07 - Handling - Fundamentals Notes

8/10/2019 07 - Handling - Fundamentals Notes

1/24

Vehicle Dynamics, Control and Suspensions

7 - Handling - Fundamentals

David Crolla

Contents

7.1 Analysis .. 2

ddddd7.1.1 Steady state responses ... 3

ddddd7.1.2 Stability . 7

ddddd7.1.3 Frequency responses ... 11

7.2 Practical implications 14ddddd7.2.1 Vehicle CG position .. 16

ddddd7.2.2 Tyre cornering stiffnesses . 16

ddddd7.2.3 Load transfer . 17

ddddd7.2.4 Camber angle 18

ddddd7.2.5 Compliance steer effects .. 19

7.3 A complete example .. 19

ddddd

7 - Handling - Fundamentals - 1 -

D.A. Crolla


2/24

7.1Analysis

The equations developed in Module 5 are first order, simultaneous, linear differential equations and

three types of solution are useful to the vehicle designer. In practice, these refer to three types of

driving condition and it is important to establish a relationship between the mathematical solution

and its equivalent practical implications.

The steady stateresponse refers to a steady cornering condition in which the vehicle is driven at a

fixed speed and steer angle, resulting in a constant radius of turn. Vehicles are rarely driven for long

in such steady conditions but the steady state responses are important because they relate to one of

the standard methods of testing and characterising the basic vehicle handling behaviour.

The stabilitysolution refers to straight running conditions with no steering wheel input applied. In

practice, it reflects moderate to high speed cruising conditions on essentially straight roads such as

motorways. Under such conditions, the vehicle is continually subjected to small disturbances, for

example from wind fluctuations or an uneven road, which perturb it from its trim (ie. equilibrium

running) condition. The transient response of the vehicle to these small disturbances is of

considerable importance. Of course, the designer always wants to ensure stability, implying that the

vehicle returns to its equilibrium condition following a disturbance. However, cases of unstable

behaviour still occur in practice; well known examples include jack-knifing of articulated lorries and

car-caravan snaking. For small vehicles, the quality of the transient response can be categorised

by effective directional stiffness and damping properties.

Thefrequency responsesolution refers to the vehicle's response following a sinusoidal steer input at

the steering wheel. This clearly does not have a direct practical relevance since it does not represent

everyday driving practice! Nevertheless, it does characterise the general dynamic response of the

vehicle to steer inputs. The full significance of the frequency response can only be appreciated

through an understanding of Fourier analysis and linear dynamic systems. Briefly, this is as follows.

Any form of input can be decomposed into its frequency components; the frequency response

function provides a complete description of the system's response to each of these components, and

finally, the output is the linear sum of all these responses. Therefore, in an indirect way, the

frequency response function provides a complete description of the vehicle's response to any form of

steering input. Nevertheless, it is widely recognised that frequency response results, whether

obtained theoretically or experimentally, are rather difficult to interpret and relate to practicalvehicle behaviour.

These three solution types are inevitably interconnected. For example, the steady state response is

the limiting value of the frequency response as frequency is reduced to zero, and the expressions

defining overall handling behaviour - understeerand oversteer- recur in each solution.

- 2 - 7 - Handling - Fundamentals

D.A. Crolla


3/24

7.1.1 Steady state responses

The steady state solution is obtained by setting the dynamic terms, and&v &r, to zero in equation

(19), Module 5, and solving to obtain the outputs, vand r, in terms of the input, f The steady state

form of this equation is:

( )

( ) ( ) [ ]

Cf

Cr

UmU

aCf

bCr

U

aCf

bCr

U

a Cf

b Cr

U

f

ff

v

r

C

aC

+

+

=

2 2 (1)

and using Cramer's Rule then gives

( )r UlC C

l C C mU bC aC

ss

f

f r

f r r f

=+ 2 2

(2)

in which lis the wheelbase, , and the subscript ssis used to denote the steady state response.a b+

Cramer's Rule

The solutions, to , of a set of nlinear simultaneous equations :x1 xn

(3)

a a

a a

x

x

b

b

n

n nn n

11 1

1

1 1L

M M

K

M

=

n

M

or

A x b= (4)

have the form

xD

Djj

j= =( to1 n) (5)

whereDis the determinant ofAand is the determinant ofAwith its column replaced by b.Dj jth

