# Normal Oscillations

## Normal Oscillations

(or normal mode), free harmonic oscillations that could exist in linear oscillatory systems if there were no energy dissipation in the systems. In each normal oscillation, all points of the system oscillate at the same frequency, which, like the amplitude and phase distribution of the normal oscillation among the points of the system, is determined by the parameters of the system. The number of normal oscillations inherent in a given oscillatory system is equal to the number of oscillatory degrees of freedom in the system; in particular, for a continuous oscillatory system in which the number of degrees of freedom *n* = ∞, there is an infinitely large number of normal oscillations (in this case, generally speaking, the frequencies of all the normal oscillations are different; only in special “degenerate” cases can the frequencies of some of the normal oscillations be equal).

All normal oscillations are independent in the sense that it is possible to excite only one (any one) of the natural normal oscillations in a system by choosing particular initial conditions. However, with arbitrary initial conditions, in the general case all *n* normal oscillations are excited simultaneously, and all *n* oscillatory degrees of freedom participate in each of the oscillations. The resultant oscillation, which is the sum of all the excited normal oscillations, is no longer harmonic. The values of the amplitudes and initial phases of all the normal oscillations are determined by the initial conditions.

Any nonharmonic free oscillation in a linear system—that is, an oscillation that is generated under any initial conditions—is a superposition of the free normal oscillations of this system. However, resonance can only arise in an oscillatory system when the frequency of a harmonic external force coincides with the frequency of one of the normal oscillations in the system. Thus, the composition of the normal oscillations inherent in a given system essentially determines the features of both the free and forced oscillations in the system. The number of oscillatory degrees of freedom, and hence the number of normal oscillations inherent in the system, is equal to or less than the total number of degrees of freedom of the system.

### REFERENCES

Gorelik, G. S.*Kolebaniia i volny*, 2nd ed. Moscow, 1959. Chapter 6, sec. 9.

Strutt, J. W. (Lord Rayleigh.)

*Teoriia zvuka*, 2nd ed. Moscow-Leningrad, 1955. Chapter, 6, sec. 86. (Translated from English.)

S. E. KHAIKIN