A Treatise on the Theory of Screws

114] FREEDOM OJF THE FIRST ORDER. 105 Ihen X must be that screw on the cylindroid which is reciprocal to a, for a wrench on X, and the given wrench on t), must compound into a wrench on p, whence the three screws must be co-cylindroidal*; also X must be reciprocal to a, so that its position on the cylindroid is known (§ 26). Finally, as the impulsive intensity i/" is given, and as the three screws p, X, /x are all known, the impulsive intensity X'" becomes determined (§ 14). 114. Small Oscillations. We shall now suppose that a rigid body which has freedom of the first order occupies a position of stable equilibrium under the influence of a system of forces. If the body be displaced by a small twist about the screw « which prescribes the freedom, and if it further receive a small initial twist velocity about the same screw, the body will continue to perform small twist oscillations about the screw a. We propose to determine the time of an oscillation. is The kinetic energy of the body, when animated by a twist velocity MS')’«- da' dt The potential energy due to the position attained by giving the body a twist of amplitude a away from its position of equili- brium, is Fv^a" (§ 102). But the sum of the potential and kinetic energies must be constant, whence fdax1 „ . \dt) + MuS const. Differentiating we have d2a' Fv^ , _ dF Mua1 a Integrating this equation we have “ A sin 1 B C,,S where yl and B are arbitrary constants. The time of one oscillation is therefore ua /M 217 va\/F‘ Regarding the rigid body and the forces as given, and comparing inter se the periods about different screws a, on which the body might have been constrained to twist, we see from the result just arrived at that the time for each screw a is proportional to . We shall often for convenience speak of three screws on the same cylindroid as co-cylindroidal.