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Vehicle dynamics Suspension effects on cornering Pierre Duysinx LTAS – Automotive Engineering University of Liege Academic Year 2013-2014 1
Bibliography T. Gillespie. « Fundamentals of vehicle Dynamics », 1992, Society of Automotive Engineers (SAE) W. Milliken & D. Milliken. « Race Car Vehicle Dynamics », 1995, Society of Automotive Engineers (SAE) R. Bosch. « Automotive Handbook ». 5th edition. 2002. Society of Automotive Engineers (SAE) J.Y. Wong. « Theory of Ground Vehicles ». John Wiley & sons. 1993 (2nd edition) 2001 (3rd edition). 2
Suspension effects on cornering So far our first theory has shown that the behaviour is dependent on the load / cornering coefficient W/Ca (called cornering compliance [deg/g]) of the front and rear axles Weight and cornering coefficient can be affected by suspension mechanics due to Lateral load transfer Change in camber angles Roll steer Lateral force compliance steer Aligning torque Effect of tractive force on cornering 3
Roll Moment distribution For virtually all pneumatic tires, the cornering forces are nonlinearly dependent on the vertical load (load sensitivity) Straight line: Fy= 760 lb Cornering: Fy=680 lb 4
Roll Moment distribution Lateral load transfer occurs in cornering because of the elevation of the CG with respect to the ground. Because of the load sensitivity of tyre cornering force, the lateral force developed by the axle is affected in cornering. Thus to sustain the lateral forces, the slip angle has to be increased. For front tyres, greater slip angles lead to an understeer effect For rear tyres, greater slip angles lead to an oversteer behaviour 5
Roll Moment distribution The load transfer is in action on both front and rear axles. The relative importance of load transfers will depend on the balance of roll moments distributed over the front and the rear axles. More roll on front axles contributes to understeer More roll on rear axles contributes to overtseer Auxiliary roll stiffeners (stabilizer bars) alter the handling performance through the redistribution of the relative importance of the roll between the axles 6
Antiroll bars Hillier Vehicle And engine Technology Fig 7.38 8
Roll Stiffness of an Axle The mechanics governing the roll moment All suspensions are functionally equivalent to two springs. The stiffness of the axle is Springs of stiffness Ks Separated by a distance s Kf roll stiffness of the suspension f 0.5 ² K K s s
Roll Center Definition: the Roll Center is the point at which the lateral forces are transferred from the axles to the sprung mass The roll centre can also be regarded as the point on the body at which a lateral force applied produces no roll angle. The roll centre is also the point around which the axle rolls when subjected to a pure roll moment. 10
Roll Center The notion of instantaneous center of rotation (CIR) is very useful to determine the kinematic parameters of a suspension mechanism At a given time t, the motion is equivalent to a rotation about a fictitious hinge located at the CIR One can replace the complex suspension mechanism by a rigid bar rotating about the CIR When the mechanism undergoes large motions, the CIR is also modified! CIR
Roll Center The concept of instantaneous rotation center is basically two dimensional. For 3D problem, one can come to plane problems by using projections onto projection planes including the wheel center: front view and lateral view. Mlliken Fig 17.6 : CI in the front And lateral vue planes 12
Roll Center In 3D, the concept of CIR is replaced by the concept of instantaneous rotation axis. This axis includes the CIR in the different view planes The instantaneous rotation axis is the axle about which the wheel rotates with respect to the chassis Independent suspension mechanisms have 1 instantaneous rotation axis, while rigid axle suspensions have 2 instantaneous rotation axes, one for the rebound and one for the roll degree of freedom. The IC axes are changing with the suspension travel 13
Roll Center How can we calculate / determine the roll centre? By using the methods of instantaneous rotation centers and making use of the Kennedy theorem The CIR of 3 bodies undergoing relative motions are aligned By using the virtual work principle By using the curve of half track modification with the wheel travel 14
Roll Center Determining the roll centre using the curve of half track modification with the wheel travel (Reimpel, p 165) 15
Roll Center Graphical determination of the roll center using the methods of instantaneous roll centers 16
Roll Center Graphical determination of the roll center for a double wishbone suspension 17
Roll axis In 3D, the roll center is replaced by the roll axis. Front and rear axles have different roll centers. Roll axis passes through the two roll centers 18
Load transfer Newton’s second law (applied to axle) t track width hr roll centre height Relation between the wheel loads, the lateral forces and the roll angle. t Kf ( )2 F F F h zo zi y r F F F y yi yo ff 2 / 2 / 2 F F F h t K t F zo zi y r z
Load transfer Load transfer Comes from two mechanisms Lateral load transfer due to the cornering forces. Independent of the roll angle of the body and roll moment distribution. Lateral load transfer due to vehicle roll. It depends on roll dynamics and may lag the changes in cornering conditions. It depends on front / rear roll moment distribution ff 2 / 2 / 2 F F F h t K t F zo zi y r z 2 / F h t y r ff 2 / K t 20
