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Multibody systems simulation software 125
The integration step size is variable and if a solution is not achieved at the
next prediction point the solver can reduce the integration time step and alter
the order of the polynomial to attempt another solution. This typically occurs
when there is a sudden change in an equation associated with a physical
event such as an impact or clash of parts. A general rule in modelling is never
to program an equation that instantly changes value at a certain time. Problems
will be avoided by using, for example, a function that allows a smooth
transition from one value to another over a physically small time interval.
Consideration of the predictor–corrector approach will indicate that at the
start of a transient solution the solver may step forward and backward initially
as it establishes a suitable scheme to progress the solution. Experienced users
will again be aware of this and avoid programming important events to
occur immediately after the start. For example, the simulation of a 5 second
lane change manoeuvre may be best accomplished by an initial static analysis
followed by the transient simulation for, say, 1 second of straight line driving
before any steering inputs are made.
Advanced applications may also involve the incorporation of control algorithms
based on a discreet time step. An example discussed later in this book
will include the modelling of an Antilock Braking System (ABS) using a
fixed time cycle. A possible solution here is to fix the integration minimum
and maximum step size to ensure the solver computes solutions at the fixed
time steps of the ABS algorithm. This may of course result in inefficient
computation at some stages of the simulation. On the other hand the fixed
step scheme must be refined enough to deal with any sudden non-linear
events during the analysis.
Commercial multibody systems codes often have a range of integrators available
that will adopt variations on the solution process discussed here. The
most commonly used in MSC.ADAMS is referred to as the Gstiff integrator
(Gear, 1971). Rather than storing past values of the polynomial a Taylor
expansion (3.83) is used to store the polynomial in a form using the current
value of the state variable x and the integration step size h. This is known as
a Nordsieck vector. At a current time t the Nordsieck vector [N t ] has the form
[ N ] x h x 2 2 3 3
k k
T ⎡ d h d x h d x h d x ⎤
t ⎢
2
3
...
k ⎥
(3.83)
⎣ dt
2 dt
3!
dt
k!
dt
⎦
The components of [N t ] are added together to predict the next value of x at
a time step h forward to a new time t h. As the simulation progresses the
Nordsieck vector is updated by pre-multiplying by the Pascal triangle matrix
to give a new vector [N th ]. An example of this is shown for a polynomial
with an order k equal to 5 in (3.84):
⎡1 1 1 1 1 1⎤
⎢
0 1 2 3 4 5
⎥
⎢
⎥
[ Nth] ⎢0 0 1 3 4 5⎥
⎢
⎥ [ Nt]
⎢
0 0 0 1 4 5
⎥
⎢0 0 0 0 1 5⎥
⎢
⎥
⎣0 0 0 0 0 1⎦
(3.84)