Dynamics of non-linear automobile shock-absorbers

Inr. J. Non-Lmear

Mechanm,

Vol. 25. No. 2;). PP. 299-308.

1990

Pnnted in Great Bntain.

DYNAMICS

0

0020-746390 s3.ao+o.al 1990 Pergamon Press plc

OF NON-LINEAR AUTOMOBILE SHOCK-ABSORBERS JORGWALLASCHEK

Institut fur Mechanik, Technische Hochschule Darmstadt, HochschulstraBe 1, D-6100 Darmstadt, West Germany Abstract-In this paper the methods of harmonic and stochastic linearization are discussed with respect to applications in shock-absorber dynamics. It is shown that the parameters of an equivalent linear system can be obtained directly from experimental data. In a second step an attempt is made to give a simple physical interpretation of the experimental results, which were obtained for a typical passenger car’s shock-absorber.

INTRODUCTION Active and semi-active control of vehicle dynamics has become a subject of major interest during recent years [l-4], and today simple “damping control systems” are incorporated in serial constructions of passenger cars [S]. Rapid progress in the analysis, design and technology of the control systems leads to the need for an accurate description of the dynamics of all components involved, such as, for instance, the tire or shock-absorber. It is well known that automobile shock-absorbers are non-linear dynamical systems [6,7]. The models used to analyze vehicle dynamics, and especially ride-comfort, are mainly linear, with only some isolated non-linear elements [S,9]. If only the system’s overall dynamic response is of interest, these isolated elements can be taken care of by equivalent linearization techniques. The shock-absorber or the tire is then replaced by linear elements whose parameters can be obtained analytically, if the governing non-linear equations are known. Most often, however, this is not the case and one must resort to experimental data. The parameters of the linear models obtained by equivalent linearization do, of course, depend on the test-signal. By using various test-signals one can sometimes deduce the non-linearities in the system simply by observing the corresponding variation of the equivalent linear system’s parameters. It is then possible to determine the coefficients of an ad-hoc postulated non-linear differential equation for the original non-linear system. In what follows we first briefly review equivalent harmonic and stochastic linearization; we then show how the linearization can be performed by using experimental data and give some numerical results for a typical shock-absorber. We finally discuss some simple dynamical models for the shock-absorber with respect to the experimental results.

EQUIVALENT

LINEARIZATION

It is clear, and this has been well accepted, that a mathematical model-linear or non-linear-of a real dynamic system yields, at best, an appproximation to the “true” system behavior. Although for certain applications a non-linear model is indispensable to describe a system, there are cases when linear models are of sufficient accuracy. Many linearization techniques have been proposed. Here, however, only harmonic and stochastic linearization-are considered. These techniques are based on the method of Krylov et al. [lo]. Harmonic linearization is limited to a description of the system’s dynamics for harmonic motion or nearly harmonic motion only. Stochastic linearization, which is an extension of Krylov et al.‘s method to problems with random excitation, was developed independently by Booton [ll], Kazakov [12] and Caughey [13,14]. It is usually restricted to stationary ergodic random excitations. As the idea behind both methods is the same and the techniques are very similar, we will discuss them together. Let us assume that a dynamical system is described by the non-linear differential equation

g(t) + gCx(tX2 @)I = F(r), 299

(1)

300

J.

WALLASCHEK

where F(r) is a harmonic function or a stationary ergodic random process, x(r) describes the system response and y (. , .) is a non-linear function of the state variables. The basic idea of equivalent linearization is to replace the non-linear differential equation (1) by a linear one: j(r) + aj (0 + by(c) = F(c) (2) and to choose the coefficients a and b so as to minimize the difference between x(c) and y(t) in a certain sense. In the general case we will have to take into account that g (. , .) is not an odd function and that the mean of F(t) is not zero. Correspondingly, we first separate the fluctuating parts x,(c) = .x(c) - E(x(c)~, (3)

Fi (0 = F(r) - E {F(r)) from (1) by subtracting the mean E{gCx(r),i where E { } denotes an operator. defined as

(4)

@)I) = EIF(O)>

In the case of harmonic linearization

E{x(c)j

this operator is

x(r)dr,

=;

(5a)

with

being the period of vibration associated with the circular frequency R of the external force. In the stochastic case, E ( } denotes the expectation operator 7. x(c) dr , E{x(r)} = lim -L (5b) i--,2T s -T and as we assume stationarity

and ergodicity, it may also be defined as T xp (x) ds , Efxtr)) = s -II

where p(x) is the probability density function of x(t), which is independent stationary processes. After separating the mean (4) from (1) we finally obtain

