4569846498

Multibody systems simulation software 123
k
x
m
c
Fig. 3.40 Simple 1 degree-of-freedom mass, spring, damper system
2
d
m
x d
c x kx 0
(3.76)
2
dt
dt
The implementation of this as first order differential equations is simply a
matter of introducing a new variable, for example z, for the velocity and
writing (3.76) as two implicit first order differential equations:
dx
z 0
dt
(3.77)
d
m z dt
cz
kx
0
(3.78)
The predictor phase of the solution can be explained with the help of the
plot of the value of the state variable x as a function of time t as shown in
Figure 3.41. The current solution point, or nth integration time step, is shown
to be occurring at time t n . Previous successful solutions have been computed
at times t n1 , t n2 , t n3 , …, t nk . The next value to be predicted x n1 will
occur at time t n1 . The time between each solution point is the integration
time step h. Note that this should not be confused with output steps that are
generally defined before the analysis and are used to fix the time interval at
which results will be calculated for printing and plotting. The integration
time step must be at least as small as the output time step to compute solutions
but will generally be smaller in order to obtain a solution. In some
programs the integration time steps may be fixed but usually, as with programs
like MSC.ADAMS, they will be variable and will use programmed
logic to determine the optimum step size for the problem in hand.
For the solution at the next time step to lie on the polynomial both the
value of the state variable x n1 and the derivative dx n1 /dt must satisfy the
polynomial.
The derivative dx n1 /dt at the next time step t n1 can be related to the
unknown future value x n1 and the past computed values using
dxn
dt
1
( )
Pn x , x , x , x , x , K,
x
n1 n n1 n2
n3
nk
(3.79)
For the solution at the next time step to lie on the polynomial both the value
of the state variable x n1 and the derivative must satisfy the polynomial.