ROAD RESEARCH LABORATORY

Ministry of Transport

RRL REPORT LR 314

VEHICLE BEHAVIOUR IN COMBINED CORNERING AND BRAKING

by

A.J. Harris and B.S. Riley

Vehicles Section

Road Research Laboratory

Crowthorne, Berkshire

1970

Ownership of the Transport Research Laboratory was transferred from the Department of Transport to a subsidiary of the Transport Research Foundation on 1 st April 1996.

This report has been reproduced by permission of the Controller of HMSO. Extracts from the text may be reproduced, except for commercial purposes, provided the source is acknowledged.

CONTENTS

Abstract

i. Introduction

2. Notation

3. Steady state motions of a simple two wheeled vehicle subjected to non -linear sideways forces

3.1 Vehicle model, equations of motion and solutions

3.2 Description of the graphical representation of solutions

3.2.1 General description

3.2.2 The critical line and critical velocity

3.2.3 Effect of small forces or couples

3.2.4 The number of solutions

3.3 Steer characteristics

3.3.1 General

3.3.2 Stability of steady motion

3.3.3 The linear case

3.4 Effect of various factors on the steer characteristics

3.4.1 Introductory remarks

3.4.2 Change of load

3.4.3 Radial- and cross-ply tyres

3.4.4 Smooth and patterned tyres

3.4.5 Braking and drive forces

3.4.6 Compliance and roll steer effects

4. Behaviour of a four wheeled vehicle subjected to non-linear combined sideways and braking forces

4.1 Vehicle model and equations of motion

4.2 Sideways force /slip angle curves and the conditions for stability

4.3 Vehicle behaviour in combined cornering and braking on a constant radius path

S. Conclusions

6. Appendix

6.1 References

6.2 Derivation ofstability conditions

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I0

I0

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ii

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16

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6 . 3 Four w h e e l e d v e h i c l e model and road s u r f a c e d a t a

6 . 4 E q u a t i o n s o f mo t ion f o r t he f o u r wheeled v e h i c l e

6 . 5 T a b l e s o f r e s u l t s

24

25

29

~) CROWN COPYRIGHT 1970 Extracts from the text may be reproduced

provided the source is acknowledged

VEHICLE BEHAVIOUR IN COMBINED CORNERING AND BRAKING

ABSTRACT

A graphical method is first given which facilitates the discussion of vehicle behaviour and stability in steady cornering motion. Then follows a detailed discussion of steadily decelerated motion at the limit of cornering ability.

The vehicle model first considered is charac- terised by the nonlinear sideways force/slip angle relationships of its wheels. By modifying these the graphical treatment ca~ take into account suspen- sion flexibilities, roll steer, load transfer etc., and the necessary drive force.

In the later discussion of decelerated cornering on a path of fixed radius, the effects of load trans- fer, braking force, etc., which in the simpler treatment are dealt with in general terms, are calculated for a specific vehicle and road surface for which empirical tyre force data under combined cornering and braking are available. The limiting cornering acceleration for given braking deceleration is found.

An interesting feature emerging is the very sudden and large fall off in the combined sideways force curve for a pair of wheels which can occur immediately beyond the peak as a result of brake force and load transfer. It may help to explain some features of breakaway when cornering with deceleration.

i. INTRODUCTION

Emergency conditions, usually involving heavy braking or acceleration or sudden manoeuvres, are not strictly those in which steady state solutions, or linearised systems of forces for small angles of slip, apply. To deal com- pletely with vehicle behaviour under such conditions it is necessary to com- pute the transient and steady state motions for a whole range of initial and control conditions.

It is felt, however, that steady motions, when taken close to the limiting conditions available for the vehicle, do give valuable insight into stability problems.

The paper considers first a simple two wheeled vehicle in which the

sideway forces acting on the wheels are completely non-linear in relation to the slip angles. A graphical presentation of both steady state problems and their solutions is used and stability boundaries are discussed. The second part is a more detailed study of a four wheeled vehicle sub- jected to combined sideways and braking forces. The combined forces are again non-linear with respect to the slip angles but this time they are actual tyre and surface characteristics for combinations of sideway force, braking force and wheel vertical load.

An important aspect of both studies is the simple graphical solution to the steady state problems.

C°M°

M

W

a

b

f

h cm

h

A

K

D

D F

C D

D A

V

P

m

n

P

C

k

g

L

2. NOTATION 0

Total vehicle centre of mass

Total vehicle mass

Total vehicle weight

Distance of vehicle C.M. behind front axle

Distance of vehicle C.M. in front of rear axle

Wheel base, ~ = a + b

Wheel track

C.M. height above ground

C.M. height above roll axis

Vehicle frontal cross sectional area

Brake force ratio - front wheels/rear wheels

Total vehicle rolling drag

Rolling drag of two front wheels

Vehicle aerodynamic drag coefficient

Aerodynamic drag acting on vehicle

Vehicle speed along direction of travel

Vehicle acceleration along direction of travel (negative when decelerating)

Vehicle path radius

V/g (negative when decelerating) Non-dimensional vehicle acceleration

V2/pg Non-dimensional vehicle cornering acceleration

Rate of increase of roll angle with (n)

Mass moment of inertia of vehicle about vertical axis through C.M.

Radius of gyration, C = Mk 2

Acceleration due to gravity

Wheel vertical load

2

S

B

BFC

SFC

~F ' ~R 'y and f (y)

B

aH

A F

AR

F

R

F1

F2

R1

R2

Wheel sideways force

Wheel braking force

Brake force coefficient = B/L

Sideways force coefficient = S/L

Functions relating SFC and slip angle, all defined in text,

Wheel cornering slip angle

Vehicle body slip angle, relative to direction of travel

Steered road wheel angle

Reference steer angle

Decrease in front wheel slip angle due to compliance and roll steer

Decrease in rear wheel slip angle due to compliance and roll steer

The following suffices are used:-

Front wheels

Rear wheels

Outside front wheel

Inside front wheel

Outside rear wheel

Inside rear wheel

. STEADY STATE MOTIONS OF A SIMPLE TWO WHEELED VEHICLE SUBJECTED TO NON-LINEAR SIDEWAYS FORCES

3.1 Vehicle model, equations of motion and solutions

To retain a simple graphical solution to steady state problems, the vehicle is two wheeled and subjected only to sideways forces at the wheels. It is also assumed that in the equations of motion the sine of any angle can be replaced by the angle itself and the cosine by unity, which effectively means that the angles should not be too large. It may be assumed however that within the restricted angular range that this implies any type of sideways force/slip angle relationship may hold.

The most general steady motion of a vehicle on a horizontal plane is motion in a circle and with the symbols and conventions of Fig. la the equations of radial and rotational motion are:-

3

S F c o s ( 6 + 8 ) + S R c o s B = MV 2

I . . . . . . . . . . (i) 0

and S F a cos 8 - S R b - J ......... (2)

where I and J are an arbitrary force and couple applied at the centre of mass and used in illustrating the effect of small forces in changing the equilibrium conditions but otherwise generally ignored.

If the side forces S F and S R are expressed as multiples of the loads

carried at the wheels under static conditions we may write:

Mgb Mga SF = aYb ~F and S R - a+b ~R (3)

where ~F and ~R are functions of the slip angles a F and aR and are analogous

to coefficients of friction. If these are inserted in (i) and (2), and if cosines are replaced by unity and sines by the angles, and the terms I and J are now ignored, we get:

V 2 CF = ~R = ~ ( 4 ) g p . . . . . . . . . . . . . . .

Thus if~ F and~ R are plotted as functions of slip angle as in Fig. 2, any

pair of points such as F and R which are at the same height above the horizontal axis represent a solution of the problem.

To complete the solution we need the geometrical relations between the slip angles which, with angles approximated as before, are from Fig. i,

a F = 8 + B - E ) 0 ).

b ) ~R = ~ + -- ) P )

(s)

a+b so that aR - aF = ~ ................ (6)

P

As already pointed out any horizontal line such as FR in Fig. 2 represents a solution of equations (4). .For a given line, that is, for a given value of V2/gp or cornering acceleration there are an infinity of solutions depending on the p~rticular values of V and P which combine to produce the given value of V~/gp. Each point P on the line FR may represent one of these solutions by being chosen in the following way. Let P be chosen so that

a + b F P - • . . . . . . . . . . . . . . . . . . . . . . . ( 7 )

P

then it follows from eqn (6) that

4

RP = 8 . . . . . . . . . . . . . . . . . . . . . . . . (8)

since FR = o R - ~F ......................... (9)

It should be noted that the order of the letters is important e.g. FP means the displacement that carries F to P (positive in fig. 2) and there- fore PF would mean the negative of FP.