(5c) of c for

ii (0 + g [(x(r), .t (01 - E (9 Cx(0, i @)I> = 4 (0

(7)

for the fluctuating parts. The differential equation (7) is then replaced by (8)

~~(O+aj,,(O+by,(O=F,(O and the parameters a and b will be used to minimize the error e(r): = aa, which arises when the to note that we define problem. The error e(t) can minimize the expected leads to

Cl= E{ii(r)gCx(O,i E(~;-:W b

=

+ bx,(t) - gGx(O,21 (03 + E(gCx(r),i

@)I)

non-linear function g (. , .) is replaced by a linear one. It is important the error (9) based upon the solution x(t) of the original non-linear be minimized in various ways; the most common practice is to value of its square E {e’(t)}. In this case, straightforward calculation

@)I) = E{~,(t)gCx(O,~ (01) E(i*

(0)

(10)

’

E{x,(t)gCxtt),~ (03) = E{x,tt)gCxtt),i @)I) - E{x@))ECgC.~@),~ (01)

W:(O)

(9)

E(x*(r)f

for the parameters of the linearized equation [IS].

- E* (x(t))

301

Non-linear automobile shock-absorbers

In the case of stochastic linearization it can be proved [16] that the second-order moments of the solution of the linearized equation (8) are identical to those of the original non-linear equation (7) if and only if the expectations indicated in (10) are evaluated for the statistics of the original non-linear problem (the “true” or “honest” statistics). It is for this reason that the error (9) was defined for x(t) and not for y(t). However, except for some very special cases, the statistics of the solution of the original non-linear problem are not known, otherwise one would not have to apply linearization. In a further step of approximation it is common to define the error (9) based on y(t), which can easily be obtained, as the governing equation (8) is linear. One is thus led to a set of non-linear equations for a and b which can eventually be solved if g (_, .) is given (see [ 17,181 for examples). The major drawback of this latter method is that for a real system (such as, for example, shock-absorbers) g (. , .) is not known a priori in most cases.

AN EXPERIMENTAL

TECHNIQUE

be able to apply equivalent linearization also in cases when g (. , .) is not known, we will now derive a method which uses experimental data for the determination of the parameters of the linearized system. We start with equation (1) and multiply it by i(t ). If then the operator E ( .} is applied, To

E{$t) 2(t)) + E{l(r) g[x(t),Zi(t)])

= E{F(t)~(t))

(11)

rest&s. With the help of E{<(t) i(t)) = &E{i’(t)}

(12)

we conclude that E {F (0 a(r) > a =

E{_?(t))

(13)

’

because in the stochastic case, E {i’ (t)) is constant as a result of the stationarity and, in the harmonic case, E {i2 (t)) is constant as a result of the periodicity of the solution. Equation (13) can be interpreted as a simple power balance: in the mean, all power imparted to the original system by the external force F(t) is dissipated in the damper of the equivalent linear system if it is undergoing the same motion as the original non-linear system. One of the main features of the non-linear system, namely its power-flow characteristic, is thus conserved by the linearization. This makes equivalent linearization very attractive for many applications and especially for shock-absorber dynamics. We now multiply (1) by x(t) and obtain E {x (0 g(r) } + E {x (t) g Cx(0, a(r) I)

= E{f-(t)x @I},

(14)

which yields b = EV+)x(r))

+ E{i’(r)} E {x’(t)}

- E{F(t)} Q(r)}

- E2 {x(t)}

’

(15)

where $E{x(r)i(t)

> = E{i2(t)}

+ E{x(t)%(t)}

= 0

(16)

was used. The result (15) can be interpreted as an energy balance, stating that, in the mean, kinetic and potential energy of the linearized system are equal to each other. The parameters a and b can now easily be evaluated by simply measuring F(t) and the state x(t), i(t) of the system for the appropriate excitation. As the observations are made for the real non-linear system, they will automatically yield the correct values in the sense of “true” or “honest” statistics.

SOME

EXPERIMENTAL

RESULTS

Based on the method described in the previous section, a test facility for automobile shock-absorbers was built 1193. It consists of a hydropulse machine to control the damper

302

J.

