Computer-aided analysis of rigid and flexible …Fundamentals of Vehicle Dynamics, T.D. Gillespie, SAE publications, 1992 The Multibody Systems Approach to Vehicle Dynamics, Mike Blundell

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GraSMech – Multibody 1

Computer-aided analysis of rigid and

flexible multibody systems (Part II)

Simulation of road vehicles

Prof. O. Verlinden (FPMs)

GraSMech course 2006-2007


Simulation of vehicles as MBS

The tyre

is the

typical

element

of a road

vehicle


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 !!

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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)

� ..


Components of a road vehicle

Chassis/carbody

(often rigid body)

Rear

suspension

Front

suspension Tyres

(force element)

Steering

mechanism

Natural multibody

systems !


Tyres

Bias-ply tyre

Several plies oriented

35 to 40 deg wrt to tyre

plane

Invented by Dunlop about 1877 for the bicycle of his father (veterinary)

Two major types of construction: bias-ply and radial

3


Radial tyre

Invented by Michelin in 1947

� Carcass: radial

parallel plies

� Circumferential belt:

steel or fabric wires

Most common tyre

in Europe


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

Physical phenomena

� friction/sliding in the

contact area

� deformation of the tyre

Relative motion Forces


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

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ISO tyre axis system

� Same as SAE but Y and Z in opposite direction


Tire efforts

Forces

� Tractive (X)

� Lateral (Y)

� Normal (Z)

Moments

� Overturning

(X)

� Rolling resistance

(Y)

� Self-aligning

(Z)


Normal (vertical) force

The normal force is derived by considering the tyre as a spring-

damper system

k: radial stiffness

c: radial damping

sometimes determined from

with

ξ: the damping ratio

m: mass of tyre

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Rolling resistance

The rolling resistance force FR (opposite to velocity) is defined from

with µR the rolling resistance coefficient and FV the vertical force.

A rolling resistance moment (opposite to spin velocity) can be

defined equivalently

Typical values of µR

0.200.040.02Tractors

0.250.060.012Heavy trucks

0.30.080.015Passenger car

sandmedium hardconcrete


Rolling resistance

The rolling resistance coefficient increases with speed


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

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Pure cornering


Lateral force vs slip angle

Typical curve lateral force – slip angle


Adhesion and slip areas in the contact

There is always a part of the tyre which is sliding !

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Fromm model

Main hypotheses

� Parabolic distribution of

the pressure along the

longitudinal axis

(uniform laterally)

� Lateral distributed

elasticity of

the rubber tread


Fromm model: kinematics

Successive positions of the tyre


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 deflection δ on a piece of rubber

of width ∆x is worth

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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

areaadhesion

area


Principle to calculate the force

The total force is given by

Transition

point

sliding

areaadhesion

area


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

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Total force

The total force exerted by the ground on the tyre is given by

or

with Cα the

cornering stiffness

(slope at the origin)


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)


Evolution of the self-aligning torque

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Friction coefficient

Characteristic values of the friction coefficient


Cornering stiffness

Typical ratio between cornering stiffness and normal load (/deg)

=> For a radial tyre, Cs (N/rad)=8.5 vertical load (about 20000

N/rad for a classical passenger car of 1000 kg)

Cα/Fz (N/deg/N)


Cornering stiffness and normal force

The cornering stiffness is not constant and depends namely on

normal force

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Fromm refinements: Sakai model

Friction coefficient fs in adhesion area and fd in the sliding area


Fromm refinements: UA model

University of Arizona: evolution

of friction coeff. with slip


Application: soap box vehicle

� (Very) Simplified vehicle

� Major assumptions: vGx=V >> vGy, θ <<

(small perturbations wrt dominant motion)

Dominant motion

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Slip angles and lateral force

The slip angle is derived from the velocity

For wheel 1


Linearized equations of motion

� Lateral equilibrium

� Rotational equilibrium (about G)

� In matrix form


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 !

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Stability – Poles

� The poles are the roots of the characteristic polynomial

� Two cases

� If bCr>aCf unconditional stability wrt to V (understeer)

=> 2 real poles at low speed

=> 2 complex conjugate poles at high speed

� if bCr<aCf stability if V < Vlim (oversteer)

=> always two real poles (exponential behaviour)


Root locus – Unconditionally stable case


Root locus – Conditionally stable case

Very small poles => the vehicle reacts very slowly !

