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Hindawi Publishing CorporationAdvances in Acoustics and VibrationVolume 2012, Article ID 863061, 7 pagesdoi:10.1155/2012/863061

Research Article

Development of a Refined Quarter Car Model forthe Analysis of Discomfort due to Vibration

A. N. Thite

Department of Mechanical Engineering and Mathematical Sciences, Faculty of Technology, Design and Environment,Oxford Brookes University, Wheatley, Oxford OX33 1HX, UK

Correspondence should be addressed to A. N. Thite, [email protected]

Received 24 April 2012; Accepted 26 May 2012

Academic Editor: Joseph C. S. Lai

Copyright © 2012 A. N. Thite. This 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.

In the automotive industry, numerous expensive and time-consuming trials are used to “optimize” the ride and handlingperformance. Ideally, a reliable virtual prototype is a solution. The practical usage of a model is linked and restricted by the modelcomplexity and reliability. The object of this study is development and analysis of a refined quarter car suspension model, whichincludes the effect of series stiffness, to estimate the response at higher frequencies; resulting Maxwell’s model representation doesnot allow straightforward calculation of performance parameters. Governing equations of motion are manipulated to calculatethe effective stiffness and damping values. State space model is arranged in a novel form to find eigenvalues, which is a uniquecontribution. Analysis shows the influence of suspension damping and series stiffness on natural frequencies and regions ofreduced vibration response. Increase in the suspension damping coefficient beyond optimum values was found to reduce themodal damping and increase the natural frequencies. Instead of carrying out trial simulations during performance optimizationfor human comfort, an expression is developed for corresponding suspension damping coefficient. The analysis clearly shows theinfluence of the series stiffness on suspension dynamics and necessity to incorporate the model in performance predictions.

1. Introduction

In the vehicle suspension design, dependent on the usagepattern, handling and ride comfort performance havecontradicting requirements [1]. The parameter range foroptimum suspension design can vary for different vehicles;often complex combinations of parameters form a solution.In the industry, numerous expensive and time-consumingexperimental trials are used to “optimize” the performance.Ideally, a reliable virtual prototype is a solution. The practicalusage of a vehicle model is linked and restricted by modelcomplexity; a refined model without complexities is arequirement. One of the modelling difficulties is the presenceof various compliances and their connections within thevehicle suspension system. The object of this research is theinvestigation of a quarter car model with the aim of analysingcomfort considering the compliances in shock absorber andsuspension top mount; there is series stiffness in the model.

Models of varying complexity, considering only lineardynamic elements to systems involving nonlinear elements,

have been developed to predict, specifically, vehicle handlingand, to some extent, ride comfort. Shock absorbers havebeen modelled to represent hydromechanical behaviour forexample, [2, 3] the complexities do not allow easy integrationof the model into vehicle dynamics applications. It is difficultto perform parametric studies and visualize the outcome.In passive simplified models, shock absorber is treated asa viscous damper, simplest being a linear model. Often,elements connecting the shock absorbers to the suspensionsystem and the effects of various noise and vibration designsare ignored [4, 5].

Simplest vehicle model used to assess discomfort due tovibration is of two degrees of freedom (DOF) [1], commonlyknown as a quarter car model. Only vertical motion is con-sidered for discomfort quantification [6]. In the model, tyreis represented by a stiffness, wheel and associated elementsare represented by a mass, and suspension is representedby a spring and a damper working in parallel (Kelvin-Voigtmodel, hence forth called as an ordinary model) and vehiclebody by a mass. The representation is very simplistic and

2 Advances in Acoustics and Vibration

may not capture the dynamic response accurately; impulsiveforces would make suspension spring and damper combi-nation act practically like a rigid element. In the industry,to eliminate adverse force transmission, a stiffness in series(generally called the top mount) to the shock absorber andspring combination is introduced; the presence of series stiff-ness also benefits noise, vibration, and harshness (NVH) per-formance at higher frequencies, above about 100 Hz. How-ever, introduction of the series stiffness may have a complexinfluence at lower frequencies, below about 30 Hz, affectingvehicle ride comfort. Further, compliance within the shockabsorber is not considered in the Kelvin-Voigt model. Toan extent, these compliant elements limit the damping forceby relieving the pressure inside shock absorber. Overall, anappropriate representation of series stiffness requires the useof the Maxwell model or a viscoelastic element.

