varying as harmonic oscillation, with amplitude decreasing in the same Resistances ratio in equal successive intervals of time. But if the re- velocities. sistance equals or exceeds the critical value, the system when displaced from its position of equilibrium, and left to itself, returns gradually towards its position of equilibrium, never oscillating through it to the other side, and only reaching it after an infinite time. In the unresisted motion, let n' be the rate of acceleration, when the displacement is unity; so that (§ 57) we have : and let the rate of retardation due to the resistance corresponding to unit velocity be k. Then the motion is of the increased by the resistance from T to T n (n2 — 4k2) 3 or Effect of is varying as and the rate at which the Napierian logarithm of the amplitude diminishes per unit of time is k. If a negative value be given to k, the case represented will be one in which the motion is assisted, instead of resisted, by force proportional to the velocity: but this case is purely ideal. The differential equation of motion for the case of one degree of motion is resistance velocity in a simple motion. equal roots. A and B in one case, or C and C' in the other, being the arbitrary constants of integration. Hence the propositions above. In the Case of case of k2 = (2n)* the general solution is = (C+C't) € ̄škl ̧ small dissipative 342. The general solution [§ 343 a (2) and § 345] of the Infinitely problem, to find the motion of a system having any number, i, of motion of a degrees of freedom, when infinitely little disturbed from a position system. of stable equilibrium, and left to move subject to resistances proportional to velocities, shows that the whole motion may be resolved, in general determinately, into 2i different motions each 24 VOL. I. small Infinitely either simple harmonic with amplitude diminishing according motion of a to the law stated above, or non-oscillatory and consisting of system. equi-proportionate diminutions of the components of displacement in equal successive intervals of time. dis-ipative Cycloidal system defined. 343. It is now convenient to cease limiting our ideas to infinitely small motions of an absolutely general system through configurations infinitely little different from a configuration of equilibrium, and to consider any motions large or small of a system so constituted that the positional* forces are proportional to displacements and the motional* to velocities, and that the kinetic energy is a quadratic function of the velocities with constant coefficients. Such a system we shall call a cycloidal + system; and we shall call its motions cycloidal motions. A good lecture il and instructive illustration is presented in the motion of one two or more weights in a vertical line, hung one from another, and the highest from a fixed point, by spiral springs. Easy and instructive lustration. Positional 343 a. If now instead of 4, 4,... we denote by V,,,,... the generalized co-ordinates, and if we take 11, 12, 21, 22..., 11, 12, 21, 22,... to signify constant coefficients (not numbers as in the ordinary notation of arithmetic), the most general equations of motions of a cycloidal system may be written thus : Much trouble and verbiage is to be avoided by the introduction of these and Motion- adjectives, which will henceforth be in frequent use. They tell their own meanings as clearly as any definition could. al Forces. A single adjective is needed to avoid a sea of troubles here. The adjective 'cycloidal' is already classical in respect to any motion with one degree of freedom, curvilineal or rectilineal, lineal or angular (Coulomb-torsional, for example), following the same law as the cycloidal pendulum, that is to say:-the displacement a simple harmonic function of the time. The motion of a particle on a cycloid with vertex up may as properly be called cycloidal; and in it the displacement is an imaginary simple harmonic, or a real exponential, or the sum of two real exponentials of the time In cycloidal motion as defined in the text, each component of displacement is proved to be a sum of exponentials (Ce+ C'et + etc.) real or imaginary, reducible to a sum of products of real exponentials and real simple harmonics [Cemt cos (nt − e) + C'em't cos (n't - e') + etc.]. Positional forces of the non-conservative class are included by not assuming 12 = 21, 13 = 31, 23 = 32, etc. The theory of simultaneous linear differential equations with constant coefficients shows that the general solution for each co-ordinate is the sum of particular solutions, and that every particular solution is of the form equations of complex cycloidal motion. tion. Assuming, then, this to be a solution, and substituting in the Their soludifferential equations, we have where da, +λ(11a ̧ + 12a ̧ + ...) + 11a, + 12a ̧+... = 0 +λ(21a, + 22a, +...) + 2 ra, + 22a,+...=0 etc. etc. (3), denotes the same homogeneous quadratic function of a1, a............, that T is of ,,,,.... These equations, i in number, determine A by the determinantal equation (11)λ2 + 11λ + 11, (12)λ2 + 12A + 12,... =0.....(4), where (11), (22), (12), (21), etc. denote the coefficients of squares and doubled products in the quadratic, 27; with identities (12)=(21), (13) (31), etc............... .(5). The equation (4) is of the degree 2i, in A; and if any one of its roots be used for λ in the i linear equations (3), these become harmonized and give the i-1 ratios a/a,, a/a,, etc.; and we have then, in (2), a particular solution with one arbitrary constant, a,. Thus, from the 2i roots, when unequal, we have 2i distinct particular solutions, each with an arbitrary constant; and the addition of these solutions, as explained above, gives the general solution. Solution of differential equations of complex cycloidal motion. Algebra of linear equa tions. 343 b. To show explicitly the determination of the ratios a/a,, a/a,, etc. put for brevity (1 1 ) A3 + 1 là + 1 1 = I'I, (12) λ2 + 12λ + 12 = 1°2, etc., (32) A2+32λ +32=3'2, etc. ......(5)'; and generally let jk denote the coefficient of a, in the jth equation of (3), or the kth term of the 5th line of the determinant (to be called D for brevity) constituting the first member of (4). Let M (jk) denote the factor of jk in D so that jk. M (jk) is the sum of all the terms of D which contain jk, and we have Minor determinants. because in the sum Σ each term of D clearly occurs i times: D= I'I M (11) + 1'2 M (12) + 1.3 M (13) + etc. = 1'1 M (1'1) + 2'1 M (2·1) + 3'1 M (3.1) + etc. = 1.2 M (12) + 2·2 M (2·2) + 3·2 M (3.2) + etc. = = 1°3 M (1·3) + 2·3 M (2·3) + 3°3 M (3·3) + etc. in all 2i different expressions for D. ....(5)" Farther, by the elementary law of formation of determinants we see that a determi The quantities M(11), M (12), ...... M(jk), thus defined Minors of are what are commonly called the first minors of the determi- nant. nant D, with just this variation from ordinary usage that the proper signs are given to them by the factor in 5 so that in the formation of D the ordinary complication of or or and when D=0, which is required to harmonize them, they among the minors of an evanescent determinant. The remarkable relations here shown among the minors, due Relations to the evanescence of the major determinant D, are well known in algebra. They are all included in the following formula, M (j·k). M (l'n) — M (j'n). M (l·k) = 0.........................(5) vill, which is given in Salmon's Higher Algebra (§ 33 Ex. 1), as a consequence of the formula ......... M (j·k). M (l'n) — M (j·n). M (l·k) = D . M (j, l·k, n).......(5)1, where M(j,lk, n) denotes the second minor formed by suppressing the jth and 7th columns and the kth and nth lines. |