Forced Vibrations, Damping, Resonance.—On account of the dissipation of energy by forces of a frictional nature, a vibrat ing body if left to itself, is gradually brought to rest. Its vibra tions may be maintained, however, by the application of a suit able periodic force, which supplies the energy dissipated by friction and sound radiation. It is important to consider the relation between the frequency of such a force and the frequency of the free or natural vibration of the body. It is generally assumed that for small oscillations the frictional forces are pro portional to the velocity of the particle. Consequently the equa tion of forces becomes dependent both on the intensity of the sound and the sensitive ness of the ear under the particular conditions. Near the limits of audibility loudness may be very feeble although the intensity be very great. The sensation of loudness varies over a relatively small range for enormous variations of intensity. An ear which can detect a feeble watch tick remains undamaged by a neigh bouring explosion—although the range of intensity involved in these examples may be greater than Ion to I. The relation be tween sensation (loudness) and stimulus (intensity or amplitude) is generally expressed by Weber's law—` `The increase of stimulus where F cospt represents the external periodic force of maximum value F and frequency p/27r, r is the resistance per unit velocity and s the restoring force per unit displacement. Writing s/m F/m=f and r/2m = k it can be shown that the displacement x is given by representing the forced vibration of period p/2ir and amplitude f sine / 2kp. In the case where there is no friction, the damping constant k=o, sine= tane =o, E = 0 or r, and In the case where there is no damping (k = o), and the free frequency n is the same as the forced frequency p, it will be seen that the amplitude A becomes infinite. Such a case of course never occurs in practice, for the damping k is never zero. For moderate values of damping the forced amplitude A (equation [I i]) is greatest when the forced frequency coincides with that of the free vibration, and may become very large when k is small. This condition is known as resonance. When dealing with sources of sound, we shall have occasion to refer frequently to resonant vibrations.
Phase of Forced Vibrations.—The force and the resultant forced vibrations are not necessarily in phase. From equation (8) we see that tane is always positive when p is less than n, that is E lies between o and 7r/2 when the forced frequency is less than the free frequency, and tan E is always negative when p is greater than n, that is, E lies between 7r/2 and 7r when the forced fre quency is greater than the free frequency. At resonance, when n = p, tane = 00 and E= r/2, i.e., the force and the displacement • are ,‘ m quadrature" whilst the force and velocity are "in phase." Away from resonance, if the damping is small, the phase difference E will, in general, be nearly equal to o or to r, that is, the dis placement will be in phase or out of phase with the force according as the frequency of the force is less or greater than the frequency of the free vibrations. These phase effects are beautifully demon
strated by means of a frequency meter of the vibrating reed type, viewed intermittently at, or very near, the frequency of excitation. The graduated series of reeds appear as in fig. 2 (a) when viewed in the ordinary way, but as in fig. 2 (b) when viewed intermittently. In accordance with the theory, the reeds on opposite sides of the resonant one are seen to be in opposite phase, with the resonant reed intermediate.
Power Dissipation.—In order to maintain vibrations against damping forces, a certain rate of energy supply is necessary. This is measured by the product of the force f cos pt and the i.e., when the frequency of the force differs from that of the resonator by the fraction k/n. This ratio k/n therefore constitutes a measure of the sharpness of resonance. The reciprocal of the ratio, i.e., n/ k is sometimes referred to as the " persistence " of the vibrations. The smaller the damping k and the higher the natural frequency n, the sharper will be the tuning, and the greater the persistence of the vibrations. This principle has many striking and important applications. Thus in frequency standardisation, e.g., of tuning forks or quartz resonators, very great accuracy of tuning is essen tial, and the damping must be extremely small. In other cases, e.g., the faithful reception or reproduction of sound vibrations over a range of frequencies, resonance is distinctly undesirable, and the system must have a natural frequency (n/ 27r) removed as far as convenient from any possible values of the forced frequency (p/ 27r), or, alternatively, the system must be heavily damped (k large).
The damping constant k of a vibrating system is determined by direct measurement of the rate of decay of its free oscilla tions (from 1 or log A. 2 = kT where and A 2 are successive amplitudes on the same side and T is the periodic time —kr is known as the logarithmic decrement of the oscilla tions). The damping coefficient k may be determined alter natively from observations of the sharpness of resonance.