Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

46, 3, pp. 581-596, Warsaw 2008

NONLINEAR VIBRATION MODES OF THE DOUBLE
TRACKED ROAD VEHICLE

Yuri Mikhlin

Sergei Mytrokhin

National Technical University ”Kharkov Polytechnic Institute”, Kharkov, Ukraine

e-mail: muv@kpi.kharkov.ua

Free damped oscillations of a double tracked roadvehiclewith a nonline-
ar responseof the suspensionareconsidered.A7-DOFnonlinearmodel is
used to analyze dynamics of the suspensionwith smooth characteristics.
Vibrationmodes of the systemand the corresponding skeleton curves are
obtained. Nonlinear vibrations of the quarter-car model are considered
for the case of non-smooth characteristics of the shock absorber.

Key words: vehicle suspension, nonlinear normal modes, smooth and
non-smooth characteristics

1. Introduction

VEstimation of the vehicle dynamical state is essential for all designing auto-
motive systems that enhance safety and handling characteristics. Many works
have been carried out on the dynamic response and the dynamics control with
linear vehicle models. However, substantially, the system is nonlinear because
it consists of elastic components with nonlinear characteristics. A vehicle can
bemodelled as a complexmulti-body dynamic system. The degree of comple-
xity dependson the aimofmodelling.Asa rule, the simplestmodels of vehicles
are considered, in particular, the ”quarter-car” model (two-DOF system) for
studyingheavemotion (Robson, 1979;Williams, 1997; Haroon et al., 2005), or
”half-car”model (four-DOF system) for studying the heave andpitchmotions
(Hrovat, 1991; Vetturi et al., 1996; Marzbanrad et al., 2004).

A 7-DOF model of the vertical and axial vehicle dynamics is considered
here for the case of an independent-solid axle suspension to predict behaviour
of the vehicle body andwheels. It is possible to study, by using thismodel, all



582 Yu. Mikhlin, S. Mytrokhin

principal vehiclemotions, namely, heave, pitch and roll (Wong, 1993; Hyo-Jun
Kim et al., 2002; Pilipchuk et al., 2006). Correctness of thismodel is confirmed
by comparisonwith some experiments. It is known that nonlinear effects in the
suspension dynamics are important and must be considered if displacements
are equal to 0.05-0.1m, or larger.Here, smoothnonlinear springcharacteristics
in the front and rear suspensions are taken into account.

Nonlinear normal modes are a generalization of linear normal vibration
modes.We can suppose that inmany cases, for example after an impact, such
regimes are or may be the most important to describe the vehicle nonlinear
dynamics, because a transient here is very short in viewof the fact that there is
a strong dissipation in the systemunder consideration.Moreover, an approach
to the nonlinear normal modes can be successfully used for smooth and for
non-smooth characteristics of the car suspension. As we can predetermine,
such analysis of nonlinear normal modes for the vehicle model was not made
and published previously.

In this paper, the principal 7-DOF model is described in Section 2 at
first. In Section 3, normal vibrations are obtained in the linearised model by
using a solution to the eigenvalue and eigenvector problem. Then, a concept
of the nonlinear normal vibration modes (NNMs) is presented. These NNMs
and corresponding skeletons are obtained for the nonlinear 7-DOF system
withnonlinear smooth characteristics. In Section 4, nonlinear vibrations of the
quarter-carmodel are considered for the case of a non-smooth characteristic of
the shockabsorber.Both thepiecewise linear characteristic of theabsorberand
more realistic piecewise cubic shock absorption characteristic are considered.
Nonlinear elastic characteristics of the suspension are taken into account in
that Section.
Note that analysis of the nonlinear normalmodes is crucial for the system

because damping is strong here and any transients is a short-term process.

2. Principal model and equations of motion

In order to describe the vertical dynamics of a double-tracked road vehicle,
the 7-DOF model (Fig.1) is used, which is based on works by Wong (1993)
and Hyo-Jun Kim et al. (2002). The car body is represented as a rigid bo-
dy. The heave, roll and pitch motions are considered. Here, z is the vertical
displacement, α is the pitch angle, β is the roll angle, xi are the vertical di-
splacements of the ith suspendedmass which are equivalent to the wheel, d1,
d2 are front and rear track widths and l1, l2 are front and rear wheel bases.



