ACTIVE REDUCTION OF OVERSTEER AND UNDERSTEER USING SUSPENSION ELEMENTS

SVOČ – FST 2013

Martin Vlček, University of West Bohemia, Univerzitni 8, 306 14 Pilsen

Czech Republic

ABSTRACT This paper, which is a part of my Ph.D. thesis, consists of a basic explanation of car yaw behaviour and shows examples of design solutions reducing car oversteer and understeer. There is a software model of a small passenger car created with Adams Car, which is verified by a mathematical model of steady state cornering. The main emphasis of the paper is placed on my proposed solution reducing oversteer and understeer using active steering elements.

KEYWORDS Oversteer, Understeer, Suspension elements, MSC Adams Car, Steering.

INTRODUCTION There are three situations which may occur during cornering. When the car is moving on a static radius during cornering, it means neutral steer. Oversteer means that the car turns more than is required and understeer means that the car turns less than is required. The problem with oversteer and understeer relates to car safety. The processes that occur during cornering are described using a software model, which is verified by a mathematical model of steady state handling. This paper gives an idea of the car behaviour during cornering and shows possibilities for improving vehicle handling stability. BASIC PRINCIPLES OF OVERSTEER AND UNDERSTEER There are three possibilities of car behaviour, which may occur during cornering. The first situation is neutral steer. This is an ideal situation when the car keeps required direction (moving on static radius R). We can say that the car is behaving according to ideal Ackermann theory. The second situation is understeer when the car turns less than is required. This is typical for cars with front engine and front wheel drive when its cornering speed is too fast. The driven wheels are more loaded by the additional forces and tyres cannot fully transfer this load on to the road. The third situation is oversteering. Oversteer means that the car turns more than is required. This is typical for too fast cornering in cars with rear wheel drive. Oversteer is the most dangerous situation for car handling because it can cause uncontrollable slip. [6] We usually use Stability factor “K” to explain car behaviour in cornering. This factor is given by the formula:

lCClClC

mKzp

ppzz

⋅⋅⋅−⋅

⋅=αα

αα

''

(1)

R [m] – turning radius v [m/s] – wheels velocity a [°] – slip angle (p = front axle, z = rear axle) m [kg] – car weight Cαz [N/rad] – cornering stiffness of rear axle (by virtue of tyre stiffness) C’αp [N/rad] – cornering stiffness of front axle (by virtue of tyre stiffness and pneumatic caster) l [m] – axle base lz , lp [m] – horizontal distance between the rear/front axle and the vehicle centre of gravity The stability factor K depends on the weight distribution and on cornering stiffness of axles. When K is equal to 0, it shows neutral steer. When the stability factor is less than zero, it shows oversteer. The car turning radius is less than radius for neutral steer. The third situation is understeer, when K is greater than zero and car turning radius is greater than radius for neutral steer.

Figure 1: Forces at tyre contact area [6] Figure 2: Kamm circle of frictional forces [6]

The following forces acting on the tyre contact area are important for good handling stability:

H [N] – driving force B [N] – braking force S [N] – cornering (lateral) force R [N] – resultant force at tire contact area Z [N] – vertical wheel force m [-] – adhesion coefficient Resultant force is resultant of lateral force and braking force (Figure 1) or lateral force and driving force (in case of accelerating). Figure 2 shows the Kamm circle of frictional forces. The radius of this circle is the product of vertical wheel force and adhesion coefficient (between tyre and road). We must ensure that the resultant force R is not bigger than the radius of the Kamm circle of frictional force to avoid a slip. DESIGN SOLUTIONS REDUCING OVERSTEER AND UNDERSTEER There are many common solutions that improve car yaw behaviour and car handling stability. Here are few selected active devices:

- active body roll - which changes weight distribution in cornering - active and adaptive suspension unit - improves tyre contact with road - active suspension geometry - for example active wheel camber which changes lateral force - active aerodynamics - increases vertical wheel load in high speed - steering of rear wheels - decreases slip angles and improves car handling stability - electronic stability programme - brakes selected wheels so that it eliminates moment of inertia - active differential (torque vectoring) - divides the driving torque among the wheels with the help of multi-disk

clutches - active roll stabilization - device which changes stiffness of suspension stabilizer and eliminates body roll

during cornering - active steering - changes front wheel angles depending on velocity and adhesion - electronic brake distribution or cornering brake control – divides braking pressure among the wheels in

cornering - and many another solutions...

