Computer-aided analysis of rigid and flexible multibody systems (Part II) Simulation of road vehicles. Prof. O. Verlinden (FPMs)

Transcription

1 GraSMech course Computer-aided analysis of rigid and flexible multibody systems (Part II) Simulation of road vehicles Prof. O. Verlinden (FPMs) GraSMech Multibody 1 Simulation of vehicles as MBS The tyre is the typical element of a road vehicle GraSMech Multibody 2 References Fundamentals of Vehicle Dynamics, T.D. Gillespie, SAE publications, 1992 The Multibody Systems Approach to Vehicle Dynamics, Mike Blundell and Damian Harty, Elsevier, 2004 Vehicle Handling Dynamics, J.R. Ellis, Mechanical Engineering Publications, 1994 Tyre and Vehicle Dynamics, H.B. Pacejka, Buterworth- Heinemann, 2002 Tires, Suspensions and Handling, J.C. Dixon, SAE publications, 1996 Race car vehicle dynamics, W.F. Milliken and D.L. Milliken, SAE Publications, 1995!! UK: tyre USA:tire!! GraSMech Multibody 3 1

2 Simulation software s The most widespread MBS simulation codes have features to simulate road vehicles ADAMS (ADAMS/car) Simpack LMS Virtual motion (former DADS)... Independent simulation tools exist CarSim/TruckSim (University of Michigan, UMTRI) ASM/Vehicle Dynamics Simulation Package (dspace, Matlab toolbox).. GraSMech Multibody 4 Components of a road vehicle Natural multibody systems! Chassis/carbody (often rigid body) Steering mechanism Rear suspension Front suspension Tyres (force element) GraSMech Multibody 5 Tyres Invented by Dunlop about 1877 for the bicycle of his father (veterinary) Two major types of construction: bias-ply and radial Bias-ply tyre Several plies oriented 35 to 40 deg wrt to tyre plane GraSMech Multibody 6 2

3 Radial tyre Invented by Michelin in 1947 Carcass: radial parallel plies Circumferential belt: steel or fabric wires Most common tyre in Europe GraSMech Multibody 7 The tyre as a force element Input data: motion of the tyre with respect to the ground (position, orientation, translational and rotational velocties) Output data: ground forces Relative motion Forces Physical phenomena friction/sliding in the contact area deformation of the tyre GraSMech Multibody 8 SAE tyre axis system Z: perpendicular to the ground (downwards) X: intersection between tyre plane and ground plane camber angle γ: angle between wheel plane and vertical plane slip angle α: angle between wheel plane and direction of travel GraSMech Multibody 9 3

4 ISO tyre axis system Same as SAE but Y and Z in opposite direction GraSMech Multibody 10 Tire efforts Forces Tractive (X) Lateral (Y) Normal (Z) Moments Overturning (X) Rolling resistance (Y) Self-aligning (Z) GraSMech Multibody 11 Normal (vertical) force The normal force is derived by considering the tyre as a springdamper system k: radial stiffness c: radial damping sometimes determined from with ξ: the damping ratio m: mass of tyre GraSMech Multibody 12 4

5 Rolling resistance The rolling resistance force F R (opposite to velocity) is defined from with µ R the rolling resistance coefficient and F V the vertical force. A rolling resistance moment (opposite to spin velocity) can be defined equivalently Typical values of µ R concrete medium hard sand Passenger car Heavy trucks Tractors GraSMech Multibody 13 Rolling resistance The rolling resistance coefficient increases with speed GraSMech Multibody 14 Simple regime conditions Pure cornering: no acceleration/braking, no camber Slip angle => lateral force, aligning torque Pure longitudinal: no slip angle, no camber Longitudinal slip => longitudinal force Pure camber: no acceleration/braking, no slip angle Camber angle => lateral force, aligning torque GraSMech Multibody 15 5

6 Pure cornering GraSMech Multibody 16 Lateral force vs slip angle Typical curve lateral force slip angle GraSMech Multibody 17 Adhesion and slip areas in the contact There is always a part of the tyre which is sliding! GraSMech Multibody 18 6

7 Fromm model Main hypotheses Parabolic distribution of the pressure along the longitudinal axis (uniform laterally) Lateral distributed elasticity of the rubber tread GraSMech Multibody 19 Fromm model: kinematics Successive positions of the tyre GraSMech Multibody 20 Kinematics for the adhesion area Motion of the center of the wheel from t to t : CC Motion of the rubber piece with respect to the wheel: BA Longitudinal component of BA = rotation of the wheel Lateral component of BA= deformation of the tyre The force necessary to impose the deflectionδon a piece of rubber of width x is worth GraSMech Multibody 21 7

