Simulation
Numerical simulation is an important tool in the analysis and design
of dynamical systems.
Numerical simulations have a variety of uses. First, simulations can help
in visualizing the
system kinematics and dynamics, particularly when used with graphical
animations. Simulations
also permit one to test different control system designs rapidly, and
without risking damage to
the laboratory apparatus.
There are numerous computer programming languages that may be used to
develop numerical
simulations. In addition to the many general purpose programming language
languages such as C
and FORTRAN, today there are a number of programming languages in which
the program is written
using block diagram notation. We will use Simulink, a popular block
diagram programming language
for our simulation examples.
In this section, we will use a numerical simulation of the SPENDULAP
to verify the second-order dynamics
analysis presented in the previous section. Consider the Simulink model of
the SPENDULAP shown in Figure 8. In this model,
equation (27) is implemented in
Function block Compute theta_dot_dot.
Since a constant frame angular velocity is desired, the frame angular
acceleration is supplied by
Constant block phi_dot_dot, which is configured with a value of
zero. The initial condition of Integrator
blocks theta_dot integrator and phi_integrator are set
to zero. To make the initial pendulum deflection
and the frame angular velocity visible, Integrator blocks theta
integrator and phi_dot integrator are
configured with initial condition ports, which receive the output of
constant blocks configured with the
initial conditions for a particular test case. The state and acceleration
trajectories are sent to the
MATLAB workspace for post processing to compute the reactions and torques
and for plotting. Given the
state trajectory, the motor torque 
can be computed from equation (20). The reaction forces are computed using equations
(22) and (23). The state trajectory and
torque for initial pendulum deflection of 30.5 deg. are plotted in Figure 9 and the corresponding reactions at AB
are plotted in Figure 10.
From Figure 9 we
measure the oscillation period to be 1.78
sec, which agrees well with the period computed earlier using
linearization. Figure 11 illustrates computed trajectories
[
(0) = -35, 35, 45
deg, 
(0) =0
deg/sec] in the state plane superimposed on the state portrait discussed
earlier.
Figure 8. Simulink model of constant frame angular velocity
SPENDULAP.
Figure 9. Constant frame angular velocity
state trajectory.
Figure 10. Reaction forces at AB
for constant frame angular velocity trajectory.
Figure 11. State portrait for constant
frame angular velocity with sate trajectories superimposed.
Graphical animations can facilitate visualization of the system
dynamics. The Simulink model
shown in Figure 8 contains
a graphical animation
block constructed as described in [2]. The
corresponding animation window is shown in Figure
12.
Figure 12. SPENDULAP animation
figure.
