Prof. Laithwaite's Gyroscope Experiments Part IX

Negative precessional starting velocity

To conclude these investigations I had a quick look at two more cases where the gyroscope starts and finishes its 180º of precession with negative velocity in the direction of precession.

1. Looping Nutation

The looping nutation case shown in this video occurs for a wheel speed of 244.7 rad/s. Its precession is started at -2.0 rad/s, (i.e. it starts off in the negative-y-direction) and it completes 180º of precession, including four loops of nutation, in 2.23 seconds. Even though its net precession is still over the positive-y-direction as before, it now exerts a net negative force in the y-direction. The integrated value of its Fy graph is -72.01 N-s. This essentially balances out the 71.9 N-s found for a 30kg "dead" mass completing another forward 180º of revolution at the same radius and rotational speed of -2.0 rad/s. The combined locus for this is shown below:—

Locus for a gyroscope wheel, "live" and undergoing looping nutation over 180º,
then acting as a "dead" mass for the remaining 180º of a full cycle.
Both half-cycles are in the forward (positive-y) direction, but there is no net force.

It's obvious enough that force from the half-loops at the start and finish of the cycle cancel out the forward force generated over the remainder of the cycle, as would occur similarly for the simpler locus shown below:—

2. Gyro wheel locus on a cylinder rather than a sphere

Examination of the force graph for the negative-force looping nutation case raised the question of what would happen if the radial position of the gyro wheel from the vertical z-axis could be kept constant throughout the experiment. Would it still deliver a balancing-out net negative force?  Would it still be able to precess at all?

In the model shown above a rod (purple, of 1 gram mass) is added, with a 3-d.o.f. rotational joint to the wheel's center, and a z-direction translational joint to the central axle. This forces the wheel to keep a constant radius from the z-axis, while still transferring its weight via its shaft to the central pivot as before. The original wheel joint is modified to allow translation as well as rotation on the shaft. (Note that the model is valid as is, although I haven't bothered about avoiding superposition of object images. In a real physical model, gimbals etc would be required to avoid such clashes).

Provided this model is given a somewhat higher wheel speed than before, it can still undergo looping nutation. For a wheel speed now 410 rad/s and a starting precession of -2.0 rad/s as before, it completes 180º of precession in 2.565 seconds, with six rather than four loops. The integrated value of its Fy graph is still negative. It has the same "balancing-out" value of -72.0 N-s as before, again matching (i.e. equal and opposite to) the Fy graph for a "dead" mass completing the remaining 180º at 2 rad/s.

Once again these two results agree with the "obediance to Newton's laws re inertial propulsion" argument given previously.

Prof. Laithwaite's Gyroscope Experiments Part VIII

More experiments with zero precessional starting velocity

I repeated the cuspidal nutation experiment performed previously, for other whole numbers of nutations over 180º, with results tabulated below:—

Nutations       Wheel      Time        Minimum       Maximum        Integral of
in 180º of        rot.          reqd. to    precessional   precessional    centrifugal
precessional    speed    precess      velocity            velocity             force resolved
motion            [rad/s]    180º [s]    [rad/s]            [rad/s]               in y-dirn. [N-s]

      3                  227.4       1.898            0                  3.522                 -0.0548
      4                  271.9       2.215             0                  2.95                   -0.00067
      5                  308.7      2.486             0                  2.588                -0.0239
      6                  340.7      2.724             0                  2.36                     0.0036
  

In all these cases, there is essentially zero centrifugal force in the y-direction. To help explain why we should expect this, let's consider a simple "inertial propulsion" problem:—

An inertial propulsion problem. The mass has to
reverse direction at the locations shown dashed.

In the above image, a mass (red) is attached to one end of an arm (black line). The other end of the arm has a central pivot to a vehicle. The mass undergoes 180º of revolution, and so the centrifugal force it exerts at the pivot drives the vehicle forward. The problem comes in starting and stopping the mass's motion at the start and finish of its half-revolution. Various ways of doing that could be proposed, e.g. using springs attached to the vehicle, or by delivering/recollecting torque at the pivot-end of the arm etc. However, there is no way of using this idea to achieve any net unreacted force on the vehicle, without disobeying Newton's laws of motion, especially the third one.

