The Polar Gyroscope, Part II

Efficient extraction of energy

Last time, I suggested that the only way to extract energy efficiently using a polar gyroscope would be to permit full gyroscopic action to occur. So, as well as the energy gained as the Earth rotates beneath the initially non-rotating axle and outer gimbal, more energy will then also be obtained from rotation between the inner and outer gimbals, as the latter does start to rotate, and full gyroscopic action occurs. As before, another torsion spring can be installed at the rotational joint between the gimbals, to absorb this energy for delivery later.

Full gyroscopic action, with the inner gimbal permitted to turn

Resetting

A problem occurring now is that the inner gimbal can only turn through a limited angle, e.g. as shown above. After that, it must be reset. If not, the gyroscope will soon go into "gimbal lock", rendering it useless for any further energy extraction.

In theory, resetting could be done by decelerating the gyro rotor to rest, turning the inner gimbal and rotor back to its starting angle, with no net energy penalty, then accelerating the rotor back up to its original speed. But in practice this would be a very crude, "brute-force" method. Even a very small percentage loss in transferring the huge amount of rotational energy away from, and then back to the spinning rotor would be likely to cancel any other energy gain.

Energy and power gain

Ignoring resetting difficulties for now, let's see what energy could be gained from the 100 tonne gyroscope shown above, with details as given last time.

In my second version of the model, the fixed joint between the gimbals was replaced with a rotational joint. Both this joint and the other rotational joint between the axle and the hub were provided with torsion springs, which could be given user-assigned values as desired.

Graphs of joint angles between the axle and the hub, and between the gimbals, vs time, as the Earth turns
beneath the large polar gyroscope. Here, both joints have torsion springs of  κ = 1e6 N-m/rad

Results

I found that for a wide range of torsion spring constants, from κ = 1e4 to 1e8 N-m/rad, and in a fairly wide range of experiments, there was essentially the same power output, i.e. 0.14 watts. Tabulated values are given below for the experiment graphed above, where the axle/hub joint undergoes a complete half-sinewave of oscillation, back to zero angle, in which time the gimbal joint reaches a maximum angle, with maximum stored energy in its spring. Each time the experiment is run, both joints have torsion springs of the same κ value. (Note that below about κ = 1e4 N-m/rad, the gyroscope risks going into gimbal lock in this experiment).

κ [N-m/rad]                            1e4           1e5              1e6               1e7                 1e8
Gimbal joint angle [rad]      0.5721  0.06028  0.006039  0.0006036  0.0000601
Stored spring energy [J]      1636.5   181.68      18.234         1.8217           0.1806
Elapsed time [s]                     11753     1300.6     130.15          13.07             1.30
Power output [W]                 0.1392    0.1397      0.1401         0.1394          0.1389

The above results are of only moderate accuracy, using a mouse-positioned cursor on the graphs, rather than the more accurate but tedious method of saving simulation data for later examination.

Conclusion

From these experiments I conclude that although a large polar gyroscope can extract some energy from the rotating Earth, it is far too little to be of any practical use. So it is not worth spending more time on the resetting difficulties.