Equable elastic rotat another of two bodies rotating round axes which may be in clined to one another at any angle, and need not be in one ing joint. plane. If they are in one plane, if there is no resistance to the rotatory motion, and if the action of gravity on the wire is insensible, it will take some of the varieties of form (§ 612) of the plane elastic curve of James Bernoulli. But however much it is altered from this; whether by the axes not being in one plane; or by the torsion accompanying the transmission of a couple from one shaft to the other, and necessarily, when the axes are in one plane, twisting the wire out of it; or by gravity; the elastic central curve will remain at rest, the wire in every normal section rotating round it with uniform angular velocity, equal to that of each of the two bodies which it connects. Under Properties of Matter, we shall see, as indeed may be judged at once from the performances of the vibrating spring of a chronometer for twenty years, that imperfection in the elasticity of a metal wire does not exist to any such degree as to prevent the practical application of this principle, even in mechanism required to be durable. It is right to remark, however, that if the rotation be too rapid, the equilibrium of the wire rotating round its unchanged elastic central curve may become unstable, as is immediately discovered by experiments (leading to very curious phenomena), when, as is often done in illustrating the kinetics of ordinary rotation, a rigid body is hung by a steel wire, the upper end of which is kept turning rapidly. 622. If the wire is not of rigorously equal flexibility in all Practical directions, there will be a periodic inequality in the communi- nequalities. cated angular motion, having for period a half turn of either body or if the wire, when unstressed, is not exactly straight, there will be a periodic inequality, having the whole turn for its period. In other words, if and p' be angles simultaneously turned through by the two bodies, with a constant working couple transmitted from one to the other through the wire, - will not be zero, as in the proper elastic universal flexure joint, but will be a function of sin 24 and cos 26 if the first defect alone exists; or it will be a function of sin and cos if there is the second defect whether alone or along with the first. It is probable that, if the bend in the wire when ing joint. Elastic rotat- unstressed is not greater than can be easily provided agains in actual construction, the inequality of action caused by z may be sufficiently remedied without much difficulty: practice, by setting it at one or at each end, somewhat incline: to the axis of the rotating body to which it is attached. But these considerations lead us to a subject of much greater interest in itself than any it can have from the possibility of usefulness in practical applications. The simple cases we shall choose illustrate three kinds of action which may exist, each either alone or with one or both the others, in the equilibrium of a wire not equally flexible in all directions, and straight when unstressed. Rotation round its 623. A uniform wire, straight when unstressed, is bent t elastic cen- its two ends meet, which are then attached to one another, with tral circle, wire made of a straight the elastic central curve through each touching one straigh: into a hoop. line so that whatever be the form of the normal section, ari the quality, crystalline or non-crystalline, of the substance, the whole wire must become, when in equilibrium, an exact circle (gravity being not allowed to produce any disturbance). It is required to find what must be done to turn the whole wire uniformly through any angle round its elastic central circle. If the wire is of exactly equal flexibility in all directions,' it will, as we have seen (§ 621), offer no resistance at all to this action, except of course by its own inertia; and if it is once set to rotate thus uniformly with any angular velocity, great or small, it would continue so for ever were the elasticity perfect and were there no resistance from the air or other matter touching the axis. To avoid restricting the problem by any limitation, we must suppose the wire to be such that, if twisted and bent in any way, the potential energy of the elastic action developed, per unit of length, is a quadratic function of the twist, and two components of the curvature (§§ 590, 595), with six arbitrarily given coefficients. But as the wire has no twist,2 three terms of this function disappear in the case before us, and there remain only 1 In this case, clearly it might have been twisted before its ends were put together, without altering the circular form taken when left with its ends joined. Which we have supposed, in order that it may take a circular form: althengh in the important case of equal flexibility in all directions this condition would obviously be fulfilled, even with twist. round its tral circle, wire made three terms, those involving the squares and the product of Rotation the components of curvature in planes perpendicular to two elastic cenrectangular lines of reference in the normal section through of a straight any point. The position of these lines of reference