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11. STEADY-STATE CORNERING The cornering behaviour of a motor vehicle is an important performance mode often equated with

handling. "Handling" is a loosely used term meant to imply the responsiveness of a vehicle to driver input, or the ease of control. As such, handling is an overall measure of the vehicle-driver combination. The driver and vehicle is a "closed-loop" system – meaning that the driver observes the vehicle direction or position, and corrects his/her input to achieve the desired motion. For purposes of characterizing only the vehicle, "open-loop" behaviour is used. Open loop refers to vehicle response to specific steering inputs, and is more precisely defined as "directional response" behaviour.

The most commonly used measure of open-loop response is the understeer gradient. Understeer gradient is a measure of performance under steady-state conditions, although the measure can be used to infer performance properties under conditions that are not quite steady-state (quasi-steady-state conditions).

Open-loop cornering, or directional response behaviour, will be examined in this section. The approach is to first analyze turning behaviour at low speed and then consider the differences that arise under high-speed conditions. The importance of tyre properties will appear in the high-speed cornering case and provide a natural point for systematic study of the suspension properties influential to turning.

maneuver [U.S.]; manoeuvre [U.K.]: manevră; schimbare controlată a cursului, a direcţiei de

deplasare maneuverability [U.S.]; manoeuvrability [U.K.]: abilitate de a schimba direcţia de deplasare a unui

vehicul handling: the relative ability of a vehicle to negotiate curves The term manoeuvrability mean the maximum performance, usually the time to complete a given

manoeuvre, of a vehicle subjected only to physical limitations (like traction limits, available power etc.) but without considering the limitations of the controller (i.e., if the driver is perfect).

We term handling mean the maximum performance of the same vehicle, but considering the limitations of the control actuation subsystem (i.e., the driver).

In other words, manoeuvrability means the maximum performance that a vehicle can produce without considering the driver’s limitations, whereas handling measures how much of this potential can really be exploited by a driver who may apply only limited inputs to the system.

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11.1. CORNERING MODALITIES

Fig. 11.1. Cornering realised by inducing different circumferential speeds to the outer and inner wheels

Fig. 11.2. Cornering realised by semi-frames relative rotation upon a pivot

Fig. 11.3. Cornering realised by steering wheels

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11.2. LOW-SPEED CORNERING

The first step to understanding cornering is to analyze the low constant-speed turning behaviour of a motor vehicle. At low speed (parking lot manoeuvres without acceleration) the tyres need not develop lateral forces. Thus they roll with no slip angle, and the vehicle must negotiate a turn as illustrated in Figure 11.4 (6.1). If the rear wheels have no slip angle, the centre of turn must lie on the projection of the rear axle. Likewise, the perpendicular from each of the front wheels should pass through the same point, named instantaneous centre of turn. If they do not pass through the same point, the front tyres will "fight" each other in the turn, with each experiencing some scrub (sideslip) in the turn. The ideal turning angles on the front wheels are established by the geometry seen in the figure, and define the steering angles for the turn.

Fig 11.4 (6.1) Geometry of a turning vehicle

For proper geometry in the turn (assuming small angles), the steer angles are given by:

δo ≅ tgδo = )2/( tR

L+

(6-1)

δi ≅ tgδi = )2/( tR

L+

(6-2)

The average angle of the front wheels (again assuming small angles), is defined as the Ackerman Angle:

RL /=δ (6-3)

The terms "Ackerman Steering" or "Ackerman Geometry" are often used to denote the exact geometry of the front wheels shown in Figure 11.4 (6.1). The correct angles are dependent on the wheelbase of the vehicle and the radius of turn. Errors, or deviations, from the Ackerman in the left-right steer angles can have a significant influence on front tyre wear. Errors do not have significant influence on directional response; however, they do affect the cantering torques in the steering system. With correct Ackerman geometry, the steering torques tend to increase consistently with steer angle, thus providing the driver with a natural feel in the feedback through the steering wheel. With the other extreme of parallel steer, the steering torques grow with angle initially, but may diminish beyond a certain point, and even become negative (tending to steer more deeply into the turn). This type of behaviour in the steering system is undesirable.