Figure 3: ESP (Electronic stability programme) [5]

The most effective design solution eliminating oversteer and understeer is today Electronic Stability Programme (ESP), which is shown in Figure 3. SOFTWARE MODEL AND MATHEMATICAL MODEL

Figure 4: Vehicle assembly in MSC Adams Car 2011

I made a software model in the program Adams Car 2011 to simulate various driving situations. In the Figure 4 is shown full assembly of a small passenger car, which consists of these created subsystems: front suspension (MacPherson), rear suspension (trailing arm), steering (rack and pinion), wheels (Pacejka model – 175/70 R13), body (defined as mass properties), powertrain (front wheel drive). To evaluate software model I created a mathematical model of steady state handling. It means a model for constant radius cornering (see Figure 5). To find a velocity where the loss of adhesion occurs for turning radius 50 meters, was an objective of my calculation.

Figure 5: Illustration of selected motion trajectory for steady state handling

I used the following plane model (Figure 6) to investigate forces effects on the vehicle during cornering. In the case of steady state vehicle handling we can use these simplifications and assumptions [6]:

- Car velocity and radius of turning are constant - Acceleration is equal to zero (it means that the acceleration resistance and moment of inertia are zero) - Car model with front wheel drive with common differential (driving forces are the same and act only on the

front wheels). - Car is driving on level ground, which means that the climbing resistance is zero. - Velocity of wind is equal to zero - Negligible gyroscopic moment of the wheels - Friction coefficient is the same for all wheels

Figure 6: General plane model of vehicle cornering [6]

From the preview model, we can write these three balance conditions for steady state vehicle handling:

1) Force balance in the direction of X axis:

0sincoscos

sinsincoscos

2211

2122112211

=⋅+−⋅−⋅−

−−−⋅−⋅−⋅+⋅

αββ

ββββ

ovpfppfp

fzfzpppppppp

FOOOOOSSHH

(2)

2) Force balance in the direction of Y axis:

0cossinsin

coscossinsin

2211

2211212211

=⋅−⋅−⋅−

−⋅+⋅+++⋅+⋅

αββ

ββββ

opfppfp

ppppzzpppp

FOOSSSSHH

(3)

3) Moment balance around the Z axis:

022

sin

2cossin

2cos

2sincos

2sincos

2cossin

2cossin

21212122

22111121

22221111

22221111

=−−−−⋅−⋅+⋅⋅−

−⋅⋅−⋅⋅−⋅⋅+⋅−⋅−

−⋅⋅−⋅⋅+⋅⋅+⋅⋅+

+⋅⋅+⋅⋅+⋅⋅−⋅⋅

zzppz

fzz

fzppfp

ppfpppfp

ppfpzzzz

pppppp

pppppp

pppppp

pppppp

MvrMvrMvrMvrtOtOlO

tOlO

tOlSlS

tSlS

tSlS

tHlH

tHlH

β

βββ

ββββ

ββββ

(4)

We need to know many important unknown dependencies and relations for solving these three main equations. I solved these three equations using Microsoft Excel and MATLAB. I used the same following entered parameters for the mathematical and software model of steady state handling to ensure simple comparison. The models were different only in the tyre size because I did not find the tyre characteristic for the same size of tyres. 175/70 R13 tyres were used for the software model, and 155/70 R13 for the mathematical model.

Models parameters:

Figure 7: Models parameters [6]

The comparisons of obtained characteristics of the mathematical and software models are represented in the following charts. Graphical model of steady state handling in Figure 8 shows obtained angles, forces and Kamm circles.

Figure 8: Graphical model of obtained characteristics

Here are the resulting characteristics of steady state handling for radius 50m. The blue colours represent the mathematical model and red colours are used for the software model created in the MSC Adams Car.

