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any time, the deviation of any one of these co-ordinates from its value for the configuration of equilibrium will vary according to a complex harmonic function ($ 88), composed in general of i simple harmonics of incommensurable periods, and therefore (§ 85) the whole motion of the system will not recur periodically through the same series of configurations. There are in general, however, i distinct determinate displacements, which we shall call the normal displacements, fulfilling the condition, that if any one of them be produced alone, and the system then left to itself for an instant at rest, this displacement will diminish and increase periodically according to a simple harmonic function of the time, and consequently every particle of the system will execute a simple harmonic movement in the same period. This result, we shall see later, includes cases in which there are an infinite number of degrees of freedom; as, for instance, a stretched cord; a mass of air in a closed vessel; waves in water, or oscillations in a vessel of water of limited extent, or in an elastic solid; and in these applications it gives the theory of the so-called 'fundamental vibration,' and successive 'harmonics' of the cord, and of all the different possible simple modes of vibration in the other cases.
291. If, as may be in particular cases, the periods of the vibrations for two or more of the normal displacements are equal, any displacement compounded of them will also fulfil the condition of a normal displacement. And if the system be displaced according to any one such normal displacement, and projected with velocity corresponding to another, it will execute a movement, the resultant of two simple harmonic movements in equal periods. The graphic representation of the variation of the corresponding co-ordinates of the system, laid down as two rectangular co-ordinates in a plane diagram, will consequently (8 82) be a circle or an ellipse; which will therefore, of course, be the form of the orbit of any particle of the system which has a distinct direction of motion, for two of the displacements in question. But it must be remembered that some of the principal parts 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 not a representation of any motion actually taking place in any part of the system.
292. In nature, as has been said above ($ 250), 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 ($ 247) practically we are obliged to 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; and 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 very rapidly through it, and leaving a great deal of irregular motion, such as 'eddies,' in its wake, seems to be nearly in proportion to the square of the velocity; although, as Stokes has shown, at the lowest speeds the resistance is probably in simple proportion to the velocity, and for all speeds may, it is probable, be approximately expressed as the sum of two terms, one simply as the velocity, and the other as the square of the velocity.
293. The effect of friction of solids rubbing one against 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 dissipative agencies, give rise to resistances simply as the velocities, without the corrections referred to, when the motions are infinitely small, and can never balance the system in a configuration deviating to any extent, however small, from a configuration of equilibrium without friction. 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, terms consisting of the generalized velocities each multiplied by a constant, 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, in the first place, stated in general terms, are as follows:
294. If the resistance is less than a certain limit, in any particular case, the motion is a simple harmonic oscillation, with amplitude decreasing by equal proportions in equal successive intervals of time. But if the resistance exceeds this limit, 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 na be the rate of acceleration, when the displacement is unity; so that ($ 74) we have T=27: 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 k> 2n. In the first case, the period of the oscillation is increased, by the resistance, from T' to Ti
(ik—na)!! and the rate at which the Napierian logarithm of the amplitude diminishes per unit of time is k.
295. An indirect but very simple proof of this important proposition may be obtained by means of elementary mathematics as follows:-A point describes a logarithmic spiral with uniform angular velocity about the pole-find the acceleration. .
Since the angular velocity of SP and the inclination of this line to the tangent are each constant, the linear velocity of P is as SP. Take a length PT, equal to n SP, to represent it. Then the hodograph, the locus of p, where Sp is parallel and equal to PT, is evidently another logarithmic spiral similar to the former, and described with the same uniform angular velocity. Hence (9$ 35, 49) pt, the acceleration required, is equal to n Sp, and makes with Sp an angle Spt equal to SPT. Hence, if Pu be drawn parallel and equal to pt, and wv parallel to PT, the whole acceleration pt or Pu may be resolved into Pv and vu. Now Pvu is an isosceles triangle, whose base angles (v, u) are each equal to the constant angle of the spiral. Hence Pv and vu bear constant ratios to Pu, and therefore to SP and to PT respectively.
The acceleration, therefore, is composed of a central attractive part proportional to the distance, and a tangential retarding part proportional to the velocity.