The series stiffness, as in 1 DOF model, may influenceeffective damping; two modal damping ratios of the quartercar model can be different from those based on the Kelvin-Voigt model and the variations can be a complex functionof the suspension damping coefficient and the stiffness.Further, the effect of series stiffness may not be equalon the modes of vibration. The resonance frequencies areexpected to be dependent on the damping coefficient. Foroptimization, sufficient damping is required for both modes.The modal damping ratio calculation is not straightforward;in practice, they are not calculated explicitly; instead costfunctions are formed and trials are carried out to reduce thevibratory responses directly. The process does not provideinsight into the complex problem. It is desirable to find acombination of parameters resulting in large damping ratiosand small shift in resonance frequency, without having toperform numerous forced vibration analyses; the focus ofoptimization continues to be the response reduction butachieved through an alternative approach. Availability ofan explicit formula for effective damping can minimize theoptimization effort and provide critical information.

In what follows, the quarter car model dynamic responsewith and without series stiffness is compared to showthe effect of series stiffness. For the harmonic input,equations of motion are rearranged to calculate effectivestiffness and damping values. Using the knowledge of forcestransmitted through the series stiffness, a novel form ofstate space equations is generated so as to calculate thenatural frequencies and modal damping ratios. The effectsof combination of suspension damping coefficient and seriesstiffness are analysed showing regions of reduced vibrationresponse. A simplified expression is developed for optimumsuspension damping with the aim of mainly reducing wheelhop frequency response; the wheel hop frequency responsewill be shown being more sensitive to the presence of seriesstiffness than the vehicle body mode response. The modelclearly shows the influence of series stiffness on the modaldamping ratios and the natural frequencies.

2. Vehicle Dynamic Lumped Parameter Model

The influence of damping on shift in the resonance frequencyand variation in the corresponding amplitude for 1 DOF

mv

mu

ks cs

kt

kd

y

x1

x3

x2

Figure 1: Schematic of a quarter car model with the use ofMaxwell’s model to represent the suspension dynamics.

Maxwell’s model is well documented. The vehicle suspensionsystem with series stiffness may show similar influence, butthe influence may not be a simple function of the dampingvalue alone; relative values of series and the tyre stiffnessesmay play a significant role. Further, the behaviour due todifferent forms of input may also be complex.

Initially, in the next section two models of the suspensionsystem, with and without series stiffness, are compared fortheir effect on the steady-state frequency domain response.Later, a state space model is used to calculate the naturalfrequencies and the modal damping ratios. An expression isdeveloped for optimum damping. In the model, stiffness anddamping coefficients used are linear equivalent parameters.

3. Application of Maxwell’s Model to Representthe Quarter Car Dynamics

Figure 1 shows schematic of Maxwell’s model representingthe corner of a vehicle. The equations of motion can beobtained by applying Newton’s 2nd law, which after somemanipulations are given by

mvx3 + kdx3 − kdx2 = 0, (1)

csx1 − csx2 + ksx1 − ksx2 − kdx2 + kdx3 = 0, (2)

mux1 + cs(x1 − x2) + ks(x1 − x2) + ktx1 = kt y. (3)

There is an additional response variable x2 compared toan ordinary 2 DOF model. For harmonic excitation, thisvariable can be eliminated from the analysis by substitutingequivalent values in terms of response variables x1 and x3,reducing the system to 2 DOFs. Hence, for forced vibrationanalysis, we have

(−Mω2 + jωC + K)

X = F, (4)

Advances in Acoustics and Vibration 3

where mass matrix is given by

M =[mu 00 mv

]

, (5)

effective damping matrix is given by

C =

⎡

⎢⎢⎢⎢⎢⎣

k2dcs(

(kd + ks)2 + c2

sω2)

−k2dcs(

(kd + ks)2 + c2

sω2)

−k2dcs(

(kd + ks)2 + c2

sω2)

k2dcs(

(kd + ks)2 + c2

sω2)

⎤

⎥⎥⎥⎥⎥⎦

, (6)

and effective stiffness matrix is given by

K=

⎡

⎢⎢⎢⎢⎢⎣

kt +

(c2sω

2kd + kdks(kd + ks))

((kd + ks)

2 + c2sω2

) −kd−kd

(c2sω

2 + ks(kd + ks))

((kd + ks)

2 + c2sω2

) kd − k2d(kd + ks)(

(kd + ks)2 + c2

sω2)

⎤

⎥⎥⎥⎥⎥⎦.