8 Transition point The lateral force exerted by the ground to the piece of rubber is limited by the friction limit ( f = friction coefficient) Transition point sliding area adhesion area GraSMech Multibody 22 Principle to calculate the force Transition point sliding area adhesion area The total force is given by GraSMech Multibody 23 Transition point The transition point is the point where the adhesion and sliding elementary forces are equal The solution is meaningful (x * <a) only if If the slip angle is over the limit, the sliding area covers the whole contact patch -> the tyre skids GraSMech Multibody 24 8

9 Total force The total force exerted by the ground on the tyre is given by or withc α the cornering stiffness (slope at the origin) GraSMech Multibody 25 Total moment The total torque is obtained by which leads to The moment is positive => self-aligning torque (aligns the wheel to the direction of travel) GraSMech Multibody 26 Evolution of the self-aligning torque GraSMech Multibody 27 9

10 Friction coefficient Characteristic values of the friction coefficient GraSMech Multibody 28 Cornering stiffness Typical ratio between cornering stiffness and normal load (/deg) => For a radial tyre, Cs (N/rad)=8.5 vertical load (about N/rad for a classical passenger car of 1000 kg) C α /F z (N/deg/N) GraSMech Multibody 29 Cornering stiffness and normal force The cornering stiffness is not constant and depends namely on normal force GraSMech Multibody 30 10

11 Fromm refinements: Sakai model Friction coefficient f s in adhesion area and f d in the sliding area GraSMech Multibody 31 Fromm refinements: UA model University of Arizona: evolution of friction coeff. with slip GraSMech Multibody 32 Application: soap box vehicle (Very) Simplified vehicle Major assumptions: v Gx =V >> v Gy, θ << (small perturbations wrt dominant motion) Dominant motion GraSMech Multibody 33 11

12 Slip angles and lateral force The slip angle is derived from the velocity For wheel 1 GraSMech Multibody 34 Linearized equations of motion Lateral equilibrium Rotational equilibrium (about G) In matrix form GraSMech Multibody 35 Linearized equations of motion Mass, tangent damping and tangent stiffness matrices are given by with K is neither symmetric nor positive definite C decreases with V! GraSMech Multibody 36 12

13 Stability Poles The poles are the roots of the characteristic polynomial Two cases If bc r >ac f unconditional stability wrt to V (understeer) => 2 real poles at low speed => 2 complex conjugate poles at high speed if bc r <ac f stability if V < V lim (oversteer) => always two real poles (exponential behaviour) GraSMech Multibody 37 Root locus Unconditionally stable case GraSMech Multibody 38 Root locus Conditionally stable case Very small poles => the vehicle reacts very slowly! GraSMech Multibody 39 13

14 Pure longitudinal slip GraSMech Multibody 40 Longitudinal force The longitudinal force is mainly related to the longitudinal slip defined as Longitudinal stiffnessc S : slope at the origin GraSMech Multibody 41 Effective rolling radius The effective rolling radius R e (or rolling radius) is defined by In practice with R f the radius of the undeformed tyre GraSMech Multibody 42 14

15 Pure camber GraSMech Multibody 43 Camber thrust Camber generates a lateral force (thrust) (important for motorbikes) Camber stiffnessc γ =slope at the origin of the curve GraSMech Multibody 44 Camber stiffness order of magnitude Classical value (radial-passenger car): 1500 N/rad (about 0.6 times the normal load) GraSMech Multibody 45 15

16 Overturning moment Mainly due to the lateral deflection of the tyre GraSMech Multibody 46 Comprehensive models GraSMech Multibody 47 Comprehensive models Comprehensive models generate the forces and moments when different slips are combined Example: lateral and longitudinal slips Notion of friction circle The total contact force is limited by friction GraSMech Multibody 48 16

17 Comprehensive models Data transfer of a comprehensive model Input data radial deflection longitudinal slip speed of revolution slip angle vertical spin camber angle Output data normal load longitudinal force rolling resistance moment lateral force self-aligning torque overturning moment GraSMech Multibody 49 University of Arizona model Analytical model derived from the Fromm-Fiala model Simple: needs only 10 parameters geometry of the tyre (R 1,R 2 ) coefficients C α, C S, C γ rolling resistance coefficient f R tyre radial stifness and damping k z, c z friction coefficients f s, f d Useful when too few data are available (truck or bus tyres,...) Reference: Vehicle Dynamic Simulation with a Comprehensive Model for Pneumatic Tires, G. Gim, Phd, University of Arizona, 1988 GraSMech Multibody 50 Model of the ground The ground is generally modelled as a set of triangles GraSMech Multibody 51 17

