Simple Pendulum Double Pendulum Spherical Pendulum Driven Pendulum Coupled Pendulum Spinning Tops

Spinning tops need to be visualised in reference to 2 frames - the space frame ( x , y , z ) that is effectively the 'real life' frame that we have been using up until now when describing the position of the pendulums; and the body frame ( x' , y' , z' ) that is basically another set of ( x , y , z ) axes taken with the z'-axis pointing up through the middle of the spinning top. This body frame obviously changes its orientation with respect to the space frame as the spinning top moves.

There are 3 degrees of freedom needed to describe the position of the spinning top - theta, which is the angle between the z-axis of the space frame and the z'-axis of the body frame; phi, which is the rotation angle of the z'-axis of the body frame around the z-axis of the space frame; and psi, which is the rotation angle of the spinning top around its own z'-axis (that of the body frame).

A diagram of the symmetric spinning top is shown below:

The Symmetric Spinning Top



Lagrange's equations are needed again to derive the equations of motion used to model the spinning top. The equations of motion for the symmetric top are shown below:
(Click Here to see the proof)

The Symmetric Spinning Top Equations of Motion



The applet below simulates the motion of the symmetric spinning top:

If you can read this writing then either you do not have Java installed, or your Java is not new enough to view the applets on this site. You need Java 1.4 to view these applets.

It is possible to adjust the initial values for theta, thetadot, phi, phidot, psi and psidot. It is also possible to alter the values for the mass m of the top, and l, which is the length of the top along the z'-axis from the origin (point of the top) to the centre of the circle at the other end of the top.

In addition to this, you can activate/deactivate gravity; adjust the framerate of the animation (if you want to watch the applet in slow motion, for example); and turn a reference dot on/off. This reference dot highlights a fixed point on the top, and is designed to help visualise how fast the top is spinning.

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© Michael Hart 2004