The other significant aspect of low-speed turning is the off-tracking that occurs at the rear wheels. The off-tracking distance, ∆, may be calculated from simple geometry relationships as:

∆ ≅ R [1 - cos(L/R)] (6-4a)

Using the expression for a series expansion of the cosine, namely:

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...!6!4!2

cos642

1zzzz −+−=

Then

∆ ≈ L2/(2 R) (6-4b)

For obvious reasons, off-tracking is primarily of concern with long-wheelbase vehicles such as trucks and buses. For articulated trucks, the geometric equations become more complicated and are known as "tractrix" equations.

11.3. TRACTRIX EQUATIONS AND TURNING REQUIREMENTS

The turning circle dimension (twice the turning radius) represents the diameter of the circle created by the outer front wheel when making a full turn. There are two ways of measuring the turning radius: curb to curb and wall to wall. The latter is always larger because it takes into account front-end overhang. As the vehicle turns, the inside wheels make a smaller circle than the outside tyres.

Off-tracking is the difference in radii from the turning centre to the vehicle centreline at the foremost and rearmost axles of the vehicle or combination and represents the increase beyond the tangent track occasioned by a turn. The spiral pattern generated will determine the off-tracking and turning path of the vehicle or combination of vehicles. The track will determine whether or not the roadbed is of sufficient width to accommodate the vehicles and is a significant factor in determining the ability of a vehicle or combination to negotiate turns safely and compatibly with other traffic. Off-tracking increases with length of wheelbase for a single vehicle; however, on the overall wheelbase for a doubles or triples combination, the off-tracking is about equal to or less than a shorter overall wheelbase tractor-semitrailer combination.

Fig 11.5. ISO Requirements regarding articulated vehicles turning circle and off-tracking

Even the tractor-fulltrailer combination consist of three rigid bodies, theirs off-tracking can be smaller than the off-tracking of tractor-semitrailer combination. This is possible due to the shorter length of individual vehicles and to the optimised steering system of the fulltrailer.

Two systems are used to steer the fulltrailers: • the drawbar and the trailer’s front axle form a rigid body which is named dolly; this can pivots

upon a vertical axis passing through the middle of the trailer’s front axle; • the trailer’s front axle is non-pivoting but is equipped with a steering mechanism that direct outer

the trailer’s front wheels, reducing therefore the off-tracking.

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Fig 11.6. Tractor-trailer combinations in transitory turn:

left – with dolly; right – with trailer steering system

The designers of articulated vehicle have to deal not only with the problem of turning circle and off-tracking, but must also to avoid the contact between the bodies of tractor and trailer. As can be seen in the right side of the previous figure, this problem is not solved. This concern is more stringent if considers the possible longitudinal and lateral declivities of the road.

The trailer length influences directly the off-tracking so that vehicles with longer trailers need more space to turn. The position of the articulation point and the trailer’s front overhang affect the off-tracking and also the wall to wall turning circle. To simplify the driver’s vehicle control during cornering, the most outer point of the road train must be in the front side of the tractor (the most outer point of the trailer must not exceed the trajectory of the front outer corner of the tractor).

Fig 11.7. Trailer length influence over the off-tracking

11.4. LATERAL BEHAVIOUR OF TYRES

Under cornering conditions, in which the tyre must develop a lateral force, the tyre will experience lateral slip as it rolls. The angle between direction of heading and its direction of travel is known as slip angle α. These are illustrated in Figure 11.8 (6.2).

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Fig. 11.8. (6.2) Tyre cornering force properties

Fig. 11.9. (6.3) Variables affecting tyre cornering stiffness

The lateral force, denoted by Fy, is called the "cornering force" when the camber angle is zero. At a given tyre load, the cornering force grows with slip angle. At low slip angles (5 degrees or less) the relationship is linear. Hence, the cornering force is described by:

Fy = Cα α (6-5)

The proportionality constant, Cα, is known as the "cornering stiffness" and is defined as the slope of the curve for Fy versus α at α=0. In SAE convention, a positive slip angle produces a negative force (to the left) on the tyre, implying that Cα must be negative; however, SAE defines cornering stiffness as the negative of the slope, such that Cα takes on a positive value. The same definition for the cornering

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stiffness is adopted in the ISO convention, because the slip angle in Figure 11.8 is positive but the lateral force is negative.