WALLASCHEK

end-point motion and a computer-controlled signal-analyzer, which is used to evaluate the expectations in (13) and (15). Figure 1 is a photograph of the set-up. All experiments were carried out for the standard twin-tube shock-absorber (BOGE) from the suspension of the rear-axle of the Ford Sierra. Some results for stochastic linearization have already been given in [ZO]. Here we consider only the harmonic case and determine a. To this end, the force signal F(t) and the velocity i(t) were measured and digitized. Then the expectations in (13) were calculated using the cross-correlation mode of the signal-analyzer. During the theoretical considerations in the previous sections the mass of the system was normed to be equal to one, so that F(t) represented an acceleration rather than a force, and a was the inverse of the decay time constant of free damped oscillations of the linear system. In the experiment, however, F(t) represents a force, so that Q now is a damping coefficient with dimension N s/m. Figure 2 shows the measured damping coefficient for a 2 Hz sinusoidal test-signal, whose amplitude Z? was varied between 3 and 45 mm. The damping coefficient is plotted over the root-mean-square velocity t’,rr = &j=0Ti2(i)dr].

(17)

The curve starts with a positive slope for small velocities, reaches its maximum and then descends monotonically. For higher velocities it approaches a horizontal tangent. If the tests are repeated for higher frequencies, the qualitative behavior of the curves remains the

Fig. 1. The test facility.

Non-linear automobile shock-absorbers

303

0.8 --

0.6 f

260 Velocity

veff,

(mm/s

1

Fig. 2. Damping coefficient for a 2 Hz sinusoidal test-signal.

same, but the absolute values of the damping coefficients decrease as shown in Fig. 3. The velocity at which the maximum damping coefficient is attained is shifted towards higher values for increasing frequency, whereas the final tangent which is approached at high velocities remains almost unchanged.

SIMPLE

MATHEMATICAL

MODELS

FOR THE SHOCK-ABSORBER

The dynamic behavior of a shock-absorber strongly depends on its constructive details and there is little information available on shock-absorber characteristics. Here we try to give only a very basic description of the shock-absorber investigated: we will be interested only in an accurate modeling of the power-flow characteristic which is associated with a, but the model should give satisfactory results for the whole range of velocities (0 < bff < 500 mm/s) and frequencies (1 Hz c f < 10 Hz) investigated. The shock-absorber used in the experiments is a twin-tube shock-absorber schematically shown in Fig. 4. It basically consists of a piston with valves, which separates the chambers of the inner tube and moves in a fluid, usually oil. The inner and outer tubes are connected by a second set of valves, the outer tube merely being an oil-reservoir for the inner tube. In principle, the shock-absorber can be reduced to the model of a piston moving in a closed cylinder containing a fluid, as shown in Fig. 5. Assuming that the two chambers are interconnected by a single channel with constant circular cross-section of diameter d and length 1 and that the liquid is incompressible, the piston’s equation of motion can be derived: m:(t) + d, i(t) + d *1* (t) sgn [i(t) ] + d, sgn [i(r) ] = F(t),

(18)

with

d,=pN,

(19)

where A is the effective piston area, q and p are the dynamic viscosity and the mass density of the fluid and c is a constant. Coulomb friction between piston and cylinder has been taken into account, PN being the friction force. The three damping terms in (18) describe energy dissipation due to laminar tlow, throttle losses and friction, respectively. We will, furthermore, assume that the forces acting at the two ends of the shock-absorber are equal to each other, so that the shock-absorber is a two-force element, which is massless, observed from the outside [21].

304

J. WALLASCHEK

12--

aa--

Ok--

Velocity

Fig. 3. Damping

veff,

(mm/s

coefficient for a 2, 4, 6 and

&!_

1

10 Hz sinusoidal

piston

test-signal.

rod

/membrane

oil , ,piTton with

valves

inner, tube

bottom with valves

outer, tube

Fig. 4. Principle

of the twin-tube

shock-absorber.

This basic model is not sufficient to describe the power-flow behavior of the real shock-absorber which was investigated. This immediately becomes obvious, when the theoretical values of a, which will be denoted by h, are calculated for (I8). Straightforward calculation yields ri=d,+f$Q+

4d,

xi-’

(20)

where now the approximation mentioned in the section on equivalent linearization, above, namely the assumption of purely harmonic motion, was made. The curve (20) cannot describe the behavior of the measured damping coefficients at low velocities. For higher velocities, however, a good approximation is achieved, as can be seen from Fig. 6. There the approximating curve (20) was fitted to the experimental data using a Gaussian error minimization based on measurements for high velocities only. It turned out that d,, describing the throttle losses, is so small that it may be neglected for practical purposes.

Non-linear

auromobile

305

shock-absorbers

-X

-i

Fig. 5. Basic model for a shock-absorber.

!

--sr-%Velocity

Fig. 6. Comparison

of measured

veff.

and calculated

(mm/s)

damping

coefficients

for model (18).