Since the precessional speed of the cuspidal-nutating gyroscope discussed above is indeed brought to zero at times; in particular when it coincides with the ±x-axis, then it too cannot be expected to deliver any net force in the y-direction over its half-revolution of precession. If it did, it would have to disobey Newton's laws just as much as the simple mass in the inertial propulsion example would.

Maximum precessional starting velocity

Next, I looked at a case where the cuspidal-nutating gyroscope starts and finishes its 180º of precession with maximum, rather than minimum velocity:—

The four-cusp case modelled above now occurs for a wheel speed of 290 rad/s. Its precession starts at 2.75 rad/s, and it completes 180º of precession in 2.22 seconds. The integrated value of its Fy graph is 98.797 N-s. This essentially balances the -99.0015 N-s found for a 30kg ("dead") mass completing the other (backward) 180º of revolution at the same radius and rotational speed of 2.75 rad/s. (Small discrepancies in all these results are only because of my trial-and-error initial "tuning" of the models). Once again, this modelling shows only orthodox behaviour.

Prof. Laithwaite's Gyroscope Experiments Part VII

Is there a reduction in time-varying centrifugal force?

So far, my computer modelling of smoothly precessing gyroscopes has given orthodox results. Nevertheless, I believe that Professor Laithwaite was quite correct in claiming a reduction in centrifugal force exerted by a precessing gyroscope, and I'll now explain why.

Nutation

Those physicists and others who were quick to denigrate Prof. Laithwaite and his gyroscope experiments apparently failed to notice that, according to strictly orthodox physics, he must have been correct in claiming a reduction in centrifugal force for a precessing gyroscope. That is because in the experiments where he made this claim, his gyroscopes would have been nutating as well as precessing. This can make a large change to the centrifugal force.

It's also possible that this effect was not fully appreciated by Prof. Laithwaite himself. In my opinion, it explains the significant reductions he reported in the centrifugal force exerted by a precessing gyroscope. As we'll see, this force can not only be reduced: hard though it may be to imagine, under the right conditions it can drop to zero, and even beyond, to a net negative value!

A high speed air-driven gyroscope on a stand which can tip over.
Nutation is not clearly visible in this experiment, but it must be occurring,
as long as there was hardly any added precessional motion at the start.
A gyro spinning at high speed will undergo many nutations per revolution of precession,
which may be so small (low-amplitude) that they are difficult or impossible to see.

(In the following discussion, I assume that over no more than a single revolution of precession, friction tending to damp out nutations is negligibly small).

Nutation can cause zero net centrifugal force in a given direction

The experiment shown above starts at about 28:00 in the video at http://richannel.org/christmas-lectures/1974/1974-eric-laithwaite#/christmas-lectures-1974-eric-laithwaite--the-jabberwock

We can quibble about how clearly it is demonstrating a lack of centrifugal force, versus the ability of the gyro to shift its weight back to its central pivot. Nevertheless, if a gyroscope is undergoing cuspidal nutation, as Prof. Laithwaite's must have been (at least at the start of his experiments when he simply released them, without any added precessional motion) then if there is an integral number of cusps over 0º to 180º of precession, the net force resolved along a 90º line is not merely reduced; it must be reduced to zero. In more detail:—

A gyroscope undergoing four cycles of cuspidal nutation in 180º of precession.
The graphs show joint forces in x-direction (red) and y-direction (green).

The image above shows the gyroscope already modelled before, with a 30kg wheel at 0.6m radius from the central axle. Its wheel's rotational speed is now 271.9 rad/sec, and it starts aligned with the x-axis, at zero precessional speed. This causes it to undergo four cuspidal nutations in 180º of precession, as shown by the locus of the wheel center (purple). Only the centrifugal force resolved in the 90º direction on this arc of precession, i.e. in the y-direction, is of interest now (green trace on graph). When the mean value of this graph is found, [using "Mean" from UM's Processor of Variables] it is only -0.00589 N. When the graph is integrated, [using "Integral"] it is only -0.00067 N-s. So there is essentially zero net force in the y-direction, as should be expected (I'll explain that further next time).

Video

See the video of this experiment below.

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.