Download - Rill Stable Implicity Euler
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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
[email protected] Halle, 2004
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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