Nonlinear vibration modes... 583

In this model, the tires are presented as elastic elements with linear charac-
teristics. The suspension is characterised by nonlinear elastic characteristics
of the front and rear springs and linear damping characteristics (the shock
absorber will be considered later). Typical elastic characteristics are shown in
Fig.2 (some information on such characteristics can be found, for example, in
Zhu and Ishitobi (2004)). Elastic forces appearing in the springs f1(x) and
f2(x) can be correctly approximated by polynomials of the 7-th degree.

Fig. 1. Model of a double-tracked road vehicle under consideration

Fig. 2. Typical nonlinear characteristics of the front f1(x) and rear f2(x) tracks

One has seven generalized coordinates to describe vibrations of the mo-
del. The displacements of the body mass center are described by the vector
q = [z,α,β]⊤, and the displacements of the suspended masses, which are
equivalent to the wheels, by the vector x = [x1,x2,x3,x4]

⊤. The matrices
of the body inertia MC, of the suspension elements MS, of the tire stiff-
ness C, and of the damping K are diagonal. The angular displacements of the



584 Yu. Mikhlin, S. Mytrokhin

bodyare connectedwith the displacements of themass center by the following
matrix

H=











1 −l1 d1
1 −l1 −d1
1 l2 d2
1 l2 −d2











(2.1)

One writes the difference between the displacements of the body and the
elements of suspension as U = Hq−x, and the difference of velocities as
V =Hq̇− ẋ. A vector of the nonlinear characteristics can be written in the
form

CNL = [f1(U1),f1(U2),f2(U3),f2(U4)]
⊤ (2.2)

where Ui are components of the vector U.

Finally, one has the following ODE system (in the matrix form) which
describes free nonlinear vibrations of the car

MCq̈+H
⊤
CNL+H

⊤
KV =0

(2.3)

MSẍ−CNL+Cx−KV =0

Values of the car parameters which were used in the calculations are pre-
sented in Table 1.

Table 1.Assumed car parameters

Parameter Value Parameter Value

M 2369kg k2 900kg/s

Ix 4108kgm
2 ct 258000n/m

Iy 938kgm
2 l1 1.459m

m1 77kg l2 1.486m

m2 108.5kg d1 0.868m

k1 700kg/s d2 0.837 m

3. Nonlinear normal modes and transient (smooth characteristics
of the suspension)

First, normalvibrationmodesof the linearised systemwereobtainedbysolving
the eigenvalue andeigenvector problem.Theamplitudesof vibrationmodes for
some concrete parameters of the vehicle are shown inFig.3. The frequencies of



Nonlinear vibration modes... 585

modes are given at the bottom of this figure. The amplitudes of displacements
and angles are presented in m and rad, respectively. Besides, all eigenvectors
are normalized. Comparison of motions in time of the full nonlinear system
and the corresponding linearised one will be shown later.

Fig. 3. Amplitudes of seven linear normal modes

Thenonlinear normal vibrationsmodes (NNMs) are a generalization of the
normal vibrations of linear systems. In the normal mode, a finite-dimensional
system behaves like a conservative one having a single degree of freedom, and
all position coordinates can be analytically parameterised by any of them
(Rosenberg, 1966; Mikhlin, 1996; Vakakis et al., 1996). In general, NNMs tra-
jectories in a configuration space are curvilinear instead of straight lines as for
linear systems.

In Shaw and Pierre (1991, 1993), the authors reformulated the concept
of NNMs for a general class of nonlinear discrete oscillators. The analysis is
based on computation of invariant manifolds of motion on which the NNMs
take place. Some mechanical applications of this approach are presented in
Shaw and Pierre (1991, 1993) and other publications by the same authors.

