4569846498

124 Mutibody Systems Approach to Vehicle Dynamics
x
Next predicted
value
Last successful
solution
h
dx n1
dt
t nk
t n3
t n2
t n1
t n
t n1
time t (s)
Fig. 3.41
Use of predictor to fit a polynomial through past values
For successful computation of the next solution at time t n1 there are
therefore two unknowns, x n1 and dx n1 /dt, that must be found. This
requires two equations, the first being the polynomial in (3.79) and the second
being the state equation G(x, ẋ, t) 0. Using the Newton–Raphson approach
described in the previous section the solution takes the form in (3.80):
∂G
∂x˙
∂G
x˙n1 xn1G x˙ n1, xn1,
t
∂x
( )
(3.80)
If we substitute a term that represents the ratio ẋ n1 x n1 we end
up with the following two equations on which the solution is based:
⎡ ∂G
∂G
⎤
⎣
⎢
x G x x t
∂x
∂x
⎦
n 1
˙
˙
n 1, n 1,
ẋ
x
n1 n1
⎥ ( )
(3.81)
(3.82)
The integration process can be thought of as having two distinct phases.
The first of these is the predictor phase that results in values of x n1 and
ẋ n1 that satisfy a polynomial fit through past values. The second is the corrector
phase that iterates using the process shown in Figure 3.39 until the
error is within tolerance and the solution can progress to the next time step.
Parameters can be set that control the solution process. Examples of these
are the initial, maximum and minimum integration step sizes to be used and
the order of the polynomial fit. Parameters used during the corrector phase
include the acceptable error tolerance, the maximum number of iterations
and the pattern or sequence to be used in updating the Jacobian matrix during
the iterations. In general these will default to values programmed into
the software but with experience users will find the most suitable settings
for the analysis in hand.