n (r

admiddotsminus

2)

minus3

minus2

minus1

0

1

2

3

1 2 3 4 5 6 7 8 9 100Time (s)


NonlinearOptimized

Roll

angu

lar a

ccel

erat

ion

(rad

middotsminus2)

minus5

0

5

1 2 3 4 5 6 7 8 9 100Time (s)






Competing Interests


Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Forc

e (kN

)

1 2 3 4 5 6 7 8 9 100Time (s)

(a) 05Hz


minus15

minus10

minus5

0

5

10

15

Forc

e (kN

)

005 01 015 02 025 03 035 040Time (s)

(b) 12Hz


k1

f

bfc

k2c



11986510

= 1198961(1198851minus 11988510) + 1198962(1198851minus 1198851198871) + 119888 (

1minus 1198871)

1198962(1198851minus 1198851198871) + 119888 (

1minus 1198871)

= 1198910sign (

1198871minus 10) + 119887 (

1198871minus 10)

+ 1198621(1198871

minus 10)2

+ 1198622(1198871

minus 10)

11986520

= 1198961(1198852minus 11988520) + 1198962(1198852minus 1198851198872) + 119888 (

2minus 1198872)

1198962(1198852minus 1198851198872) + 119888 (

2minus 1198872)

= 1198910sign (

1198872minus 20) + 119887 (

1198872minus 20)

+ 1198621(1198872

minus 20)2

+ 1198622(1198872

minus 20)

11986530

= 1198961(1198853minus 11988530) + 1198962(1198853minus 1198851198873) + 119888 (

1minus 1198873)

1198962(1198853minus 1198851198873) + 119888 (

3minus 1198873)

= 1198910sign (

1198873minus 30) + 119887 (

1198873minus 30)

+ 1198621(1198873

minus 30)2

+ 1198622(1198873

minus 30)

11986540

= 1198961(1198854minus 11988540) + 1198962(1198854minus 1198851198874) + 119888 (

4minus 1198874)

1198962(1198854minus 1198851198874) + 119888 (

4minus 1198874)

= 1198910sign (

1198874minus 40) + 119887 (

1198874minus 40)

+ 1198621(1198874

minus 40)2

+ 1198622(1198874

minus 40)

(20)





Suspension 1

Suspension 2

Suspension 3

Suspension 4

X

Y

Z

d

z10 F10

z20F20

120579za

ma

120593z30

F30

z40F40

m1

m2

m3

m4

z1

z2

z3

z4

kt

kt

kt

kt

Q1

Q2

Q3

Q4

lf lr




[

1205721015840

119894

1205731015840

119894

] = 119880 (120579119894) [

120572119894

120573119894

] (21)

where [1205721015840

1198941205731015840


119894]119879 is the



119880(120579119894) = [

cos (120579119894) minus sin (120579

119894)

sin (120579119894) cos (120579

119894)] (22)









(23)


min 119869 = 1199081

119860

1198600

+ 1199082

119861

1198610

+ 1199083

119862

1198620

(24)





0is




le

[119891119889] where 120590




le 119866where 120590



Table 3



119894(119905) = minus2120587119891cut119876119894 (119905) + 2120587radic119866

0119881119908 (119905) (25)









119886kg 770




119891m 0955







1kNsdotmminus1 201






6 Conclusion


NonlinearOptimized

Vert

ical

bod

y ac

cele

ratio

n (m

middotsminus2)

minus3

minus2

minus1

0

1

2

3

1 2 3 4 5 6 7 8 9 100Time (s)


NonlinearOptimized

Pitc

h an

gula

r acc

eler

atio

n (r

admiddotsminus

2)

minus3

minus2

minus1

0

1

2

3

1 2 3 4 5 6 7 8 9 100Time (s)


NonlinearOptimized

Roll

angu

lar a

ccel

erat

ion

(rad

middotsminus2)

minus5

0

5

1 2 3 4 5 6 7 8 9 100Time (s)






Competing Interests


Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Suspension 1

Suspension 2

Suspension 3

Suspension 4

X

Y

Z

d

z10 F10

z20F20

120579za

ma

120593z30

F30

z40F40

m1

m2

m3

m4

z1

z2

z3

z4

kt

kt

kt

kt

Q1

Q2

Q3

Q4

lf lr




[

1205721015840

119894

1205731015840

119894

] = 119880 (120579119894) [

120572119894

120573119894

] (21)

where [1205721015840

1198941205731015840


119894]119879 is the



119880(120579119894) = [

cos (120579119894) minus sin (120579

119894)

sin (120579119894) cos (120579

119894)] (22)









(23)


min 119869 = 1199081

119860

1198600

+ 1199082

119861

1198610

+ 1199083

119862

1198620

(24)





0is




le

[119891119889] where 120590




le 119866where 120590



Table 3



119894(119905) = minus2120587119891cut119876119894 (119905) + 2120587radic119866

0119881119908 (119905) (25)









119886kg 770




119891m 0955







1kNsdotmminus1 201






6 Conclusion


NonlinearOptimized

Vert

ical

bod

y ac

cele

ratio

n (m

middotsminus2)

minus3

minus2

minus1

0

1

2

3

1 2 3 4 5 6 7 8 9 100Time (s)


NonlinearOptimized

Pitc

h an

gula

r acc

eler

atio

n (r

admiddotsminus

2)

minus3

minus2

minus1

0

1

2

3

1 2 3 4 5 6 7 8 9 100Time (s)


NonlinearOptimized

Roll

angu

lar a

ccel

erat

ion

(rad

middotsminus2)

minus5

0

5

1 2 3 4 5 6 7 8 9 100Time (s)






Competing Interests


Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119886kg 770




119891m 0955







1kNsdotmminus1 201






6 Conclusion


NonlinearOptimized

Vert

ical

bod

y ac

cele

ratio

n (m

middotsminus2)

minus3

minus2

minus1

0

1

2

3

1 2 3 4 5 6 7 8 9 100Time (s)


NonlinearOptimized

Pitc

h an

gula

r acc

eler

atio

n (r

admiddotsminus

2)

minus3

minus2

minus1

0

1

2

3

1 2 3 4 5 6 7 8 9 100Time (s)


NonlinearOptimized

Roll

angu

lar a

ccel

erat

ion

(rad

middotsminus2)

minus5

0

5

1 2 3 4 5 6 7 8 9 100Time (s)






Competing Interests


Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













Competing Interests


Acknowledgments


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









research article modeling and optimization of vehicle

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