As an example, consider the second order equations:

(6)a a

a a

x

x

b

b

11 12

21 22

1

2

1

2

=

In this case,

D a aa a

D b ab a

D a ba b= = =

11 12

21 221

1 12

2 222

11 1

21 2

, , , (7)

and then

xb a b a

a a a ax

b a b a

a a a a11 22 2 12

11 22 12 21

2

2 11 1 21

11 22 12 21

=

=

, (8)

This solution represents the vehicle in a steady cornering condition with a yaw rate, . Since the

vehicle forward speed is constant, there are some simple relationships with other parameters used to

describe the vehicle motion, see

rss

Figure 7.1.


D.A. Crolla


4/24

For a radius of turn,R, the path curvature, ss , is defined as

ss R=

1 (9)

the steady state yaw rate is

r URss =

(10)

and the steady state lateral acceleration is

aU

Rss=

2

(11)

Figure 7.1- The basic vehicle describing a steady turn

Equation (2) can be rewritten in terms of the path curvature per unit steer angle, as follows:

ss

f l KU=

+

12 (12)

in which lis the wheelbase, a+b, and

( )Km bC aC

lC Cr f

f r

=

(13)

The expression , within K, is called the stability marginand its sign determines the sign

of K. In fact, the sign of K is central to the description of the handling behaviour in terms of

understeer or oversteer. There are three conditions of interest, shown in

bC aC r f

Figure 7.2:

1.

, Neutral steer- HereK= 0 ss f l= 1 , corresponding to a pure rolling vehicle for which

f l R (see Figure 7.3).


D.A. Crolla


5/24


6/24

be predicted from more sophisticated models, or obtained practically from measurements. Equation

(12) can be rearranged to give

fl

RK

U

R= +

2

(15)

in which l R is the steer angle, at the roadwheels, required to drive a pure rolling vehicle (nosideslip) along a circular path of radiusR, and U R

2is the steady state lateral acceleration. If Ris

fixed, as in a constant radius steerpad test, then Kis the gradient of the graph of f against U R2

and is called the understeer gradientor understeer parameter. Its units are ( )rad m s2 but it iscommonly expressed in / g (g = 9.81m / s2).

In linear models, the graph of steer angle against lateral acceleration is linear and Kis the constant

gradient. In non-linear models, although the graph is approximately linear at low levels of lateral

acceleration (less than 0.3g, say), its deviation from linearity at higher lateral accelerations can still

be used to characterise the steady state behaviour.

Figure 7.3- The relationship between steer angle, turn radius and wheelbase for a pure rolling vehicle. Sincel R


7/24

Because the steady state solutions are so important, research workers have, over the years,

experimented with every possible method of presenting them. These differing ways of presenting

steady state results can often lead to confusion, but it is worth noting that they only contain a limited

amount of information - albeit important information - and so it is preferable to standardise on a

single method of presentation.

Figure 7.4- The relationship between understeer parameter and vehicle weight for a selection of Americanand European cars (from SAE paper 840561).

Finally, it is worth reiterating the key reasons behind the importance of steady state results:

The concepts of understeer and oversteer carry over into dynamic behaviour,

They are cheap and relatively simple for practical testing on a steering pad.

7.1.2 Stability

The solution of the equations of motion (19), Module 5, with the input, f , set to zero, represents

the case of the vehicle travelling in a straight line, for example under motorway conditions. Under

such straight running conditions, the vehicle is continually subjected to small perturbations arising

from wind gusts, road camber, road irregularities, etc, and the transient response to these small

disturbances can be assessed by looking at the roots of the equations. In other dynamical systems,

this solution is often referred to as the free vibration response.

With no input, the equations (20), Module 5, become homogeneous,:

P Q&x x 0+ = (16)and solutions have the form


D.A. Crolla


8/24

(17)x x= 0 et

where is a constant vector andx0 is a complex number. It follows that , and

equation (

&x x= = 0 et x

16) becomes

( )P Q+ =x 0 (18)This has a non-zero solution only if the determinant of matrix P Q+ is zero.

