4569846498

Multibody systems simulation software 119
[L][y] [b] as given in (3.66) we get
⎡L11
0 0 0
⎢L21 L22
0 0
⎢L31 L32 L33
0
⎢
⎣L L L L
41 42 43 44
therefore
⎤ ⎡y1
⎤ ⎡b1
⎤
⎥ ⎢y2
⎥ ⎢b2
⎥
⎥ ⎢y
⎥ ⎢
3 b ⎥
3
⎥ ⎢
⎦ ⎣y
⎥ ⎢
4 ⎦ ⎣b
⎥
4 ⎦
(3.69)
y 1 b 1 /L 11
y 2 (b 2 L 21 y 1 )/L 22
y 3 (b 3 L 31 y 1 L 32 y 2 )/L 33
y 4 (b 4 L 41 y 1 L 42 y 2 L 43 y 3 )/L 44
The next step is a back substitution, utilizing the terms found in the [U]
matrix to find the overall solution for the terms in [x]. For this example if
we expand [U][x] [y] as given in (3.65) we get
⎡1 U12 0 U14
⎤ ⎡x1
⎤ ⎡y1
⎤
⎢0 1 0 U
(3.70)
24 ⎥ ⎢x2
⎥ ⎢y2
⎥
⎢0 0 1 U ⎥ ⎢
34 x ⎥ ⎢
3 y ⎥
3
⎣
⎢0 0 0 1 ⎦
⎥ ⎢x
⎥ ⎢
4 y ⎥
⎣ ⎦ ⎣ 4 ⎦
therefore
x 4 y 4
x 3 (y 3 U 34 x 4 )
x 2 (y 2 U 24 x 4 )
x 1 (y 1 U 12 x 2 U 14 x 4 )
It can be seen that in the solution of (3.69) it is necessary to divide by the
diagonal terms in [L]. These are referred to as ‘pivots’ and must be nonzero
to avoid a singular matrix and failure in solution. There will also be
problems if the pivots are so small that they approach a zero value. In these
circumstances the condition of the matrices is said to be poor. The answer
to this is to rearrange the order of the equations, referred to as pivot selection,
until the best set of pivots is available. This process is also called refactorization.
The mathematics has been developed so that the process will also
attempt to minimize the number of fills to assist with a faster solution. The
sequence of operations can be stored to speed solution as the simulation
progresses unless the physical configuration of the system changes to a
point where the matrix changes sufficiently to justify the reselection of a set
of pivots. It should be noted that in general solution of these equations will
involve much larger matrices than the four by four example shown here and
the sparsity of the matrix will be more apparent for these larger problems.
3.3.3 Non-linear equations
In the case of non-linear equations an iterative approach must be undertaken
in order to obtain a solution. A set of non-linear equations may be
described in matrix form using
[G][x] 0 (3.71)