MECA0492 : Vehicle dynamics Pierre Duysinx Research Center in Sustainable Automotive Technologies of University of Liege Academic Year 2017-2018 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». 5 th edition. 2002. Society of Automotive Engineers (SAE) J.Y. Wong. «Theory of Ground Vehicles». John Wiley & sons. 1993 (2 nd edition) 2001 (3rd edition). M. Blundel & D. Harty. «The multibody Systems Approach to Vehicle Dynamics» 2004. Society of Automotive Engineers (SAE) G. Genta. «Motor vehicle dynamics: Modelling and Simulation». Series on Advances in Mathematics for Applied Sciences - Vol. 43. World Scientific. 1997. 2
INTRODUCTION TO HANDLING 3
Introduction to vehicle dynamics Introduction to vehicle handling Vehicle axes system Tire mechanics & cornering properties of tires Terminology and axis system Lateral forces and sideslip angles Aligning moment Single track model Low speed cornering Ackerman theory Ackerman-Jeantaud theory 4
Introduction to vehicle dynamics High speed steady state cornering Equilibrium equations of the vehicle Gratzmüller equality Compatibility equations Steering angle as a function of the speed Neutral, understeer and oversteer behaviour Critical and characteristic speeds Lateral acceleration gain and yaw speed gain Drift angle of the vehicle Static margin Exercise 5
Introduction In the past, but still nowadays, the understeer and oversteer character dominated the stability and controllability considerations This is an important factor, but it is not the sole one In practice, one has to consider the whole system vehicle + driver Driver = intelligence Vehicle = plant system creating the manoeuvering forces The behaviour of the closed-loop system is referred as the «handling», which can be roughly understood as the road holding 6
Introduction Model of the system vehicle + driver 7
Introduction However because of the difficulty to characterize the driver, it is usual to study the vehicle alone as an open loop system. Open loop refers to the vehicle responses with respect to specific steering inputs. It is more precisely defined as the directional response behaviour. The most commonly used measure of open-loop response is the understeer gradient The understeer gradient is a performance measure under steady-state conditions although it is also used to infer performance properties under non steady state conditions 8
AXES SYSTEM 9
Reference frames O X Y Z Inertial coordinate system OXYZ Local reference frame oxyz attached to the vehicle body - SAE (Gillespie, fig. 1.4) 10
Reference frames Inertial reference frame X direction of initial displacement or reference direction Y right side travel Z towards downward vertical direction Vehicle reference frame (SAE): x along motion direction and vehicle symmetry plane z pointing towards the center of the earth y in the lateral direction on the right hand side of the driver towards the downward vertical direction o, origin at the center of mass 11
Reference frames y x z Système SAE z Comparison of conventions of SAE and ISO/DIN reference frames x Système ISO y 12
Local velocity vectors Vehicle motion is often studied in car-body local systems u : forward speed (+ if in front) v : side speed (+ to the right) w : vertical speed (+ downward) p : rotation speed about a axis (roll speed) q : rotation speed about y (pitch) r : rotation speed about z (yaw) 13
Forces Forces and moments are accounted positively when acting onto the vehicle and in the positive direction with respect to the considered frame Corollary A positive F x force propels the vehicle forward The reaction force R z of the ground onto the wheels is accounted negatively. Because of the inconveniency of this definition, the SAEJ670e «Vehicle Dynamics Terminology» names as normal force a force acting downward while vertical forces are referring to upward forces 14
VEHICLE MODELING 15
