of course, be the form of the orbit of any particle of the system Graphic representawhich has a distinct direction of motion, for two of the displace- tion. ments in question. But it must be remembered that some of the principal parts [as for instance the body supported on the fixed axis, in the illustration of $ 319, Example (C)] may have only one degree of freedom; or even that each part of the system may have only one degree of freedom, as for instance if the system is composed of a set of particles each constrained to remain on a given line, or of rigid bodies on fixed axes, mutually influencing one another by elastic cords or otherwise. In such a case as the last, no particle of the system can move otherwise than in one line; and the ellipse, circle, or other graphical representation of the composition of the harmonic motions of the system, is merely an aid to comprehension, and is not the orbit of a motion actually taking place in any part of the system. systems. 340. In nature, as has been said above ($ 278), every system uninfluenced by matter external to it is conservative, when the ultimate molecular motions constituting heat, light, and magnetism, and the potential energy of chemical affinities, are taken into account along with the palpable motions and measurable forces. But (S 275) practically we are obliged to Dissipativo admit forces of friction, and resistances of the other classes there enumerated, as causing losses of energy, to be reckoned, in abstract dynamics, without regard to the equivalents of heat or other molecular actions which they generate. Hence when such resistances are to be taken into account, forces opposed to the motions of various parts of a system must be introduced into the equations. According to the approximate knowledge which we have from experiment, these forces are independent of the velocities when due to the friction of solids: but are simply proportional to the velocities when due to fluid viscosity directly, or to electric or magnetic influences; with corrections depending on varying temperature, and on the varying configuration of the system. In consequence of the last-mentioned cause, the resistance of a real liquid (which is always more or less viscous) against a body moving rapidly enough through it, to leave a great deal of irregular motion, in the shape of Views of to a solid through a bable law. Friction of solids. “eddies," in its wake, seems, when the motion of the solid has resistance been kept long enough uniform, to be nearly in proportion to moving the square of the velocity; although, as Stokes has shown, at liquid. the lowest speeds the resistance is probably in simple proportion to the velocity, and for all speeds, after long enough time of one speed, may, it is probable, be approximately expressed as Stokrs' pro- the sum of two terms, one simply as the velocity, and the other as the square of the velocity. If a solid is started from rest in an incompressible fluid, the initial law of resistance is no doubt simple proportionality to velocity, (however great, if suddenly enough given;) until by the gradual growth of eddies the resistance is increased gradually till it comes to fulfil Stokes' law. 341. The effect of friction of solids rubbing against one another is simply to render impossible the infinitely small vibrations with which we are now particularly concerned; and to allow any system in which it is present, to rest balanced when displaced, within certain finite limits, from a configuration of frictionless equilibrium. In mechanics it is easy to estimate its effects with sufficient accuracy when any practical case of finite oscillations is in question. But the other classes of dis sipative agencies give rise to resistances simply as the velocities, Resistances without the corrections referred to, when the motions are invelocities. finitely small; and can never balance the system in a con figuration deviating to any extent, however small, from a configuration of equilibrium. In the theory of infinitely small vibrations, they are to be taken into account by adding to the expressions for the generalized components of force, proper ($ 343 a, below) linear functions of the generalized velocities, which gives us equations still remarkably amenable to rigorous mathematical treatment. The result of the integration for the case of a single degree of freedom is very simple; and it is of extreme importance, both for the explanation of many natural phenomena, and for use in a large variety of experimental investigations in Natural Philosophy. Partial conclusions from it are as follows: If the resistance per unit velocity is less than a certain critical value, in any particular case, the motion is a simple varying as n n harmonic oscillation, with amplitude decreasing in the same Resistances ratio in equal successive intervals of time. But if the re- veloc.ties. 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 no be the rate of acceleration, when the displacement is unity; so that (8 57) we have 27 T = : and let the rate of retardation due to the resistance corresponding to unit velocity be k. Then the motion is of the oscillatory or non-oscillatory class according as k* < (2n)' or Effect of h> (2n)? In the first case, the period of the oscillation is varying as velocity in a simple increased by the resistance from T to T and the rate motion. (no – 3k?) at which the Napierian logarithm of the amplitude diminishes per unit of time is Ik. 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 ý + ký + not = 0; of which the complete integral is y = {4 sin mot + B cos nt}<-ka, where n = (no - 4k), or, which is the same, V = (C:-*/ + Cen) - 4k, where n = (1 ko – Ý), 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 k* = (2n) the general solution is v = (C + C't) e-tke. equal roots. 342. The general solution (S 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 VOL. I. 2+ Infinitely either simple harmonic with amplitude diminishing according mnotion of a to the law stated above, or non-oscillatory and consisting of equi-proportionate diminutions of the components of displacement in equal successive intervals of time. dis-ipative system. SV tem 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 Cycloidal system so constituted that the positional* forces are proportional detined. 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 + Easy and system; and we shall call its motions cycloidal motions. A good cture il and instructive illustration is presented in the motion of one lustration. two or more weights in a vertical line, hung one from another, and the highest from a fixed point, by spiral springs. instructive 343 a. If now instead of 4, 6,... we denote by 4,, ,... 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 : Positional * 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 al Forces. meanings as clearly as any definition coulů. + 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 (CEM + C'edt +etc.) real or imaginary, roducible to a sum of products of real exponentials and real simple harmonics [Cemt cos (nt - e)+C'em't cos (n't - e') + etc.). at dį, at dý, d /dT Differential ( +114, +124, + ... +114, + 124, + ... = 0 equations of complex cycloidal .(?)+214, +224, + ... + 214, + 224, + ... = 0 motion. (1). 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 4. = a, edt, 4n = a.cat.... ...(2). Assuming, then, this to be a solution, and substituting in the Their soludifferential equations, we have tion. where T denotes the same homogeneous quadratic function of |(11) XP + 111 + II, (12)1+ 121 + 12,... = 0.....(4), where (11), (22), (12), (21), etc. denote the coefficients of squares .(5). The equation (4) is of the degree 2i, in ; and if any one of its roots be used for d 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, aj. 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. |