Proceedings of the Institution of Mechanical Engineers,Conference Proceedings

The present paper combines older work by the first author with more recent research by the second author. A basic simplifying assumption of the entire investigation is that the height of the centre of gravity of the vehicle above the road surface has been neglected, such that the vertical wheel reactions remain constant.

In the earlier analysis it was also assumed that the drift angle of rolling wheels under lateral forces may be neglected, and the possibility of lateral skidding of rolling wheels was also ignored. The equations of motion may then be reduced to a comparatively simple non-linear second-order system of ordinary differential equations. The equations may be given a non-dimensional form in several ways, and two such methods are employed to reveal some characteristic properties of the skidding phenomenon.

The more recent analysis has been refined considerably by proper allowance for the drift angle of rolling wheels under the action of lateral forces, for the possibility of sideways skidding of rolling wheels, and for braking action on the rolling wheels. The equations of motion then reduce to a third-order system which may be solved in closed form in the linear case of small deviations from a rectilinear motion. It appears that the drift angle of rolling wheels under the action of lateral forces leads to a small increase of the critical speed and to a significant reduction of the danger when the critical speed is exceeded. The non-linear equations for larger deviations from a rectilinear motion cannot so easily be discussed qualitatively, but detailed results have been obtained by means of a digital computer.

This paper considers first the steady-state motions of a simple two-wheeled vehicle model having non-linear sideway force relationships with respect to tyre slip angle. It is shown that any steady-state conditions may be represented and their solutions found by simple graphical means, using only the non-linear curves. The curves can be modified to take into account the influence of vehicle parameters such as compliance, roll steer, wheel camber, and load transfer.

Stability boundaries are discussed and criteria are presented showing that stability of the motion depends only on the slopes of the curves and the speed of the manoeuvre at the cornering acceleration being considered.

A more involved four-wheeled vehicle model is then considered when subjected to braking while cornering on a fixed radius path of 45·8 m on a wet Bridport macadam surface. Actual sideway force–slip angle curves for combined braking and cornering, as presented by Holmes and Stone (see reference (6))^†, are used with the equations of motion derived for the quasi-steady state conditions of decelerating while cornering. The effects of front wheel steered angle and body slip angle on the forces necessary for the manoeuvre are also considered.

An envelope of maximum cornering acceleration at various braking decelerations is presented. This shows that for those particular conditions up to about 70 per cent of maximum deceleration may be obtained before there is more than about 10 per cent loss in maximum cornering ability. Outside the envelope the vehicle fails to maintain the path. At the lower deceleration the car spins, and at higher values it continues tangentially to its original path without spinning.

It is also shown that the total sideway force–slip angle curve for a pair of front or rear wheels, when one or both wheels have a high braking force coefficient, can have a sharp peak, such that for small increase in slip angle there is a rapid fall in sideway force. It is suggested that this is why a rear wheel skid which occurs while braking and cornering is more difficult to correct than one which occurs when only cornering.

The pure braking and cornering forces of a given tyre on a given wet surface at a given speed are functions of braking slip and slip angle respectively. These functional relationships are of great importance in connection with the handling and stability of vehicles and the behaviour of brakes and antilocking devices. When cornering and braking occur simultaneously the forces interact and the functional relationships are modified, with consequent effects on vehicle characteristics.

The empirical data available at the Laboratory on these relationships, both for pure forces and for forces in combination, are presented. The known effects of such factors as speed, road surface texture, tread pattern, tread resilience, and tyre construction are reported, but since the work was carried out at various times and for various purposes the coverage is incomplete and uneven.

The test vehicle, test procedure and the methods of measuring and evaluating the data are briefly described. Some data have been analysed statistically to discover the dependence of the various frictional coefficients on factors such as those mentioned above. It is believed that greater use should be made of these statistical techniques in future work. Raw data can also be converted into such practical quantities as braking distances, wheel locking times, etc., by computing theoretically the behaviour of model braking systems. An example of such a study is given.