The set of linear simultaneous equations Ax= b, with b = 0, has a unique solution x=A-1

b if and

only if the inverseA-1

exists, ie. if the determinant ofAis not zero. If b = 0 then there is always the

trivial solution x = 0, but another solution exists if the determinant ofAiszero. Compare the scalar

equation ax = 0, which must have a = 0 or x = 0, with the matrix equation Ax= 0, in which both A

and xmay be non-zero.

This determinant is a quadratic in , called the characteristic equation, which will yield two

solutions, 1and 2,say, and can be reduced to the form:

( ) ( )

( )

22 2

2

2 0

+ + + +

+

+ +

+

=

I C C m a C b C

mIU

a b C C

mIU

bC aC

I

f r f r

f r r f

(19)

or

(20) 2 0+ + =D S

Note that D is always positive but Scan be positive or negative. A useful analogy is the damped

simple harmonic oscillator with no forcing (Figure 7.5), which is governed by the equations:

mx cx kx&& &+ + = 0 (21)leading to a characteristic equation:

2 0+ + =c

m

k

m (22)

or(23) 2 2+ +n n

2 0=

Thus, the coefficient,D, can be viewed as a damping term and Sas a stiffness term. Damping as

a proportion of critical is D S2 and is always positive if Sis positive. From the expression forD,

damping is seen to decrease as Uincreases.


D.A. Crolla


9/24

Figure 7.5- A one degree of freedom mass-spring-damper

For the stiffness term there are two distinct cases, depending on the sign of the stability margin,

bCr- aCf.

1.

bCr> aCf

Sis always positive but its magnitude decreases as Uincreases.

The system is unconditionally stable.

The system exhibits damped oscillatory behaviour.

Damping decreases as U increases, and the system may exhibit markedly oscillatory

behaviour.

2. bCr< aCf

Sdecreases to zero as Uincreases to the critical speed, Ucrit, given in equation (14), and when

U > Ucrit, Sis negative, giving divergent, unstable motion.

Damping is always high. It increases as Uincreases, and even at moderate values of Uit can

be greater than critical.

The two solutions of equation (19) are either both real: 1= 1, 2= 2, or complex conjugates:

1= + i, 2= i. In either case, the values of determine stability - if any is positive,

then the system is unstable, otherwise it is stable.

The solutions, , are called eigenvalues. They can be plotted in the complex plane and their

positions tracked as some parameter is changed over a range of values. The natural frequency and

damping ratio are related to the eigenvalues as follows (refer to Figure 7.6):

The undamped natural frequency is the distance of the eigenvalue from the origin.

The damped natural frequency is the imaginary part of the eigenvalue.

The damping ratio (damping as a proportion of critical) is the sine of the angle between the

eigenvalue and the imaginary axis.


D.A. Crolla


10/24

Figure 7.6- Typical eigenvalues in the complex plane, and their relationship with natural frequency and

damping ratio

As an illustration, consider an idealised parameter set given in Table 7.1. Since a = band Cr= Cf,

the stability margin is zero and the vehicle has neutral steer. The two eigenvalues are 1= -5.30 and

2= -5.52, giving a stable system which happens to be critically damped in this special case.

Now consider variations in aand b(with a + bconstant) which alter the stability margin, as given in

Table 7.2. In Figure 7.7 the eigenvalues are plotted in the complex plane for values of forward speed

increasing in geometric progression from 10 m/s to 50 m/s, denoted by increasing symbol size.

Table 7.1 -Handling parameter data for an arbitrary vehicle.

Parameter Symbol Value Units

Total mass m 1

Total yaw inertia I 1.5 tm2

CG to front axle a 1.25 m

CG to rear axle b 1.25 m

Front axle cornering stiffness Cf 53 kN/rad

Rear axle cornering stiffness Cr 53 kN/rad

Forward speed U 20 m/s

Table 7.2 - Modifications to the parameter set given inTable 7.1.