3 .2 D e s c r i p t i o n o f t h e g r a p h i c a l r e p r e s e n t a t i o n O f s o l u t i o n s

3.2.1 General description.

(A) Any point P in the plane of the diagram therefore represents a solution of the steady state problem. Given such a point P, (8) determines the steer angle 8, (7) determines the radius of curvature p, and (4) therefore determines V.

(B) All solutions for a constant cornering acceleration lie on a horizontal line whose height is the cornering acceleration as a multiple of g. Thus the ~ 's can be regarded as cornering accelerations.

(C) All solutions for a constant steer angle 8 are represented by a curve identical with that of ~R but displaced 8 to the right, since RP = 8 for every R, see fig. 2. Such a curve will be called "parallel" in subsequent references. ~,-

(D) All solutions for a constant radius of curvature 0 are represented by a line parallel to that of %F but displaced a45 to the right. Incidentally, a+b P

is the ideal steer angle. O

(E) All solutions for a constant velocity V are represented by a curve which is displaced from ~F by an amount proportional to the height above

the horizontal axis, see fig. 2. If we denote the vertical distance of FRP by y, then by (4)

V 2 y =

go

and therefore by (7)

FP = a+b g(a+b) y (10) . . . . . . . . . . o o .

0 V 2

so that FP is proportional to y for a constant value of V. The further the curve is to the right the smaller the velocity. For infinite velocity the curve coincides with ~F" Points to the left of ~F do not represent

real states since they require V 2 to be negative. Such points meally represent backward movements of the vehicle, which we ignore.

~) Given any two of 8, p and V the solution of the problem is found by constructing the appropriate lines and finding their intersection, as in Fig. 2, where, incidentally, because of the maxima in the curves, two solutions appear to be possible, such as P and P', given p and V, or P

A ~

and P" given p and 6.

Algebraically it is simplest to regard y the vertical coordinate as a fundamental variable or parameter. In terms of y, and of FR as a function f(y) of y, we have:

given y and 6

V 2 _ g(a+b)y and O = a+b - 6+£(y) 6+f (y)

given y and p

V 2 = gyp and 6 = -f(y)+ a+b p

given y and V

V 2 p = and 6 =-f(y) + (a+b)gy

gY V 2

. . . . . . . . . . . . ( l l a )

and of course if V and p are given y is immediately determined.

(11b )

( 11c )

3.2.2 The critical line and critical velocity. Suppose P in Fig. 3 represents a solution, then other neighbouring solutions for the same velocity V are given by points on PV', and for the same steer angle 6 by points on PR'. In general these two lines diverge so that given V and 6 there are no other points near to P also representing solutions for the same conditions. But if the two lines coincide for some distance from P there is a range of points, that is, a range of radii of curvature p, which are compatible with the given V and 6. A very small disturbance might therefore shift the system from one equilibrium point to another. This suggests that points for which this is true represent an unstable equilibrium and it will be shown later that this is so. Even if the two lines through P do not coincide over a finite length but simply have the same slope, the same instability can be expected, and it is clear that on each horizontal line there is a critical point Pc where the curves for V = const.and 6 = const, have the same slope i.e. touch. A critical point is therefore one at which the steer angle is stationary for constant velocity and varying radius of path.

The critical point on any horizontal line can be obtained by a very simple construction shown in Fig. 3. Draw FL,the tangent to CF at F, to cut the horizontal axis at L and draw LP c parallel to the tangent to cR at R. LP c cuts the horizontal FR in the critical point Pc. The locus of the critical point Pc may be Called the critical line. It represents, as will be shown later,the boundary between the stable and the unstable regions of the plane. ~.

From the construction

c (12)

With each critical point there ~is associated a "critical" velocity Vc; and for each horizontal line this critical velocity is given by (I0) and (12).

V 2 = g (a+b) y = g (a+b) . . . . . . ( 15 )

R- F

w h e r e t h e d e r i v a t i v e s a r e t a k e n o f c o u r s e a t t h e p o i n t s F a n d R on t h e horizontal line. One might also say that the radius of curvature Pc, or steer angle 8c associated with a critical point are "critical" values since they all represent in one sense the arrival at the critical line, the boundary of the instability region.

3.2.3 Effect of small forces or couples. Had the quantities I and J, which appear in Eqns. (I) and (2), been retained the solution of the equations would have become:

J(a+b) _ V 2 (I + J/b~ (14) ~F = ~R Mg ab gp Mg / .........

The solution can be reduced to the old form, eqn. (4), by shifting the ~R

J(a+b) and then shifting the origin downward by curve downwards by Mgab

I + J/b The effect of these small shifts on the position of the point P Mg

representing the solution for a given set of conditions can generally be made quite small if I and J are small. But if P is a critical point Pc, where the curves V = const, and 8 = const, simply touch, the slight dis- placement of the curves makes a large displacement in their point of inter .... section P. This again illustrates the association of the critical point P~ with instability by showing how a very small disturbing force makes a relatively large change in the radius of curvature which gives a steady state solution.

3.2.4 The number of solutions. Typical ~F and ~R curves for rubber

tyred wheels rise to a maximum and then fall off and can lead to combined patterns of many different kinds. It has been said that each point of the plane represents a solution of the steady state problem, but in fact a point may correspond to more thab one solution when there is more than one branch of the ~F and ~R curves to be considered. Typical examples

of the situation are shown in Figs. 4 and 5. It will be shown that solutions involving branches of the curves where the slopes are negative are mostly unstable.

3.3 Steer Characteristics

3.3.1 General. The steer characteristics of the vehicle have a simple geometrical significance in the solution diagram. Through any point P as

i n F i g . 3 p a s s t h e t h r e e c u r v e s o f c o n s t a n t 0 (o r c o n s t a n t i d e a l s t e e r a n g l e ) , c o n s t a n t 6 and c o n s t a n t V, t h e f i r s t two b e i n g p a r a l l e l t o CF and

CR a t t h e same l e v e l as P. In F i g . 3 sma l l e l e m e n t s o f t h e s e c u r v e s P F ' .

PR' and PV' a r e drawn upwards c o r r e s p o n d i n g t o a sma l l p o s i t i v e change i n t h e c o r n e r i n g a c c e l e r a t i o n .

The following relations hold:

F' to the left of R' ~ oversteer

F' to the right of R' ~ understeer

F' coinciding with R' ~ neutral steer

d~ d(~)R

(is)

This is easily seen to be in accordance with the usual definitions, see references i, 2, 3 and 4, Page 21.

(A) Increasing the velocity while maintaining the same radius of curvature means moving along PF' and if F' is to the left of R' then F' is closer to the ~R curve than P is, i.e. 6 is decreasing which according to one

definition of the term indicates oversteer. It should be noted that if P were to the left of R in Fig. 3 so that 6 was negative, the uhrase"~ decreasing" would mean that ~ was increasing in absolute magnitude

(B) If the velocity is kept the same and the steer angle 6 changed so that the cornering acceleration is increased the solution point moves along PV'. The increase in the steer angle is R'V', the increase in the ideal steer angle

a+b i.e. in , is P*V~and if F' is to the left of R~ the smaller increase is

0 in the steer angle - another definition of oversteer. When P lies to the left of R care is needed with the signs of the angular changes.

Fig. 6 shows how the plane of a typical diagram is divided into under- steer and oversteer regions by a horizontal line of neutral steer points, and how the critical line emerges to the right of ~F only in the oversteer

region. Had the ~F and ~R curves been of the same slope over a finite

length of arc (i.e. been parallel) there would have been a neutral steer region and not just a line of neutral steer points. \

3.3.2 Stability of steady motion. It is shown in Appendix 6.2 that the steady motion represen£ed by a typical point P is stable if the following two conditions are satisfied.

d~ da d~ -

(16)

1 _

d~ U~ d~ ~d~J R

where k is the radius of gyration of the vehicle about a vertical axis through its centre of mass, and the derivatives are taken at the same value of ~, i.e. at the same vertical height (cornering acceleration) on the curves.

