PAMM · Proc. Appl. Math. Mech. 11, 343 – 344 (2011) / DOI 10.1002/pamm.201110163

On the influence of design parameters and nonlinearities in vehicle

suspensions on road deformation

Heike Vogt1,∗, Hartmut Hetzler1, and Wolfgang Seemann1

1 Karlsruhe Institute of Technology (KIT) – Institute of Engineering Mechanics (ITM) – 76131 Karlsruhe, Germany

Among others, two main objectives of modern vehicle design are road friendliness and ride comfort. Both aspects are strongly

related since the dynamical tire forces depend on the vertical acceleration of the vehicle. In order to investigate the influence

of design and operation parameters, different car models are considered which move with constant velocity on a rippled road.

First, a linear half car model is examined and the influence of different design parameters is discussed. Second, nonlinear

suspensions with Coulomb friction due to sealings as well as with bilinear shock absorbers are taken into account. The vertical

dynamics of the vehicle model and the dynamic tire forces between vehicle and road are calculated using analytical methods.

c© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Modeling

In order to investigate the vertical dynamics of vehicles and their influence on the deformation of roads, simple vehicle

models are considered. The vertical displacements of their suspension points from static equilibrium is denoted by y. Since

the deformations of the road are much smaller than the oscillations of the vehicle and take also place on a much slower

time scale, the current road surface is assumed to be rigid during one passage and given as a general harmonic function

un(x) = un cos(Ωwx+∆ϕn). Thus, in an analytical way the normal contact forces Ni(x) between vehicle and road can be

calculated. Their influence on the evolution of the surface deformation can be determined by assuming a functional relation

∆u(Ni) between the loading Ni and the permanent deformation ∆u(x).

2 Linear 2DoF model

First, a linear half car model moving with constant velocity v on a rippled road is investigated (see Fig. 1) in order to determine

the influence of some design parameters on the deformation of roads. The linear spring-damper elements of the 2DoF oscillator

(mass m, moment of inertia J , wheelbase sf + sr) represent the suspensions and tires. The spring and damping constants

are denoted by ci and di(i = r, f). Assuming a linear relation ∆ui = p · Ni between the permanent deformation ∆ui and

the normal tire forces Ni, the road surface after K vehicles have passed over the initial surface u0(x) = u0 cos(Ωwx) can

be calculated by uK(x) = V K−1u0 cos[Ωwx + (K − 1)∆ε]. Here the amplification function V and the phase shift ∆ε are

functions of the design, operation parameters and the factor p. By determing V and ∆ε the evolution of the road surface

concerning amplification of the amplitude and the direction of the shift of the ground waves can be predicted.

For an undamped, symmetric half car model it is found that only amplification of the road amplitude and no shift of the

ground waves occur, i.e. ∆ε = 0 (cf. [1]). Breaking the symmetry (e.g. by sf 6= sr) changes the magnitude of the amplification

of the road amplitude, but still the road waves are not shifted. For the undamped oscillator the influence of the asymmetry is

depicted in Fig. 2, where ξ = cr/cf and γ = sr/sf hold.

Introducing damping has a significant influence as it leads to a shift of the road waves as well as to a smaller change of

the road amplitude. The damping parameter β = 2κβ (κ2 = cf/m and β = df/cf = dr/cr) plays a crucial role for the

amplification of the road amplitude as can be seen in Fig. 3 and Fig. 4, where the amplification as a function of the excitation

parameter η = vΩw/2κ and the mass distribution parameter µ = J/ms2f is depicted. Whereas in Fig. 3 with β = 0.4 still

parameter regions with a decrease of the amplitude (sgn(V − 1) < 1) can be found, for higher damping (e.g. β = 1) only

regions with an increase of the amplitude (V > 1) can be found (Fig. 4).

Fig. 1 Linear 2DoF oscillator.

1.051.06

1.061.061.07

1.071.07

1.08

1.081.08

1.08

1.09

1.091.09

1.091.1

1.1

1.1

1.121.13

ξ

γ

V

symmetry

0.8 1 1.2

0.7

0.8

0.9

1

1.1

1.2

Fig. 2 Amplification of road

amplitude due to asymmetry.

0 1 20

1

2

3

4

5 sgn(V −1)

η

µ

−1

−0.5

0

0.5

1

Fig. 3 In-/decrease of the

amplitude for β = 0.4.

0 1 20

1

2

3

4

5 V

η

µ

1

1.2

1.4

1.6

1.8

2

Fig. 4 Amplification of

the amplitude for β = 1.

