Gyroscopes

 

 

 

 

 

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Overview

Gyroscopes are a simple toy to many, yet they are poorly understood. This paper derives the mathematics of gyroscopes.

Gyroscope concepts

A gyroscope has three axes. First, a spin axis, which defines the gyroscope strength or moment. Let us call the other two the primary axis and the secondary axis. These three axis are orthogonal to each other.

The spin axis rotates around the vertical line. The primary axis rotates the whole gyroscope in the plane of the page, and the secondary axis rotates the gyroscope up-and-over into the page.

The spin axis is the source of the gyroscopic effect. The primary axis is conceptually the input or driving axis, and the secondary the output. Then if the gyroscope is spun on its spin axis, and a torque is applied to the primary axis, the secondary axis will precess. The primary axis appears infinitely stiff to the applied torque and does not give under it. This is the generally recognised characteristic of gyroscopic behaviour.

It is important not to confuse the concepts of angular momentum and gyroscopic moment. When a mass ‘m’ moves in a straight line at velocity ‘v’ it exhibits linear momentum (m.v ). It is trivial to predict that if it is constrained to travel in a radius ‘r’ it will produce an angular momentum (m.v.r). However with the angular momentum an effect that could not have been predicted turns up - gyroscopic behaviour. The fact that in the larger world the two effects occur together and in simple proportion to each other does not mean that this is always the case - gyroscopic behaviour occurs without angular momentum in electron behaviour, even though the terms ‘spin’ and ‘spin angular momentum’ are still used for historical reasons, even though there is no direct evidence that the electron’s mass or charge spins on its own axis. It may simply be that rotating an object exposes the gyroscopic moments of the elementary particles that make it up, possibly through the asymmetric relativistic effects created by the centripetal acceleration; some major experimental work is required in this area.

Angular momentum has the form “kilogram-meters2 per second”. Gyroscopic moment has the form “Newton-meters per Hertz”, or torque required to produce a precession rate of one Hertz. For those familiar with dimensional analysis, both have the dimensions ‘L2M/T’, which means only that they are related by a simple scalar number. However (as far as the author has been able to determine) the actual value has never been researched; it may be unity, it may not. Whatever the case, from here on I will ignore angular momentum and consider only the gyroscopic moment, regardless of how it is generated.

Basic gyroscope equations

The strength of a gyroscopic effect is termed the gyroscopic moment. I use the symbol ‘G’, in units “Newton-meters/Hertz”. A higher moment requires more torque to precess at the same frequency, or for the same torque precesses at a lower rate

Where a gyroscope receives torque on the primary axis and precession on the secondary, no work is being done. The torque ‘TP’ on the primary axis has no precession associated with it, while the precession rate ‘vS’ on the secondary axis is...

vS = TP / G

...and has no torque associated with it. Since the rate of doing work on each axis is the torque times the precession on that axis, it follows that in this simple case no energy is involved.

Gyroscopes do not differentiate between primary and secondary axes - this is a purely artificial definition of my own.  A torque on the secondary axis creates precession on the primary axis. Simultaneous torque on both axis will result in simultaneous precession. In this case each axes will have both torque (creating precession on the other axis) and precession (created by torque on the other axis). Then the rate of doing work ‘PP’ on the primary axis is...

PP = TP.vP / G

...and on the secondary...

PS = TS.vS / G

Now by applying the conservation of energy...

PP = - PS

i.e. the work done on one axis must appear on the other.

So far I have dealt purely with behaviour, but to go further we need to look at why it behaves this way - what mechanism is at work?  Let us go through the basic operation where torque on one axis creates precession on the other (those familiar with electric motor theory will be familiar with the following ideas).

First apply a forcing torque to the primary axis; at this stage in the argument imagine that the primary axis presents no stiffness against the forcing torque. The secondary axis would precess at an infinite frequency, but for a limiting mechanism that comes into play; just as torque creates precession, so precession creates torque. So as the secondary axis precesses it creates a reverse torque TPF on the primary axis...

TPF =  -vS.G

The precession rate always runs at that point where TPF is exactly equal and opposite to TP. At this point...

TPF = -vS.G
= - ( TP / G ).G
= - TP

The reverse torque generated by the precession exactly opposes the applied torque so that the net torque is zero. If it was more the work would be done by the gyroscope. If it was less the primary axis would give way under the applied torque and work would be done with no outlet for it. Both conditions violate conservation of energy principles.

To sum up, the gyroscope precesses the right rate on the secondary axis to exactly oppose the applied torque on the primary axis. This leads directly to an aspect of gyroscopic behaviour that is seldom experienced in conventional gyroscopes, but is important in the behaviour of electrons:- If the secondary axis is locked against rotation and the primary axis is driven, no opposing torque will appear on the primary axis - it is free to rotate without hindrance. No work is transferred through the gyroscope - there is motion without torque on the primary axis. The secondary axis has no motion - it is locked - but instead experiences a torque TSF...