It is also shown that (17) may be written

d(~,l~ l d(~m~_ F P - FP > 0 , . . . . . . . . . . . . . . . . . (18)

\ ~ / F \ - T R

Thus i f b o t h t h e d e r i v a t i v e s and ¢ a r e p o s i t i v e t h e two c r i t e r i a a r e s a t i s f i e d f o r any p o i n t P l y i n g to t h e r i g h t o f t h e c r i t i c a l p o i n t Pc f o r t h a t v a l u e o f ¢. I t i s a l s o c l e a r t h a t i f cR has a lower maximum t h a n ~F'

t h e n (17) w i l l f a i l and t he m o t i o n become u n s t a b l e b e f o r e t h e maximum i s r e a c h e d , s i n c e t h e n e g a t i v e t e rm w i l l become t h e dominan t one a t some p o i n t . I f ~F has t h e lower maximum t h e n (17) i s s t i l l s a t i s f i e d even a t

the maximum.

If one derivative is positive and the other negative and large, as occurs just beyond the maximum of the lower ~ curve, then (16) will be satisfied and (18) will be satisfied for points P which lie to the left of the critical point Pc appropriate to these branches of the ~ curves. If CR

has the lower maximum the appropriate P lies well to the left of the C

diagram and meaningful points P still correspond to unstable motion. If the lower maximum belongs to ~F the appropriate P is well to the right

C

and meaningful points still correspond to stable motion except for low values of V.

When both derivatives are negative (16) does not hold and the motion is unstable.

3.3.3 The linear case. When the forces are linearly related to the slip angles the steady state solutions are very easily obtained by the present method. The second equation of (llb) takes the well known form:

• %

•5

F- Z _-I a+b a -l-b = - E y ÷ - 1 1

P P L g(a+b)J (19)

where E is a constant. In terms of the @F and @R curves it is

p g (a+b) (2o)

the derivatives now being constants.

Any case in which f(y) took the form Ey would give the same result even if the actual @F and ~R curves were not straight lines.

3.4 Effect of Various factors on the steer characteristics and cornerin~ capability

3.4.1 Introductoryremarks. It is clear from the discussion of the stability conditions that anything that tends to lessen the slope of @R

in comparison with that of ~F will tend towards oversteer and instability.

Also that if the curves have maxima, the important question from the point of view of stability and cornering ability is whether the lower maximum belongs to the front or the rear wheel. The effect of a number of factors can therefore be easily understood in a qualitative way.

3.4.2 Change of Load. If the loading on a wheel is increased the side- ways force for a given angle of slip is not greatly affected at small angles of slip but increases more or less in proportion to the load where the sideways force is near its maximum. Since the @ curve is obtained by dividing the sideways force by the static loading the following changes are to be expected.

If an increase of load is due to a rearrangement of mass in the vehicle or to the addition of mass, this is an increase involvingthe static loading. Therefore, the initial slope of the curve tendsto fall but the maximum value remains more or less unchanged. On a dry road at least the maximum for the two curves should be at much the same height above the horizontal axis.

An increase of load on a rear wheel would tend towards oversteer and instability, and the reverse would be true of a front wheel.

If an increase of loadingwere due to air pressure or to load transfer resulting from acceleration or braking (though in this latter case we should not be dealing with a steady state) the static loading (as defined) would be unchanged and therefore the initial slope of the @ curve would also remain unchanged while the maximum would be raised. Such an increase on the rear wheel would therefore tend towards stability.

i0

During cornering, a four wheel vehicle suffers lateral load transfer to an increasing degree as cornering acceleration increases. In such a case a ~ curve represents the combined force of the pair of wheels and it will be affected by this load transfer if the dependence of tyre sideways force on vertical load is non-linear. A typical result would be that for a given wheel slip angle the total sideways force for a pair of wheels would be less than if no load transfer was present. The effect will be to decrease the slope of the curve, but the maximum ~ value may not be affected very much.

Increasing the load transfer on the front wheels of a vehicle will in general lead to more understeer and stability. Increasing it on the rear wheels will lead to more oversteer and towards instability.

3.4.3 Radial- and cross-ply tyres. Since radial-ply tyres are stiffer laterallythan~cross-ply they tend to have curves of greater initial slope, with perhaps slightly higher maxima. Fitted to the front wheels, with cross-ply tyres on the rear, they therefore tend to produce marked over- steering, and unstable motion as the maximum is approached. Fitted on the rear they tend to give marked understeering and stable motion as the maximum is approached. Compliance in the steering linkage may modify these expectations to some extent as explained later in 3.4.6.

3.4.4 Smoot ~ and patterned tyres. A smooth tyre on a dry road can be as good as or even better than a patterned tyre from the point of view of producing frictional forces. On a wet road however especially if the road is fine textured a smooth tyre will give a lower maximum sideforce than one with a good pattern. The initial slope may be less affected, but obviously, for a given ordinate the slope must soon be less for the smooth tyre. Smooth tyres at the front tend to stability as seen in Fig. 7. Smooth tyres on the rear may therefore convert a vehicle from understeer to oversteer on a wet fine textured road surface as in Fig. 8 and a wet patch on a sufficiently sharp curve might take the vehicle over the instability boundary.

/.

3.4.5 Braking and drive forces. The sideways force available is, in general, reduced if there,s a demand on the tyre for brake or drive force. At small angles of slip the sideways force is less affected than at near the maximum. The general result is however, to lower the ~ curve and make it less steep. Although drive forces have been ignored in the present theory, the effect on the ~ curves has been included implicitly. Thus, if the drive needed for steady motion is on the rear wheel the ~R curve is

lower and flatter than in the absence of drive and so tends towards over- steer and instability. Conversely front wheel drive tends towards under- steer and stability. Since the amount of drive required increases with speed the effect should be more pronounced at higher speeds.

Although the steady state theory applies only to equilibrium states it may be expected that there will be some analogy between steady and non steady states even though stability etc. for non steady states has sot been clearly defined. One might imagine that the ~ curves instead of being independent of the motion of the vehicle are continually changing

ii

as a result of load transfer, demand for brake or drive forces, and road conditions and that the criteria of stability at least have useful meaning even under the non steady conditions. When braking isapplied heavily however certain unstable motions may not proceed very far because the vehicle comes to rest under the braking before the unstable divergent motion has time to develop.

Drive forces also have an important effect on cornering capability because the peak sideways force is reduced. There are several cases to be considered.

(i) If, in the absence of drive, the maxima of the ~ curves would have been equal, the cornering capability is reduced whatever the axle to which the drive is applied, and is probably reduced more if the drive is applied at the more lightly loaded end. In addition, if drive is applied at the rear, stable motion will not be obtainable right up to the maximum of the new ~R since the vehicle will become unstable at the point

where according to Eqn. (17) the slope (-~)Rd~ equals (~)F/E v 2 d ~ i + g(a+b) . ,~.~C~)~ q.

For front wheel drive a stable motion would be obtainable right up t the peak of the ~F curve.

(2) If, in the absence of drive, the maxima of the ~ curves would have been unequal, then drive applied to the axle having the lower maximum would reduce the cornering capability and not change the character of the limiting motion, for example, if the front axle curve was the lower the motion would remain stable.

(3) If, in the unequal case, the drive is applied to the axle having the greater maximum, the initial effect depends on the axle involved. With drive applied to the front axle the cornering capability is actually

increased at first since ~)F/E + g(a+b)V2 (~-~)F~diminishes as d~(~)F

diminishes, and the motion therefore remains stable to a higher point on the ~R curve. With drive applied to the rear axle the cornering c~pability

at first remains unchanged. If drive is increased, either because of higher speed or sharper turns, until the maxima of the two curves become equal, then, as in (I), cornering capability decreases and the limiting motion is stable or unstable according as the drive is on the front or the rear axle.

3.4.6 Compliance and roll steer effects. Nothing has been said so far about compliance steer, the change in wheel angles caused by forces and moments on the flexible suspension and steering linkages, or about roll steer, the change in wheel angles due to rolling of the vehicle body. These affect the steady state motions and if they are to be included in the present discussion, equations (5) and (6) must be modified and become, referring to Figur% ib:-

12

a a

cz F = ($ + IB - --p = ~$H - A F * IB - --p . . . . . ( 2 1 )

b = B + --

P

a+b so that (~R + ~R ) - (~F + AF) - P 6H ..... (22)

6His the reference steer angle, that is the road wheel angle that would be

given by an infinitely stiff linkage between the steering handwheel and road wheel with no rolling of the vehicle.(1)(2) A F and A R are the total decrease

in wheel slip angle, front and rear respectively, due to compliance and roll steer. Thus 6 H is a function of the steering hand wheel angle and AF, A R

are both functions of the cornering acceleration. In equation (22), (~R + AR) is the angle between the direction of motion of the rear wheel

.and the original ~leel plane on the vehicle. (aF + AF) is the angle

between the direction of motion of the front wheel and the plane the wheel would have been in had there been no compliance or roll steer.