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Pure longitudinal slip


Longitudinal force

The longitudinal force is mainlyrelated to the longitudinal slip defined as

Longitudinal stiffness CS: slopeat the origin


Effective rolling radius

The effective rolling radius Re (or rolling radius) is defined by

In practice

with Rf the radius of the

undeformed tyre

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Pure camber


Camber thrust

� Camber generates a lateral force (thrust)

(important for motorbikes)

� Camber stiffness Cγ=slope at the

origin of the curve


Camber stiffness – order of magnitude

Classical value (radial-passenger car): 1500 N/rad (about 0.6 times

the normal load)

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Overturning moment

Mainly due to the lateral deflection of the tyre


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

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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


University of Arizona model

� Analytical model derived from the Fromm-Fiala model

� Simple: needs only 10 parameters

� geometry of the tyre (R1,R2)

� coefficients Cα, CS, Cγ

� rolling resistance coefficient fR� tyre radial stifness and

damping kz, cz

� friction coefficients fs, fd

� Useful when too few data are available (truck or bus tyres, ...)

Reference: Vehicle Dynamic Simulation with a ComprehensiveModel for Pneumatic Tires, G. Gim, Phd, University of Arizona, 1988


Model of the ground

The ground is generally modelled as a set of triangles

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Semi-empirical models

Based on the « magic formula » (Pacejka, Bakker, Nyborg)

Last version: Delft Tyre model (1997)


Lateral – longitudinal force fitting


Moment fitting

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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 !


Advanced tyre models


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

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Relaxation length

� A first-order filter is introduced in the data flow

with s the running distance and L the relaxation length, or

with V the forward velocity

� Depending on authors, the filter is applied either on the slips, or

on the tyre forces


Relaxation length

The effect of the relaxation length L can 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


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

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Vehicle dynamics


Axes of a vehicle

� SAE vehicle axis system


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)

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Solid axle suspensions

Hotchkiss

Four link rear suspension

de Dion


Independent suspensions

Mac Pherson strut

(front or rear)

Short long arm

Double wishbone

(front or rear)


Independent suspensions

Multilink

(front or rear)

Semi-trailing arm

(rear)

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Semi-independent

� Torsion beam rear suspension (Fiat Punto, golf,...)


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


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)

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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)


Roll axis

The roll centers of the front and rear suspensions define the roll

axis: instanteneous axis about which the vehicle rolls


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

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Springs and dampers

Springs and dampers are naturally involved in multibody

systems, even if nonlinear

� Spring is defined by its stiffness k and rest length l0or a force-length curve

� Damper is defined by its damping coefficient c or a force-

velocity 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


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 CD 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 !


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 3888-1:1999: Passenger cars – Test track for a severe

lane-change maneuver – Part 1: double lane change

� ISO 3888-2:2002: 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

� ...

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Typical vehicle modes

Lateral Bounce Roll

Pitch Hop


Cornering behaviour

Understeer/oversteer


Ackermann construction

The steering mechanism should respect the Ackermann rule

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Steering mechanism


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


Understeer/oversteer coefficient

During cornering, steering angle depends not only on the radius of

turn but also on speed (lateral acceleration)

with δK the kinematic (Ackermann) steering angle and KU the

understeer coefficient

� KU>0: understeer vehicle (the steering angle increases with

lateral acceleration/speed)

� KU<0: oversteer vehicle (the steering angle decreases with

lateral acceleration/speed)

� KU=0: neutral vehicle

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Linear model

For small angles, we have

Soap box: unconditionnal stable if Cr c-Cf b >0 (understeer)


Danger of an oversteer vehicle

� An oversteer vehicle

can become unstable

� The behaviour is not

natural


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

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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 !)


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


Example: the kart

Constant velocity test

Velocity (m/s)

yaw rate (1/s)

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Example: motorbike


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


Principal vibration modes

Capsize (real pole)

bounce

(1.52 Hz, damping 25%)

weave


wobble


front hop


rear hop


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Root locus

Evolution of poles with speed


Example: long bus


Long bus

� GLT: guided light transit vehicle

� Vehicle built by Bombardier (Nancy, Caen,...)

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Layout of the bus

� Articulated bus with 3 carbodies

� 25 m long

� Powered by electric network (reserved track) or Diesel engine

(normal traffic)


Guiding mechanism

Each axle has its own steering

mechanism

On reserved track, each axle is

independently guided by a

central rail


Steering without rail

The steering mechanism of the axles is driven by the articulations

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Footprint of the vehicle

Important issue: what is the

footprint of the bus in free

mode ?


Evolution of the deviation with speed


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

Documents

Computer-aided analysis of rigid and flexible …Fundamentals of Vehicle Dynamics, T.D. Gillespie, SAE publications, 1992 The Multibody Systems Approach to Vehicle Dynamics, Mike Blundell