The cornering stiffness is dependent on many variables. Tyre size and type (radial- versus bias-ply construction), number of plies, cord angles, wheel width, and tread are significant variables. For a given tyre, the load, inflation pressure and camber are the main variables. Speed does not strongly influence the cornering forces produced by a tyre. The plots in figure 11.9 (6.3) illustrate the influence of many of these variables.

Because of the strong dependence of cornering force on load, tyre cornering properties may also be described by the "cornering coefficient" which is the cornering stiffness divided by the load. Thus the cornering coefficient, CCα, is given by:

CCα = Cα /Fz (6-6)

Cornering coefficient is usually largest at light loads, diminishing continuously as the load reaches its rated value. At 100% load, the cornering coefficient is typically in the range of 0.2 daNy/daNz/deg ≈ 11.5 daNy/daNz/rad or 0.2 (lb cornering force per lb load per degree of slip angle).

The lateral force is resisted by the lateral cornering forces at the wheels. Cornering forces can only

be generated between a rubber tire and the road surface when the tire rolls at an angle to its longitudinal plane; that is, a certain wheel slip angle is required.

Fig. 11.10. Tyre under slip: components of velocity and friction force in the tyre-ground contact patch

(upper view, right turn)

The degree of lateral cornering force which a pneumatic tire can provide depends upon numerous factors, such as wheel slip angle, wheel load, tire design and dimensions, tire pressure and the amount of grip (friction) afforded by the road surface.

Fig. 11.11. Cornering force as a function of wheel slip angle

The pneumatic tire’s specific response characteristics mean that, at a constant slip angle, higher

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wheel loads will not evoke proportional increases in cornering forces. In the example presented in Figure 11.11, doubling the wheel load increases the cornering force by a factor of only 1.5…1.7. The slip angle must also be increased if the cornering force is to be doubled. This explains why the axle supporting the greater load assumes a larger slip angle than its less heavily loaded counterpart, assuming that identical ratios of lateral (side) force to tire contact force act on both axles.

Fig. 11.12. Cornering forces Fs and slip angle α on a 3-axle vehicle with non-steered tandem axle

11.5. HIGH-SPEED CORNERING

The most important reasons for steering a vehicle are to produce lateral acceleration (for example, to avoid an obstacle) and yaw velocity (to quickly change the heading angle, for example to reorient the vehicle in a new direction of travel).

At high speed, the turning equations differ from the low speed geometrical equations because guiding forces will be present to develop lateral acceleration. But the lateral forces are a consequence of the slip angles that are present at each wheel. Note that slip angles appear also, even al low speed, when vehicle is subjected to longitudinal forces that induce longitudinal acceleration.

11.5.1. Plan Model for Cornering

Cornering and handling qualities of a motor vehicle constitute important aspects of the active safety, directly related with the traffic accidents. Consequently, even for the design stage, knowing the handling characteristics and controlling the means that can influence them have very big importance for the automotive engineers. Computer simulation permits to obtain useful information about vehicle’s dynamic behaviour, easily and rapidly.

Fig. 11.13. Plan model for vehicle’s cornering dynamics

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11.5.2. “Bicycle Model” for Cornering The dynamic model presented here – an extension of the so called “bicycle” model – is able to

describe the basic cornering behaviour in various travelling conditions of a two-axle vehicle and can include other influences for steering, traction and braking systems. It can lead to a better understanding of the automotive dynamics.

Normally, experimental data, achieved in real conditions tests, is necessary first to calibrate the model (to realise fine adjustment of the model parameters values) and validate the model (to confirm the correctness of the results).

