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EUROMECH, March 1-4, 2004

A modified implicit Euler Algorithmfor solvingVehicle Dynamic Equations

Georg RillFachhochschule RegensburgUniversity of Applied SciencesGalgenbergstr. 3093053 Regensburg

GeorgRill@aol.com Halle, 2004

2/16

Contents

1. Vehicle Dynamic Equations Modelling Aspects Structure Stiff Ratio Force Characteristics Damper Friction Brake Torque Conclusion

2. Modified Implicit Euler Algorithm divide et impera Use Physical Knowledge Implicit Displacement Estimation Global Derivatives Driving and Braking on Rough Road Gear - Euler - Gear

3. Final Conclusion

Quarter-Car-Model

damper

spring

bushingknuckle

wheel

chassis

zR

xRz0

x00

R

zC

xCC

KuB

wB

K

uC

wC

W

B

roadQ

P

SD

Etop mount

uT

Standard Implicit Euler

xk+1 = xk + h f(xk+1

)

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Vehicle Dynamic Equations Modelling Aspects

Vehicle FrameworkAxle ModulesSteering SystemDrive TrainForce ElementsTireDriverRoad many d.o.f

damper

spring

bushingknuckle

wheel

chassis

zR

xRz0

x00

R

zC

xCC

KuB

wB

K

uC

wC

W

B

roadQ

P

SD

Etop mount

uT

uCwC

vehicle

uBwBK

axle

W wheel rot.

uT top mount dis.

6 d.o.f + 1 f.e.

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Vehicle Dynamic Equations Structure

V ehicle Framework)

K(y) y = z

M(y) z = q(y, z,, w)

Wheel Rotation / Drive Train

=

= r(y, z,)

Dynamic Force Elements

w = s(y, z, w)

lose

physical

knowlegde

x = f (x)

*) Rill, G.: Simulation von Kraftfahrzeugen. Vieweg-Verlag, Braunschweig / Wiesbaden 1994.

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Vehicle Dynamic Equations Stiff Ratio

Linearized Quarter Car Model at v = 30m/s

K(y) y= z

M(y) z = q(y, z,, w)

=

= r(y, z,)

w= s(y, z, w)

x = Ax ; eig(A) =

0

0

1162.749.832+284.07i28.666+85.063i5.5836+63.309i29.2976.8472+ 5.281i

0

stiff ratio:max(|Re(eig(A))|)min(|Re(eig(A))|) =

1162.7

5.5836 200 mildly stiff

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Vehicle Dynamic Equations Force Characteristics

Bushing B Top Mount

-0.02 -0.01 0 0.01 0.02-30

-20

-10

0

10

20

30

m

kN

longitudinal

vertical

-0.01 -0.005 0 0.005 0.01

-40

-20

0

20

40

m

kN

Tire) (longitudinal) Damper

-0.5 0 0.5

-8

-6

-4

-2

0

2

4

6

8

Fz

F x [kN

]

sx [-] -1 -0.5 0 0.5 1-2

-1

0

1

2

m/s

kN

*) Hirschberg, W; Rill, G. Weinfurter, H.: User-Appropriate Tyre-Modelling for Vehicle Dynamics in Standard andLimit Situations. Vehicle System Dynamics 2002, Vol. 38, No. 2, pp. 103-125. Lisse: Swets & Zeitlinger.

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Vehicle Dynamic Equations Damper Friction

s

u

damperfriction

top mountspring: FS = FS(s)

damper: FD = FD(u s)friction: FF = FF (u s)

force balance:

FD(u s) + FF (u s) FR(u s)

= FS(s) ;

s = u F1R (FS(s))

+

=

FD FF FR

v v v

=>F

FR-1

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Vehicle Dynamic Equations Brake Torque

RB

Q

FT

TB

TD

TT

knuckle

wheel+

rim=W

.

braking torque:

TB = TB(),

|TB| TBM

Coulomb simple approximation enhanced approximation

+TBM

TBM

TB

dnum+TBM

TBM

TBdnum

+TBM

TBM

TB

TBS

jump at v=0 TB = dnum TB = TBS dnumlockable not lockable lockable

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Vehicle Dynamic Equations Conclusion

We have: moderate stiff differential equations

strong nonlinear force characteristics

We need: implicit integration formulas

Euler, ..., Gear, ...