∗ Corresponding author: email heike.vogt@kit.edu, phone +49 721 608 46823, fax +49 721 608 46070

c© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

344 Section 5: Nonlinear oscillations

3 Nonlinear 1DoF model

In order to investigate the influence of nonlinearities, a quarter car model is considered, which is connected via a linear spring

(spring constant c) and a nonlinear damper to the ground. The damper consists of a linear part (damping constant d) and

a bilinear damping with different damping constants for the pressure and tension region: hence, the damping force reads

Fbd = d(y − u+ ζ|y − u|), where ζ is a non-dimensional constant. Coulomb friction can be captured by an additional force

FCd = Rsgn(y − u). If scaled displacements ( ) = ( )/u, scaled time τ = ω0t, ω2

0= c/m (thus d

dτ() = ()′) as well as the

transformation x = y − u are introduced, the system can be described in dimensionless form as

x′′ + βx′ + ζβ|x′|+ rsgn(x′) + x = −u′′. (1)

With the general base excitation un = un

ucos(ητ + ψn), η = Ω/ω0 a first order harmonic balance approach xn = a0 +

a1 cos(ητ + ψn − ε) yields the approximate solution xn. It is found, that bilinear damping, i.e. ζ has no influence on the

amplitude a1 and the phase shift ε of the approximate solution xn. However, bilinear damping leads to a shift of the mean

value in negative direction, e.g. a0 < 0. For physically realstic solutions additional Coulomb friction changes the magnitude

of a0, but not the sign. The normalized dynamic normal tire force

n = N/(cu) = x+ βx′ + ζβ|x′|+ rsgn(x′) (2)

can be calculated with the solution x. Fig. 5 and Fig. 6 illustrate the influence of bilinear damping and Coulomb friction on the

normal force, whereas the linear normal force (dashed grey line), the normal force calculated with the approximate solution x(grey line) and a numerically calculated solution for the motion (black line) are shown. As can be seen the nonlinearities lead

to distorted nonharmonic normal tire forces with different maxima than in the linear case and higher harmonics, which will

be carved into the road surface. As a consequence, the spectral components of the road surface will develop in a cascade like

way with every passage. Developing the approximate normal forces into a Fourier series shows that with bilinear damping the

normal forces and thus the deformed road surface contain only the base frequency and even multiples whereas with Coulomb

friction only the base frequency and odd multiples occur. The prediction of the evolution of the road surface can be compared

with numerical results. In Fig. 7 and 8 the numerically calculated evolution of the amplitudes of the spectral components

of the road surface for bilinear damping and Coulomb damping respectively is depicted. Whereas the analytical approach

predicts the evolution of the same spectral components of the road as the numerical simulation does for Coulomb damping,

the numerical results in case of bilinear damping differ from the analytical prediction.

0 10 20 30

−2

−1

0

1

2

x [m]

n[1

]

Fig. 5 Normal forces with bilinear

damping.

0 10 20 30

−2

−1

0

1

2

x [m]

n[1

]

Fig. 6 Normal forces with

Coulomb damping.

1 2 3 4 5 6 7 8 9 10

20

40

60

80

100

·η

ui

#ve

h

Fig. 7 Spectral components

with bilinear damping.

1 3 5 7 9

20

40

60

80

100

·η

ui

#ve

h

Fig. 8 Spectral components

with Coulomb damping.

4 Conclusion and Outlook

Based on analytical reflections the influence of design parameters and nonlinearities in vehicle suspensions on the normal tire

forces and the displacement of the road surface has been investigated. It has been found that bilinear damping and Coulomb

friction cause distorted nonharmonic normal tire forces with different and shifted maxima compared to linear suspensions.

The higher harmonics in the normal forces due to the nonlinearities are reflected in the road surface whose spectral content

develops in a cascade like way. The lowering of the vehicle due to bilinear damping and the resulting shift of the operating

point of the spring suggest to investigate vehicle models with nonlinear spring characteristics. In addition, extensive parameter

studies can be carried out with semi-analytical solutions for the vertical tire forces at hand. Finally, the approximate approach

should be improved using higher harmonics, especially when including nonlinearities in the half car model.

References[1] H. Vogt and W. Seemann, On Dynamic Aspects and Pattern Generation in Vehicle-Road Interaction, Proc. Appl. Math. Mech., Karls-

ruhe, Germany, 10(1), pp. 267–268, 2010.[2] W. V. Wedig: Vertical Dynamics of Riding Cars Under Stochastic and Harmonic Base Excitation. IUTAM Symposium on Chaotic

Dynamics and Control of Systems and Processes in Mechanics, 122:371-381, 2005.[3] G. Genta, et.al.: An Approximated Approach to the Study of Motor Vehicle Suspensions with Nonlinear Shock Absorbers, Meccanica,

24:47-57, 1989.

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