And, if the resolved part of P's motion parallel to any line in the plane of the spiral be considered, it is obvious that in it also the acceleration will consist of two parts—one directed towards a point in the line (the projection of the pole of the spiral), and proportional to the distance from it; the other proportional to the velocity, but retarding the motion.'
· Hence a particle which, unresisted, would have a simple harmonic motion, has, when subject to resistance proportional to its velocity, a motion represented by the resolved part of the spiral motion just described. ..296. If a be the constant angle of the spiral, w the angular velocity of SP, we have evidently
PT. sin a = SP.W. But PT = n SP, so that n = 7
. Pv=Pu=pt=nSp=nPT =n?. SP
and vu=2Pv.cos a= 2n cos a PT=k.PT (suppose.) Thus the central force at unit distance is n-=- , and the co
2w cos a efficient of resistance is k= 2n cos a = in .
The time of oscillation in the resolved motion is evidently – if there had been no resistance, the properties of simple harmonic motion show that it would have been 25; so that it is increased by
the resistance in the ratio cosec a to 1, or n to V neThe rate of diminution of SP is evidently
PT. cos a=n cos a SP= SP; that is, SP diminishes in geometrical progression as time increases, the rate being per unit of time per unit of length. By an ordinary result of arithmetic (compound interest payable every instant) the diminution of log. SP in unit of time is .
This process of solution is only applicable to resisted harmonic vibrations when n is greater than When n is not greater than the auxiliary curve can no longer be a logarithmic spiral, for the moving particle never describes more than a finite angle about the pole. A curve, derived from an equilateral hyperbola, by a process somewhat resembling that by which the logarithmic spiral is deduced from a circle, may be introduced; but then the geometrical method ceases to be simpler than the analytical one, so that it is useless to pursue the investigation farther, at least from this point of view.
297. The general solution of the problem, to find the motion of a system having any number, i, of degrees of freedom, when infinitely little disturbed from a position of 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 either simple harmonic with amplitude diminishing according to the law stated above ($ 294), or non-oscillatory, and consisting of equi-proportionate diminutions of the components of displacement in equal successive intervals of time.
298. When the forces of a system depending on configuration, and not on motion, or, as we may call them for brevity, the forces of position, violate the law of conservatism, we have seen (§ 244) that energy without limit may be drawn from it by guiding it perpetually through a returning cycle of configurations, and we have inferred that in every real system, not supplied with energy from without, the forces of position fulfil the conservative law. But it is easy to arrange a system artificially, in connexion with a source of energy, so that its forces of position shall be non-conservative; and the consideration of the kinetic effects of such an arrangement, especially of its oscillations about or motions round a configuration of equilibrium, is most instructive, by the contrasts which it presents to the phenomena of a natural system.
299. But although, when the equilibrium is stable, no possible infinitely small displacement and velocity given to the system can cause it, when left to itself, to go on moving either farther and farther away till a finite displacement is reached, or till a finite velocity is acquired; it is very remarkable that stability should be possible, considering that even in the case of stability an endless increase of velocity may, as is easily seen from § 244, be obtained merely by constraining the system to a particular closed course, or circuit of configurations, nowhere deviating by more than an infinitely small amount from the configuration of equilibrium, and leaving it at rest anywhere in a certain part of this circuit. This result, and the distinct peculiarities of the cases of stability and instability, is sufficiently illustrated by the simplest possible example,—that of a material particle moving in a plane.
300. There is scarcely any question in dynamics more important for Natural Philosophy than the stability or instability of motion. We therefore, before concluding this chapter, propose to give some general explanations and leading principles regarding it.
A conservative disturbance of motion' is a disturbance in the motion or configuration of a conservative system, not altering the sum of the potential and kinetic energies. A conservative disturbance of the motion through any particular configuration is a change in velocities, or component velocities, not altering the whole kinetic energy. Thus, for example, a conservative disturbance of the motion of a particle through any point, is a change in the direction of its motion, unaccompanied by change of speed.
301. The actual motion of a system, from any particular configuration, is said to be stable if every possible infinitely small conservative disturbance of its motion through that configuration may be compounded of conservative disturbances, any one of which would give rise to an alteration of motion which would bring the system