18 Semi-empirical models Based on the «magic formula» (Pacejka, Bakker, Nyborg) Last version: Delft Tyre model (1997) GraSMech Multibody 52 Lateral longitudinal force fitting GraSMech Multibody 53 Moment fitting GraSMech Multibody 54 18

19 Semi-empirical model Actual implementation Each effort is expressed by a magic formula whose coefficients are themselves expressed by a magic formula (dependance on normal force, combined slips,...) More than 100 coefficients determined from measurements on a specific tyre The data are valid for only one tyre! GraSMech Multibody 55 Advanced tyre models GraSMech Multibody 56 Dynamic models Models only represent the steady-state behaviour. In some cases, the dynamics of the tyre itself must be taken into account. Relaxation length: first-order filter in inputs or outputs String type tyre models The deflection of the tyre is introduced through one or several strings GraSMech Multibody 57 19

20 Relaxation length A first-order filter is introduced in the data flow withsthe running distance and L the relaxation length, or withvthe forward velocity Depending on authors, the filter is applied either on the slips, or on the tyre forces GraSMech Multibody 58 Relaxation length The effect of the relaxation lengthlcan be well represented by introducing some compliance between the rim and the rolling tread. For the lateral force, choose the stiffness For the longitudinal force, choose a rotational stiffness as GraSMech Multibody 59 Other advanced models String-based tyre models models the deflection of the tyre through one or more strings (automatically accounts for the tyre dynamics) Advanced modelling of geometry (durability analysis) Finite element models GraSMech Multibody 60 20

21 Vehicle dynamics GraSMech Multibody 61 Axes of a vehicle SAE vehicle axis system GraSMech Multibody 62 Suspensions Initial role of the suspension Reduction of vertical wheel load variations Isolation of road inputs from the body linked to the spring-damper system of the suspension But also Load transfer control in cornering or acceleration/braking Handling (behaviour and feel) control by adjusting kinematics of the wheel during suspension travel Types of suspensions Solid-axle suspensions (trucks) Independent suspensions (cars) GraSMech Multibody 63 21

22 Solid axle suspensions Hotchkiss de Dion Four link rear suspension GraSMech Multibody 64 Independent suspensions Mac Pherson strut (front or rear) Short long arm Double wishbone (front or rear) GraSMech Multibody 65 Independent suspensions Multilink (front or rear) Semi-trailing arm (rear) GraSMech Multibody 66 22

23 Semi-independent Torsion beam rear suspension (Fiat Punto, golf,...) GraSMech Multibody 67 Kinematic analysis The multibody approach naturally allows the kinematic study of the suspension => Evolution during the bump motion of the camber angle the toe angle (steer) the roll center: center of rotation of vehicle wrt ground the equivalent stiffness (damping)... cf. SAE J670e «Vehicle dynamics terminology» for complete rigourous definition of terms The joints of the suspension can be introduced as kinematic joints, linear bushings or non linear bushings GraSMech Multibody 68 Roll center Roll center: instantaneous point about which the vehicle rolls Construction of the roll center R must be on the symmetry axis application of Kennedy s theorem: on the same line as F (wheel/ground) and E (wheel/vehicle) GraSMech Multibody 69 23

24 Roll center The roll center is also the location where the lateral forces developed by the wheels are transmitted to the sprung mass A lateral force applied at the height of the roll center doesn t induce any roll of the vehicle The roll center affects the distribution of normal forces of the tyres (load transfer) GraSMech Multibody 70 Roll axis The roll centers of the front and rear suspensions define the roll axis: instanteneous axis about which the vehicle rolls GraSMech Multibody 71 Anti-dive / Anti-squat Anti-dive (front) and anti-rise (rear) control pitch during braking Anti-lift (front) and anti-squat (rear) control pitch during traction GraSMech Multibody 72 24