Parameter Units

Understeer

US

Neutral steer

NS

Oversteer

OS

a m 1.15 1.25 1.35

b m 1.35 1.25 1.15

bC aC r f kNm/rad +10.6 0 -10.6

K deg/g +0.85 0 -0.85

Ucrit m/s 41


D.A. Crolla


11/24

In Case US the eigenvalues are seen to be oscillatory with damping ratio decreasing as speed

increases. In Case OS the eigenvalues are real and one of them crosses the origin to instability at

U = Ucrit i.e. at 41 m/sec. In Case NS (not shown) the eigenvalues are also real but remain to the left

of the origin.

Eigenvalue plots usually show only the upper half of the complex plane since values below the real

axis are mirror images of those above.

Figure 7.7- The eigenvalues of the basic vehicle as forward speed is increased from 10 m/sec (smallestsymbols) to 50 m/sec (largest symbols).

7.1.3 Frequency responses

The frequency response gives a full description of the small perturbation, dynamic behaviour of the

vehicle. The steady state response is the limiting case of frequency response at zero frequency.

The input is assumed to be sinusoidal and in general terms, referring to equation (20), Module 5, has

the form

(24)u U= ei t

in which is the frequency (in rad/sec) and Uis a constant vector. The solution will then have a

similar form:

(25)x X= ei t ,

where X is, in general, complex and represents the gain and phase shift which x undergoes in

response to the input. It follows that &x x= i and equation (20) Module 5, can be rearranged to

give:


D.A. Crolla


12/24

D.A. Crolla

U (26)( )X U= + =

i P Q R H 1

Matrix H is called the transfer function of the system of equations. In this case there is a single

input, f= fei

, and the transfer function is a (2 x 1) column matrix:

H

H

H

v

r=

(27)

so that

( ) ( )X H X Hv f v r f r = =, (28)

Using Cramer's Rule these expressions can be written as:

XV iV

D iDX

R iR

D iDv fr i

r i

r f

r

r i

= +

+ = i

+

+, (29)

in which subscripts rand idenote real and imaginary parts, and

(D mI l C CU m bC aC rf r

r= + + 2

2

2 )f (30)

( ) ( )2 2f r f ri

I C C m a C b CD

U

+ + += (31)

VlbC C

UmaC U r

f r

f= (32)

V ICi f= (33)

RlC C

Urf r

= (34)

R maCi f= (35)

The complex quantity, Xv / f is most meaningfully expressed in polar form, resulting in a lateral

velocity gain per unit steer angle and an associated phase shift, both of which are functions of

frequency, . Similarly, Xr / f results in a yaw rate gain per unit steer angle and its associated

phase.

Once Xv and Xr have been evaluated, any linear combination is also available. In particular, the

response of the lateral acceleration, , is iX&v uR+ v+ UXr.

Since yaw rate relates to what the driver sees and lateral acceleration relates to what the driver feels,

they are generally considered to be the two important output parameters. Graphs of their gains and

phases, plotted as functions of frequency, give a complete summary of the linear response of the

vehicle.

Using the example parameters in Tables 7.1 and 7.2 above, Figures 7.8 and 7.9 show frequency

responses of yaw rate and lateral acceleration. Note that:

The response gains at zero frequency are the same as the steady state responses calculated

earlier.



13/24

The understeering vehicle maintains a flat response up to a higher frequency. This is

highlighted in Figure 7.10, in which the response gains are normalised to their steady state

values.

Figure 7.8- Yaw rate frequency response gains and phases for three cases: understeer, neutral steer andoversteer.


D.A. Crolla


14/24

Figure 7.9- Lateral acceleration frequency response gains and phases for three cases: understeer, neutral

steer and oversteer.

7.2 Practical implications

The term bC , referred to as the stability margin, has been shown to be a critical design

parameter. Moreover, its importance in controlling understeer and oversteer behaviour has been

reinforced in both the steady state and stability forms of solution of the equations of motion. This

link between the steady state behaviour and the transient response of the vehicle is an important

element in vehicle dynamics studies. It means, for example, that a vehicle classified as understeering

or oversteering from a steady state test will exhibit certain types of transient behaviour. This

correlation between steady state and transient response of the vehicle provides the justification for

aCr f


D.A. Crolla


15/24

the continuing reliance on steady state test procedures throughout the industry to characterise

vehicle dynamic behaviour.

Figure 7.10- The gains of the previous two figures, normalised to their steady state (zero frequency) values.