These effects can be included in the present graphical presentation by modifying the ~, a curves. Each point of the front w ,eel curve is moved sideways by A F and each point of the rear wheel curve by A R. If A F and

A R are positive then both curves move to the right, but there will be no

change in the maximum values of ~. With the new curves ~F and ~R as

functions of(aF + 6F) and(aR + AR)everything is now as before and the effect.

on the stability of ~he vehicle can be judged by the stability criteria which now become (compare with equations (16)and (17))

t, ,X t, > 0 ..... (23)

> 0 ..... (24)

I f t h e s l o p e s o f t h e ~F" ~R c u r v e s a r e p o s i t i v e ~ a e n i n t h e s a m e

m a n n e r a s e q u a t i o n ( 1 5 ) t h e f o l l o w i n g r e l a t i o n s h o l d

13

understeer

oversteer

neutral steer

and instability occurs when

(d(.+Ah d~ ffF

dC~÷Ah

)

>\ d~ /R )

<d(~+ A)~ ) < de 2R ) )

)

=\d~ )F )

(H(~+A)% + ~g ) - d~ )R V -~ "< 0 !

.f

Equation (22) may be rewritten:

= + 6 H V-~ ~ + (eL F A F) -

(25)

(c~ R ÷ A R) (26)

and if V is kept constant,

d~ V 2 + k, dtp , I F - (27)

Thus at constant speed equation (27) shows that the steer criteria (25) become

understeer d6H gg ) >

de V 2 ) • )

)

o v e r s t e e r dCH gg ) < - - ) d~ V 2 )

) ..... (28) neutral steer d~H £g )

~ ~ )

de V 2 )

)

and instability dSH ) occurs when d~ ~ 0 )

)

where £g/V 2 is a constant for all cornering accelerations. are in agreement with the definitions in references 1 and 2.

E q u a t i o n s (28)

If now the radius of turn 0 is kept constant, equation (22) gives #

d~- k d, JF d~ A . . . . . . . . . . . ( 2 9 )

14

and the steer characteristics are now:

understeer d5 H

d4 > 0

oversteer d5 H

d4 < 0

neutral steer d~ H - 0

d4

and instability d6H _ £g occurs when d~- ~ V 2

(or ~<-~- )

(30)

where in this case ~g/V 2 is not constant.

Compliance steer is more prominent on the front axle due to steering linkage flexibility and the rubberised mounting of these links, and is likely to have its largest effect at low cornering accelerations. Examples of possible compliance effects are shown in Figures (9a,b,c).

Avehicle having no compliance and fitted with say, radial ply tyres on the front wheels and cross ply tyres on the rear will have 4,~ curves

as shown in Pigures (9a,b) where[.~-L~.~Fd(~.~ _ ( h is kd4]D i negative even at zero

cornering accelerations. Consequently, as can be seen from equations (25), there will be a critical speed above which straight line travel will be unstable. This speed could be as low as 50 km/h with typical radial and

crossply tyres. With the addition of compliance as shown,E(d(~)~_ ~fd(a+A)~d4 ~d

may easily be made positive at low cornering accelerations (4 values) and the vehicle will then be stable at all speeds in straight ahead travel, and behaviour at higher acceleration will depend on which of the 4F or 4R maxima

is the lower, and on the shape of the compliance curve. If the 4 F peak

is lower, which for modern cars is likely on some surfacest5),r then over most of the range of cornering accelerations the instability is eliminated and the critical speeds for the rest of the range are increased. Where the 4 R peak is lower the cornering acceleration at which instability first occurs will be increased and may be brought a lot nearer to the peak 4R value.

For the more likely vehicle configuration of similar tyres and static loadings,front and rear, the 4 F and 4R curves will be closer as in Figure (9c).

The effect of compliance in the region of instability is then not so notice- able. In the example shown the understeer is increased throughout most of the range of 4, especially at the lower accelerations, but there is only a small change in the cornering acceleration at which instability occurs, which will be just before peak 4 R is reached. For the case where peak 4F

15

is lower the understeer is increased throughout the whole range of Corner- ing accelerations.

The roll steer effect is likely to be less, and its amount may, for example, be proportional to the cornering acceleration as shown in Figure (9d) where it affects only the front suspension. In the form shown, its effect will be to increase the understeer characteristics over the whole range of cornering accelerations.

4~° BEHAVIOUR OF A FOUR WHEELED VEHICLE SUBJECTED TO NON-LINEAR COMBINED SIDEWAY AND BRAKING FORCES

4.1 Vehicle model and eqUations of motion

The rear wheel driven model is shown in Fig. (iO) and its full specification appears in appendix 6.3. The vehicle parameters chosen are typical of a modern medium sized car.

The vehicle travels on a constant radius path and is decelerated at a constant rate. The body slip angular acceleration B is assumed negligible for this manoeuvre and the equations of motion thus formed are for quasi- steady state motion. The appendix 6.4 gives full details of the derived equations and other assumptions that are made. The equations give the sideways forces and braking forces necessary at each axle of the vehicle to maintain the constant radius path at the combination of deceleration and cornering acceleration. With the brake force coefficient and vertical load known for each wheel, it is possible to estimate graphically from the curves of Fig. (ii) the required cornering slip angles at the front and rear pairs of wheels. At this stage a test for stability may also be made. The curves and their stability criteria are discussed more fully in section 4.2. Steered road wheel angle ~ and body slip angle B then follow from knowledge of the slip angles.

The calculations are in two parts. The first is an approximation of the forces neglecting the components of the equations due to sin 6 and sin B. Thus, approximate slip angles aF and aR' and then approximate 6 and 8 are

found. The 6 and 8 terms are reintroduced into the equations and the forces recalculated in the second part of the calculation to give more accurate values of ~,B. It is possible to repeat the second part of the solution but the subsequent changes in the slip angles are very small.

Thus for each axle or pair of wheels it is possible to say, for the combination of deceleration and cornering acceleration, whether the required sideways force can be obtained, and whether the system as a whole is stable. If the stability criteria are not satisfied then the vehicle spins off its path. If the required front axle sideforce exceeds that available then it leaves the constant radius path but does not spin.

4.2 Sideways force/_slip angle curves and the conditions for stability

Figure (ii) shows the curves of sideways force against cornering slip angles for various constant braking force coefficients. Each set of curves

16

is for a constant vertical load. The complete set, taken from reference (6), is for a cross-ply tyre on a wet Bridport macadam surface. This is a slippery surface for which there is little variation of characteristics with speed.

The looping effect at the higher brako force coefficients and lower slip angles is implicit in the character of the BFC/SFC and BFC/braking slip data from which these curves are derived.

For a pair of wheels at the same slip angle the curve of total available sideways force is made up from the individual curves of the two wheels. As cornering acceleration and braking deceleration increase, lateral and longitudinal load transfer also increase. Each point on the combined curve will then derive from the sideways force characteristics of Figure ii. at the wheel loads and braking force coefficients appropriate to the conditions. For the complete vehicle there are two curves, one for each axle, and if the total required sideways forces are known both at the front and rear, the slip angles may also be found. Although the required sideways forces may be less than those available, the vehicle condition must be tested for stability.

Referring back to sections 3.3.2 and 3.4.6, stability conditions were derived for the graphical presentation of steady state conditions on the ~,a plane. In the present example it is not possible in general to work in terms of the non-dimentional ~ until after the curves have been derived for each axle. Even then, for correct presentation the curves would have to be moved up or down the vertical axis slightly because of the sideways force increments due to deceleration and the ~,8 effects. Since the stability of the system depends on the slopes of the ~,~ curves at the cornering acceleration considered, this vertical shifting of the curves would not be important as the slopes would remain the same.

The curves have been left in terms of sideways force and slip angles and the stability conditions of equations (16) and (17) have been modified as follows:

Wb Since S F = ~--

Wa and S R = i- ~R'

Wb daF

£ dS F

and d(. _~ Wa daR = ~ dS R ............... (32) \ VR

(d) F = . .............. (31)

and the condition for stability becomes, for positive slopes:

\dSF/- ~-- ~dSR/+ > 0 .......... (33)

17

de F de R where dS F and ~ are measured at the sideways force S F and S R values

on the curves.

In the present example, in the caseswhere the vehicle becomes unstable, rather than the required front axle sideways force exceeding that available, the instability occurs when the rear sideways force is very close to the maximum available at that axle. With other vehicle configurations this may not be so.

