Ideal Longl Force Distribution for a 4WD Vehicle

8/20/2019 Ideal Longl Force Distribution for a 4WD Vehicle

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Ideal Vehicle Longitudinal Force Vectoring

Ideal Distribution of Braking / Drive ForcesAlbert Loh - 27 November, 2015

Introduction

It is normal practice to proportion braking forces front-to-rear in accordance to the

predicted friction available on the front/rear axles.

When turning into a corner, there is a lateral weight transfer which changes the

friction available left-to-rignt. Therefore, the brake forces acting on the individual wheels

should also be proportioned laterally as well as longitudinally.

Theoretically, the drive force should be proportioned among the 4 wheels in the

same way as the brake force, since they are the same thing, but only in opposite direction.

(Assume individual wheel thrust vectoring is possible on all 4 wheels for this study.)

Optimum distribution of longitudinal drive/braking forces

What is the optimum distribution of longitudinal drive/brakingforce?

For a plane uniform friction surface, the ideal longitudinal force

distribution is in proportion to the friction on each wheel , which

is proportional to the vertical load on each wheel.

Reasons:-

1) At the friction limit, the 4 individual wheel forces act in the same direction;

therefore, the total force is maximised.

2) If the friction coefficient is the same for each wheel, the combined force acts

through the CG. (see appendix B)

3) Since the forces from all 4 wheels act in the same direction and their resultant

also acts directly through the CG, it is the maximum total force that can be obtained

from the 4 wheels.

IDEAL VEHICLE LONGITUDINAL FORCE PROPORTIONING 1



Conclusion

1) Lateral vectoring is required in addition to longitudinal proportioning for the

braking/drive forces in order to obtain the best possible combined force from the 4

wheels.

2) For a vehicle on a uniform friction surface, the ideal proportioning is in direct

proportion to the vertical load on each wheel.

3) Differentials operate on relative motion / force between wheels rather than the

absolute force and are thus not ideal for force vectoring. It would be better to use a

system of clutches or electrical drive which would allow for thrust control on

individual wheels.

Notes

1) The quasi-static wheel load distribution is used instead of the instantaneous

wheel loads to prevent sudden disruptions to the vehicle stability. (i.e. the vehicle and

individual wheel transients are ignored).

2) This strategy works for uniform friction on all 4 corners. The friction limit can be

either a circle or an ellipse.

3) This strategy does not work for side-to-side split ! cases, nor for where the front

and rear tires have significantly different friction coefficients.

4) Cross weighting has no effect on the strategy.

5) Longitudinal forces are defined to act along the car axis and lateral forces are

perpendicular to the car axis. Therefore:-

A. Steering angle changes the direction of the tire forces and thus the tire forces

must be resolved to act relative to the car axis.

B. Slip angles only change the direction of motion of the vehicle but not the tire

forces relative to the car axis. Therefore slip angles do not change the optimum

force distribution strategy.




APPENDIX

A. QUASI-STATIC LOADING

B. WEIGHT TRANSFER EFFECT

C. NUMERICAL CHECK




Appendix A - Quasi-Static Loading

The quasi-static wheel load distribution is used instead of the instantaneous wheel

loads to prevent sudden disruptions to the vehicle stability. (i.e. the vehicle and individualwheel transients are ignored).

This is probably valid for most cases of normal road and even race track conditions.

Most drivers, especially on the race track are conditioned to operate all controls smoothly

and avoid jerking (i.e. transients).

Particularly in the case of steering, it takes a significant period of time to actually

turn the steering wheel in normal road or race track driving. For braking and throttle

application, it is possible to have a sudden input, but this would only be under emergency

conditions. Race drivers are trained to hit the brake pedal quickly but also smoothly in

order not to elicit transient overshoot. Therefore, step inputs leading to transients are

probably only significant during extreme avoidance manoeuvres.

The study of ramp or single pulse sinusoidal

inputs would be much more useful for

normal vehicle dynamics.


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Appendix B - Weight Transfer Effect

Weight Shift

The total Lateral force Fxyv causes a lateral shift in

the centroid of the resultant force on the ground

plane (i.e. lateral weight transfer) by a distance of

! y = Fyv/W * H

Similarly, the total Longitudinal force Fxv causes a

shift in the centroid of the resultant force on the

ground plane (longitudinal weight transfer) of ! x = Fxv/W * H

Yaw Moment

The blue forces are the resultant forces from the

tires acting on the ground plane. The red forces are

the opposing inertial forces from the CG. Note the

following:-

1. The total Longitudinal force Fxv causes a yaw

moment Fxv* ! y about the CG.

2. The total Lateral force Fyv causes an

opposite yaw moment -Fyv* ! x about the CG.

3. Note Fxv*!y = Fxy*{Fyv/W * H} = -Fyv*!x,

These 2 yaw moments exactly balance out!, i.e. the

resultant of the wheel forces act through the CG.

Put in another way, the lateral forces cause a weight transfer to the outside of the

corner. If the longitudinal forces are vectored such that they act at the point of the

weight transfer, the yaw moments caused by the longitudinal and lateral forces balance

out.


TOP VIEW

Fyv

Fxv

Fxv

Fyv "x

"y

H

F v

W

"yFyv

W



Appendix C - Numerical Check

1) Spreadsheet Calculation

To verify the effectiveness of the proposed the longitudinal force distribution in

proportion to the individual wheel load, a spreadsheet which calculates the individual

wheel vertical loads, friction forces and moments of the individual wheel forces about the

CG was constructed.

The calculations confirm that

1) total yaw moment about CG is zero

2) total combined friction force utilisation factor is 1.000, showing that the

maximum friction capacity is utilised.




Appendix C - Numerical Check

2) Vehicle Simulation

Using the CAR4.BAS simulator to calculate the total forcesavailable on each wheel, the g-g diagram shows a perfect circle

equal to the friction circle. This is the holy grail I have previouslyspent countless hours trying to approach without success byplaying around with all sorts of vehicle parameters. YES!


Calculated G-G Diagram

Available Friction

Ideal Longl Force Distribution for a 4WD Vehicle

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