may be into a hoop. conveniently chosen so as to make the product of the components of curvature disappear: and the planes perpendicular to them will then be the planes of maximum and minimum flexural rigidity when the wire is kept free from twist. There is no difficulty in applying the general equations of § 614 to express these circumstances and answer the proposed question. Leaving this as an analytical exercise to the student, we take a shorter way to the conclusion by a direct application of the principle of energy. Let the potential energy per unit of length be (Bk2+Cλ3), when and are the component curvatures in the planes of maximum and minimum flexural rigidity: so that, as in § 617, B and C are the measures of the flexural rigidities in these planes. Now if the wire be held in any way at rest with these planes through each point of it inclined at the angles and - to the plane of its elastic central circle, the radius of this 2 1 1 circle being r, we should have cos, λ=sin. Hence, since 2r is the whole length, B r r (1). Let us now suppose every infinitely small part of the wire to be 1 When, as in ordinary cases, the wire is either of isotropic material (see § 677 below), or has a normal axis (§ 596) in the direction of its elastic central line, flexure will produce no tendency to twist: in other words, the products of twist into the components of curvature will disappear from the quadratic expressing the potential energy or the elastic central line is an axis of pure torsion. But, as shown in the text, the case under consideration gains no simplicity from this restriction. Rotation round its elastic cen tral circle, of a straight wire made into a hoop. Which shows that the couple required in the normal pla through every point of the ring, to hold it with the planes f greatest flexural rigidity touching a cone inclined at any ang o, to the plane of the circle, is proportional to sin 26; is in t direction to prevent from increasing; and when Rotation round its elastic central circle, wire equally directions, when un strained. B-C amounts to 2 = From per unit length of the circumference. this we see that there are two positions of stable equilibrium --being those in which the plane of least flexural rigidity lies in the plane of the ring; and two positions of unstable equili- ' brium,-being those in which the plane of greatest flexura! rigidity is in the plane of the ring. 624. A wire of uniform flexibility in all directions, so shaped as to be a circular arc of radius a when free from stress, is bert of a hoop of till its ends meet, and these are joined as in § 623, so that the flexible in all whole becomes a circular ring of radius r. It is required to but circular find the couple which will hold this ring turned round the central curve through any angle in every normal section, from the position of stable equilibrium (which is of course that in which the naturally concave side of the wire is on the concave side of the ring, the natural curvature being either increased or diminished, but not reversed, when the wire is bent into the ring). Applying the principle of energy exactly as in the preceding section, we find that in this case the couple is proportional to sin o, and that when 7, its amount per unit of length of the circumference is B if B denote the ar If every part of the ring is turned half round, so as to bring the naturally concave side of the wire to the convex side of the ring, we have of course a position of unstable equilibrium. equally flexi ent direc circular strained, other circle 625. A wire of unequal flexibility in different directions is Wire unormed so that, when free from stress, it constitutes a circular ble in differre of radius a, with the plane of greatest flexural rigidity at tions, and ach point touching a cone inclined to its plane at an angle a. when unts ends are then brought together and joined, as in §§ 623, 624, bent to an- so that the whole becomes a closed circular ring, of any given by balancing radius r. It is required to find the changed inclination, o, to plied to its the plane of the ring, which the plane of greatest flexural rigidity assumes, and the couple, G, in the plane of the ring, which acts between the portions of matter on each side of any normal section. The two equations between the components of the couple and the components of the curvature in the planes of greatest and least flexural rigidity determine the two unknown quantities of the problem. 1 r 1 ture in the principal planes, and therefore cos - cos a, and sin 4 r 1 a a sin a, are the changes from the natural to the actual curvatures in these planes maintained by the corresponding com- The problem, so far as the position into which the wire turns. round its elastic central curve, may be solved by an application of the principle of energy, comprehending those of §§ 623, 624 as particular cases. Let L be the amount, per unit of length of the ring, of the couple which must be applied from without, in each normal section, to hold it with the plane of maximum flexural rigidity at each point inclined at any given angle, 4, to the plane of the ring. We have, as before (§§ 623, 624), for the potential energy of the elastic action in the ring when held so, couples ap ends. |