The effects of compliance and roll steer were discussed in section 3.4.6 and it is felt that in this example their effects on the maximum cornering accelerations at the combinations of speed and deceleration would be small.

It is interesting to note the two basic shapes of the sideways force, slip angle curve for either axle, as shown in Figure 12. At lowbraking force coefficients the sideways force available will take the form shown in Figure 12a, a smooth curve having a peak at some slip angle. At higher braking force coefficients the looping effect is present and the sideways force available can be different in character. If a is exceeded, the

m inside wheel will lock and the sideways force contribution from that wheel reduces to its locked wheel value at that slip angle. With further in- crease in slip angle the inside wheel remains locked and the total sideways force may again continue to increase. Thus for a pair of wheels it is possible to get very sudden loss of sideways force for a small increase in cornering slip angle.

4.3 Vehicle behaviour in combined cornering an d braking on a constant aius ath,

The manoeuvre considered is braking on a 45.8m (150 ft.) constant radius path. Braking is applied to give a constant vehicle deceleration, and the sideways forces and braking forces necessary to keep the vehicle on the path at combinations of deceleration and cornering acceleration are shown in Table i, appendix 6.5.

For each deceleration, there is a cornering acceleration, and an equivalent speed, above which the vehicle is unable to maintain the path. This loss of control is due to either the stability condition of equation (33) not being satisfied, or else the peak available sideforces are such that the required front wheels sideways force exceeds that available. In the former case the vehicle spins off its path and in the latter case the front wheels lose control and the vehicle still leaves its path butdoes not spin.

In the example given here, there is little loss of maximum cornering acceleration'for decelerations up to m = -0.15. Fig. 13 shows that at higher decelerations the maximum cornering speed rapidly decreases until at decelerations just greater than m = -0.25 it is not possible to maintain control except at very low speeds when the vehicle will in any case quickly stop.

18

It is interesting to note the change in the way control of the vehicle is lost over the range of deceleration. Referring to Table i, at constant velocity the vehicle finally becomes unstable due to the loss in rear wheels available sideways force caused by the power needed at the rear wheels. At deceleration values up to m = -0.15 the vehicle continues to become unstable at the limiting cornering acceleration although the front wheels are not far from their peak-sideways forces. In this deceleration range the rear wheels require sideways forces slightly greater than those at the front, and the rear wheel loads are lower due to load transfer. The brake force coefficients are small and do not have a large effect at these decelerations.

At higher decelerations the braking force coefficients are larger and their effect is more apparent in that it now makes the front wheels lose control first by reducing the available sideways force at that axle. With a lower brake ratio the rear axle peak sideways force may be lower due to the high BFC then present and the sudden loss of sideways force may occur as discussed in section 4.2. This could lead to a skid which is more difficult to correct than a skid which occurs at low deceleration when the sideways force, slip angle curve does not have this sudden loss of sideways force.

S t e e r e d r o a d wheel ang le 6 and body s l i p ang le 8 v a r i a t i o n s w i t h d e c e l e r a t i o n and c o r n e r i n g a c c e l e r a t i o n a r e shown in Fig . (14) . The s t e e r e d wheel ang le i n c r e a s e s wi th c o r n e r i n g a c c e l e r a t i o n , f o r t h o s e d e c e l e r a t i o n s where the f r o n t ax l e l o s e s c o n t r o l f i r s t and d e c r e a s e s wi th c o r n e r i n g a c c e l e r a t i o n when the v e h i c l e becomes u n s t a b l e .

The e f f e c t o f 6 and B on the c a l c u l a t i o n o f t h e r e q u i r e d f o r c e s i s shown in Table 2, append ix 6 .5 . The most n o t i c e a b l e e f f e c t i s on t h e b rake o r power f o r c e s . Taking the power case f i r s t , t h e component o f t h e c e n t r i f u g a l f o r c e in the d i r e c t i o n o f mot ion f a c t o r s t he r e q u i r e d power by n e a r l y t h r e e a t t he h i g h e s t c o r n e r i n g a c c e l e r a t i o n . Added to t h i s i s t h e power n e c e s s a r y to overcome t h e component o f t h e f r o n t whee l s s ideways f o r c e . S ince t he body s l i p ang le 8 i s dependen t on t h e r e a r whee l s c o r n e r i n g s l i p ang le o n l y , t h i s e x t r a power may be c o n s i d e r a b l y r e d u c e d when say r a d i a l p l y t y r e s a r e f i t t e d , which need s m a l l e r c o r n e r i n g s l i p a n g l e s f o r g iven s ideways f o r c e s . When a v e h i c l e i s d e c e l e r a t e d t h e s e drag e f f e c t s mean t h a t l e s s b r a k i n g i s needed .

Both the front and rear sideways forces are increased by small amounts due to 6 and 8. The effect on the front wheels is larger because the front wheels braking force has a component perpendicular to the vehicle body. The vertical loads are also affected by a small amount, the front wheel loads being decreased and the rear loads increased. The overall effect when considered with the braking forces are lower brake force coefficients except in the power case.

As i s shown in Table 1 the maximum c o r n e r i n g a c c e l e r a t i o n s a r e changed by smal l amounts . Also the f r o n t whee ls s t a r t to l o s e c o n t r o l f i r s t a t a h i g h e r d e c e l e r a t i o n than when the ~,~ components a r e n o t c o n s i d e r e d .

19

An important factor not included in the example is the effect of wheel camber angle. Large camber angles are present on most cars at high cornering accelerations, the amplitudes depending on the suspension systems at each end of the vehicle. The effect on the vehicle handling capability will of course depend on the influence of camber on the peak sideways forces available. For some tyres, perhaps radial ply, the camber influence may be small. For a car fitted with cross ply tyres, live rear axle and independent front suspension for instance, the maximum available sideways force at the front axle will probably be reduced. This may lead to lower maximum cornering speeds and a tendency for the front axle to lose control first more easily.

S . CONCLUSIONS

The study of steady state or quasi steady state motions of a vehicle can lead to useful conclusions about the stability of such motions and of decelerated motions such as are common in emergencies.

A graphical treatment of the steadystate motions of a simple two or four wheeled vehicle can be given. With wheel slip angle and cornering acceleration as x, y coordinates, the sideways forces at the two axles can be represented by two curves and each steady motion by a single point. The sideways force curves are those of the tyres, modified, if necessary, by such factors asroll or compliance steer, load transfer, and drive or braking force. Extremely simple graphical means are then available to find the curves representing steady motions corresponding to constant steer angle, constant path radius and constant speed, or the boundaries of the regions of stable and unstable motion and oversteer or understeer behaviour.

Among the aspects of vehicle stability and cornering ability which can conveniently be discussed in general terms in this way are: the relationship of stability to the slopes and heights of the basic curves, the effect of changes of wheel loading due to redistribution or changes of mass or to dynamic or aerodynamic load transfer, the influence of radial ply or smooth tyres when fitted to front~or rear wheels, the effect of front or rear drive, and the modifying effects of roll steer and vehicle compliance.

A more detailed treatment of these effects, in particular the determination of their magnitudes in specific cases, requires numerical treatment of a more elaborate model.

Such a study, making use of empirical data on combined braking and cornering forces, has been made for a four wheeled vehicle subjected to simultaneous braking and cornering. The maximum cornering acceleration was found as a function of vehicle braking deceleration for a path of fixed radius, 46 m (iS0 ft). For the particular rear driven car assumed and tyre data appropriate to a slippery type of surface (wet Bridport macadam) about 70 per cent of the maximum braking deceleration could be obtained with only a i0 per cent loss of maximum cornering acceleration. At low decelerations the vehicle became unstable and spun off its course,

20

at higher decelerations it left the path tangentially without spinning.

For high braking forces on one or both of a pair of front or rear wheels, the curve of resultant sideways force against slip angle for the pair could have a sharp fall off beyond the peak value though no such fall off occurs in the absence of braking. A small increase in slip angle would lead to a sudden loss of sideways force. This may be one reason why a rear wheel skid is more difficult to correct when one is braking and cornering than when cornering only.

6. APPENDIX

6.1 References

io SAE J67Oa '~ehicle Dynamics Terminology" issued 1965).

(Handbook supplement

, HALES, F.D. MIRA Report No. 1965/1 "The handling and stability of motor vehicles. Part i. Handling, stability and control response definitions".

.

.

NORDEEN, D.L. General Motors Research Laboratories Report No. GMR-423. '~ehicle handling: its dependence upon vehicle parameters".

OLLEY, M. 1946-47. Proc. Inst. Automobile Eng. Vol. 51, "Road manners of the Modern Car".