The bicycle model b T f t Velocity x,u,p F xf f F yf When the behaviours of the left and right hand wheels are not too different, one can model the vehicle as a single track vehicle known as the bicycle model or single track model. L c T r z,w,r y,v,q r F yr The bicycle model proved to be able to account for numerous properties of the dynamic and stability behaviour of vehicle under various conditions. F xr 16
The bicycle model t Velocity f Geometrical data: Wheel base: L Distance from front axle to CG: b T f F yf Track: t Distance from rear axle to CG: c L b c T r z,w,r x,u,p F xf y,v,q r F xr F yr Tire variables Sideslip angles of the front and rear tires: f and r Steering angle (of front wheels) Lateral forces developed under front and rear wheels respectively: F yf and F yr. Longitudinal forces developed under front and rear wheel respectively: F xf and F xr. 17
The bicycle model t Velocity f Assumptions of the bicycle model Negligible lateral load transfer Negligible longitudinal load transfer L b c T f T r z,w,r x,u,p F xf y,v,q r F yf F yr Negligible significant roll and pitch motion The tires remain in linear regime Constant forward velocity V Aerodynamics effects are negligible Control in position (no matter about the control forces that are required) No compliance effect of the suspensions and of the body F xr 18
The bicycle model Remarks on the meaning of the assumptions Linear regime is valid if lateral acceleration<0.4 g Linear behaviour of the tire Roll behaviour is negligible Lateral load transfer is negligible Small steering and slip angles, etc. Smooth floor to neglect the suspension motion Position control of the command : one can exert a given value of the input variable (e.g. steering system) independently of the control forces The sole input considered here is the steering, but one could also add the braking and the acceleration pedal. 19
The bicycle model Assumptions : Fixed: u = V = constant No vertical motion: w=0 No roll p=0 No pitch q = 0 Bicycle model = 2 dof model : r=w z, yaw speed v, lateral velocity or b, side slip of the vehicle Vehicle parameters: m, mass, J zz inertia bout z axis L, b, c wheel base and position of the CG 20
The bicycle model h Velocity f Velocity f T f F yf F yf x,u,p F xf b F xf u b V z,w,r y,v,q r L c r r v m, J T r F yr F yr F xr F xr 21
LOW SPEED TURNING 22
Low speed turning At low speed (parking manoeuvre for instance), the centrifugal accelerations are negligible and the tire have not to develop lateral forces The turning is ruled by the rolling without (lateral) friction and without slip conditions If the wheels experience no slippage, the instantaneous centres of rotation of the four wheels are coincident. The CIR is located on the perpendicular lines to the tire plan from the contact point In order that no tire experiences some scrub, the four perpendicular lines pass through the same point, that is the centre of the turn. 23
Ackerman-Jeantaud theory 24
Ackerman-Jeantaud condition One can see that This gives the Ackerman Jeantaud condition Corollary e i 25
Ackerman-Jeantaud condition The Jeantaud condition is not always verified by the steering mechanisms in practice, as the four bar linkage mechanism 26
Jeantaud condition The Jeantaud condition can be determined graphically, but the former drawing is very badly conditioned for a good precision In practice one resorts to an alternative approach based on the following property Point Q belongs to the line MF when the Jeantaud condition is fulfilled The distance from Q to the line MF is a measure of the error from Jeantaud condition 27
Ackerman theory v f L u b V r v v r R CG R centre du virage 28
Ackerman theory The steering angle of the front wheels The relation between the Ackerman steering angle and the Jeantaud steering angles 1 and 2 v f L r u b V v v r R CG R centre du virage R=10 m, L= 2500 mm, t=1300 mm 1 = 15.090 2 = 13.305 = 14.142 ( 1 + 2 )/2=14.197 29
