4569846498

Modelling and analysis of suspension systems 233
⎡230⎤
[ FF21xFF21yFF21
z]
⎢
0
⎥
0 N mm
⎢ ⎥
⎣⎢
0 ⎦⎥
(4.306)
230F F21x 0 (4.307)
For this particular suspension system the line E–F is parallel to the model
x-axis yielding the trivial result F F21x being equal to zero. In this case we
can therefore ignore F F21x in the following matrix solution of the system
equations.
The axis A–B for the upper wishbone is not parallel to a model axis and
therefore applying the vector dot product to ensure that {F B31 } 1 is perpendicular
to the line A–B yields the final equation needed to solve the remaining
19 unknowns:
{F B31 } 1 • {R AB } 1 0 (4.308)
⎡230⎤
[ F F F ]
⎢
(4.309)
B31x B31y B31
z 0
⎥
0 N mm
⎢ ⎥
⎣⎢
14 ⎦⎥
Equation 19 230F B31x 14F B31z 0 (4.310)
The 19 equations can now be set up in matrix form ready for solution:
⎡ 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 ⎤ ⎡FA31x⎤
⎡ 0 ⎤
⎢
0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0
⎥ ⎢F
⎥ ⎢
⎢
⎥ A31y
0
⎥
⎢ ⎥ ⎢ ⎥
⎢ 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 ⎥ ⎢ FA31z
⎥ ⎢ 0 ⎥
⎢
⎥ ⎢ ⎥ ⎢ ⎥
⎢
0 0 0 0 0 0 0 0 0 0 9 275 9 275
0 0 0 0 0
⎥ ⎢
FB31x⎥
⎢
0
⎥
⎢ 0 0 0 0 0 0 0 0 0 9 0 115 0 115 0 0 0 0 0 ⎥ ⎢F
⎥ ⎢ B31y
0 ⎥
⎢
⎥ ⎢ ⎥ ⎢ ⎥
⎢ 0 0 0 0 0 0 0 0 0 275 115 0 115 0 0 0 0 0 0 ⎥ ⎢ FB31z⎥
⎢ 0 ⎥
⎢ 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 3 ⎥ ⎢F
⎥ ⎢
D34x
0 ⎥
⎢
⎥ ⎢ ⎥ ⎢ ⎥
⎢ 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 9
⎥ ⎢FD34y⎥
⎢ 0 ⎥
⎢
0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 436
⎥ ⎢
F
⎥ ⎢
⎢
⎥ D34z
0
⎥
⎢ ⎥ ⎢ ⎥
⎢ 0 15 239 0 1 239 0 0 0 0 0 0 0 0 0 0 0 0 42 521⎥
⎢FE21x⎥⎢
0 ⎥
⎢
⎥ ⎢ ⎥ ⎢ ⎥
⎢ 15 0 115 1 0 115 0 0 0 0 0 0 0 0 0 0 0 0 69
⎥ ⎢FE21y⎥
⎢ 0 ⎥
⎢239 115 0 239 115 0 0 0 0 0 0 0 0 0 0 0 0 0 294 ⎥ ⎢ F ⎥ ⎢ E21z
0 ⎥
⎢
⎥ ⎢ ⎥ ⎢ ⎥
⎢ 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 ⎥ ⎢FF21y⎥
⎢ 0 ⎥
⎢ 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 228 0 ⎥ ⎢ F ⎥ ⎢
F21z
0 ⎥
⎢
⎥ ⎢ ⎥ ⎢ ⎥
⎢ 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
8 0 ⎥ ⎢FG21x⎥
⎢ 10 000 ⎥
⎢
0 0 0 0 0 0 0 216 31 0 0 0 0 0 0 0 0 60 276 0
⎥ ⎢
F
⎥ ⎢
⎢
⎥ ⎢ G24y⎥
580 000
⎥
⎢ ⎥
⎢ 0 0 0 0 0 0 216 0 19 0 0 0 0 0 0 0 0 1304 0 ⎥ ⎢FG24z⎥
⎢ 70 000 ⎥
⎢
⎥ ⎢ ⎥ ⎢ ⎥
⎢ 0 0 0 0 0 0 31 19 0 0 0 0 0 0 0 0 0 37 164 0 ⎥ ⎢ fS1
⎥ ⎢ 0 ⎥
⎣
⎢ 0 0 0 230 0 14 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎦
⎥
⎣
⎢ fS2
⎦
⎥
⎣
⎢ 0 ⎦
⎥
(4.311)
Examination of the square matrix in (4.311) indicates a large number of
zero terms, hence the matrix is referred to as sparse. As discussed in
Chapter 3 this is a typical characteristic of the matrices generated in MBS
and is one of the reasons why fast and efficient matrix inversion techniques
can be deployed. The overall result is that MBS programs appear to solve
quite complex engineering problems with a much lower requirement for
computational effort than comparable other CAE methods such as non-linear