A Treatise on the Theory of Screws

106 THE THEORY OF SCREWS. [115-117 115. Property of Harmonic Screws. As the time of vibration is affected by the position of the screw to which the motion is limited, it becomes of interest to consider how a screw is to be chosen so that the time of vibration shall be a maximum or minimum. With slightly increased generality we may state the problem as follows: Given the potential for every position in the neighbourhood of a position of stable equilibrium, it is required to select from a given screw system the screw or screws on which, if the body be constrained to twist, the time of vibration will be a maximum or minimum, relatively to the time of vibration on the neighbouring screws of the same screw system. Take the n principal screws of inertia belonging to the screw system, as screws of reference, then we have to determine the n co-ordinates of a screw a bv the condition that the function shall be a maximum or a J va minimum. Introducing the value of ua (§ 97), and of va (§ 102), in terms of the co-ordinates, we have to determine the maximum and minimum of the function Jnax2 + ... + Anna,2 + 2 A^a., + 2/1 ^a,. + ... = U^ai1 + ... + Un«-n Multiplying this equation by the denominator of the left-hand side, differentiating with respect to each co-ordinate successively, and observing that the differential coefficients of x must be zero, we have the n equations:— (-Ajl — 0£j + -ZLi2®2 •••4" A~ 0? &c., &c. AjuOti + A^ct-j... "I- x) — 0. We hence see that there are n screws belonging to each screw of the wth order on which the time of vibration is a maximum or minimum, and by comparison with § 104 we deduce the interesting result that these n screws are also the harmonic screws. Taking the screw system of the sixth order, which of course includes every screw in space, we see that if the body be permitted to twist about one of the six harmonic screws the time of vibration will be a maximum or minimum, as compared with the time of vibration on any adjacent screw. If the six harmonic screws were taken as the screws of reference, then u2 and v2 would each consist of the sum of six square terms (§§ 89, 102). If the coefficients in these two expressions were proportional, so that u2 only differed from va2 by a numerical factor, we should then find that every screw in space was an harmonic screw, and that the times of vibrations about all these screws were equal.