-
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
mvẍ3 + kdx3 − kdx2 = 0, (1)csẋ1 − csẋ2 + ksx1 − ksx2 − kdx2 +
kdx3 = 0, (2)muẍ1 + cs(ẋ1 − ẋ2) + 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 + c2sω2)
−k2dcs((kd + ks)
2 + c2sω2)
−k2dcs((kd + ks)
2 + c2sω2)
k2dcs((kd + ks)
2 + c2sω2)
⎤
⎥⎥⎥⎥⎥⎦
, (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 − k
2d(kd + ks)(
(kd + ks)2 + c2sω2
)
⎤
⎥⎥⎥⎥⎥⎦.
(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
-
4 Advances in Acoustics and Vibration
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:
ẍ3 = − kdmv
x3 +kdmv
x2,
ẋ2 = ẋ1 − 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
muẍ1 + kd(x2 − x3) + ktx1 = 0
=⇒ ẍ1 = − ktmu
x1 − kdmu
x2 +kdmu
x3.(9)
Let y1 = x1, y2 = ẋ1, y3 = x2, y4 = x3, and y5 =
ẋ3.Therefore,
y =
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
y1y2y3y4y5
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
=
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
x1ẋ1x2x3ẋ3
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
, ẏ =
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
ẏ1ẏ2ẏ3ẏ4ẏ5
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
. (10)
In the state space form, equations of motion are writtenas
ẏ = 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
-
Advances in Acoustics and Vibration 5
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
Damping coefficient (Ns/m)
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
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 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
Damping coefficient (Ns/m)
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
Damping coefficient (Ns/m)
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.
-
6 Advances in Acoustics and Vibration
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 k2dcs/(kd + ks)
2,which for large series stiffness value gives cs and (b)
forlarger suspension damping coefficients it tends to k2d/csω
2.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:
dceffdcs
=d(k2dcs/
((kd + ks)
2 + c2sω2))
dcs= 0
=⇒ k2d(
(kd + ks)2 + c2sω
2)− 2k2dc2sω2 = 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
-
Advances in Acoustics and Vibration 7
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.
-
International Journal of
AerospaceEngineeringHindawi Publishing
Corporationhttp://www.hindawi.com Volume 2010
RoboticsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Hindawi Publishing Corporation http://www.hindawi.com
Journal ofEngineeringVolume 2014
Submit your manuscripts athttp://www.hindawi.com
VLSI Design
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Shock and Vibration
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
The Scientific World JournalHindawi Publishing Corporation
http://www.hindawi.com Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Modelling & Simulation in EngineeringHindawi Publishing
Corporation http://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
DistributedSensor Networks
International Journal of