Figure 9: Dependence of steering angles bi vs. car velocity v

Figure 10: Dependence of slip angles ai vs. car velocity v

Figure 11: Dependence of resultant forces Ri and frictional forces Zi*m for axles vs. car velocity v

The last dependence shows resultant forces and frictional forces (radiuses of the Kamm circles) for the front and rear axle. The velocity, where the resultant force of an axle is higher than frictional force means loss of adhesion. In Figure 11 you can see that loss of adhesion for front axle occurs at 70 km/h for the mathematical model and at 73 km/h for the software model. These similar results show that the mathematical model and software model in the Adams Car are probably correct. Small differences may be caused by slightly wider tyres in the Adams Car.

PROPOSED SOLUTION REDUCING OVERSTEER AND UNDERSTEER I created a new device in Adams Car that can help reduce oversteer and understeer. I implemented rotation actuators to the wheel carriers of the front suspension (see Figure 12). These rotation actuators change toe angle according to lateral acceleration ay, car velocity v and steering angles b when the loss of adhesion is near.

Figure 12: Rotation actuator in the front suspension

The functions shown in the Figure 13 were developed to control two actuators. The control functions use “arctg” dependence, which I obtained by a long time optimization.

Figure 13: Actuators control functions

The actuator control functions are based on the Ackermann theory, which says that the rotation axis of all wheels should be aimed at one point. But it is not possible in fact due to slip angles, steering geometry and suspension travel. Therefore I developed the actuators changing steering angles (toe angles) independently on the front wheels, which can improve safety and handling stability at high speeds. I performed a few tests and manoeuvres with the actuators and without the actuators to demonstrate actuators effects. I tried out these tests: - Steady state handling (constant radius cornering)

- ISO lane change manoeuvre (AVTP 03 – 160) - SINE wave steering (SIN steer) - Sudden steering-angle change (ramp steer) - Drifting test

I performed the tests for various input parameters. The testing car with the actuators had faster steering reactions in all cases. Slip occurred at higher speeds and the car was stabilised earlier when the loss of adhesion occurred. In the following picture is an example of the results for steady state handling test. The test with and without the actuators was performed for initial radius 50 meters.

Figure 14: Results of steady state handling test with and without actuators

Dependence in Figure 14 shows lateral acceleration, cornering radius and actuator behaviour depend on velocity. The velocity at which lateral acceleration and cornering radius suddenly changes shows loss of adhesion. In the detail (Figure 15) you can see that the loss of adhesion occurs at higher speed with the actuator. The difference is approximately 0.5 km/h. But it is also important that the car is stabilized earlier with the actuator. The difference in cornering radius is greater than 4 meters.

Figure 15: Detail of the results of steady state handling test

CONCLUSION AND RECOMMENDATIONS The obtained characteristics give us an idea of car behaviour during cornering and show possibilities for improving vehicle handling stability. My proposed solution independently changes steering angles (toe angles) on the front wheels thereby improving car driving stability especially in cornering. The solution may be created by various design concepts ant it is possible to combine this solution with other existing solutions improving the active vehicle safety.

REFERENCES A Book Publication: [1] E. Tönük, Y. S. Ünlüsoy: Prediction of automobile tire cornering force characteristics by finite element modelling and analysis, Ankara, 2001. [2] S. Sadeghi, M. T. Ahmadian: Tire Modeling with Nonlinear Behavior for Vehicle Dynamic Studies, Sharif University of Technology, 2001. [3] Reimpell, J., Helmut, S., Betzler, J. W.. The Automotive Chassis: Engineering Principles, Reed Elsevier and Professional Publishing Ltd, 2001. ISBN 0 7506 5054 0 [4] F. Vlk: Dynamika motorových vozidel, Brno, 2003. [5] Z. Jan, B. Ždánský: Výkladový automobilový slovník, Brno, 2007. B Conference Proceedings [6] M. VLČEK, Optimization of Car Yaw Behaviour Using Active Suspension Elements. Proceedings SVOČ 2011. Plzeň: ZČU-FST, 2011, s. 211-221. ISBN 978-80-7043-995-1.

The article has been supported by the form of professional advice within framework of the Project No. CZ.1.07/2.3.00/35.0048 ‘Popularization of research and development in mechanical engineering and its results (Popular)’ and co-financed by European Social Fund and a state budget of the Czech Republic.