Having accepted that the stability margin is a key design parameter, the question arises of what the

designer can do to influence it in order to promote under or oversteer effects. This design thinking

can, however, be usefully extended by broadening the definition of the cornering stiffnesses,

and . Strictly, these terms are functions of the slip angles only. However, an effective cornering

stiffness can be viewed as consisting of the primary effect due to slip angle, plus a smaller,

secondary effect due to other features such as camber, compliance steer, etc. These effective

cornering stiffnesses will be denoted by

Cf

Cr

Cf and Cr . Hence, the effective stability margin becomes

.bC aC r f


D.A. Crolla


16/24

These additional small effects can be used in practice to tune the vehicle by adjusting the stability

margin by small amounts. For example, the effect of camber angle, , is to generate a lateral force

which for small angles may be expressed as:

Fy (due to camber) C= (36)

where is the tyre camber stiffness. This force may be assumed (again for small angles) to add to

the slip angle term by linear superposition. Thus, an effective cornering stiffness, , could be

viewed as the combined effects of C and a camber effect. The important result of this argument is

that the simple 2dof model can be used as a basis for considering the effect of camber, even though

camber is not included in the model.

CCf

f

Some examples of design features which influence the stability margin, bC aC r f , are discussed

below.

7.2.1

Vehicle CG position

Altering the weight distribution of the vehicle will change aand band affect the stability margin

directly. However, the situation is not so straightforward because the vertical loads on the tyres will

also be changed and these will affect the cornering stiffness values. If and are relatively

insensitive to vertical load - which is often the case for relatively heavily laden tyres - then the effect

on the stability margin of changing aand bis easily predicted. In other cases, the combined effects

of the CG position and the cornering stiffnesses must be calculated. The general trend is that moving

the CG rearwards - the extreme case being a rear engine vehicle - promotes oversteer.

Cf Cr

7.2.2

Tyre cornering stiffnesses

The cornering stiffnesses, C and , can be altered by a substantial amount by changing the tyre

size. However, although equipping the car with different tyres front to rear is not practical for most

cars, it is used on some powerful rear engined cars, Porsche, for example, as a means of

substantially increasing to avoid oversteer.

f Cr

Cr

More modest changes in C and can be effected by altering tyre pressures. Over the normal

working load range of the tyre, an increase in pressure results in an increase in cornering stiffness.

The vehicle designer invariably has to use this tyre pressure effect to compensate for the variousloading conditions of the vehicle. For a typical passenger car, going from unladen to full laden

results in a substantial change in aand b. The designer, however, would prefer a car whose handling

qualities remain consistent over the loading range and the practice of recommending different tyre

pressures for various conditions is a standard way of trying to compensate the stability margin.

f Cr

Historically, another problem existed which is now disappearing with the widespread adoption of

radial tyres. Cross-ply tyres have a significantly lower cornering stiffness than radials of the same

size. Hence, if both radials and cross-plys are used on a vehicle, there is a danger of affecting the


D.A. Crolla


17/24

stability margin substantially. Legislation still exists in the UK to prohibit the use of cross-plys on

the rear axle in these circumstances.

7.2.3 Load transfer

During cornering, load transfer across an axle takes place so that the load on the outer wheel is

increased and the load on the inner one is decreased by the same amount. Under the assumptions

used for the 2dof model, which ignores suspension deflections, the calculation of this load transfer is

straightforward (Figure 7.11).

Figure 7.11- A simple calculation of load transfer in a cornering vehicle, in which denotes the lateral

acceleration.

ay

If the cornering stiffness were proportional to load, then the net effect of this load transfer would be

zero. The decrease in lateral force at the inner wheels would be exactly compensated by an

equivalent increase at the outer wheels, so when the forces on the two wheels were summed the total

lateral force would remain constant. However, the cornering stiffness is not constant with vertical

load and a typical graph of lateral force against vertical load for a given value of slip angle is shownin Figure 7.12. It can be seen from this curve that there is a net reduction in total lateral force due to

the load transfer effect. The lateral force generated by the inner plus outer wheels during cornering

must always be less than the lateral force generated by the two wheels at their static loads. The

designer can use this load transfer effect to reduce the effective cornering stiffness at one end of the

vehicle.

