4569846498

136 Multibody Systems Approach to Vehicle Dynamics
⎡ 1 1 1 1
⎢
AG BG CG DG
⎢
y y y y
⎢AG BG CG DG
⎢
⎣ ? ? ? ?
x x x x
⎤ ⎡F
⎥ ⎢
F
⎥ ⎢
⎥ ⎢F
⎥ ⎢
⎦ ⎣F
⎤ ⎡mg
2 ⎤
⎥ ⎢
0
⎥
⎥ ⎢ ⎥
⎥ ⎢ 0 ⎥
⎥ ⎢ ⎥
⎦ ⎣ ? ⎦
(4.10)
If a modified formulation is adopted, the problem can be posed more completely.
If the rigid platform is presumed to be sprung in the manner of a
normal road vehicle with an effective wheel rate, k A , k B , k C and k D , at A, B,
C and D then a new formulation is possible. Writing A 2z , B 2z , C 2z and D 2z
for the height of corners A, B, C and D on Body 2 and A 1z , B 1z , C 1z and D 1z
for the ground height corners at A, B, C and D we can then define a preload
Fp A , Fp B , Fp C and Fp D on each spring such that if the z co-ordinates of
Body 2 at the corner are equal to the z co-ordinates of the ground, Body 1,
at each corner then the spring load is equal to the preload. This leads to the
following equations for the spring force at each corner:
F Az k A (A 2z A 1z ) Fp A (4.11)
F Bz k B (B 2z B 1z ) Fp B (4.12)
F Cz k C (C 2z C 1z ) Fp C (4.13)
F Dz k D (D 2z D 1z ) Fp D (4.14)
At first sight it would appear this is simply a more elaborate formulation of
the previous problem; there are still four unknown quantities (the corner
heights of Body 2) and three equations (4.4), (4.8) and (4.9). However, consideration
of the rigid platform yields a fourth relationship:
D 2z C 2z (B 2z A 2z ) (4.15)
This leads to, after some manipulation of (4.10) substituting in the spring
equations (4.11–4.13) for F Az , F Bz , F Cz and relying on (4.15) to solve for
F Dz , the equations in (4.16):
⎡ kA kD kB kD kC kD
⎤ ⎡A2
z ⎤ ⎡1
⎤
⎢
kAAGy kDDGy kBBGy kDDGy kCCGy k DG
⎥ ⎢
B
⎥
⎢
D y⎥
⎢ 2z
⎥
⎢
⎢
⎥
2 ⎥
⎣⎢
kAAGx kDDGx kBBGx kDDGx kCCGx kDDGx⎦⎥
⎣⎢
C2
z ⎦⎥
⎣⎢
3
⎦⎥
(4.16)
where
1 m 2 g k A A 1z Fp A k B B 1z Fp B k C C 1z Fp C k D D 1z Fp D
(4.17)
2 (k A A 1z Fp A )AG y (k B B 1z Fp B )BG y (k C C 1z Fp C )CG y
(k D D 1z Fp D )DG y (4.18)
3 (k A A 1z Fp A )AG x (k B B 1z Fp B )BG x (k C C 1z Fp C )CG x
(k D D 1z Fp D )DG x (4.19)
The equations in (4.16) may be solved by Gaussian elimination and substitution.
It can be seen that preloads (Fp A , Fp B , Fp C and Fp D ), the ground
Az
Bz
Cz
Dz