Ackerman theory Curvature radius at the centre of mass Relation between the curvature and the steering angle Side slip b at the centre of mass 30
Ackerman theory The off-tracking of the rear wheel set v f L u b V r v v r R CG R centre du virage 31
HIGH SPEED STEADY STATE CORNERING 32
High speed steady state cornering At high speed, it s required that tires develop lateral forces to sustain the lateral accelerations. The tire can develop forces if and only if they are subject to a side slip angle. Because of the motion kinematics, the CIR is located at the intersection to the normal to the local velocity vectors under the tires. The CIR, which was located at the rear axel for low speed turn, is now moved to a point in front. 33
High speed steady state cornering v r r f v f f 34
Dynamics equations of the vehicle motion Newton-Euler equilibrium equation in the non inertial reference frame of the vehicle body e J x y = 0 e t J y z = 0 Velocity f Model with 2 dof b & r a F xf u b F yf V nt : Equilibrium equations in F y and M z : F y = M ( _v + r u) N = J zz _r Operating forces L b r r v M, J Tyre forces F yr Aerodynamic forces (can be neglected here) F xr 35
Equilibrium equations of the vehicle Equilibrium equations in lateral direction and rotation about z axis Solutions The lateral forces are in the same ratio as the vertical forces under the wheel sets 36
Behaviour equations of the tires Cornering force for small slip angles C = @F y @ = 0 < 0 Gillespie, Fig. 6.2 37
Gratzmüller equality Using the equilibrium and the behaviour condition, one gets One yields the Gratzmüller equality 38
Compatibility equations The velocity under the rear wheels are given by v r r f v f f The compatibility of the velocities yields the slip angle under the rear wheels 39
Compatibility equations The velocity under the front wheels are given by v r r f v f f The compatibility of the velocities yields the slip angle under the front wheels 40
Steering angle Steering angle as a function of the slip angles under front and rear wheels The steering angle can also be expressed as a function of the velocity and the cornering stiffness of the wheels sets 41
Understeer gradient The steering angle is expressed in terms of the centrifugal acceleration So With the understeer gradient K of the vehicle 42
Steering angle as a function of V Gillespie. Fig. 6.5 Modification of the steering angle as a function of the speed 43
Neutralsteer, understeer and oversteer vehicles If K=0, the vehicle is said to be neutralsteer: The front and rear wheels sets have the same directional ability If K>0, the vehicle is understeer : Larger directional factor of the rear wheels If K<0, the vehicle is oversteer: Larger directional factor of the front wheels 44
Characteristic and critical speeds For an understeer vehicle, the understeer level may be quantified by a parameter known as the characteristic speed. It is the speed that requires a steering angle that is twice the Ackerman angle (turn at V=0) For an oversteer vehicle, there is a critical speed above which the vehicle will be unstable 45
Lateral acceleration and yaw speed gains Lateral acceleration gain Yaw speed gain 46
Lateral acceleration gain Purpose of the steering system is to produce lateral acceleration For neutral steer, lateral acceleration gain is constant For understeer vehicle, K>0, the denominator >1 and the lateral acceleration is reduced with growing speed For oversteer vehicle, K<0, the denominator is < 1 and becomes zero for the critical speed, which means that any pertubation produces an infinite lateral acceleration 47
Yaw velocity gain The second raison for steering is to change the heading angle by developing a yaw velocity For neutral vehicles, the yaw velocity is proportional to the steering angle For understeer vehicles, the yaw gain angle is lower than proportional. It is maximum for the characteristic speed. For oversteer vehicles, the yaw rate becomes infinite for the critical speed and the vehicles becomes uncontrollable at critical speed. 48