The mechanism which the designer can use to control this effect is the roll stiffness balance between

the front and rear of the vehicle. The total load transfer must be reacted across both ends of the

vehicle and the ratio in which this occurs turns out to be an important factor in controlling vehicle


D.A. Crolla


18/24

handling. This ratio, or front/rear balance as it is commonly called, is controlled by the relative roll

stiffnesses between front and rear. For example, if the roll stiffnesses are in the ratio 60:40 between

front and rear, then the load transfer will also be split 60:40 between front and rear (assuming that

the body is rigid). The roll stiffness is a combination of the primary suspension stiffness plus any

additional effects such as anti-roll bars. Hence, by specifying the anti-roll bars, the designer cancontrol the load transfer balance and thus control the lateral force reduction. Common practice, for

example, is to fit anti-roll bars at the front of the vehicle so that, as load transfer takes place, the net

lateral force at the front reduces, thereby promoting an understeering effect. This mechanism is quite

a subtle one because its effect increases in proportion to the lateral acceleration, which is a measure

of the cornering severity. Thus it can be used to ensure that the car has stable behaviour throughout

the range up to its cornering limit.

Figure 7.12- The effect of load transfer in reducing the total lateral force generated at one end of the vehicle.

7.2.4 Camber angle

For small angles, it can reasonably be assumed that the lateral force due to slip angle and the lateral

force due to camber angle can be simply added by assuming that linear superposition applies. The

camber effect is invariably of secondary importance for two reasons. First, the tyre camber stiffness

is much lower than the cornering stiffness, and second, camber angles arising from suspension

displacement are normally very small (< 2). Nevertheless, these small camber effects can be used to

make minor adjustments to the handling balance. An example is shown in exaggerated form in

Figure 7.13. The camber angles lean outwards from the corner in this case and result in a slight

reduction in the net lateral force available for cornering.

The designer can exercise control of the camber angles through the suspension kinematic design.

However, scope is limited because in practice large camber angles are unacceptable for both tyre

wear and aesthetic reasons.


D.A. Crolla


19/24

Figure 7.13- An example of a cornering vehicle indicating a net reduction in total lateral force due to cambereffects.

7.2.5 Compliance steer effects

The relationships between the wheel orientation and the suspension deflection are discussed in detail

in Module 8 - Handling - extensions. These relationships are governed by both kinematic and

compliance effects. However, an important feature of suspensions is their ability to cause small steer

angles when a lateral force acts at the wheel. Often this may be an undesirable side effect caused by

the rubber bushes needed to reduce suspension harshness and high frequency transmission problems,

but it can also be exploited to improve the vehicle handling properties. Rear compliance steer has

been claimed to offer advantages by, for example, Mercedes and Vauxhall. An example of a

compliance steer effect at the rear of a cornering vehicle is shown in Figure 7.14. The rear lateral

tyre force has resulted in a small steer angle, r, which, in this example, is in the same direction as

f . This results in an understeering effect since the front steer angle required to negotiate a given

turn now needs to be increased. The effect may be viewed as modifying the effective cornering

stiffness values. In the example shown, the effective rear cornering stiffness has been increased,

thereby promoting understeer in the stability margin term.

Cr

7.3

A complete example

A linear analysis is conducted to compare two contrasting vehicles: a 1949 Buick and a Ferrari

Monza. The parameters are given in Table 7.3 and highlight the fundamentally different designs of

these vehicles. For example, the Buick, with its large overhanging masses outside its wheelbase, has

a large yaw inertia compared with the Ferrari, which has its wheels at each corner of the body. The

Ferrari, as might be anticipated, has much higher cornering stiffnesses relative to its mass; the

vehicle's force-generating ability relative to its mass governs its ability to generate lateral

acceleration.


D.A. Crolla


20/24

The understeer gradients indicate that the Buick is a strongly understeering vehicle - typical of

American cars of the period - while the Ferrari is slightly understeering but very close to neutral

steer.

Table 7.3 -Vehicle parameters for a 1949 Buick and a Ferrari Monza.