25 Springs and dampers Springs and dampers are naturally involved in multibody systems, even if nonlinear Spring is defined by its stiffnesskand rest lengthl 0 or a force-length curve Damper is defined by its damping coefficient c or a forcevelocity curve (at least two coefficients as a vehicle damper is always more resistant in extension (rebound) than in compression (bounce) A damper involves fluid flow so that the nonlinear force-velocity curve is often necessary GraSMech Multibody 73 Aerodynamic forces Aerodynamic forces come from wind and the motion of the vehicle, and generate principally a drag force, determined by with ρ the air density, V the vehicle speed and A the frontal area of the vehicle and C D the drag coefficient (about 0.3 for cars) but also forces and moments in all directions (side force, lift force, pitching moment, yawing moment, rolling moment) References: Gillespie, Milliken The aerodynamic forces are generally ignored except for winged or very rapid vehicles (formula 1), or for trucks! GraSMech Multibody 74 Typical analyses Geometric analysis of suspensions Linear analysis (root locus vs velocity) Ride: transmission of road vibrations (linear or nonlinear) Typical maneuvers (same as tests that vehicle engineers carry out with prototype vehicles) ISO :1999: Passenger cars Test track for a severe lane-change maneuver Part 1: double lane change ISO :2002: Passenger cars Test track for a severe lane-change maneuver Part 2: Obstacle avoidance ISO 4138:1996: Passenger cars Steady state circular driving behaviour Open loop test procedure ISO 7975:1996: Passenger cars Braking in a turn Open loop test procedure... GraSMech Multibody 75 25

26 Typical vehicle modes Lateral Bounce Roll Pitch Hop GraSMech Multibody 76 Cornering behaviour Understeer/oversteer GraSMech Multibody 77 Ackermann construction The steering mechanism should respect the Ackermann rule GraSMech Multibody 78 26

27 Steering mechanism GraSMech Multibody 79 Some typical angles The steering axis is not vertical the inclination gives an aligning torque wih gravity the caster angle reinforces the aligning torque of the tyre GraSMech Multibody 80 Understeer/oversteer coefficient During cornering, steering angle depends not only on the radius of turn but also on speed (lateral acceleration) withδ K the kinematic (Ackermann) steering angle and K U the understeer coefficient K U >0: understeer vehicle (the steering angle increases with lateral acceleration/speed) K U <0: oversteer vehicle (the steering angle decreases with lateral acceleration/speed) K U =0: neutral vehicle GraSMech Multibody 81 27

28 Linear model For small angles, we have Soap box: unconditionnal stable if C r c-c f b >0 (understeer) GraSMech Multibody 82 Danger of an oversteer vehicle An oversteer vehicle can become unstable The behaviour is not natural GraSMech Multibody 83 Factors influencing the under/oversteer Distribution of masses Tire properties Dependance cornering stiffness/normal force Camber change due to suspension Steer change due to suspension (including compliance) Effect of self-aligning torque Effect of tractive forces (2WD/4WD, differential)... => With a simulation tool, the best is to measure the understeer coefficient from virtual tests on the model GraSMech Multibody 84 28

29 Constant radius test On a circular trajectory, the steer angle is measured for different velocities the understeer coefficient is given by the slope (depends on speed!) GraSMech Multibody 85 Constant velocity test The lateral acceleration is measured for different steering angles at a constant velocity The understeer coefficient is the difference of slope between the measured case and the ideal case (Ackermann steering) The Ackermann curve is obtained by a low speed test with eventuallly majored cornering stiffnesses GraSMech Multibody 86 Example: the kart Constant velocity test yaw rate (1/s) Velocity (m/s) GraSMech Multibody 87 29

30 Example: motorbike GraSMech Multibody 88 Model of the motorbike 8 bodies Main frame Front fork Front wheel trim Front tyre tread Swing arm Rear wheel trim Rear tyre tread pilot 14 dof Special elements lateral compliance of the tyre lateral flexibility of the fork GraSMech Multibody 89 Principal vibration modes Capsize (real pole) bounce (1.52 Hz, damping 25%) weave (3.02 Hz, damping 22%) wobble (7.71 Hz, damping 86%) front hop (11.4 Hz, damping 30%) rear hop (13.1 Hz, damping 24%) GraSMech Multibody 90 30

31 Evolution of poles with speed Root locus GraSMech Multibody 91 Example: long bus GraSMech Multibody 92 Long bus GLT: guided light transit vehicle Vehicle built by Bombardier (Nancy, Caen,...) GraSMech Multibody 93 31

32 Layout of the bus Articulated bus with 3 carbodies 25 m long Powered by electric network (reserved track) or Diesel engine (normal traffic) GraSMech Multibody 94 Guiding mechanism Each axle has its own steering mechanism On reserved track, each axle is independently guided by a central rail GraSMech Multibody 95 Steering without rail The steering mechanism of the axles is driven by the articulations GraSMech Multibody 96 32

33 Footprint of the vehicle Important issue: what is the footprint of the bus in free mode? GraSMech Multibody 97 Evolution of the deviation with speed GraSMech Multibody 98 Conclusions Any multibody code equipped with tyre models can deal with road vehicles The specialized software tools help in defining the system (suspensions, steering system) finding coherent initial conditions defining typical simulations (including the driver) interpreting the results Simulation has become an inescapable tool for the design of road vehicles GraSMech Multibody 99 33