To calculate the load transfer To calculate the load transfer We need to compute: The roll center height hr Geometrical property of the suspension mechanism The roll angle under the action of the lateral acceleration This one must be determined for the vehicle. It depends on front / rear roll moment distribution ff 2 / 2 / 2 F F F h t K t F zo zi y r z 21
Roll Moment distribution The roll moment distribution requires the total vehicle model We define the roll axis as the line connecting the roll centres of the front and rear suspensions. h1: height of the centre of mass of the rolling body with respect to the roll axis position 22
Roll Moment distribution Moment due to the forces acting at CG about the roll axis For small angles, it comes 1 M Wh f ² V gR f f sin cos cos M Wh W h f 1 1 ² V gR f 23
Roll Moment distribution The roll moment of suspension Solving for f: W h V K K f f f f M M M K K f f f f f f r r ²/( ) gR W h f 1 f r 1 The derivative of the roll with respect to the lateral acceleration is the roll rate of the vehicle: The roll rate is usually in the range of 3 to 7 degres/g on typical passenger cars f W h K f R 1 f a K W h f f r 1 y 24
Roll Moment distribution Combining the expression of the roll angle with the expression of the load transfer in each axle, One gets the expression of the roll moments on the front axle 1 f f f 1 r K K Wh f f ff ² 2 / 2 / 2 F h t K t F f f f f y z ² c V L R V gR F m W f f y ²/( ) ² Wh V gR V gR ' M K W h F t f f f f zf f 25
Roll Moment distribution One gets the roll moments on the front and rear axles ²/( ) ² Wh V K f gR Wh V gR ' M K W h F t 1 f f f f f f zf f K f f 1 r ²/( ) ² Wh V K f gR Wh V gR ' M K W h F t 1 f f r r r r zr r K f f r 1 /2 ( /2) F F W F W zf zfo f zfi f /2 ( /2) F F W F W zr zro r zri r 26
Roll Moment distribution In general the roll moment distribution tends to be biased towards the front wheels: Relative to the load, the front spring rate is usually slightly lower than the rear (for flat ride) bias towards higher roll stiffness at the rear. However on independent rear suspension, the higher distance between the suspension springs enhance the front roll stiffness. Designers usually strive for a higher front roll stiffness to ensure understeer in the limit cornering Stabilizer bars are often used on the front axles to obtain higher front roll stiffness. If stabilizer bars are needed to reduce the body roll, they must be installed on the front or on the front + the rear. Warning: a stabilizer bar on the rear only can cause unwanted oversteer. 27
Roll Moment distribution We have determined the roll moments for front and rear axles and the difference in load between the left and right wheels To translate the lateral load transfer into an effect on understeer gradient, we need data about the sensitivity of tires with respect to the slip angle and load. The difference in the change of slip angles between the front and the rear when load transfer is present represents the understeer effect. 28
Understeer gradient due to lateral load transfer Express the cornering forces of each tires The sensitivity to the load can be modeled using a second order polynomial for the cornering coefficient a : first coefficient in the cornering stiffness polynomial [N/N/deg] b: second coefficient in the cornering stiffness polynomial [N/N²/deg] Caa y F aa a 2 ( ) F C aF bF y z z 29
Understeer gradient due to lateral load transfer For a vehicle cornering, the lateral force of the two tires of the axle is given by: Introduce the lateral load transfer It comes and 2 2 ( ) F aF bF aF bF a y zo zo zi zi F F F F F F zo z z zi z z b F a 2 2 (2 2 2 ) F aF bF y z z z b F a 2 ( 2 ) F C a 2 2( ) C aF bF a y z z z 30
Understeer gradient due to lateral load transfer Remind that the equation between the steering angle and the slip angles on front and rear axles is: For the front and the rear tire, we can write Substituting, we gets ²/( ) 2 z R C b F a L R a a f r b F a a 2 f 2 r ( 2 ) ²/( ) F C W V gR a f f f f y z ( 2 ) ²/( ) F C b F W V gR a r r r r y z ²/( 2 ) WV gR WV C a gR b F L f r 2 f 2 r f r z 31
Understeer gradient due to lateral load transfer This equations can simplify by using a Taylor expansion The steering angle equation becomes 2 2 1 2 1 2 1 b F C 1 z b F C 2 2 C b F C a a a 1 C z z a a 2 f 2 r 2 2 ² W C W C W C b F C a W C b F C a L R V gR f r f r z z a a a a f f f r r r Understeer gradient contribution From lateral load transfer Understeer gradient from Nominal cornering stiffness 32
Understeer gradient due to lateral load transfer The gradient from lateral load transfer The lateral load transfer are given by the equilibrium equations All the six variables are positive, so that the contribution from front axle is always understeer while the contribution from rear axle is always oversteer 2 f 2 r 2 2 W C b F C a W C b F C a K f r z z LLT a a f f r r 33
Camber change The inclination of the wheel outward from the body is called the camber angle. Camber on a wheel will produce a lateral force known as camber thrust. The camber thrust is oriented toward to the intersection of the wheel rotation axis with the ground 34
Camber thrust Origin of the camber thrust: The contact patch is deformed and takes a banana shape The tyre particles would like to follow a circular trajectory which creates a double shear stress distribution. The net resultant of the shear stress distribution is oriented to the centre of rotation of the cambered wheel Distorted contact patch because of camber Milliken : Fig 2.23 35