TSF = vP.G

This is the identical situation to basic gyroscope operation, but viewed from the other side. Instead of saying that torque on the primary axis leads to precession on the secondary we say that precession on the secondary axis leads to torque on the primary axis. It is exactly the same thing.

Using the equations

Now look at the special case of a gyroscope operating in an external linear field that serves to invert it through exactly 180 degrees. For example, when a tabletop gyroscope whose bottom pivot point is fixed starts pointing upwards, the linear vertical gravitational field of the Earth acts to invert it.

 

Figure 2

 

The gyroscope mass ‘m’ acts through the centre of gravity of the gyroscope. The vertical force is...

 m.g.sin(a)

 

...where ‘g’ is the gravitational constant of 9.81 meters per second. The torque around the gyroscope pivot point ‘P’ - and hence around the centre of mass - is...

TP = m.g.r.sin(a)

 

...where ‘r’ is the radius from the pivot point to the centre of mass.

This creates an instantaneous precession FS that in Figure 2 will cause the gyroscope to precess out of the page at the top of the picture. The instantaneous rate of precession is...

vS = m.g.r.sin(a) / G

 

Now the reason for emphasising that this is the instantaneous rate, is that TP is not a simple torque under the external linear field. As the gyroscope precesses out of the page, the angle the field makes with the gyroscope rotates with the precession. The end result of this is to cause the gyroscope’s precession to be truncated into a circle whose circumference is normal to the field. The circumference of this circle is shortened to...

r.sin(a)

 

...while the full precessional circumference would be simply ‘r’. This means that it takes the gyroscope less time to trace out the circular path, so the actual foreshortened precession rate is corrected to...

vS = (m.g.r.sin(a) /G) / (r.sin(a) / r)
= m.g.r / G

= TPmax / G

 

In other words the revised precession rate is constant.

Now if you experiment with a table gyroscope you will find that it will slowly droop as it precesses, so that the free end - opposite the pivot point ‘P’ in Figure 2 - will tilt more and more towards the table surface. It may seem at first that it is simply falling under gravity, but if you examine the motion you will find that the gravitational energy used up is not being translated into kinetic energy since the rate of droop remains slow and controlled. So we need  to find out where the energy is going to understand the behaviour.

In fact it is simply precession! What is happening is this:- The torque produced by the gravitational field on the primary axis causes the precession on the secondary axis. However, the secondary axis has friction associated with its motion, and this creates a frictional torque on the secondary axis that in turn causes precession on the primary axis. Energy conservation tells us that torque times precession on one axis must equal the negative of torque times precession on the other, so that the sum of the two energy rates is zero.

The work done by the gyroscope in going from the straight-up to the horizontal position may be found by integrating the torque over this rotation, when it will be found to be numerically equal to TP, but in joules of energy, rather than the TP Newton-meters of torque that creates the motion. So in the table gyroscope example, TP joules of energy will have been lost to friction on the secondary axis when the gyroscope has dropped from the straight-up to the 90-degree orientation on the primary axis.

Full inversion from straight-up to straight-down involves 2.TP joules. This leads to an important relationship. For any gyroscope the precession on the secondary axis is...

vS = TP / G

 

...and the work done in inverting the gyroscope from straight-up to straight-down in an external field is...

E = 2.TP

...so the ratio of energy to frequency is...

E / vS = 2.TP / (TP / G)
= 2.G

In other words, a given gyroscope moment G will always result in the same ratio of energy to frequency.

Another example is an electron in an external magnetic field. The electron has a gyroscopic moment and a magnetic field.

Now the tabletop gyroscope has frictional losses and operates in a linear external gravitational field that serves to invert it, while an electron has electromagnetic radiative losses and operates in a linear external magnetic field that serves to invert it. In the former case the external field acts on the mass, while in the latter case it acts on the magnetic field of the electron. However, the maths is identical, with the electron’s gyroscopic moment being h/2 (‘h’ being Plank’s constant)...

E / v = 2.G
= 2.(h/2)
= h

...so as you can see Plank’s constant owes nothing to the electromagnetic world - it is a purely gyroscopic property. The concept that the electron spin is 1/2 is related to its gyroscopic moment being h/2.

Although it is difficult to do for the table gyroscope, it is easy to reverse the process for an electron, to cause it to return from the straight-down (field alignment) position to the straight-up (field opposition) position. Just as a loss torque on the secondary axis causes the precession on the primary axis from straight-up to straight-down, so a gain or forcing torque on the secondary axis will cause precession on the primary axis back to the straight-up position. How this operates with the electron is beyond the scope of this paper, but it is possible to employ an electric motor integrated into the secondary axis of sophisticated gyroscopes to overcome and even reverse frictional losses.

Gyration

An interesting characteristic appears when you put a spring on the secondary axis. As you drive the primary axis the secondary axis will at first precess and the primary axis will be stiff, but as the spring winds up you will find the precession slow down and stop, and as a result the primary axis will give way more and more until it rotates freely. This behaviour is that of an inertial torque on the primary axis. Equally, if you put an inertial torque on the secondary axis the primary axis will behave like a spring torque. This behaviour, where inertial torque is converted into spring torque and vice-versa, is termed gyration.

