I A V S D EXTENSIVE SUMMARIES

A Simplified Theory for Non-Hertzian Contact

43

J. J. KALKER*

In the present paper a simplified theory is proposed both for contact formation and for frictional rolling, in which the contact area is not known in advance, and, indeed, may be non-Hertzian. The program first determines the contact area on the basis of the given total normal force, and then applies the FASTSIM algorithm in the contact area on the basis of the given creepages and the spin, to find the total tangential force.

In the Hertzian case the program was compared with the 1982 FASTSIM program [1], and it is found that the algorithms are equally accurate, but that the new program is about three times slower than the old one.

Consider two elastic bodies in contact, and rolling over each other. They are pressed together by a normal compressive force N, and roll with creepages vx, vy,. We refer to [2] for a definition of these concepts, and for background information. The bodies are approximated by half-spaces. A Cartesian coordinate system x(xh x2, x3) (underlined symbols: vectors) is introduced in which the half spaces are x3^0 (lower body), x33s 0 (upper body). The plane of contact is the plane of x, and x2, where x, points in the rolling direction. x 3 points normally into the upper body. In the contact a surface traction (= load per unit area) p (pu p2, p}) acts on the upper body. It is required to find the traction distribution when the normal force, the creepages and the spin are given together with geometrical information and material constants.

Discretisation The surface of the half-spaces is descretised by a net of equal rectangles, numbered/, / ( t w o indices for each rectangle). T h e is taken constant over each mesh of the net. I f (/, T) are the numbers of a mesh, then we take as representation of p in mesh (IT) the value pu at its midpoint X/y.

In the same manner u denotes the value of the surface displacement difference (upper body displacement minus lower body displacement) in the center x,-, of mesh (IT).

Constitutive relations The surface displacement UJJ may be expressed in the surface tractions/7, v . , which, it will be recalled, is the (constant) value of/; in the entire mesh ( / ) . We have

ufjj= X F , J i T r r p T f f (1) r. J\ i

where superscirpt H indicates that the influence numbers F follow from the halfspace integral representation of Boussinesq-Cerruti, see [2]. These so-called constitutive relations are approximated in the Simplified Theory (see [1], [2]) by setting

* Deparlment of Mathematics and Informatics, Delft University of Technology.

44 J.J. K A L K E R

uUi=WipUi + ai,i=l,2,3 (2)

where W, and a, are constant troughout the net, and, for the sake of the FASTSIM algorithm [1] Wy = W^

Principle of the determination of Wt, 0; Given a load distribution pm we calculate u^jji ' Wj and a, are now determined by means of least squares:

Minimize M(Wi, at), with

M(Wi, ad = X (u% - W f i u - of (3) u

which leads to linear equations.

Determination of the contact area Let L(x) be the distance between opposing points on the surfaces of the bodies (upper minus lower) in the undeformed state. Lfx) is a function of the profile of the surfaces; it is a given quantity. D(x) is similarly defined as L(x), but refers to the deformed state; it may be shown that D(x) = L(x) + u3(x). The contact formation conditions are

or

D(x) = L(x) + W3 p3 (x) + a3 = 0 i f p3 (x) > 0 (contact)

= L(x) +a3>0 i f /> 3 (x) = 0. (no contact) ^

Hence the contact area and the contact pressure are given by

Area:Z,(x) +

46 G. P A L M K V I S T A N D O L L E NORDSTROM

A pure computer simulation requires a correct mathematical description of the anti-lock system, which can be difficult to obtain and very time consuming and expensive to program. It is also a considerable drawback that no hardware is actually tested in the simulation.

The problems of the two methods mentioned above can be largely eliminated with a hybrid method where the most important components from anti-lock system testing point of view are actually used in the test by integrating them in a computer-simulated test. Such a method which has been tested with good results is presented in this paper.

Hybrid simulation of this type has been used before in anti-lock research. New in this context is to our knowledge the evaluation of steerability and stability in addition to braking efficiency and the use of pure digital simulation.

Method The hardware used for the test consists of four pressure transducers with amplifiers, an interface unit for transmission of computer-simulated sensor signals to the logic control unit of the anti-lock system, a brake switch and a fast digital computer SEL 32 including A / D - D / A converters.

The pressure transducers are installed near or on the brake cylinders, the transducer outputs are fed into the computer.

The computer is programmed with a simple four-wheel vehicle model. The brake pressure signal from the real vehicle is converted into brake torque by the

computer program. The brake torque changes the simulated rotational wheel speed. This wheel speed is transformed into a simulated wheel sensor signal, which is fed back to the control logic of the real anti-lock system. If the signal indicates a tendency to wheel lock, the anti-lock system reacts and the pressure is lowered. The loop is thus closed.

The initial speed is preset in the computer and the simulated test starts when the brake pedal switch is actuated.

