The Polar Gyroscope, Part I

Extracting energy with a polar gyroscope

Other than exploiting the Earth's tides, the simplest, most often suggested method for extracting energy from the Earth's rotation is to exploit the torque that occurs when it turns on its axis against some other device which is not turning. The device usually suggested is a large gyroscope. For simplicity, the gyroscope would be installed at either the north or the south pole, as in the UM model shown below (although that would not be essential, provided it was oriented so that its outer gimbal axis remained parallel with Earth's polar axis):—

Universal Mechanism model of a very large gyroscope with input/output axis
aligned with the Earth's polar axis = the z-ordinate (blue)

The basic idea of using a gyroscope to extract energy from the rotating Earth is common enough on various websites and has been patented, e.g. in German patents DE 102006057073 and 102006053069, and US patent 5313850. However, I have never seen anyone do any computer modelling of it, to give estimates of expected energy output for realistic cases. One reason for that, I suspect, is because there is a serious problem with this idea as it is usually presented. I'll explain that shortly.

Note: Planetary energy extraction can be taken much further theoretically than I do here. For example, see "A Mechanical Principle for Acquisition of Useful Power on a Celestial Body Through Utilisation of its Planetary Precession" by Kyril Vulkov, Journal of the British Interplanetary Society, 55, pp394-397, 2002; abstract at http://www.jbis.org.uk/paper.php?p=2002.55.394

Gyroscope details

The gyroscope that I used in my model is as shown above. It is of "traditional" design with a metallic rotor, rather than a more modern carbon-fibre, polyester-fibre or fused-silica type, which would all have higher energy densities. Its rotor has the following details:—

Material:                 Maraging steel
Diameter:               4.5 meters
Width:                     1.5 meters
Mass:                       100 tonnes = 100000 kg
Inertia tensor:        Ixx (red): 331230 kg-m²
                                   Iyy (green) and Izz (blue): 179270 kg-m²
Rotational speed:   125 rad/sec, ≈ 1200 rpm

In this model, the hub (dark green) is forced to rotate constantly at the Earth's rotational speed of 0.00007292115 rad/sec. The axle (yellow) is rigidly joined to the outer gimbal. A torsion spring is installed between the hub and the axle. The spring absorbs energy as it is twisted, for delivery later, as the Earth rotates the hub with respect to the (initially) non-rotating axle and gimbals.

In my first version of this model, following the usual way in which this idea is presented, the axle and outer gimbal were also rigidly joined to the inner gimbal, to form a single component. (In UM, rigid joints like these are achieved with a 6 d.o.f. joint with all degrees of freedom switched off).

A non-gyroscopic result

However, as I had already suspected, there was no gyroscopic effect at all in this model. Whatever rotational speed was given to the rotor made zero difference to the result. The rotor might just as well have been a dead (non-rotating) mass rigidly joined to the gimbals. 

As Prof. Gray says in his book A Treatise on Gyrostatics and Rotational Motion (p10), "...gyrostatic resistance to change of direction of the axis of rotation is bound up with freedom of the gyroscope to turn as a whole with precessional motion." With the gimbals and the axle all rigidly joined there can be no such freedom for precessional turning. Note that all joints in the model are the mathematically "perfect" joints of finite-element computer analysis.

This situation is shown as the fourth case (d) in the above figure, from Analysis and Design of the Gyroscope for Inertial Guidance, by Ira Cochin (p36). As seen in this case, the gyroscope's angular momentum H plays no part in the result, which is non-gyroscopic.

A gyroscope with a single gimbal able to turn only about a vertical axis

But doesn't it work in practice?

When I set up my small gyroscope as shown, and try to turn the rotor plus gimbal with respect to the base, there is significantly more resistance to turning when the gyro rotor is spinning, than when it is at rest. This resistance to turning was the property exploited by the French experimentalist Léon Foucault, as one of his methods of determining the rotation of the Earth. (Reference Comptes Rendus no 35, 1852).

I think what is happening here is that although the gyroscope is theoretically unable to turn as a whole with precessional motion, in practice even a very small amount of play or deformation in the bearing between the gimbal assembly and the base does allow a very small amount of such precessional turning to occur. This is sufficient to cause the added resistance to turning about the vertical axis. However, I think there would be problems, or at least serious inefficiencies in trying to extract energy without permitting full gyroscopic action to occur. I'll look at that next time.