My Gyroscope Experiments Part II

Torque output for rate input

This is a very speculative post, concerning the ability of a gyroscope to produce a torque output when given a rate input.

Fig 1.  Steady state conditions for a single axis gyroscope.
Ref: Analysis and Design of the Gyroscope for Inertial Guidance, Ira Cochin.

I've posted the above image before, when discussing the non-gyroscopic case (d). I now draw attention to the first case (a), which notes that a gyroscope produces a torque output when given a rate input (i.e. a change of angle over time). Obviously if the torque output is allowed to turn through some finite angle, then an output of energy must be delivered. The question is: could it be possible to separate this effect out from the "converse" effect noted in the second case, i.e. that a torque input gives a rate output?

Fig 2. A gyroscope with a connection between its gimbals

This question is shown in the 3D drawing above, which has a connection, shown in very schematic form, between the inner and outer gimbals of the gyroscope. Arms are joined to these gimbals, which are connected together via components in a "black box" (or a "black sphere" in this case). So, is it possible that this connection could be made using only passive components, such as links, springs, dampers, energy storing/delivering flywheels etc, in such a way that a net energy output could be delivered with only a rate input — i.e. without much torque input?

I haven't got much further with this idea beyond confirming that there is no net energy production for simple mechanical links between the gimbals.

Review

I'll now review, with some brief final comments, some of the discrepancies already discussed between actual versus predicted gyroscopic performance.

1. Torque applied to the output axis of a gyroscope
     (post of 17 January 2015).

Textbooks (e.g. case (d) in Fig 1 above) and finite-element computer analysis agree that the result should be non-gyroscopic, i.e. that it should make no difference whether the gyro wheel is spinning or not. This does not agree with experiment. (I suspect, but cannot confirm that finite-differences analysis, in which joints and bearings generally have some "flexibility", would give a more realistic result).

2. Prof. Laithwaite's "double-joint" experiment
     (posts of 14 and 21 February 2015).

The result obtained by computer analysis is different from either the result predicted by Prof. Laithwaite, or the result he obtained in his experiment.

3. Force-precessing, then lifting a heavy gyroscope
     (post of 31 January 2015).

It would be easy enough to show, from a strict energy-conservation point of view, that a strong enough experimenter could initially expend sufficient energy to force-precess a gyroscope significantly faster than its natural precessional speed, in which case the subsequent easy lift would be explained. But I'm far from convinced that either Prof. Laithwaite, or the experimenter in the Australian replication video was really doing that. A well-instrumented physical experiment would be needed to make progress in this area.

I have the building and testing of a heavy gyroscope on my "to-do" list, but unfortunately that will be well into the future.

4. Reduced centrifugal force for a precessing gyroscope
     (posts of 7 and 14 March 2015).

I have shown that Prof. Laithwaite's claims of reduced centrifugal force were quite correct, provided that his gyroscopes were nutating as well as precessing, as they surely would have been. However I don't agree that this effect could be developed into a net imbalanced-force propulsion system.

This concludes my series of posts on gyroscopes.

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.

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.