Compound gyroscopic action

The foregoing discussion presupposes that the primary and secondary axes are massless, so that when they precess they have no gyroscopic moment. This is never actually true. As soon as the primary or secondary axis starts to precess its inherent mass creates a gyroscopic moment on that axis.

When you take a spinning gyroscope and apply torque to the primary axis it causes the secondary axis to precess. This causes the secondary axis to develop a small gyroscopic moment of its own. This axis then acts as an alternative spin axis coupling torque on the primary axis to the original spin axis (acting here as an alternative secondary axis). Because this gyroscopic moment is normally very low, torque on the alternative primary axis generates very high precession rates on this alternative secondary axis. The resultant picture is inevitably difficult to visualise...

The following table shows the relationship of the first alternative axes to the original ones in this particular example...

Axis

Original

First
Alternative

Second
 Alternative

1

Spin

Secondary
/Primary

Primary/
Secondary

2

Primary/
Secondary

Primary/
Secondary

Spin

3

Secondary
/Primary

Spin

Secondary
/Primary

If we accept that there is no real difference between the primary and secondary axes there are just three gyros, each corresponding to one of the three possible spin axes. Comparing the original and the first alternative...

  • Original: The high-rate spin on axis 1 generates a high gyroscopic moment so that torque on axis 2  generates a low-rate precession on axis 3.
  • Alternative: The low-rate spin on axis 3 generates a low gyroscopic moment so that torque on axis 2 generates a high-rate precession on axis 1.

Note that there is no fixed relationship between the spin rates for each viewpoint. In normal circumstances our viewpoint is locked to the spin axis with the highest gyroscopic moment, but this does not mean nothing else is going on.

Let us use the notation where the suffix has two characters, the first being either ‘O’ for original or ‘A’ for first alternative from the above table, and the second being the axis number. Then for the specific case where the gyroscope mass is a sphere so that the angular inertia is the same on all axes, the gyroscopic moment is always directly proportional to the spin rate, regardless of the axes combination. Then...

TO2 = vO3.GO1

TA2 = vO1.GO3

But for a sphere, where the inertial moment is the same for all axis, the gyroscopic moment is proportional to the spin rate, so...

v01.GO3 = vO3.GO1

...so that for stable precession/spin in this compound case...

TO2 = TA2

That is, the torque supplied to the ‘A’ set must be matched by an identical torque applied to the ‘O’ set. Since this is the same axis you must supply double the expected torque as the sum of the two.

If the rate of precession vO3 (= the spin rate sA3) is fixed by the frequency that the forcing torque rotates in the above image, then if TO2 is increased beyond TA2 the spin rate sO1 will increase, while if it is dropped below TA2 it will fall. Since the spin has angular momentum energy must be supplied in the former case to spin it up, while in the latter case energy will be released from the gyroscope as it spins down.

In this manner it is possible to “pump up” the orginal spin axis rate by utilising an alternative gyroscopic moment that treats it as a secondary (precessing) axis. There is an extremely effective application of this on the market - the “Power Ball” wrist exerciser. Given the difficulty of visualising compound motion in three dimensions it may be worth getting hold of one if you are sufficiently interested.

You can then store flywheel energy in a gyroscope’s original spin axis, charging and discharging that energy by means of an alternative spin axis. Suppose there are losses on axis 3 rotation so there is torque and precession (and hence work being done). The original gyro set maps the energy from O3 to O2, which therefore also has precession and torque. The precession on O2 couples to A2, and from there into A1, so that work lost on the spin axis A3 couples to work supplied on the spin axis O1. During these charge or discharge cycles the orginal spin axis can be seen turning end-over-end as the alternative spin axis A3 revolves. The clever thing about this approach - as opposed to simply tapping the original spin axis directly for its flywheel energy - is that the output frequency and torque can be held constant during the discharge without gearing even though the flywheel is slowing down.

Removing the Gyroscopic Moment

If the equations for angular momentum and gyroscopic moment are truly independent, can we separate angular moment from gyroscopic moment in a flywheel? The answer is “Yes” - simply ensure the mass travels in a circular path, but does not rotate. Imagine a flywheel which is simply a frame that carries two masses which are free to rotate independently on their own axles...

Then add some gearing (not shown) that makes the two masses counter-rotate slowly in the opposite sense to the frame’s rotation. This is at a much lower rotation rate than the frame’s rotation, just enough to offset the small  gyroscopic moment of the frame. With the right gearing the frame’s gyroscopic moment will be annulled by the masses’ counter-moment and you will have a flywheel with plenty of angular moment, but no gyroscopic moment. Many other arrangements are possible - two contra-rotating flywheels close to each other on the same axis will do the trick; the gyroscopic moments cancel out but the angular moments add.

The application of such a device is to engines in high-performance machines. For example the engines in small aircraft have such a high gyroscopic moment that they can affect the handling of the aircraft, and longitudinally- mounted engines in racing cars can affect manoeuverability.