The principal features of the mutual effect of braking and cornering forces may be summarized as follows:

(a)

If curves of sideway force against slip angle are plotted for constant values of braking force, there is little change of initial slope, but the maximum sideway force and the slip angle at which it occurs are reduced. If, however, the curves are plotted for constant braking slip, then the initial slope is decreased and the slip angle for maximum sideway force is increased; at this same angle the sideway force at zero braking slip will only be slightly greater.

A mathematical model of an automobile is described, which permits the study of simultaneous cornering and ride motions on irregular terrain. A major departure from previous analytical treatments of vehicles is abandonment of the concept of a vehicle-fixed ‘hinge’ to approximate the changing virtual axis about which roll takes place.

Eleven degrees of freedom and all major non-linearities are included in the equations of motion, which are programmed for time-history solutions on a digital computer. Empirical relationships used to generate tyre forces over extreme ranges of operating conditions are presented in detail. To ease the task of interpretation of the extensive output information, a computer-graphics display technique has been developed to produce detailed perspective drawings of the vehicle and terrain at selected intervals of time during a simulated manoeuvre.

Comparisons are presented of analytically predicted vehicle responses and test results. Future applications of the described mathematical model, in research related to highway safety, are briefly discussed.

The main content of the paper is concerned with the way people compensate their driving behaviour for degraded vehicle stability. It also outlines briefly the background of man–machine simulation and the use of driving simulators in the U.S.A. The problem was investigated by means of a fixed base driving simulator at the Davidson Laboratory as part of a larger programme concerned with driver behaviour.

The simulator has a two-degree-of-freedom analogue simulation of the vehicle dynamics. The display system is photographic using a real road film, and the vehicle is subjected to random appearing disturbances of the wind gust type. The disturbance, vehicle, and driver behaviour are recorded both as analogue and digital signals for experimental control and later data analysis.

The paper describes and gives the derived results from an experiment with nine subjects, driving two vehicles, one understeer and one oversteer, simulated to be travelling at 60 mile/h. While driving, the vehicles were subjected to a random appearing gust spectra having a maximum amplitude equivalent to 10 mile/h. Methods of assessing overall driving performance and what the drivers did in the two vehicles are developed and the results, in terms of a driving error and a control response function, are presented.

The driving error results show that, in general, drivers had significantly higher error scores with the over-steering vehicle. The driver control response function gives a measure of the reaction to the various disturbing frequencies of the disturbance. The results show that seven out of nine drivers had a control response function that was higher with the oversteering car. The remaining two subjects had the highest error scores with the understeering vehicle.

A driving effort function is derived and it is shown that, with exceptions which are discussed, the driving error may be regarded as a function of driving effort. The results substantiate the hypothesis that the driver will try to compensate for degraded vehicle stability, but show that the compensation is only partial and increased errors result.

The implications for real driving situations are that degraded vehicle stability, either of a basic vehicle or occurring during manoeuvring, will be only partially compensated for by the driver, and increased driving errors from the driver–vehicle system are likely to result.

The techniques used for the simulator experiment are suitable with only minor modifications for real driving situations.

Differences of car-handling behaviour between normal and emergency conditions have been investigated by mathematically simulating manoeuvres of change of lane, entry into a curve, going out from a curve, and straight running in gusty crosswinds.

A new ‘loose inverse procedure’ has been utilized to simulate these manoeuvres: the trajectory of each is followed, just as in actual road tests, by a tolerance which depends not only on the characteristics of the tyres and the car but also on the behaviour of the driver.

Calculations have shown that stronger tyres (i.e. of greater cornering stiffness) make the emergency manoeuvres less dangerous, since they increase the stability and the response of the car. Nevertheless, tyres of greater cornering stiffness may increase driver-stress in normal manoeuvres, since the same steer errors produce larger trajectory errors. This inconvenience may be eliminated if front roll understeer of the car is increased when these tyres are to be utilized.