A Treatise on the Theory of Screws

156] PLANE REPRESENTATION OF DYNAMICAL PROBLEMS. 143 155. Determination of the Wrench evoked by a Twist. The theorem just enunciated provides a simple means of discovering the wrench which would be evoked by a small twist which removes the body from a position of equilibrium. Let A (Fig. 33) be the given screw; join AO", and find 77; then the required screw A' must be reciprocal to H, and is, accordingly, found by drawing the chord HA' through 0. Fig. 33. The axis 00" is of course the hotnographic axis of § 151. We need not here repeat the demonstration of § 141, which will apply, mutatis mutandis, to the present problem. We see that the ratio of the intensity of the wrench to the amplitude of the twist is proportional to HO H0"‘ The other constructions of a like character can also be applied to this case. 156. Harmonic Screws. If after displacement the rigid body be released, and small oscillations result, the present geometrical method permits us to study the resulting movements. It has been shown (§ 130) that there are two special screws on the surface, each of which possesses the property of being a harmonic screw. If a body be displaced from rest by a small twist about a harmonic screw, and if it also receive any small initial twist velocity about the same screw, then the body will continue for ever to perform harmonic twist oscillations about the same screw. The two harmonic screws are X and Y, where the circle is intersected by the axis passing through the pole of the axis of inertia O', and the pole of the axis of potential 0" (Fig. 34).