(7)

The elements of matrices are as follows: mu is the effectivemass of wheel hub, mv is a quarter of vehicle body mass, csis the suspension damping coefficient, ks is the suspensionstiffness, kt is the tyre stiffness, kd is the series stiffness, andω is the excitation frequency. The stiffness matrix, K, hasnonlinear elements; it is now dependent on the dampingcoefficient and excitation frequency. In the suspensionsystem, for effective isolation at higher frequencies, the seriesspring stiffness is likely to be of the order of tyre stiffness [7]for a range of displacements; the effect on second resonance(the wheel hop frequency) is expected to be significant. Therelative values of stiffness and damping coefficient determinethe resonance frequency sensitivity. If the series stiffness isrelatively large, the system would act as an ordinary two-DOFmodel and the spring and the damper being connected inparallel, irrespective of damping in the system.

The effective damping (6) is also a complex functionof series stiffness, suspension damping coefficient, andfrequency, showing nonlinear behaviour. For large seriesstiffness values, it tends to the suspension damping coeffi-cient. On the other hand, for small values, the damping isa complex function of series stiffness and frequency. Theeffective value determines the response peaks of wheel huband vehicle body motion. Later in this paper an approximateexpression is developed to estimate the optimum value ofsuspension damping coefficient.

Using (4), for a harmonic input, vehicle body and wheelhub responses can be calculated. The parameters used for thecalculation are listed in Table 1, which can be obtained, forexample, using the methods of [8]. Figure 2(a) shows wheelhub motion; there are two peaks, one each for the vehiclebody bounce mode (around 2 Hz) and the wheel hop orhub mode (around 14 Hz). Also shown in the figure is theresponse where series stiffness is very large such that it canbe treated as a rigid link. There is significant difference in theresponse at the wheel hub frequency; the resonance is shiftedand amplitudes are different. The vehicle body responsealso shows a similar pattern of behaviour (Figure 2(b)); the

Table 1: Parameter values used for the quarter car model.

System parameter Parameter value

Vehicle parameters

Tyre stiffness (N/m) 2e5

Suspension stiffness (N/m) 3e4

Suspension damping coefficient (Ns/m) Varying

Hub mass, front (kg) 40

Quarter of a vehicle body mass (kg) 250

Stiffness in series (N/m) 2e5

0 5 10 15 20 250

0.5

1

1.5

2

2.5

3

Frequency (Hz)

Tran

smis

sibi

lity

(a)

0 5 10 15 20 250

0.5

1

1.5

2

2.5

3

Frequency (Hz)

Tran

smis

sibi

lity

(b)

Figure 2: (a) Wheel hub response comparison and (b) vehicle bodyresponse comparison (damping coefficient is 1500 Ns/m), (. . .):ordinary model, (—): Maxwell’s suspension model.

difference in wheel hub mode contribution to the response ismore pronounced. The change in frequency and amplitude,as observed for the model with series stiffness, can havesignificant influence on the vehicle comfort.

The shift in resonance frequency and the change inamplitude are complex functions of series stiffness, sus-pension damping coefficient, and suspension stiffness. Forcomplete analysis, forced responses may have to be calculatedfor all combinations of these parameters. The process canbe expensive and may not provide insight into the problem.The aim is to find a combination resulting in large dampingratios and small shift in resonance frequency, without having


to perform numerous forced vibration analyses. In the nextsection, a state space representation is explored to findcomplex eigenvalues from which natural frequencies anddamping ratios can be extracted.

4. Natural Frequencies and ModalDamping Ratios

Equations of motion can be rearranged for state spaceformulation as given below:

x3 = − kdmv

x3 +kdmv

x2,

x2 = x1 − kd + kscs

x2 +kscsx1 +

kdcsx3.