To use this approach, original ODE system (2.3) must be presented in the
standard form

dx

dt
=y

dy

dt
= f(x,y) (3.1)

where x = [x1, . . . ,xN]
⊤ is the vector of generalized coordinates, y =

= [y1, . . . ,yN]
⊤ is the vector of generalized velocities, and f = [f1, . . . ,fN]

⊤

is the vector of forces. One chooses a couple of new independent variables



586 Yu. Mikhlin, S. Mytrokhin

(u,v), where u is some dominant generalized coordinate, and v is the cor-
responding generalized velocity. By the Shaw-Pierre approach, the nonlinear
normal mode is such a regime when all generalized coordinates and velocities
are univalent functions of the selected couple of variables. Choosing the selec-
ted couple of variables as the coordinate and velocity with the index 1, one
writes the nonlinear normal mode of the form

























x1
y1
x2
y2
...
xN
yN

























=

























u

v

X2(u,v)
Y2(u,v)
...

XN(u,v)
XN(u,v)

























(3.2)

Computingderivatives of all variables in system (3.2), and taking into account
that u = u(t) and v = v(t), then substituting the obtained expressions to
system (3.1), one obtains the following system of partial derivative equations
(i=1, . . . ,N)

∂Xi

∂u
v+
∂Xi

∂v
f1(x,y)=Yi(u,v)

(3.3)

∂Yi

∂u
v+
∂Yi

∂v
f1(x,y)= fi(x,y)

One can present the solution to the system in the form of a power series
with respect to the new independent variables u and v

xi =Xi(u,v) = a1iu+a2iv+a3iu
2+a4iuv+a5iv

2+ . . .
(3.4)

yi =Yi(u,v) = b1iu+ b2iv+ b3iu
2+ b4iuv+ b5iv

2+ . . .

Series (3.4) are introduced to equations (3.3), then the coefficients of inde-
pendent variables of the same degree, are equated. So, a system of recurrent
algebraic equations can be written. The coefficients of series (3.4) can be de-
termined from these equations, and as a result, the corresponding nonlinear
normal mode is obtained. This procedure permits one to obtain seven NNMs
of the system under consideration. Few surfaces, which characterise the first
NNM, and the corresponding trajectories ofmotion on these surfaces for some
assumed coordinates are shown in Fig.4. Here the coordinate z describing the
vertical displacement is chosen as the independent variable u.



Nonlinear vibration modes... 587

Fig. 4. First vibrationmode of the nonlinear system (independent coordinates are
u= z and v= ż; initial values: z=0.075, and ż=0); change of the angle α (a),
displacement x1 (b), angle velocity α̇ (c), displacement velocity ẋi (d) depending on

the variables u and v

If the NNM in form (3.4) is obtained, these series are introduced to equ-
ations of motion, and the functions u = u(t) and v = v(t) can be obtained
too. Numerical calculations show good exactness of the obtained analytical
results. To construct skeletons of the NNMs, a harmonic linearisationmethod
together with a continuation procedure are used. Skeletons of the nonlinear
system are shown in Fig.5.

Fig. 5. Skeletons of nonlinear normal modes for the body center (a) and for
unsprungmasses (b)



588 Yu. Mikhlin, S. Mytrokhin

Transient and normal vibrationmodes of the linearised and nonlinear sys-
tems for different initial conditions of the car body are shown in Figs. 6-8.

Fig. 6. Comparison of transients for linearised and nonlinear systems for the initial
z=100mm; (a) pitch angle α, (b) displacement of the unsprungmass x1

Fig. 7. Displacement of the unsprungmass x1 (a) and x3 (b). Comparison of
transients for linearised and nonlinear systems for the initial α=3.5◦



Nonlinear vibration modes... 589

Fig. 8. Displacement of the unsprungmass x1. Comparison of transients for
linearised and nonlinear systems for the initial β=6◦

Onecanobserve that in thecase of large initial displacements, thevibration
frequency of the unsprung masses quickly lowers and all nonlinear vibration
modes tend to low-frequency vibration modes of the car body.

4. Nonlinear normal modes and transients (shock absorber,
non-smooth characteristic)

4.1. Quarter-car model. Equations of motion

To investigate the suspension dynamics taking into account a non-smooth
characteristic of the shock absorber, the quarter-car model is considered
(Fig.9).

