External torque on gyro

This looks like a good method, and with the code, ideas like this are of immediate interest. The general or canonical method of defining torque is

Tq = (dL / dt) (X, Y, Z) = (dL / dt + omega x L)(1, 2, 3)

leading to Eqs. (10.120) of Marion and Thornton, the Euler equations with torque. Here L is angular momentum. These are Eqs. (2) to (4) of UFT368, and the code can be used to build up any torque from the Euler angle trajectories of UFT368.

To: EMyrone@aol.com
Sent: 31/01/2017 09:58:53 GMT Standard Time
Subj: External torque on gyro

To model an external torque around the Z axis, we could add this in the
Lagrange equation for phi. Since this equation is of first order, I
suggest to make an additional time derivative. Then we can write:

phi dot dot = … + Tq/I_phi

where I_phi is the momentum of inertia aroud the Z axis. Perhaps this
can be approximated by

I_phi approx. (X^2+Y^2)*m

What is the “canonical” method to introduce a torque? It must appear in
the kinetic energy then, before applying the Lagrange mechanism.


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