(8)

The equation of motion for mu contains velocity of connec-tion point (Figure 1) between the suspension spring-dampercombination and the series stiffness. In the present form of(3), it prohibits straightforward calculation of eigenvalues.To overcome the difficulty, as the force transmitted is samethrough the suspension spring-damper combination and theseries stiffness, term corresponding to the suspension spring-damper in (3) is replaced by that of series stiffness; therefore,for free vibration analysis the equation becomes

mux1 + kd(x2 − x3) + ktx1 = 0

=⇒ x1 = − ktmu

x1 − kdmu

x2 +kdmu

x3.(9)

Let y1 = x1, y2 = x1, y3 = x2, y4 = x3, and y5 = x3.Therefore,

y =

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

y1

y2

y3

y4

y5

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

=

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

x1

x1

x2

x3

x3

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

, y =

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

y1

y2

y3

y4

y5

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

. (10)

In the state space form, equations of motion are writtenas

y = Ay, (11)

where

A =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 1 0 0 0

− ktmu

0 − kdmu

kdmu

0

kscs

1 −ks + kdcs

kdcs

0

0 0 0 0 1

0 0kdmv

− kdmv

0

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (12)

Natural frequencies and damping ratios of the system canbe obtained by solving eigenvalue problem det(A − Iλ). Theresulting eigenvalues are such that

λb = −ζbωb ± jωdb; λh = −ζhωh ± jωdh, (13)

0 1000 2000 3000 4000 5000 600010

12

14

16

18

20

22

24

26

28

30

Damping coefficient (Ns/m)

Nat

ura

l fre

quen

cy (

Hz)

5e4 N/m17e4 N/m29e4 N/m

53e4 N/m77e4 N/m113e4 N/m

Figure 3: Wheel hub natural frequency variation as a function ofthe suspension damping coefficient and the series stiffness.

where λb is the vehicle body eigenvalue, ωdb is the dampedvehicle natural frequency, ωb is the vehicle body naturalfrequency, ζb is the vehicle body modal damping ratio, λhis the wheel hop or hub eigenvalue, ωdh is the wheel hopor hub damped natural frequency, ωh is the wheel hop orhub natural frequency, and ζh is the wheel hop or hub modaldamping ratio.

Parametric studies were carried out to find variationof damping ratios and natural frequencies. The base datalisted in Table 1 is used. Figure 3 shows variation of wheelhub frequency as a function of the suspension dampingcoefficient and the series stiffness. There is very little spreadat low suspension damping values; in the extreme case of nodamping, first element of the stiffness matrix (7) tends tokt + kdks/(kd + ks), which for large-series stiffness becomeskt + ks. As the suspension damping coefficient increases,so does the frequency spread. For very large values ofsuspension damping coefficient, first element of the stiffnessmatrix now tends to kt + kd, resulting in an increase in thenatural frequency. The increase is gradual for lower seriesstiffness. In contrast, for larger series stiffness, the transitionfrom lower to higher natural frequency occurs within a smallrange of suspension damping coefficients. Ideally, in a gooddesign the change in natural frequency should be minimal.

The wheel hub modal damping ratio (Figure 4) variationas a function of suspension damping coefficient and seriesstiffness is more pronounced than corresponding naturalfrequency. There is a clear maximum damping ratio for eachof the series stiffnesses, increasing as the stiffness increases.The variation in damping ratio is smooth for lower seriesstiffness; detrimentally the maximum damping ratio reachedcan be practically very small. For the values of practicalimportance (series stiffness in the range of 1 × 105 N/m),the maximum damping ratio achieved can be as low asabout 0.2, which can result in large response amplitudeon the vehicle body due to wheel hub mode contribution.The sharp decrease in the damping ratio after reaching themaximum for larger stiffness is due to sudden increase in


0 1000 2000 3000 4000 5000 60000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


Dam

pin

g ra

tio,ζ h

5e4 N/m17e4 N/m29e4 N/m

53e4 N/m77e4 N/m113e4 N/m

Figure 4: Wheel hub modal damping ratio variation as a functionof the suspension damping coefficient and the series stiffness.