Theoretical model The basic model which has earlier been used for a driving simulator developed by V T I describes essentially a four-wheeled vehicle with freedom to move in the plane and to yaw. Neither roll nor pitch are included but a semi-static load transfer between the wheels will occur. The distribution of this load transfer between the wheels is specified among the input data thus obtaining some of the effects of different spring lay-outs. This is very important since the horizontal tyre forces depend on the vertical load in a nonlinear manner.

The equations of motion are solved numerically with a time step At of 0.0055 sec, i.e. data are exchanged between the computer and the vehicle equipped with anti-lock system 182 times per second. The integration method for solving the equations of motion is explicit Euler.

Brake model Brake pressure signals are converted into corresponding brake torques by means of a brake model that includes pushout pressure and hysteresis. It is also possible to set different gradients when pressure is rising and decreasing.

IAVSD EXTENSIVE SUMMARIES 47

Wheels and tyre friction models The tyre model is of a matrix type which makes it possible to use experimental tyre data. When considering longitudinal forces Fx input values are:

HX = F (8, s,Fz)

6 = slip angle

s = slip

Fz = normal force

fix = longitudinal friction force coefficient

Input values for lateral forces Fy are:

uy = g(8, s)-h(d,Fz)

Hy = lateral friction force coefficient

Both px and fiy are defined at distinct points and by linear interpolation actual values of tyre forces are computed. Typical characteristics are shown in Fig. 1.

Frt>T7 - - r r v r z

- _ l T1SEJ1PPLL l rz- u i f e . K

S L I P

Fig. 1. Tyre characteristics

The rotational wheel speeds (/) at time t = At n (n = 1,2, 3...) are determined by integrating the wheel acceleration n (r) = A (/) (r) + B (t)P(t), where pressure P (t) and parameters A (/) and B (t) have constant values during the timestep At.

At low vehicle speeds the fast wheel dynamics demands a shorter time step than possible to obtain (with the now available computer) during real time simulation. In spite of these limitations considering timestep and model complexity the results of the simulations are quire acceptable when comparing with field tests.

Simulated test procedures The computer program can simulate - straight braking on homogeneous surface - straight braking on split-friction surface

48 G R A N P A L M K VIST and O L L E N O R D S T R O M

- braking in a turn with fixed steering control - J-turn braking with fixed steering control The tests can also be performed with two axle drawbar trailers. The motion of the pintle hook is then pre-determined. Some examples of data presentation are shown in Figs. 2a-c.

Br? r u i !

L?C'CI6 I? I t H l C L C P O R o n C T f R S fitiiiillililtlmriuili uriom. f ^ n c . r c iMiLLFBsr- 3 et n KLICT Of C . C - 1 2B t l TPOCKUJOTM fH P X L f - 1 . S O ' I . Z-rwss no-i o I K T R T - T S E D B . L O A D OH r V X 1 X - 3J794.S 2JS7B. N POT- .30 PO*' .7B W H E E L rwss r u v o r I K T R T - 1 3 . 3 S Z 7 . e r C " H 7 IMtEL fO0ILTS= .50 f l s . r . u H u x s ^ i s e e ^20[>e. n f i a * OLS PARn-lETERS lirutJfi itiiiiitjiiiitTi

t w a x P U L S t S ' f ' o i . - ) ec

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I N I T I A L OOIUES t t T r i t l t t l l l l t l t i i t i f t f f l S P E E D - n en rv-s

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2 c . T l n SEC Fig.2a-c. Examples of data presentation.

In order to validate the hybrid simulation method a truck and a drawbar trailer, both with air brakes, were used to simulate tests that had earlier been made with these vehicles on ice, split-friction and high friction surfaces. Both vehicles were tested with three different

IAVSD EXTENSIVE SUMMARIES 49

anti-lock systems. During these tests all important variables were measured and could be compared to the simulations.

The results indicate that it is possible to reproduce real tests both qualitatively and quantitatively with satisfactory accuracy. (See Figs. 3a-c).

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50 G. R I L L

The Influence of Correlated Random Road-Excitation Processes on Vehicle Vibrations

G. R I L L *

An improved mathematical description of the interaction between vehicle and roads requires complexity in vehicle modeling as well as in road modeling. As the equations of motion for systems with many degrees of freedom can be derived from computerized algorithms, see Kreuzer [1], a good approximation to real vehicle dynamics is possible. The modeling of the road unevennesses, however, is mostly simplified to independent random excitation proceses. In this paper a special kind of shape filter approach is presented, which takes all correlations between the road-excitation processes on a two- or multi-axle vehicle into consideration. To show the influence of correlated random road-excitation processes on vehicle vibrations, the vehicle is modeled as a multibody system with 19 degrees of freedom, and the road unevennesses are described by first order shape filters.

