Prof. Laithwaite's Gyroscope Experiments Part VI

I'll now check out a couple of variations of the smoothly-precessing gyroscope modelled last time:—

Variation 1

This variation further investigates a gyroscope modelled in the "falling-away" laboratory frame, as first discussed in my post of 7 February 2015.

I used exactly the same model described in Part V, except that its tower was no longer fixed to the static reference frame [Base0 in UM], but was now following the vertical (z-axis) locus it would have in a "falling-away" laboratory frame at the Earth's equator, over the 3.15 seconds required to complete one precessional revolution.  [A generalized translational t-function joint was used between the tower and Base0, using a theoretically calculated curve (below) for the locus to control its movement].

Vertical locus (z-axis position vs time) for a laboratory frame at the equator

A gyroscope shown after 360º of precession, in a laboratory frame at the equator.
Note that the whole gyroscope, in the laboratory frame, has dropped as seen in the absolute frame.

Results

In 3.15 seconds the tower in the laboratory frame, accelerating downwards in an "absolute" frame, has dropped by -0.1683m. In the model, with the same 1.995 rad/sec starting precession as before, the gyro wheel undergoes very slight initial nutation, and after 3.15 seconds it has dropped by -0.1694m. Throughout this, the gyro's shaft remains very nearly perpendicular to the tower. The model shows no sign of any potential energy gain in the falling-away laboratory frame.

So, computer modelling gives an orthodox result. 


Video

I have added a video of this Variation 1 below.

Variation 2

This variation checks whether there is a reduction in the centrifugal force exerted by a precessing gyroscope of a particular modified design.

In the paper "A System for the Transfer of Mass Derived from the Principle of Conservation of Momentum" by Professor E. R. Laithwaite and W. R. C. Dawson, the following statement is made:—

"To get an impressive reduction in centrifugal force you need a large heavy wheel with a very thin rim, virtually no spokes or middle, running very fast and being precessed slowly round a small radius.

Experimentally, with a high quality wheel, precessing on a radius about twice that of the wheel, with a spin in excess of 100 times the rate of precession, we have recorded centrifugal forces of less than one tenth of that anticipated from calculation."

To check this idea, I re-modelled the 30 kg gyro wheel as an "ideal high quality" wheel, with all its mass concentrated in a thin hoop of radius 0.3m. So its inertia became:—

I = MR² = 30 × 0.3² = 2.7 kg-m²  about its axis of rotation, and

I = ½MR² = 1.35 kg-m²  about the orthogonal axes. 

[These values were manually input into the wheel's inertia tensor data, replacing the original values of 0.4426 and 0.2394 kg-m² auto-calculated for the original wheel shown (0.3m radius × 0.1m wide,  material: steel). I didn't bother to modify any images of the gyroscope.]

Results

This time, with the wheel rotating at 200 rad/s as before, an initial precessional speed of 0.327 rad/s was required to avoid nutation. Precession through 180º took 9.607 seconds. Once again the joint forces Fx and Fy were sinusoidal, of 1.9249N maximum value; i.e. there was no reduction from the the mω²r = 30 × 0.327² × 0.6 = 1.9247N theoretically expected.

Again, computer modelling gives an orthodox result, i.e. no reduction occurs in the centrifugal force exerted by a smoothly precessing gyroscope at its central pivot, no matter how "high quality" it is. But there is significantly more to this issue, as will be seen.