Yaw velocity gain Gillespie. Fig. 6.6 Yaw rate as a function of the steering angle 49
Sideslip angle Definition (reminder) Value Value as a function of the speed V Becomes zero for the speed independent of R! 50
Sideslip angle b > 0 b < 0 Gillespie. Fig. 6.7 Sideslip angle for a low speed turn Gillespie. Fig. 6.8 Sideslip angle for a high speed turn This is true whatever the vehicle is understeer or oversteer 51
Static margin The static margin provides another (equivalent) measure of the steady-state behaviour Gillespie. Fig 6.9 Neutral steer line e>0 if it is located in front of the vehicle centre of gravity 52
Static margin Suppose the vehicle is in straight line motion (=0) Let a perturbation force F applied at a distance e from the CG (e>0 if in front of the CG) Let s write the equilibrium The static margin is the point such that the lateral forces do not produce any steady-state yaw velocity That is: 53
Static margin It comes So the static margin writes A vehicle is Neutral steer if e = 0 Under steer if e<0 (behind the CG) Over steer if e>0 (in front of the CG) 54
Static margin Gillespie. Fig. 6.10 Maurice Olley s definition of understeer and over steer 55
Exercise Let a vehicle A with the following characteristics: Wheel base L=2,522m Position of CG w.r.t. front axle b=0,562m Mass=1431 kg Tires: 205/55 R16 (see Figure) Radius of the turn R=110 m at speed V=80 kph Let a vehicle B with the following characteristics : Wheel base L=2,605m Position of CG w.r.t. front axle b=1,146m Mass=1510 kg Tires: 205/55 R16 (see Figure) Radius of the turn R=110 m at speed V=80 kph 56
Exercise Rigidité de dérive (dérive <=2 ) : 1800 1600 1400 Rigidité (N/ ) 1200 1000 800 600 400 200 175/70 R13 185/70 R13 195/60 R14 165 R13 205/55 R16 0 0 1000 2000 3000 4000 5000 6000 Charge norm ale (N) 57
Exercise Compute: The d Ackerman angle (in ) The cornering stiffness (N/ ) of front and rear wheels and axles The sideslip angles under front and rear tires (in ) The side slip of the vehicle at CG (in ) The steering angle at front wheels (in ) The understeer gradient (in /g) Depending on the case: the characteristic or the critical speed (in kph) The lateral acceleration gain (in g/ ) The yaw speed velocity gain (in s -1 ) The vehicle static margin (%) 58
Exercise 1 Data b 0, 562 m c L b 2,522 0,562 1,960 m b L 0,2228 m 1431kg c L 0,7772 c b W f mg 10909, 8714 N Wr mg 3127,6909 N L L V 80 kph 22,2222 m/ s a y V ² R 4,4893 m / s² R 110 m 59
Exercise 1 Ackerman angle L arctan( ) arctan(0,0229) 0,0229 rad 1, 3134 R Tire cornering stiffness of front wheels c W f mg 10909, 8714 N L F z 5454, 93 N (1) C f 1550 N / deg C 3110 N / deg 177616,98 N / rad f Tire cornering stiffness of rear wheels b (1) Wr mg 3127,6909 N C L r 500 N/ deg 1563,84 N C r 1000 N / deg 57295,8 N / rad 60 Fz
Exercise 1 Side slip angles under the front tires a y V ² R 4,4893 m / s² c V ² F yf m 54992, 879 N L R ma y 6424, 1883 N C f f F C 3110 / deg f N yf Fyf 1,6106 0, 0281rad f C f 61
Exercise 1 Side slip angles under the rear tires b V² Fyf m 1431,3092 N L R C F C 1000 N/ deg r r yr r Fyr 1, 4313 0,0250 r C r Side slip angle at CG rad b cr V r c R r 1,960 b 0,0250 0,0072 rad 0,4105 110 62
Exercise 1 Steering angle at front wheels L mc / L mb / L V ² L R C C f r R f r R Understeer gradient K 1,3134 1,6106 1, 4313 1, 4927 mc / L mb / L 1112,1732 318,8268 C f C r 3100 1000 2 K ' * 0,3918 deg/ 0,0399 / ms K K g g 63
Exercise 1 Understeer gradient: check! K W f r Characteristic speed f gc Wr gc 1,3134 K 4,4893 1,4925 2 K L R V carac 0,0399 deg/ ms 2 L K L V 57,3 K ² R R 6,9639E 4 rad / m/ s² L 2,522 Vcarac 60,1793 m / s 216,64 kph K 6,9639E 4 64
Exercise 1 Lateral acceleration gain G ay ay 4,4893/ 9,81 1,4927 0,3066 g / deg Yaw speed gain G r r V / R 22, 222/110 1, 4927 7,7543 deg/ s/ deg 65
Exercise 1 Neutral maneuver point e bc C f f cc C r r 0,562.3100 1,9600.1000 3100 1000 e 0, 0531m Static margin e L 2,11% 66