Fig. 9. Quarter-carmodel

The equations of motion for the quarter-car model are

Mẍ1+f(x1−x2)+d(ẋ1− ẋ2)= 0
(4.1)

mẍ2+f(x2−x1)+d(ẋ2− ẋ1)+ ctx2 =0



590 Yu. Mikhlin, S. Mytrokhin

where f(x) is a stiffness function, d(ẋ) is a piecewise damping function of the
suspension, namely

d(ẋ)=

{

d1(ẋ) for ẋ1− ẋ2 < 0

d2(ẋ) for ẋ1− ẋ2 ­ 0
(4.2)

Values of the car parameters, which were used in the calculations, are given
in Table 2.

Table 2.Car parameters used in the ”quarter-car model”

Parameter Value

M 592.25kg

m 108.5kg

ct 258000n/m

f(x) 55000x

4.2. Piecewise linear characteristics of the absorber

The characteristic of the damping functions for the case of piecewise linear
damping is presented in Fig. 10. This is an approximation of a more exact
characteristic, which will be considered later. The stiffness characteristic of
the suspension is chosen here as linear.

Fig. 10. Piecewise linear shock absorption characteristic of the suspension

BothNNMs obtained by themethoddescribed below are shown inFigs.11
and 12 for some concrete values of the system parameters. One can observe
them as motions on places corresponding to the NNMs as well as transients
fromoneplace to another one after gap (or ”switching”) of thepiecewise linear
damping characteristic. The time representation of these nonlinear normal
modes and transients are shown in Figs.13 and 14.



Nonlinear vibration modes... 591

Fig. 11. Change of the displacement x2 (a) and the displacement velocity ẋ2
(depending on u and v). First NNM. Independent variables are: x1 and the

corresponding velocity ẋ1 (piecewise linear characteristic)

Fig. 12. Change of the displacement x2 (a) and the displacement velocity ẋ2
(depending on u and v). Second NNM. Independent variables are: x2 and the

corresponding velocity ẋ2 (piecewise linear characteristic)



592 Yu. Mikhlin, S. Mytrokhin

Fig. 13. Change of the displacement x2 (a) and the displacement velocity ẋ2. First
NNM and transient

Fig. 14. Change of the displacement x1 (a) and x2 (b). Second NNM and transient

More realistic nonlinear suspension characteristics for the stiffness and ab-
sorption (for piecewise cubic damping) are displayed inFigs.15 and 16. There,
the car principal parameters correspond to those from Table 2. The corre-
spondingNNMs, obtained in that case, and the transient from one surface to
another one after gap (or ”switching”) in the piecewise cubic damping charac-



Nonlinear vibration modes... 593

Fig. 15. Nonlinear cubic stiffness characteristic of the suspension

Fig. 16. TPiecewise cubic shock absorption characteristic of the suspension

Fig. 17. Change of the displacement x2 (a) and the displacement velocity ẋ2
(depending on u and v). First NNM. Independent variables are x1 and the

corresponding velocity ẋ1 (piecewise cubic characteristic)



594 Yu. Mikhlin, S. Mytrokhin

teristic are shown in Figs.17 and 18. The first NNM time-representation and
transient are given in Fig.19.

Fig. 18. Change of the displacement x1 (a) and the displacement velocity ẋ1
(depending on u and v). Second NNM. Independent variables are x2 and the

corresponding velocity ẋ2 (piecewise cubic system)

Fig. 19. Change of the displacement x2 (a) and the displacement velocity ẋ2. First
NNM and transient



Nonlinear vibration modes... 595

Note that in all cases, the dominant pair of phase variables is chosen as
independent coordinates in the analytical representation of NNM.

5. Conclusions

In the paper, nonlinear dynamics of a double-tracked road vehicle with a non-
linear response of the suspension is considered. A 7-DOF nonlinear model is
used in the investigation. The concept of nonlinear normal vibration modes
was used in themodel analysis for the first time.Themethod of nonlinear nor-
malmodes permits one to describedynamic behaviour of the nonlinear system
both for the case of a smooth nonlinear characteristic of the suspension and
for the case of a non-smooth characteristic of the shock absorber. In the last
case, one can observe a very fast transfer from one nonlinear normal mode
to another one after the gap in the non-smooth characteristic. It is possible
to conclude that in many cases, the nonlinear normal modes are dominant
regimes in the vehicle dynamics.