5. 'NOTORING WHICH?" Handling diagrams from October, 1965 onwards.

.

6.2

HOLMES, K.E. and R.D. STONE. Tyre forces as functions of cornering and braking slip on wet road surfaces. Ministry of Transport, RRL Report LR 254, Crowthorne, 1969 (Road i Research Laboratory).

Also in

Institution of Mechanical Engineers, Proceedings 1968-69 Vol 183, Pt.3H. Handling of Vehicles under emergency conditions. (Inst.Mech.Engrs.).

DeriVation of stability conditions

Displacements from a s t e ady s t a t e a r e c o n s i d e r e d on t h e a s sumpt ion t h a t the v e i o c i t y V and the s t e e r ang le 6 a re m a i n t a i n e d c o n s t a n t .

The e q u a t i o n s f o r non s t e a d y mot ion expressed i n the n o t a t i o n of t he main text are:

(a~R + b.~F) - (a+b)w2gp

and k 2 (a+b) ~F - ~R- - gab

. . . . . . . . . . . ( i )

. . . . . . . . . . . ( 2 )

where k is the radius of gyration of the vehicle about a vertical axis through the centre of mass and m the angular velocity about this axis.

21

The a n g u l a r , r e l a t i o n s h i p s now become:

a

a F = 6 + 8 r (3)

b a R = 8 + - - . . . . . . . . . . . . ( 4 ) r

where r is the distance from the centre of mass to the instantaneous centre of rotation, which can no longer be assumed to be the centre of curvature of the path in space.

and

We also have:

V = - . . . . . . . . . . . . ( 5 )

r

v (6) O J : " l " ~ • o o o • o o . • • e •

p

from the kinematics of the situation.

Eliminating 8 from (3) and (4) and using (5) we get:

aF _ aR = ~ (a+b) = cS (a+b)~ r V . . . . . . . . . . . .

(7)

Differentiating (3) , (4) and (5) with respect to time while assuming V to remain constant gives:

;'F ~ + ~ + a÷ = - ~ . . . . . . . . . . . ( 8 )

r

• f~ b~" aR = 2 . . . . . . . . . . . (9)

r

• V r t0 : - - - ( 1 0 ) . e o . o o i . . o e

r

Eliminating ~ and ~ from (8) and (9) with the aid of the other equations leads to:

and

V (~ -a F+a R) g (aCR+b*F) ga2b (*F-*R) = ~ + - - - - . . . . . . . . . . (11)

F (aYb) V (a+b) V (a+b) k 2

V (8,-aF+aR) g (a~R+b~F) gab 2 (~;F -~R ) ~R = (a+b) - V(a+b) + . . . . . . . . . . (12)

V(a+b)k 2

E q u a t i o n s (11) and (12) e x p r e s s t h e r a t e s o f change o f a F and a R i n t e r m s o f

a F and a R and c o u l d be i n t e g r a t e d n u m e r i c a l l y to g i v e t h e c o m p l e t e m o t i o n o f

a v e h i c l e whose v e l o c i t y was m a i n t a i n e d a t a c o n s t a n t V.

22

Now consider small displacements from a steady state defined by ~b F

= CR = C0 , and for which o F and o R take the values oFO and oRO"

°F = °FO + ~F )

o R = oRO + o R ) )

write:

and, to the first order of small quantities,

~F : ¢0 + d~d'-O)F ~F ) )

CR C0 d(~)R ~R ) = + )

)

We may

(13)

(14)

where the derivatives are taken at the steady state values.• Using equations (14) and the time derivatives of (13) in 611) and 612), and assuming that the steer angle 6 is kept constant so that ~ = O, and remembering that the constant terms cancel out, we get:

• @ E @ E v - - 1 °F - ~F gb f a 2 ÷ k 2 y d ~ q+~.. ga {ab2..k 2 d, " (1S)

: ' v - - \ - - G - ~ g 7 d ~ " v-- ,k k 2 • and

• : - - - -- i - - - IV+ V g-~b (k2-a~,dC.~q+ ~.. ~_1 ~ ~ : ga f k2+b2~d~_'~ "1 , , (16) ~R: - =F(~--~-) \ T ) t , ~ ) d ~t,sT) L" ~t, :;2 A ~ A j "

Equations (IS) and (16) are of the form:

= ClX + C2Y

y = C3x + C4Y

(17)

where CI, C2, C 3 and C 4 are constants.

x = Ae Vt, and y = Be pt

A solution is of the form

(18)

and these denote a stable motion if the real part of p is negative. Substituting from (18) into (17) shows that p must be a root of the quadratic

2 p - p(C 1 + C4) + (CIC 4 C2C3) = 0 ............ (19)

Direct consideration of the roots of (19), or use of the Routh Hurwitz criteria I for negative real parts shows that both coefficients must be positive, i.e. the conditions for stability are:

23

C I + C 4 < 0

CIC 4 - C2C 3 > 0 (20)

Inserting into (20) the values of the coefficients from (15) and (16) leads after some algebraical reduction to the two conditions:

1 [ ( a 2 ~ k 2 ) d ~ (b2+k2) d ~ ]> 0 k d ~d)Fkd ~djR R F.

(21)

and

~tc~ 1 a~ [(-~)F- (d~)R + g(a+b_______~)] > 0 . . . . . . . . . (22)

It may be seen by comparison with equations (7) and (12) in the main text that (22) may also be written:

1 (23)

REFERENCE

i. KORN, G.A. and T.M. KORN. Mathematical Handbook for Scientists and Engineers, p 17. New York , 1961, McGraw-Hill.

6.3 Four wheeled vehicle model and road surface data

The vehicle model used in the example calculations is shown in Figure i0. The specification for the vehicle is:-

W

C

a

b

f

h cm

Tyres

Tyre pressure -

Road surface

= 10680 N (24001b)

= 1768 kg m 2 (1300 slug ft 2)

= 1.22m (4 ft)

= 1.22 m (4 ft)

= 2.44 m (8 ft)

= 1.28 m (4.2 ft)

= O.52 m (1.7 ft)

cross-ply

1.38 Bar (20 p.s.i.)

h = .40 m (1.3 ft)

P = .2 rad

C D = .4

A = 1.86 m 2 (20 ft 2)

D = 191 N (42.9 ib)

D F = 95.5 N (21.5 ib)

K = 2.0

Wet 9.5 m.m. Bridport macadam carpet.

24

The vehicle suspensions are such that the vertical load transfer due to cornering accelerations is shared equally between the front and rear axles.

I t i s d r i v e n a t t h e r e a r e n d .

6.4 Equations of motion for the four wheeled vehicle

F i g u r e 10 shows t h e v e h i c l e on a c o n s t a n t r a d i u s p a t h a t some c o r n e r i n g a c c e l e r a t i o n (n) and b r a k i n g d e c e l e r a t i o n (m). The a e r o d y n a m i c d r a g i s a s sumed to a c t , f o r c o n v e n i e n c e , a t t h e v e h i c l e c e n t r e o f mass i n t h e l i n e o f m o t i o n o f t h e v e h i c l e .

Balancing the longitudinal and lateral forces and taking moments about the centre of mass gives three equations:

(mW + DA) cosB + (DR1 + DR2 + BRI + BR2 ) +(DFI + DF2 + BFI + BF2 ) cos6

+ nW sinB + (SFI + SF2) sin~ = 0 ........ (24)

nWcosB + (DFI + DF2 + BFI + BF2 ) sin~ - (mW + DA) sinB - (SRI + SR2 )

- (SFI + SF2) cos~ = 0 (25)

C (~ + mg/p) + (SRI + SR2 ) b- (SFI + SF2 ) a cos~

- - ÷ (DFI ÷ DF2 + BFi + BF2) a s i n 6

+ (SFI - SF2) (fsin~)/2

where D A

+ (DFI - DF2 + BFI BF2 ) ( F c 0 s 6 ) / 2

+ (BRI + DRI BR2 - DR2) f / 2 = O

_- ,O623V 2 AC D

(26)

The vehicle is subjected to load transfer from both longitudinal and cornering accelerations. The wheel vertical loads are given by:

LFI = W[ .25 + (ncosB - rosinS) (hcm + hP)/2f - (mcosB + nsinS) hcm/2£ ] (27)

LF2 = W [.25 - (ncos8 - msin8] (hcm+ hP)/2f - (mcosB + nsinS) hcm/2£ ] (28)

LRI = W [.25 + (ncos~ - msin8] (hcm+ hP)/2f + (mcos8 + nsin8) hcm/2£ ] (29)

LR2 = W[ .25 - (ncosB - msinS) (hcm+ hP)/2f + (mcosB + nsin8) hcm/2£ ] (30)

The following assumptions are then made to give equations of quasi- steady state.

