Starting and Interpreting the Motion

You can now start your simulation and watch the pendulum motion via the Scope and XY Graph blocks:

1. Open the XY Graph block dialog box. Set the following parameters.

2. Open the Scope block and start the model. The XY Graph opens automatically when you start the simulation.

3. View the full motion of both angle and angular velocity (in degrees and degrees per second, respectively) as functions of time in Scope. Click Autoscale if the motion is not fully visible.

Fig.9. Phase Space Plot of Simple Pendulum Motion: Angular Velocity Versus Angle

1. The motion is periodic but not simple harmonic (sinusoidal), because the amplitude of the swing is so large (180 degrees from one turning point to the other). Note that the zero of angle is the initial horizontal angle, not the vertical. The zeros of motion are always the initial conditions.

The XY Graph shows the angle versus angular velocity, with no explicit time axis. These two variables trace out a figure similar to an ellipse, because of the conservation of total energy E:

(2)

where J = I _zz + mL ²/4 is the inertial moment of the rod about its pivot point (not the center of gravity). The two terms on the left side of this equation are the kinetic and potential energies, respectively. The coordinate-velocity space is the phase space of the system.

The model is able to take into account geometrical inaccuracies, material parameter instabilities, finite mobility of fixation zones, etc., and to perform quantitative investigation of the indeterminacy of measurement results.