Hence we see that the co-ordinate Qa corresponds to a normal vibration antisymmetrical about the y-axis (x1 = x3, y1 = -y3; Fig. 29a) with frequency The co-ordinates qs1, qs2 together correspond to two vibrations symmetrical about the y-axis (x1 = -X3, y1 y3; Fig. 29b, c), whose frequencies Ws1, W82 are given by the roots of the quadratic (in w2) characteristic equation 1 When 2 x = 75, all three frequencies become equal to those derived in Problem 1. PROBLEM 3. The same as Problem 1, but for an unsymmetrical linear molecule ABC (Fig. 30). A FIG. 30 SOLUTION. The longitudinal (x) and transverse (y) displacements of the atoms are related by mAX1+mBX2+mcx3 = 0, mAy1tmBy2+mcy3= 0, MAhy1 = mcl2y3. The potential energy of stretching and bending can be written where 2l = li+l2. Calculations similar to those in Problem 1 give for the transverse vibrations and the quadratic (in w2) equation = for the frequencies wil, W12 of the longitudinal vibrations. $25. Damped oscillations So far we have implied that all motion takes place in a vacuum, or else that the effect of the surrounding medium on the motion may be neglected. In reality, when a body moves in a medium, the latter exerts a resistance which tends to retard the motion. The energy of the moving body is finally dissipated by being converted into heat. Motion under these conditions is no longer a purely mechanical process, and allowance must be made for the motion of the medium itself and for the internal thermal state of both the medium and the body. In particular, we cannot in general assert that the acceleration of a moving body is a function only of its co-ordinates and velocity at the instant considered; that is, there are no equations of motion in the mechanical sense. Thus the problem of the motion of a body in a medium is not one of mechanics. There exists, however, a class of cases where motion in a medium can be approximately described by including certain additional terms in the §25 Damped oscillations 75 mechanical equations of motion. Such cases include oscillations with fre- quencies small compared with those of the dissipative processes in the medium. When this condition is fulfilled we may regard the body as being acted on by a force of friction which depends (for a given homogeneous medium) only on its velocity. If, in addition, this velocity is sufficiently small, then the frictional force can be expanded in powers of the velocity. The zero-order term in the expan- sion is zero, since no friction acts on a body at rest, and so the first non- vanishing term is proportional to the velocity. Thus the generalised frictional force fir acting on a system executing small oscillations in one dimension (co-ordinate x) may be written fir = - ax, where a is a positive coefficient and the minus sign indicates that the force acts in the direction opposite to that of the velocity. Adding this force on the right-hand side of the equation of motion, we obtain (see (21.4)) mx = -kx-ax. (25.1) We divide this by m and put k/m= wo2, a/m=2x; = (25.2) wo is the frequency of free oscillations of the system in the absence of friction, and A is called the damping coefficient or damping decrement. Thus the equation is (25.3) We again seek a solution x = exp(rt) and obtain r for the characteristic equation r2+2xr + wo2 = 0, whence ¥1,2 = The general solution of equation (25.3) is c1exp(rit)+c2 exp(r2t). Two cases must be distinguished. If wo, we have two complex con- jugate values of r. The general solution of the equation of motion can then be written as where A is an arbitrary complex constant, or as = aexp(-Xt)cos(wta), (25.4) with w = V(w02-2) and a and a real constants. The motion described by these formulae consists of damped oscillations. It may be regarded as being harmonic oscillations of exponentially decreasing amplitude. The rate of decrease of the amplitude is given by the exponent X, and the “frequency” w is less than that of free oscillations in the absence of friction. For 1 wo, the difference between w and wo is of the second order of smallness. The decrease in frequency as a result of friction is to be expected, since friction retards motion. t The dimensionless product XT (where T = 2n/w is the period) is called the logarithmic damping decrement. 76 Small Oscillations §25 If A < wo, the amplitude of the damped oscillation is almost unchanged during the period 2n/w. It is then meaningful to consider the mean values (over the period) of the squared co-ordinates and velocities, neglecting the change in exp( - At) when taking the mean. These mean squares are evidently proportional to exp(-2xt). Hence the mean energy of the system decreases as (25.5) where E0 is the initial value of the energy. Next, let A > wo. Then the values of r are both real and negative. The general form of the solution is - (25.6) We see that in this case, which occurs when the friction is sufficiently strong, the motion consists of a decrease in /x/, i.e. an asymptotic approach (as t -> 00) to the equilibrium position. This type of motion is called aperiodic damping. Finally, in the special case where A = wo, the characteristic equation has the double root r = - 1. The general solution of the differential equation is then (25.7) This is a special case of aperiodic damping. For a system with more than one degree of freedom, the generalised frictional forces corresponding to the co-ordinates Xi are linear functions of the velocities, of the form = (25.8) From purely mechanical arguments we can draw no conclusions concerning the symmetry properties of the coefficients aik as regards the suffixes i and k, but the methods of statistical physics make it possible to demonstrate that in all cases aki. (25.9) Hence the expressions (25.8) can be written as the derivatives = (25.10) of the quadratic form (25.11) which is called the dissipative function. The forces (25.10) must be added to the right-hand side of Lagrange’s equations: (25.12) t See Statistical Physics, $123, Pergamon Press, Oxford 1969.