25

The following assumptions are then made to give equations of quasi- steady state:

= o ... (3i)

(SFI-SF2) (fsin~)12 = O ... (32)

(DFI-DF2+BFI-BF2) (fcos6)/2 = iI ... (33)

(DRI-DR2+BRI-BR2) fl 2

also let SFI+SF2 = S F ... (34)

SRI+SR2 = S R ... (35)

BFI+BF2 = B F . . . (36)

BRI+BR2 = B R .... (37)

DRI+DR 2 = D R ... (38)

DF I+DF" 2 = DF ... (39)

DR+D P = D ... (40)

B F = KB R ... (41)

Thus, equations (24), (25) and (26) become

(roW + DA) cosB + D R + B R + (DF + ~BR) cos~ + nWsinB + S F sin~ =-0 .. (42)

nWcosB + (D F + KBR) sin~ - (mW + DA) sinB - S R - S F cos~ = 0 ... (43)

Cmg!n +bS R - aS F cos~ + (D F + KBR) asin~ = 0 ... (44)

which for given m,n has the unknowns SF, SR, BR, 6, B

An approximate solution of (42), (43), (44) is found by putting

cOSB = cos~ = I.o

sin8 = sin6 = O

This gives

(SF)I" = bnW/ + Cmg/ o ...... (4s)

26

(SR) 1

(BR) 1

(LFI) 1

(LF2) 1

(LRI) 1

(LR2) 1

: nW- (SF) 1 .....

: - [roW + D A + D] /(1 + K) .....

= W [.25 + n(hcm + hP)/2f - mhcm/2~ ] .....

= W [.25 - n (hcm + hP)/2f - mhcm/2~] .....

= W [.25 + n (hcm + hP)/2f + mhcm/2~ ] .....

= W [.25 - n (hcm + hP)/2f + mhcm/2~ ] .....

(46)

(47)

(48)

(49-)

(so)

(51)

Provided that a wheel does not lock, the braking force available at each axle is divided equally between the two wheels, irrespective of vertical load. The braking force coefficients are then:

(BFCF1) 1 : K(BR) I / 2 (LF I ) I . . . . . (52)

(BFCF2) I : K ( B R ) I / 2 ( ~ 2 ) I . . . . . (53)

(BFCRI) i = (BR) 1/2 (LRI) l ..... (54)

(BFCR-2) i : (BR) I /2(LR2 ) I . . . . . (55)

Thus for each wheel there are now known vertical loads and braking force coefficients and from _the tyze characteristics of Figure fl, sfdeways force/ slip angle curves are derived for each wheel. It is then assumed that each of the front wheels is at the same cornering slip angle mF and both the rear

wheels are at slip angle a R.

Knowing the total sideways force at either axle, the slip angle for that axle is found graphically using the appropriate pair of curves. The steered wheels angle 6 and body slip angle 6 are then given by

6 = ~/P + (~F - ~R )

6 = a R - P

assuming t h a t ~ is n e g l i g i b l e . More accura te va lues of SF, S R and B R are now ob ta ined by r e t a i n i n g

the more s i g n i f i c a n t 6 and 6 terms in equat ions (42) , (43) , (44) .

Putting cos6 = cos6 : 1.0, but keeping in the sin6 and sin6 the following are obtained:

b [mW + sinB + + K (BR) 1 SF = (SF) 1 - ~- DA] [DF KnWsinB (I+K) ] sin6

terms,

(s8)

a [mW + sin6 S R = (SR) 1 - g DA] (sg)

27

nw SFS in6 BR = (BR) I (I+K) sin (I+K) ... (60)

LFI =

LF2 =

W[.25 + (n-msinB) (hcm+ hP)/2f - (m + nsin$) hcm/2t ]

W[.25 - (n-msinB) (hcm + hP)/2f - (m + nsinB) hcm/2~]

• . . [ 6 1 )

... (62)

. . . [ 6 3 ) LRI = W[.25 + (n-msinB) (hcm + hP)/2f + (m + nsinB) hcm/2~ ]

LR2 = W[.25 - (n-msin~) (hcm + hP)/2f + (m + sinB) hcm/2~] ... (64)

BFCFI = KBR/2LFI ... (65)

BFCF2 = KBR/2LF2 . . . (66)

BFCRI = BR/2LRI .. (67)

BFCR2 = BR/2LR2 ... (68)

Whe:n the initial values of 6 and ~ are used in the above, new values of ~F' ~R' ~ and B are found. By further substitution of ~ and ~ into

equations (58) to (68) a more accurate solution can be found. In practice there is little advantage in going any further than substituting the first approximate values of 6 and B back into the equations.

28

g

~D~U Z - U

~ o ~ u

m z

g ~

m o

a i m

u~

.,4

,u ~ q o . q q ' : . ~ ~

. . . . . ' ' i ' '

0o ~

~.~ ~ ~ o o , o o o o ~ o

O ~

o ~ o

~ ~ i ~ ~ i ~ ~ 0 ! ~ ~ ~ . . . . . . . . . .

o ~ ~ ~ ~ ~ ~ ~

i

O O I M M I M M I O M I N ~ M ~ R M N N ~ N

O 0

- ~ - ~ o ~ °°~ N ~ ° ~ ° ~ ° ° ~ g ° ° ~ o ~ o~ %oo:~ ~g~o~.~

• . . . .

_ ~. ~

~ ~ ~ ' 0 0 0 0 0 0 ! 0 O 0 m m l O O 0 O m O 0 0 0 ~ O ~ . . . . . . ~ g o o ~ ° ~ ° . . . . . R~

~-~ o o o o D O 0 0 0 0 0 0 0 0 0 ~ - 0 ~ N ~ 0 0 "a" i~ 0 0 ~ 0 ~,-~ . to.o o o

,--,_, ,..o o ,.~ m ~ 0-. ~ l o

~ .~, o., .,~ o - % o. ~. o .~ o g ~ .~ o i . . . . !

~ ~,~ u~ ~ooo~o~'~"~'~°~'~ ~ , ~ ~ = ~ " ~ N ~ ~ ' ~ 0 ~ . . . . . ~ ~o~,'~"~'.~ o o o ~

~ ~ o o o o o o o o o o

e-i

. ~ o o o o

O l D ~DIZ~

N

I

o o o o o o o o o o o ~ o o o o ~ o o d d o o 6

29

~,~ ~ ~ ~ + ~ + ~ o

i

i i i i i I1

u.l

~ z ° ° ~ , T ~ , '2 ° ° ~ . ~ ~" . . . . . . ~ N " ~ ' + + + + + + , + + ~ i

+

,.m o ,,D i ~

N ~ > z

÷ ,

1

O9 o + + + + + + i + + , - i i i u

i +

, ,o

~ 0 • ~ - o ~ " ~oo0~,o o ~ o , ~ o ~ ~ o ~ o ~ ,go ~

g

+

J +

+

. ~ 1 ~ ~

It 11 II

II II II

~ +

÷ .~.

÷

ii n n

o

o

, o

E

3O

CM.

Fig. la.

;F

i I ~ / F and R: points at which front .-.. ~ / and rear wheels make contact IJl--'~ I'- / with ground I" ~ SFon~- sR: side ,orces

Jk / a and b :d is tances of Fand R

,J / ~: steer ongle of front wheel SR C<p and C<R: slip angles

R p : rad ius of curvoture of path [ and J:external force and couple

PLAN VIEW OF VEHICLE MOVING IN CIRCULAR PATH, CENTRE 0

C.M.