Camber thrust The camber thrust is oriented towards the centre of rotation of the cambered wheel The tyre develop a lateral force even if there is no slip angle. Camber thrust and cornering forces due to slip are additive phenomena at least for small values The magnitude of the camber thrust depends on The camber angle The type of tyre, its constructive characteristics The shape of the tread The inflating pressure (strongly) The braking / tractive forces The slip angle The camber thrust is little sensitive to load and speed 36
Camber thrust In its linear part, (small camber angles), the camber thrust angle can be approximated by the linear relation: Fy = Cgg Cg is the camber stiffness Gillespie Fig. 6.14 37
Camber thrust The camber thrust produces much smaller lateral forces than an equivalent slip angle. About 4 to 6 degrees of camber produce the same lateral force as 1 degree of side slip for a bias tyre. For radial tyres, as much as 10 to 15 degrees are necessary for 1 degree of slip This can be understood because the side slip produces a much bigger deformation of the tyre tread. For radial tyre, the sidewall are even softer and the belts are very rigid. For passenger car, the camber thrust is fading out for camber thrust over 5 to 10 degrees For motorbikes tyres with round profiles and bias structures, the camber thrust can be generated up to 50 degrees 38
Camber thrust Camber thrust is additive to cornering forces from slip angle (at least for small angles). For large angles, both phenomena interferes and a saturation is observed a g y F C C a g Camber thrust and slip angles Milliken Fig 2.24 39
Camber thrust Camber thrust for a motorbike tyre 40
Camber stiffness Gillespie: Fig 10.17 Camber stiffness for different population Of radial, bias and bias belted tyres. 41
Camber change Camber angles are small on solid axles and, at maximum, they can change the lateral forces by 10% On independent wheel suspensions, the camber can play an important role in cornering. Camber changes result from both the body roll and from the camber change due to wheel travel to stick to the road (bounce / jounce) g g Gillespie Fig 6.15: Camber change in the cornering of the vehicle f / g w b 42
Camber change The camber angle arising from the suspension is a function of the roll angle because of the wheel travel in jounce for the inside wheel or in rebound for the outside wheel. Given a suspension geometry and type, one can obtain the derivative of the camber of the angle with respect to the roll angle from the kinematics analysis of the suspension mechanism: In turn the roll angle can be related to the lateral acceleration W h V K K f f g f f (track width,suspension geometry,roll angle) g ²/( ) gR W h f 1 f r 1 43
Camber change The influence on cornering comes from the fact that the lateral force results from both the slip angle and the camber angle Thus Both Fy and g are related to the lateral acceleration C W a C a a a g y F C C a g y F C C C g a g a a C C g f g f f W g f a a a f f f y y C a f a a f f y g r a a r r r y y C a r r y 44
Camber change Substituting the slip angles into the equation One gets The understeer coming from the camber angles L R a a f r C C C C g f f g f f ² L R W C W C V gR g g f r f r f r a a a a a a f f r y r y C C C C g f g f f g g f r K f r camber a a a f r y 45
Roll Steer When the vehicle rolls in cornering, the suspension kinematics may produce some wheel steer. Roll steer is defined as the steering motion of the front or rear wheels with respect to the sprung mass due to the rolling motion of the sprung mass. Roll steer effects on handling is lagging the steer input following the roll of the sprung roll. Roll steer affects the handling since it alters the angle of the wheels with respect to the direction of travel. 46
Roll Steer Let the roll steer coefficient on the axle [degree of steer / degree of roll] Following the reasoning that was used before, one can derive the roll steer gradient: A positive roll steer coefficient causes the wheels to steer to the right in hand right turn. So as the right hand roll occurs when the vehicle is turning to the right, positive roll steer on the front axle steers out the turn and is understeer. A positive roll steer on the rear wheels lead to oversteer. f K rollsteer f r a y 47
Roll Steer On a solid axle the suspension allows the axle to roll about an imaginary axis that is inclined with respect to the longitudinal axis of the vehicle The kinematics is envisioned as functionally equivalent to leading or trailing arms systems. The roll axis inclination is equivalent to that of the arms Given an inclination b, as the body rolls positively – turn to the left -, the arm on inside wheel rotates downwards while the arms on the outside wheel rotates upwards 48
Roll steer If the orientation of the trailing arm is angled downward, the effect of the trailing arm is To pull the inner wheel forward To push the outer wheel rearward This produces roll steer of the solid axle to contributing to oversteer The roll steer is equal to the inclination angle: On the rear trailing system, the roll understeer is achieved by keeping the transverse pivots of the trailing arm below the wheel center b 49
Roll Steer With independent suspensions, the roll steer coefficient must be evaluated from the kinematics of the suspension. On steered wheels, the interactions with the steering system must also be taken into account. 50