The planar dynamic model presented in Figure 11.14 it is called “single-track” or “bicycle” model of the vehicle. Its main characteristic is the replacement of the both wheels of an axle with only one wheel that has an equivalent kinematic and dynamic behaviour. The model disregards the effects of roll movement, considering a very stiff suspension (rigid body-axles assembly).

Fig. 11.14. “Bicycle” model for vehicle’s cornering dynamics

It considers that the vehicle moves on an even curvilinear trajectory, with his instantaneous centre of rotation Cr determined by the speed vectors fvr and svr of front and rear axles. The speed vector vr corresponds to the centre of gravity Cg, which generally adopts as the control point (the control point of a vehicle is a point who’s current position and trajectory presents maximal interest for the driving process).

It considers a fix coordinate system xOy, independent of vehicle. Also adopts an axis system linked to vehicle, which has the origin in the centre of gravity and his axis are a longitudinal axis t and a transversal axis n (perpendicular to the axis t).

The aerodynamic force is the resultant of the components Ra and Ran, which act on the two directions and focus in the centre of pressure Cp.

The influence of road declivity is taken into consideration by the forces Rp and Fdr, both acting in the centre of gravity. The force Rg, disposed on the longitudinal axis of the car, represents the grade resistance; the force Rgn, acting on the n direction, is a consequence of the road declivity in the vehicle’s transversal direction.

In the tyre-road contact surfaces, on the front and rear wheels act the tangential forces Xf and Xr, the lateral forces Yf and Yr, and the normal forces to the path Zf and Zr.

To consider the load transfer between the wheels on inner and outer sides of turn or between the axles during traction or braking it is necessary to use supplementary models that include the behaviour of suspension.

The speed vector vr of the centre of gravity makes with longitudinal axis t the angle β, named sideslip angle and indicating the lateral deviation of the vehicle. The speed vr can be decomposed in two

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perpendicular components:

ββ

sincos

n

t

vvvv

==

(11)

contained respectively in vehicle longitudinal and transversal planes. Consequence of the translational speed and angular velocity, in the centre of gravity will act:

• the tangential acceleration (on the speed direction)

dtdva =v ; (12)

• the centripetal acceleration (on the direction CgCir)

)(v)(vcp ωββ +=+= &&& Ψa =v2/R; (13)

• the yaw acceleration

ψωωε &&& ===dtd . (14)

Decomposing the translational accelerations av and acp on the directions t and n, obtains the expressions for the centre of gravity acceleration’s longitudinal and transversal components:

ββββ

cossinsincos

cpvn

cpvt

aaaaaa

+=−=

. (15)

In the hypothesis of a non-violent cornering (the transversal acceleration an under 4 m/s2), it can consider that the lateral forces on tires are proportional to the corresponding slip angles.

sss

fff

ββ

CYCY

==

. (16)

Combining relations (1…5), it can be written the system of three differential-equations (translations on directions t and n plus yaw) that describes the vehicle behaviour in any cornering situations, including transitory state:

z

z

n

t

cos)(sin

sin)(cos

JM

mF

vv

mF

vv

∑

∑

∑

=

=+−

=+−

ψ

βψββ

βψββ

&&

&&&

&&&

. (17)

The three unknowns (the speed v, the slip angle β and the yaw angle ψ) can be determined approximately by computer integration only if are known their initial values, the temporal evolutions of forces and yawing moment and the vehicle inertial characteristics (m – vehicle mass; Jz – vehicle moment of inertia about z axis).

The resultant of exterior forces and torque entering in the previous system of equations are computed using the projections on t and n directions of gravitational, aerodynamic and tyres forces. To calculate the longitudinal and lateral forces on tyres it is necessary to consider the road friction characteristics and the driving and braking torques produced by the drivetrain and brakes.

If considers small sideslip angle β (sinβ ≈ tanβ ≈ β and cosβ ≈1), then the system of equations become:

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z

z

n

t

)(

)(

JM

mF

vv

mF

vv

∑

∑

∑

=

=+−

=+−

ψ

ψββ

βψβ

&&

&&&

&&&

. (18)

The first equation shows that if the sum of the longitudinal forces remains constant during cornering, the vehicle velocity will decrease, or, in other words, to mentain the same velocity when negotiate a turn it is necessary to push more the accelerator pedal.