References

1. Haroon M., Adams D.E., WahLuk Y., Ferri A.A., 2005, A time and
frequency domain approach for identifying nonlinearmechanical systemmodels
in the absence of an inputmeasurement, Journal of Sound and Vibration, 283,
1137-1155

2. Hrovat D., 1991, Optimal suspension performance for 2-D vehicle models,
Journal of Sound and Vibration, 146, l, 93-110

3. Hyo-Jun Kim, Hyun Seok Yang, Young-Pil Park, 2002, Improving the
vehicle performancewith active suspension using road-sensing algorithm,Com-
puters and Structures, 80, 1569-1577

4. Ikenaga S., Lewis F.L., Campos J., Davis L., 2000, Active suspension
control of ground vehicle based on a full-vehiclemodel,Proceedings of America
Control Conference, Chicago, IL, USA

5. MarzbanradJ.,AhmadiG.,ZohoorH.,HojjatY., 2004,Stochastic opti-
mal preview control of a vehicle suspension, Journal of Sound and Vibration,
275, 973-990

6. Mikhlin Yu., 1996, Normal vibrations of a general class of conservative oscil-
lators,Nonlinear Dynamics, 11, 1-16



596 Yu. Mikhlin, S. Mytrokhin

7. Pilipchuk V., Moshchuk N., Chen S.-K., 2006, Vehicle state estimation
using dynamic 7 degree-of-freedom observer,Proceedings of IMECE2006, Chi-
cago, USA, 1-8

8. Robson J.D., 1979, Road surface description and vehicle response, Interna-
tional Journal of Vehicle Design, 9, 25-35

9. Rosenberg R., 1966, Nonlinear vibrations of systems with many degrees
of freedom, Advances of Applied Mechanics, 9, Academic Press, New York,
156-243

10. Shaw S., Pierre C., 1991, Nonlinear normalmodes and invariantmanifolds,
Journal of Sound and Vibration, 150, 170-173

11. Shaw S., Pierre C., 1993, Normal modes for nonlinear vibratory systems,
Journal of Sound and Vibration, 164, 85-124

12. Vakakis A., Manevitch L., Mikhlin Y., Pilipchuk V., Zevin A., 1996,
NormalModes and Localization in Nonlinear Systems,Wiley, New-York, p.552

13. VetturiD.,GadolaM.,CambiaghiD.,ManzoL., 1996,Semi-active stra-
tegies for racing car suspension control, IIMotorsports Engineering Conference
and Exposition, Dearborn, USA, SAE Technical Papers, 962553

14. WilliamsR.A., 1997,Automotive active suspensions, Part 1: basic principles,
Proceedings of the Institute of Mechanical Engineers, Journal of Automobile

Engineering, D211, 415-426

15. Wong J.Y., 1993, The Theory of Ground Vehicles, 2nd ed. New York: John
Wiley

16. Zhu Q., Ishitobi M., 2004, Chaos and bifurcations in a nonlinear vehicle
model, Journal of Sound and Vibration, 275, 1136-1146

Postacie drgań nieliniowych dwuśladowego pojazdu drogowego

Streszczenie

W pracy rozważono tłumione i swobodne drgania dwuśladowego pojazdu drogo-
wego o nieliniowej charakterystyce zawieszenia. Pojazd zamodelowano nieliniowym
układem o 7 stopniach swobody, a w analizie jego dynamiki przyjęto gładką charak-
terystykę zawieszenia. Znaleziono postacie drgań takiego układu i wyznaczono odpo-
wiednie krzywe szkieletowe. Zagadnienie drgań nieliniowych modelu ”ćwiartkowego”
(o 2 stopniach swobody) rozważono dla przypadku zawieszenia z niegładką charakte-
rystyką amortyzatora.

Manuscript received December 5, 2007; accepted for print January 30, 2008