The characteristic shape of the curves in Figs 2 and 3 somehow resembles the typical describing function behavior of a relay with backlash [17] and indeed, Karadayi and Masada [7], who investigated shock-absorbers at very low frequencies (fc 1 Hz), showed that backlash is important. If we assume that the backlash 6, which occurs at the beginning and end of each stroke (when the velocity changes its sign) can be introduced into our simple rnodei (18) by considering mj;(c) = F(c),

if x(t) - 2 < 6 and a(t ) < 0 orx(t)+9<8andi(t)>O

m%r) + dr i(r) + d 3 sgn [a(t) ] = F(t), otherwise,

(21)

instead of (18), d is then given by

-:I+-![1-i]/(l -[I -$I’>3

d=d,[l-farccor[l

+- 24

9Q7I

[

2 - s , otherwise. P1

(22)

It should be noted that there d, was set to zero from the start because of its small contribution. The approximation of the measured data is slightly better than before, but is still not good enough to be acceptable. The discrepancy between theory and experiment is partly due to the fact that the channels of the valves interconnecting the two chambers of the cylinder have a complex shape and the second set of valves in the bottom of the shockabsorber has not been taken into account in our model. Moreover, the valves contain preloaded springs which open additional channels if some critical pressure level is reached. This can be seen from Fig. 7, where the time history of the damper force for purely harmonic motion with frequency 1 Hz is shown. There are characteristic periods over which the force

306

J. WALLASCHEK

L

2

Time(s) Fig. 7. Time-history

of the damper

force for purely

harmonic

motion

with frequency

1 Hz.

200

20 Displacement Fig. 8. Force-displacement

diagram

20 (mm) corresponding

to Fig. 7.

remains almost constant at a level of approximately 50 N. Figures 8 and 9 show the corresponding force-displacement and force-velocity diagrams. These plots also show the asymmetric behavior of the shock-absorber for compression and extension. Sometimes the force-velocity characteristic of the damper is described by two constant viscous damping terms, one for compression and another for extension. Although this may yield good results in some special cases, this model is of course too poor to describe the power-flow over a broad region of operating conditions. The effects of asymmetry and of changing valve characteristics may be introduced in the model (18) by considering different parameter sets di in different operational regimes of the shock-absorber. Following this line of analysis, Fig. 10 was obtained by using C(t)

+ d,i(t)

+ d ziz (t) sgn [a(t) ] + d, sgn [i(t) ] = F(t),

m.?(t) + d;.t((t) + d~.P(t)sgn[~(t)]

C(t)

+ d;i(t)

+ d ;1’(t)sgn

if 22 < L:1,

+ d;sgn[X(t) J = F(t), if 2R > t’i and i%(t)j < ui, (23)

[a(r)] + dgsgn [i(t)]

= F(t), if 2ZR> u 1 and la(t) ( > u1

with nine coefficients. The approximation is reasonably good and yields excellent results, as far as only one frequency is considered at a time. For different frequencies, however, different parameter sets are obtained. It seems that the number of parameters involved in the model already is too large to speak of a “simple” model, and still no satisfactory results are obtained. The model might be improved further by considering elastic elements because the liquid in the damper is suspended by a pneumatic spring and for higher frequencies the foaming of oil and air may

Non-linear automobile shock-absorbers

-LOO-

I -100

Velocity

(mm/s1

307

I 100

Fig. 9. Force-velocity diagram corresponding to Fig. 7.

1

I

1

I

200 Velocity

LOO veff.

(mm/s)

Fig. 10. Comparison of measured and calculated damping coefficients for model (18) with nine parameters.

become significant. Then, however, the elasticity acts in series with the damper and an internal variable has to be introduced, resulting in a further complication of the model [22]. As a conclusion, one might say that a description of the power-flow characteristic of the investigated shock-absorber is not possible using simple mathematical models with only a few physically relevant parameters.

CONCLUSIONS It was shown that the determination of an equivalent linear model of a real non-linear system can easily be performed, if experimental data are used. The parameters of the linearized model were expressed as a function of the expectations and joint expectations of the external force and the state variables of the system. Experimental results were given for a typical automobile shock-absorber and some dynamical models were discussed to explain its power-flow characteristics. It turned out, however, that simple models with only a small number of parameters do not describe the behavior of the shock-absorber sufficiently well. The best way to describe the dynamics of a shock-absorber-at least for harmonic motion or stationary random motion-seems to be the direct use of an equivalent linear model whose parameters are estimated using the experimental technique described. If one is interested only in the second-order moments of the solution, as it is usually the case in ride-comfort investigations, this will ensure that the results obtained with the equivalent linear model are identical to those obtained by the original non-linear system. Moreover, no information on the non-linearity of the system is needed.

308

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WALLASCHEK

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