My Gyroscope Experiments Part I

Before leaving the topic of gyroscopes, I'll mention two more ideas of my own. One is very speculative. The other, the subject of this post, boils down to a warning about errors that can occur from step change transitions in finite-element analysis programs.

Negative vertical force for a looping-nutation gyroscope

The above graph, for a gyroscope undergoing looping nutation, includes the vertical reaction force Fz on the joint between the gyro shaft and the central axle (blue). Examination of this force shows that it goes negative, i.e. there is a net upwards force on the joint, over the upper portion of each loop.

An intermittently-rising looping-nutation gyroscope

I decided to check what would happen if the gyroscope as a whole was allowed to rise during these times of upwards force, while remaining held stationary for downwards force.

Graph of gyroscope tower vertical position vs time

I used a model similar to the one analysed in my post of 7 March 2015, with a generalized translational t-function joint between the tower and Base0, again using theoretically calculated curves (above) for the tower locus.

Abrupt, "step-change" transitions in tower position

Firstly, I tried the experiment with the simple "step-change" graph (black) for the tower vertical position vs time. This graph consists only of straight lines, either sloped or horizontal. Each time the vertical joint force Fz goes negative, the tower rises by 100 mm (for a full loop); otherwise it is stationary.

The results of this experiment were a surprise: although the tower rises by a total of 800 mm over one revolution of looping precession, the gyro wheel shows no overall drop in its average vertical position with respect to the tower! There is also no reduction at all in the value of the initially-assigned looping precession (of -2 rad/s). In more detail:—

r12.z (change in height between Wheel and Tower) at the top of each loop [m]:—

0, -4.484E-7, -4.027E-7, +1.710E-5, +2.305E-5, +2.900E-5, +3.494E-5, +4.088E-5, +7.764E-7 (last value recorded; not the full peak).

om.z (Hub angular velocity) at the top of each loop [rad/s]:—

-2, -2.000072, -2.000169, -2.000265, -2.000251, -2.000459, -2.00043, -2.000442, -2.0000019 (last value recorded; not the full peak).

So in this experiment there is no loss of kinetic or potential energy of the gyro wheel with respect to the rising tower during the cycle of looping nutation. Yet the 30kg gyro wheel alone would gain net potential energy of mgh = 30 × 9.80665 × 0.8 = 235.3596 joules, in 4.46 seconds. It could fall back to earth in another 0.4039 seconds, giving a power output of 235.3596 ÷ (4.46 + 0.4039) = 48.389 watts!

See the video of this experiment below.

Smooth transitions in tower position

The above result is obviously too good to be true. Sometimes results like that can be worse than clear-cut null results, because it's not always easy to find out the reason for them. But in this case I had at least a fairly obvious suspect — the "step-change" transitions in the tower position.

From my background in finite-differences dynamic analysis, I would not have expected step-changes like these to give an erroneous result. But to be fair, warnings about this issue are generally given to users of finite-element programs.

I modified the graph for the tower vertical position vs time so that each transition was now smoothly, sinusoidally curved (green) and repeated the experiment, with the tower still rising by 100 mm each time the vertical joint force Fz goes negative. This time, the results were very different, but much more credible.

I haven't analysed this experiment in detail, but the video of it (below) shows significant initial loss of energy associated with the looping behaviour as the tower rises. A gain in this energy does then occur, but that will certainly only be coming from the forced, on-going rise of the tower. I would not expect any net energy gain in this more realistic experiment.

Warnings — specific and general

A specific warning can be given here — when working with computer dynamic analysis programs, particularly finite elements ones, it is very important to avoid functions incorporating "step-change" transitions. (Mathematicians would call these functions "piecewise non-linear"). Although the curved transitions are not as simple to create and analyse as the step-changes, they are essential to achieve a true result. 

This warning can also be generalised:— whether computer analysis is involved, or just old-fashioned pencil-and-paper physical calculations, it is essential not to over-simplify the analysis; otherwise it may be impossible to achieve a true result.