0 1000 2000 3000 4000 5000 60001

1.5

2

2.5

3

3.5

4


Nat

ura

l fre

quen

cy (

Hz)

5e4 N/m17e4 N/m29e4 N/m

53e4 N/m77e4 N/m113e4 N/m

Figure 5: Vehicle body mode natural frequency variation as afunction of the suspension damping coefficient and the seriesstiffness.

the natural frequency (Figure 3); it is inversely proportionalto the natural frequency.

Figure 5 shows vehicle body frequency variation as afunction of the suspension damping coefficient and theseries stiffness. For lower suspension damping values, thesuspension stiffness and the series stiffness combined resultin equivalent stiffness of kdks/(kd + ks), which tends tothe suspension stiffness ks for large-series stiffness. Hence,there is a spread of frequencies, which is in contrast tothe frequencies of the wheel hub mode. As the dampingcoefficient increases, the effective stiffness tends to kd. Thenatural frequency values show a jump from the lower to thehigher extreme for large-series stiffness.

Vehicle body mode damping ratios (Figure 6) are largerthan the hub mode damping ratios for smaller seriesstiffnesses. For larger stiffnesses, the maximum value ofdamping ratios is reached for similar values of suspension

0 1000 2000 3000 4000 5000 60000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


Dam

pin

g ra

tio,ζ b

5e4 N/m17e4 N/m29e4 N/m

53e4 N/m77e4 N/m113e4 N/m

Figure 6: Vehicle body modal damping ratio variation as a functionof the suspension damping coefficient and the series stiffness.

5e4 N/m17e4 N/m29e4 N/m

53e4 N/m77e4 N/m113e4 N/m

0 1000 2000 3000 4000 5000 60000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


Rat

io o

f da

mpi

ng

rati

os,ζ

h/ζ

b

Figure 7: Ratio of damping ratios varying as a function of thesuspension damping coefficient and the series stiffness.

damping coefficients unlike wider range observed for thewheel hub mode (compare Figures 6 and 4). In addition,the maximum values are reached at much larger suspensiondamping coefficients, and hence the drop in damping ratiofor a further increase may not be of practical significance.

Figure 7 shows the ratio of modal damping ratios; onlyfor a small range of series stiffness values, the ratio remainsconstant for a limited range of the suspension dampingcoefficients. Hence for the majority of parameter values, acompromise has to be reached to have suitable dampingratios for both modes. Comparison of Figures 4 and 6shows the criticalness of wheel hub modal damping. Thesuspension damping coefficient giving maximum modaldamping ratio is a key performance parameter. In thenext section, a simple expression is developed to estimatethis value based on the analysis of the effective dampingcoefficient.


5. Optimal Damping Coefficient ofthe Hub Mode

The effective damping relating to hub mode (see (6))shows two clearly defined trends: (a) for lower values ofsuspension damping coefficient it tends to k2

dcs/(kd + ks)2,

which for large series stiffness value gives cs and (b) forlarger suspension damping coefficients it tends to k2

d/csω2.

As the hub mode frequency is a complex function ofvarious parameters, it is difficult to evaluate the latter trend.An appropriate approximation could be based on the factthat the interest is about variation around the wheel hubfrequency. For typical vehicle applications, this frequencyvaries between 12 to 18 Hz; the midpoint value can besuitably assumed to find an estimate of optimum dampingcoefficient.

Before developing an expression for optimum dampingcoefficient, the effect of assuming a constant hub modefrequency is analysed. Figure 8 shows variation of effectivedamping coefficient (first element in the damping matrix,represented here as ceff, of (6)) as a function of suspensiondamping coefficient. The circular frequency is held constantat 94.25 rad/s (15 Hz). Also shown are the values based onexact calculation. The approximation is reasonable repre-sentation for up to about 3500 Ns/m, which is a very largedamping coefficient for the vehicle suspension systems. Theassumption of constant hub mode frequency in calculatingoptimum damping value, therefore, should not result insignificant errors.

The maximum effective damping coefficient leading toan optimal modal damping ratio can be obtained by thefollowing process:

dceff

dcs=

d(k2dcs/

((kd + ks)

2 + c2sω

2))

dcs= 0

=⇒ k2d

((kd + ks)

2 + c2sω

2)− 2k2

dc2sω

2 = 0

=⇒ cs = kd + ksω

.

