Preliminary Results of the Computation of the Motion of a Gyroscope

Many thanks to Horst Eckardt! This is an important result, the first numerical solution without approximation of the three Euler Lagrange equations for the motion of a symmetric top with one point fixed (Marion and Thornton Section 10.10). The generalized coordinates are the three Euler angles. The first complete solution since Lagrange himself (1811 and 1815) in “Mecanique Analytique”. I agree with the interpretation, there is combined nutation and precession. This method is able to analyze the motions separately. This is a brilliant piece of work by co author Dr. Horst Eckardt, using Maxima on a desktop computer to solve three complicated simultaneous differential equations. It goes much further than the approxiamtions used by Marion and Thornton. Now we have the baseline code, and various forces in the plus Z direction of the lab frame can be added to the code, to investigate the effect on the nutation and precession of the gyro. This is the easiest way of investigating lift. The Laithwaite orientation is theta = pi / 2 or 90 degrees. So we are now in a position to try to answer why the gyro appears to be weightless. A more complete solution would be to add a translational lagrangian for the centre of mass movement. This code solves the equations of motion of the rotational lagrangian.

To: EMyrone@aol.com
Sent: 23/01/2017 19:44:36 GMT Standard Time
Subj: Re: 368(6): Final Version of Note 368(5) – preliminary results

These are the preliminary results of the gyro simulation. I think it looks good. theta varies periodically, describing a nutation (red curve of first graph). phi and psi increase continuously, showing complete rotations. This seems to be the precession. Is phi or psi the correct angle of precession?

If the results are correct, this is indeed a great improvement over the state of the art in gyro mechanics.

Horst

Am 23.01.2017 um 13:00 schrieb EMyrone:

This is the final version for checking with computer algebra. This is the motion of a symmetric top with one point fixed, first worked out by Lagrange, who was a student of Euler. In general the motion must be found by solving Eqs. (5), (6) and (7) for theta(t), phi(t) and psi(t), the time dependent Euler angles. This is a difficult problem even for contemporary supercomputers, so Lagrange worked it out by approximation in “Mecanique Analytique” (1811 and 1815). Eq. (5) must be solved for time dependent theta(t). There is no way in which the gyroscope can lift itself off the ground in this type of problem, because its point is defined as the origin of both (X, Y, Z) and (1, 2. 3). In order to consider the possibility of lifting off the ground, the more general Eqs. (11) and (12) must be used, in which the centre of mass of the gyroscope is allowed to move. The first steps towards this end were made in UFT367 and will be developed further in the next note. It may be possible to get some insight into Eqs. (5) to (7) with Maxima, as suggested in the note.

368(6).pdf

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