Parameter Symbol Units 1949 Buick Ferrari Monza

Mass m t 2.045 1.008

Yaw inertia I tm2

5.428 1.031

CG to front axle a m 1.488 1.234

CG to rear axle b m 1.712 1.022

Front axle cornering stiffness Cf kN/rad 77.85 117.44

Rear axle cornering stiffness Cr kN/rad 76.51 144.93

Wheelbase a b+ m 3.200 2.256

Stability marginbC aC

r f

kNm/rad 15.15 3.20Understeer gradient K deg/g 0.91 0.05

Figure 7.15 shows the steady state values which arise from a 0.3 g rightward (clockwise) turn at a

fixed forward speed of 20 m/sec. The fixed lateral acceleration and forward speed imply a fixed yaw

rate (8.4 deg/sec) and a fixed turn radius (136 m). Note that in each case the slip angles are negative,

giving rise to positive (rightward) lateral tyre forces. The steady state slip angles of the Ferrari's

tyres are much smaller than those for the Buick, due to the higher cornering stiffness values.

Figure 7.14- Rear compliance steer (shown exaggerated) which results in an understeering effect.


D.A. Crolla


21/24

Figure 7.16 shows the way in which the eigenvalues of the two vehicles change as forward speed is

increased from 20 m/sec to 50 m/sec. The most significant difference between the two vehicles is

that the damping associated with the Buick's responses is much lower than that for the Ferrari. The

damped natural frequencies, d , and damping ratios, , are shown for three forward speeds in

Table 7.4.

Table 7.4- Eigenvalues, damped natural frequencies and damping ratios for the Buick and the Ferrari at threeforward speeds.

Buick Ferrari

Speed Eigenvalued , Hz Eigenvalue d , Hz

20 m/s -3.711.65i 0.26 0.91 -14.50.91i 0.14 1.00

30 m/s -2.481.66i 0.26 0.83 -9.681.45i 0.23 0.99

50 m/s -1.491.67i 0.27 0.67 -5.811.65i 0.26 0.96

Figure 7.15- A comparison of steady state values for the Buick and Ferrari, negotiating a 0.3 g turn at 20 m/s.(r= 8.4 deg/s,R= 136 m)


D.A. Crolla


22/24

Figure 7.17 compares the yaw rate response of the two vehicles to a step steer input at 50 m/sec. The

step steer was adjusted in each case to give a steady state lateral acceleration of 0.3 g. The Ferrari

responds more quickly to this input, reflecting its higher cornering stiffness per unit mass. The

Buick's responses contain a transient overshoot before settling down to a steady state turning

condition - a behaviour which might be anticipated in view of its lightly damped eigenvalues.

Figure 7.16- Eigenvalues of the Buick and the Ferrari as forward speed is increased from 20 m/s (smallestsymbols) to 50 m/s (largest symbols).

Figure 7.17- Time histories of the yaw rate response of the Buick and the Ferrari at 50 m/sec, following astep steer input which results in a steady state lateral acceleration of 0.3 g.

The frequency responses, Figures 7.18 and 7.19, reinforce some of the information previously

obtained about these vehicles but also provide some further insight. At very low frequencies they are

the same as the steady state responses and the yaw rate gains of the nearly neutral steer Ferrari are

much greater than those of the understeering Buick. The lighter damping of the Buick as speed

increases is noticeable as a peak at around 0.3 Hz. This highlights one of the problems of strongly


D.A. Crolla


23/24

understeering vehicles, namely that the response can feel highly oscillatory during high speed

driving.

The additional information contained in the frequency responses relates to the frequency range over

which the vehicle is capable of responding to steering inputs. The Ferrari is markedly superior in

this respect, retaining its gains up to higher frequencies than the Buick. For example, it retains its

yaw rate gain (Figure 7.19) up to around 1 Hz, whereas the Buick's response starts rolling off at

around 0.3 to 0.4 Hz.

Figure 7.18- Yaw rate frequency response for the Buick and Ferrari at two forward speeds


D.A. Crolla


24/24

Figure 7.19- Lateral acceleration frequency responses for the Buick and Ferrari at two forward speeds.


D.A. Crolla

07 - Handling - Fundamentals Notes

Documents

Transcript of 07 - Handling - Fundamentals Notes