(14)

The result of (14) is used in estimating the optimumeffective damping coefficient with hub mode frequencyheld constant at 15 Hz. Figure 9 shows the variation of theoptimal damping coefficient for a given series stiffness. Formost practical purposes (series stiffness of the order of tyrestiffness ranging from 0.5×105 to 2×105 N/m), the estimatesare in reasonable agreement. Equation (14), therefore, canbe used to obtain a suitable value of suspension dampingcoefficient that reduces the response of the vehicle body dueto contribution of the wheel hub mode.

6. Conclusions

A vehicle quarter car suspension model was refined toinclude the effect of series stiffness. A novel form of statespace equations was used to calculate the natural frequenciesand the modal damping ratios. The effect of the suspension

5e4 N/m17e4 N/m29e4 N/m

41e4 N/m65e4 N/m113e4 N/m

0 1000 2000 3000 4000 5000 60000

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Suspension damping coefficient (Ns/m)

Eff

ecti

ve d

ampi

ng

coeffi

cien

t (N

s/m

)

Figure 8: Effective damping coefficient variation as a function ofthe suspension damping coefficient and the series stiffness (. . .):approximate effective damping coefficient for the hub mode withfrequency 15 Hz. Other lines are exact calculations based on thenatural frequencies.

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4× 105

500

1000

1500

2000

2500

3000

Stiffness in series (N/m)

Opt

imal

su

spen

sion

dam

pin

g (N

s/m

)

Figure 9: Optimal suspension damping coefficient variation as afunction of the series stiffness (. . .): calculated based on the datafrom Figure 4 and (—): calculated based on (14).

damping and series stiffness was analysed, showing regionsof reduced vibration response.

The inclusion of series stiffness reduces the effectivedamping; the damping ratios achieved at two modes ofthe quarter car model are smaller than those based on theKelvin-Voigt model. The variation of the damping ratios isa nonlinear function of suspension damping coefficient andstiffness. In extreme cases, for larger suspension dampingcoefficients the resulting damping ratios could be negligiblysmall. The effect may not be equal on the modes of vibration.

The most significant effect of series stiffness was onthe wheel hop frequency and the amplitude. Increase indamping beyond the optimal values increases the amplitudeat resonance, having a negative impact on vehicle ridecomfort. A simplified expression for the optimal suspensiondamping coefficient was developed eliminating the needfor trial simulations. Overall, the model clearly shows


the influence of series stiffness on the modal damping ratios,the natural frequencies, and hence the dynamic response.

References

[1] T. D. Gillespie, Fundamentals of Vehicle Dynamics, Society ofAutomotive Engineers, Warrendale, Pa, USA, 1992.

[2] S. Duym, “An alternative force state map for shock absorbers,”Proceedings of the Institution of Mechanical Engineers D, vol. 211,no. 3, pp. 175–179, 1997.

[3] A. Simms and D. Crolla, The Influence of Damper Properties onVehicle Dynamic Behaviour, SAE, Warrendale, Pa, USA, 2002.

[4] M. Bouazara, M. J. Richard, and S. Rakheja, “Safety andcomfort analysis of a 3-D vehicle model with optimal non-linear active seat suspension,” Journal of Terramechanics, vol. 43,no. 2, pp. 97–118, 2006.

[5] M. Bouazara and M. J. Richard, “An optimization methoddesigned to improve 3-D vehicle comfort and road holdingcapability through the use of active and semi-active suspen-sions,” European Journal of Mechanics, vol. 20, no. 3, pp. 509–520, 2001.

[6] M. J. Griffin, “Discomfort from feeling vehicle vibration,”Vehicle System Dynamics, vol. 45, no. 7-8, pp. 679–698, 2007.

[7] J. Reimpell, H. Stoll, and J. W. Betzler, The Automotive Chassis:Engineering Principles, Butterworth-Heinman, Woburn, Mass,USA, 2nd edition, 2001.

[8] A. N. Thite, S. Banvidi, T. Ibicek, and L. Bennett, “Suspensionparameter estimation in the frequency domain using a matrixinversion approach,” Vehicle System Dynamics, vol. 49, no. 12,pp. 1803–1822, 2011.

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