Fig. lb. PLAN VIEW OF VEHICLE MOVING IN CIRCULAR PATH. DIAGRAM INCLUDES COMPLIANCE-°AND ROLLSTEER EFFECTS

V 2 ~ ' , gp

or y non-

dimensiono

• . R

p ~ f

f b l S ~ o /

. ~ . ~ C o n s t a n t g

/ * ~ ~ - C o n s t a n t V

T V2// ' -7 / / '~"~-Constant" ___/__" a°b / " o

@

P

Stip angte, o¢ F oro< R

Fig.2. DIAGRAM OF~ F ANO~ R, AND POINT P REPRESENTIN6 A STEADY STATE MOTION

Understeer

. ~ - - - - --;- -- - - - , Neutrat steer

Oversteer

/ ~ " Criticct[ tine, boundary of instctb,tit y

/ ~ ' X ~ " Unstobte region (hatched)

Stabte region

Stip angle, o~

Fig.3. CRITICAL LINE,STABLE ANB UNSTABLE REGIONS,CONSTRUCTION FOR CRITICAL POINT

~F No solution

C r i t i c a l l i n e /--l, solu t ions,~( *÷* * )

.~5 ~-,x, ~o., u,,oos\-.,,.;.,xx\~x~<,)~b xx.,x.x~ ~, \

' f-.////////////////.YYY/~ F J/z~//y.~" ~ Is°luti°n' ' p°sitive/ /////~

Slip Qngle =¢

Fig./,. OIA6RAN FOR INITIALLY OVER STEER TYPE SHOWIN6 NUMBER OF SOLUTIONS IN EACH AREA AND SI6N OF

~PF

V'

tlLr R ~ N 0 solution

I / / / / I / / i / / ' , ~ \ \ \ \ \ \ . , \ \ Crit ical l i n e ' / / / / ~ " ~ " 2soIs, g(,+) ~ " ~

S

Slip angle, =¢

Fig.5. DIAGRAM FOR INITIALLY IJNBERSTEER VEHICLE

~ F

u n s t a b l e \ \ - . ,, ~.,,.. ~",,~ ' Ove rs tee r

s tab le

Line of neu t ra l s t ee r points

U n d e r s t e e r

S l i p ang le , ~<

Fig. 6. TYPICAL STEER CHARACTERISTIC ANO STABILITY REGIONS

C r i t i c a l l i n e

/ /

/

I /

R ( p a t t e r n e d )

/ / / / / I / I /

/

/ v ~F (s~mooth)

Slip angle , o<

Fig.7. SHOOTH TYRE ON FRONT TENDS TO UNOERSTEER AND STABILITY

,,.p.

-;,,t- / I

/ I I

/ /

/ v /

/

; r i t i ca t line ut-stQbte

Neut rat s tee r - -

~" F (pa t te rned)

V~'R ( smoot h)

Understeer

Slip angle,

Fig. 8 SMOOTH TYRE ON REAR WHEEL TENDS TO OVERSTEER AND INSTABILITY

~v" \&(compliance)

(o~) ~ F R

~ A

S (b) F R

j J

\[,1 a--~);(~),~ ]

[ d ~ d ~

"- _J

'¢./,," j A V~. F ," ~. R [.(~);- I~),,]

(d)

A (rol[ steer) / -,p. F*A

o~ O~

(d,,)_ (d,,~ ] [d~l t F d'r~/R

= . -Wi th rolt steer

Without roll

Fig.9.(a,b,c) EFFECT OF COMPLIANCE STEER ON ~oc DIAGRAMS. 9(d)EFFECT OF ROLL STEER ON W,o~ DIAGRAMS

L b . I o _1 E wt ,

L_.) I i L J

L wl, _

Fig. 10. FOUR WHEELEO VEHICLE HOOEL ANO FORCES ACTING ON IT

Z

"" 500 =>,.

;3 o

1 0 0 0 - Wheet vert ica[ toad : 1780N

BFC

u.O 1 ( - 0 " 1 )

/ ~ ~ o.2s / ~ - ' ; ' ~ - ~ o.28 ~ " ~ o.27 , - ~ ' ~ o . 2 6 ,

0 50 100 150 200 Cornering sl ip angte o< (m rad)

1000 Z

U

0 500

0

N 0

Whee[ ver t icat toad = 26?0N

0.31 9)0.28

I

B F C

- - - - - \ ( - 0 - I ) - - ~ " ~ 1 - 0.2 )

"0.2

~ _ 0.25

0.26

I I I

50 100 150 200

Corner ing s t ip ongte o( (m rod)

Z

(/)

e- U

0 .e-.

>,., 0

¢1 "10

(/)

Fig. 11.

1500 Wheel vert icat toad = 3560N B F C

O,

500 I- ~ ~ ~ 7 o. 26

O Y I I I I 0 50 100 150 200

Corner ing s t ip angte o<(m rod)

COMBINED SIOEWAVS AND BRAKING FORCES AT 3 VERTICAL LOADS FOR A CROSS PLY TYRE

ON A WET BRIOPORT MACADAM SURFACE

(a)

u L

2

i f)

Outside wheet

° _

Cornering slip ongte

(two wheels)

Cornering slip angle

(b)

U Outside whee[

Inside wheel

"10

sup angle

L ~m Cornering

Totat sideways force for one axle (two wheels)

slip angle

Fig.12. TWO TYPES OF TOTAL SIOEWAVS FORCE CURVES

0.3

0

¢ -

E " 0 ¢,-

0 ¢Z

IU

U

r -

0"2

0"1

I

/__ . VehicLe out of control I/

- - " " ~ " / - ~ Z / / , Front wheels ~ / / maximum sideways force

/ ~ exceeded ...

Vehicte ' ~ / / . ~ ~ ~ " under controt " ~ ~ - - ~ - ~ . " ~

~ ' - . " ~ Vehicle spins

Wet Bridport mo.cadam surface Po.th radius 65-8m

I I I I 0 0-1 0-2 0"3 0-/.

n, vehicle cornering acceleration (non dimensionaL) ,F

Fig.13. ENVELOPE OF MAXIMUM CORNERING ACCELERATION FOR COMBINED CORNERING AND BRAKING ON A CONSTANT RADIOS PATH

70

~ 60

50

~. /,0 -

"~ 3 0 - ,g ~ 2 o -

~ ~ o -

m 00

100 -

9 0 -

8 0

70

6O

so

=- 40 q;

30 0 t-t

-~ 20

o 10

-10

-20

-30

m = - - 2 0 / m, d e c eterat i ! o ~ i =-'05

I I I 0-1 "2 "3 n, c°rnerin9 acceteration (non-dimensionat)

M')

O ~ O ~ ZZ'J

,, E "E . I EmE

- ' ~ ' f I I . • 2 -3 n

corne rin 9 ¢zcce le rat ion ( non-d imensional) n,

Fig.l~,. STEERED ROAD WHEEL ANGLE AND BODY SLIP ANGLE AGAINST CORNERING ACCELERATION

(653) Dd 635272 4M 5/70 H.P. Ltd. G1915 PRINTED IN ENGLAND

ABSTRACT

Vehicle behaviour in combined cornering and braking: A. J. HARRIS and B. S. RILEY': Ministry of Transport, RRL Report LR 314: Crowthorne, 1970 (Road Research Labora- tory). A graphical method is first given which facilitates the discussion of vehicle behaviour and stability in steady cornering motion. Then follows a detailed discussion of steadily decelerated motion at the limit of cornering ability.

The vehicle model first considered is characterised by the nonlinear sideways force/ slip angle relationships of its wheels. By modifying these the graphical treatment can take into account suspension flexibilities, roll steer, load transfer etc., and the necessary drive force.

In the later discussion of decelerated cornering on a path of fixed radius, the effects of load transfer, braking force, etc., which in the simpler treatment are dealt with in general terms, are calculated for a specific vehicle and road surface for which empirical tyre force data under combined cornering and braking are available. The limiting cornering acceleration for given braking deceleration is found.

An interesting feature emerging is the very sudden and large fall off in the combined sideways force curve for a pair of wheels which can occur immediately beyond the peak as a result of brake force and load transfer. It may help to explain some features of breakaway when cornering with deceleration.

ABSTRACT

Vehicle behaviour in combined cornering and braking: A. J. HARRIS and B. S. RILEY: Ministry of Transport, 1313L Report Ll3 314: Crowth0rne, 1970 (Road Research Labora- tory). A graphical method is first given which facilitates the discussion of vehicle behaviour and stability in steady cornering motion. Then follows a de ta i led discussion of steadily decelerated motion at the limit of cornering ability.

The vehicle model first considered is characterised by the nonlinear sideways force/ slip angle relationships of its wheels. By modifying these the graphical treatment can take into account suspension flexibilities, roll steer, load transfer etc., and the necessary drive force.

In the later discussion of decelerated cornering on a path of fixed radius, the effects of load transfer, braking force, etc., which in the simpler treatment are dealt with in general terms, are calculated for a specific vehicle and road surface for which empirical tyre force data under combined cornering and braking are available. The limiting cornering acceleration for given braking deceleration is found.

An interesting feature emerging is the very sudden and large fall off in the combined sideways force curve for a pair of wheels which can occur immediately beyond the peak as a result of brake force and load transfer. It may help to explain some features of breakaway when cornering with deceleration.