11.5.3. Steady-State Cornering The steady-state cornering equations are obtained if considers constant sideslip angle ( β& =0) and

null values for translational and yaw accelerations: v&=0 (v constant) and Ψ&& =0 (Ψ& =ω constant).

z

z

n

t

JM

mFv

mFv

Σ=

Σ=−

Σ=−

0

ψ

βψ

&

&

or

0=Σ

Σ−=

Σ−=

z

n

t

MmFv

mFv

ω

ωβ

. (19)

The third equation shows that in steady-state cornering the sum of the moments over the z axis in the centre of gravity must be null. That means the tyres of the nearest axle to the centre of gravity must generate the biggest lateral force and, as consequence, will experience the biggest sideslip angle.

11.5.4. Directional behaviour (Oversteer and Understeer)

A vehicle is said to understeer if the slip angle at the front axle is bigger than the slip angle at the rear axle. The opposite applies for a vehicle which oversteers. A vehicle is said to have a neutral steering behaviour if the front slip angle and rear slip angle are equal. The circle in the Figure 11-15 passes through the centres of the bycicle-model contact patches and through the instant centre in the case of null slip angles (Ackermann conditions). Having the orientations of the real velocities of the model tyres it obtains the instant centre in the actual cornering situation. The steering behaviour of the vehicle can now be easily identified: if this point is on the circle, the vehicle will have neutral steer, if the point is inside circle the vehicle will oversteer.

Fig. 11.15. Graphical definition of vehicle directional behaviour

(understeer – outside circle; neutral steer – on circle; oversteer – inside circle)

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Another way to indicate the steering behaviour it is connected to the lateral deviation of the tyres. The oversteering vehicle present a more evident sliding tendency at the rear tyres, figure 11.16. That means it is yawing faster than an understeering vehicle and will generate bigger lateral accelerations, which means the vehicle can negotiate turns with higher speeds. That’s why the race drivers prefer oversteering attitude of their vehicles. But this kind of steering behaviour necessitates skill and experience because the yawing process it is unstable and may need steering wheel corrections in an opposite way than the bend is oriented. For example, if the driver wants to increase the velocity, that may accentuate the rear whels sliding and, to mentain the desired course, it is necessary to reduce the steering angle of the front whels, possible steering them in the other direction.

Fig. 11.16. Vehicle cornering attitude (understeer and oversteer)

This kind of driving is too tricky for the unexperinced or normal drivers, and that’s why the series motor vehicles are designed to present easy understeering: to steer more the vehicle, the driver will rotate more the steering wheel in the same way with the desired trajectory, which is natural and instinctive.

11.6. STEERING CHARACTERISTICS

A vehicle will not necessarily display the same self-steering effect at all possible rates of lateral acceleration. Some vehicles always understeer or oversteer, and some display a transition from understeer to oversteer as lateral acceleration increases, while other vehicles respond in precisely the opposite manner.

11.6.1. Centripetal Acceleration

11.6.2. Yaw

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11.6.3. Characteristic and Critical Speed

Fig. 11.17. Steering angle requirements on stationary circling

Fig. 11.18. Roll angle on stationary circling

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11.7. STEERING GEAR INFLUENCES

Fig. 11.19. Steering mechanisms: left – rigid axle; right – articulated axle

11.7.1. Steering Ratio

Fig. 11.20. Variable ratio principle (here low steering angles correspond to low steering ratio)

11.7.2. Steering Efficiency

• best possible efficiency for direct actuation (to reduce driver’s effort); • small efficiency (but not zero) for inverse actuation (to “feel” the road).

11.7.3. Assisted Power Steering

Fig. 11.21. Characteristics of power assisted- and manual steering actuation

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11.7.4. AWS – All-Wheel Steering

Fig. 11.22. 4-wheel steering

Fig. 11.23. 6-wheel steering