keep in mind: vehicle model is not accurate

there are discontinuities in derivatives

integration step size is limitedh 4xroadvvehicle

0.1230 = 4ms

dfdx cannot be calculated analytically

in consequence: use a modified implicit Euler

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Implicit Euler Algorithm divide et impera

Euler backwards formulas applied to each part

Vehicle Framework

yk+1 = yk + h K(yk+1

)1zk+1

zk+1 = zk + h M(yk+1

)1q(yk+1, zk+1,k+1, wk+1

) swapWheel Rotation / Drive Train

k+1 = k + h k+1

k+1 = k + h r(yk+1, zk+1,k+1

) swapDynamic Force Elementswk+1 = wk + h s

(yk+1, zk+1, wk+1

) semi-implicit Euler

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Implicit Euler Algorithm Use Physical Knowledge

state change ofvehicle framework

yk yk+1 , zk zk+1 slower than

state change ofdynamic force variables wk wk+1

s(yk+1, zk+1, wk+1

) s (yk, zk, wk) + sw

(wk+1wk) +

lessimportant

newdynamicforcestates

wk+1 = wk + h

(E s

w

)1s(yk, zk, wk

)state change ofvehicle framework

yk yk+1 , zk zk+1 slower than

change ofwheel ang. vel. k k+1

r(yk+1, zk+1,k+1

) r (yk, zk,k) + r

(k+1k) +

lessimportant

newwheelangularvelocities

k+1 = k + h

(E r

)1r(yk, zk,k

)

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Implicit Euler Algorithm Implicit Displacement Estimation

Vehicle Framework

zk+1 = zk + hM(yk)1 q (yk+1, zk+1,k+1, wk+1)

yk+1 = yk + hK(yk)1 zk+1

with M(yk+1

) M(yk) and K(yk+1) K(yk)q(yk+1, zk+1,k+1, wk+1

) q (yk+h zk, zk,k+1, wk+1)+

q

z

(zk+1 zk

)+

q

y

(yk+1 yk hK(yk)

1zk+1

h zk)

+ h. o. t.

zk+1 = zk

+ h(M(yk)h qzh2 qyK(yk)1)1q (yk+hzk, zk,k+1, wk+1)

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Implicit Euler Algorithm Global Derivatives

dnum

+TBM

TBM

TB

TBS1 2

globalderivative

dTB/d

1 2

globalderivative

exactderi-vativeglobal

derivativeglobalderivative

modifiedimplicit Euler

TB(k+1

) TB(k) + d TBd

(k+1k) +

globalderivative

d TBd

TB(k) TBS

k 0 no jumps !

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Implicit Euler AlgorithmDriving and Braking on Rough Road

0 2 4 6 8 10-20

0

20

40

60

80

100

120velocities

[s]

[kmh]

0 2 4 6 8 100

20

40

60

80

100

120wheel angular velocity

[s]

[rad/s

]

0 2 4 6 8 100

200

400

600

800drive torque

[s]

[Nm]

0 2 4 6 8 10-1200

-1000

-800

-600

-400

-200

0brake torque

[s]

[Nm]

vehiclewheel

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Implicit Euler Algorithm Gear - Euler - Gear

0 2 4 6 8 100

20

40

60

80

100

120wheel angular velocity

[s]

[rad/s

]

0 2 4 6 8 100

20

40

60

80

100

120wheel angular velocity

[s]

[rad/s

]

0 2 4 6 8 100

20

40

60

80

100

120wheel angular velocity

[s]

[rad/s

]

Gear): = 103, 11 170 f-callshmean = 3.3ms / hmax = 10ms

mod. impl. Euler 5 000 f-callsh = 2ms

Gear): = 106, 40 927 f-callshmean = 1ms / hmax = 5ms

0 2 4 6 8 10-1200

-1000

-800

-600

-400

-200

0brake torque

[s]

[Nm]

0 2 4 6 8 10-1200

-1000

-800

-600

-400

-200

0brake torque

[s]

[Nm]

0 2 4 6 8 10-1200

-1000

-800

-600

-400

-200

0brake torque

[s][N

m]

*) Nordsieck Form for Gears Method: Scientific Subroutine Library (SSL II), Tokyo, Japan, 1989-1998

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Final Conclusion

The presented modified implicit Euler-Algorithm is

simple, effective and robust

It is well suited for solving Vehicle Dynamic Equations)

*) veDYNA: MATLAB/Simulink Application, TESIS DYNAware, www.tesis.de

Final Conclusion