Fig. 1. Vehicle model

* Daimler-Benz A G Stuttgart, Germany

I A V S D EXTENSIVE SUMMARIES 5

Vehicle modeling The vehicle model shown in Fig. 1 represents a multibody system, consisting of rigid bodies, bearings, springs, and dashpots. Its dynamic behaviour is described by the linear state equation,

x{t)=Ax{t)+Byt). (1)

Where x is the state vector, A the time-invariant state matrix, B the excitation matrix, and ye a stochastic vector process due to road unevennesses. For a two-axle vehicle the excitation term in (1) can be written as, Mller, Popp, Schiehlen [2],

Bye(t)=Bltl(t) + B22't). (2)

Where f , = [ W\, Wlf represents the excitation at the front axle and f 2 = [ W\ PF4]rat the rear axle.

Road modeling For a two- or multi-axle vehicle, running in a straight direction, it is sufficient to consider the unevennesses of the right and left track. I f (s) and f, (s) are the scalar excitation processes at the front axle, then the scalar processes at the following k 1 axles are given by corresponding delayed processes f , (s % ) , f j (s %), i = 2 (1) k, Fig. 2.

In the frequency domain the statistical dependence of the processes f r (s) and f, (s) is characterized by the cross-spectral density Srl'SX). Dodds and Robson [3] as well as Kamash and Robson [4] assume that the road surface is isotropic, i.e. it has the same statistical properties in the longitudinal and lateral direction. For any isotropic road Sr!(Cl) can be

52 G. R I L L

evaluated from the track width bx + b2 and the spectral density Sr ( f t ) = S, ( f t ) = S ( f t ) for any single track. This road model, however, is not suitable for a shape filter approach.

Following Parkhilovskii [5] one finds, Fig. 2,

(3)

Where f M and # are uncorrelated scalar random processes. By means of shape filters f M and t? can be obtained from stationary white noise processes, with zero means and the correlation functions

E{Wj(s)Wj(s a)} 0 , i>j\ US (a), i=j

A first order shape filter approach for example results in

(4)

f (J) = J ( 5 )

r CO

(5)

CO

J8,0

0-B2

6 # M

*C0 0^2

( ^ f n f i l + G H i f s ) .

For bl=b2 = b and the values

02 =

0.3 m"

1.5 m"

S 0 = ( ? / 2 T T ) fet + &2D = 1.6 10- 5 m, (6)

0.75

the first order shape filter approach (5) fits fairly well measured spectral densities and coherency functions, Fig. 3.

Setting f j CO = f (s) the excitation process at the front axle is given by (5). The excitation process at the rear axle can be obtained by f 2 CO = f C 5 sR2), or by a shape filter, excited by delayed white noise,

bis) = Hu2

54 G. R I L L

Fig. 4. Standard deviation of carbody acceleration for different correlated excitation-processes 1) left /r ight due to measurements front /rear completely uncorrelated 2) left /r ight due to measurements front/rear completely correlated 3) left/right due to measurements front/rear due to delayed processes 4) left /r ight completely uncorrelated front/rear due to delayed processes 5) left /r ight completely correlated front/rear due to delayed processes

REFERENCES

1. Kreuzer, E.: Symbolische Berechnung der Bewegungsgleichungen von Mehrkrpersys temen. Forschr.-Ber. V D I - Z - Reine 11, Nr. 32, Dsseldorf: VDI-Verlang 1979.

2. MUller, P. C : Popp, K. ; Schiehlen, W.: Covariance Analysis of Nonlinear Stochastic Guideway-Vehicle Systems. In : The Dynamics o f Vehicles on Roads and on Tracks. Ed.: Wil lumeit , H.P., Lisse: Swets & Zeitlinger, 1980.

3. Dodds, C. J.; Robson, J. D. : The Description o f Road Surface Roughness. J. of Sound and Vibr. 31 (2) 1973, pp. 175-183.

4. Kamash, K. M . A.: Robson, J. D . : Implications of Isotropy in Random Surfaces. J. of Sound and Vibr. 54(1) 1977, pp. 89-100.

5. Parkhilovskii, I . G.: Investigations of the Probability Characteristics of the Surfaces of Distributed Types of Roads. Avtom. Prom. 8 (2968) pp. 18-22.

6. Bormann, V.: Messungen von Fahrbahnunebenheiten paralleler Fahrspuren und Anwendung der Ergebnisse. Vehicle System Dynamics 7 (1978) pp. 65-81.

7. Ri l l , G : Instationare Fahrzeugschwingungen bei stochastischer Erregung. Stuttgart, Univ., Diss., 1983. 8. Muller, P. C : Popp, K.: Kovarianzanalyse von linearen Zufallsschwingungen mit zeitlich verschobenen

Erregerprozessen. Z. angew. Math . Mech. ( Z A M M ) 59 (1979) pp. T144-T146. 9. Hirschberg, W.: Vergleich verschiedener Fahrzeugmodelle and Hand charakteristischer Kenndaten der

Systemantwort bei stationarer stochastischer Erregung. Univ. Stuttgart, Inst. B f. Mech., Studien-arbeit STUD-1, 1981.