Nonlinear oscillations, Proc. 8th int. Conf., Prague 1978, Vol. 1, 2, 511-516 (1979).
[For the entire collection see Zbl 0509.00030.]
The method of multiple scales is used to determine the response of a multi-degree-of-freedom system having quadratic nonlinearities to a sinusoidal external excitation. Let the natural frequencies be denoted by and let the external excitation of the th mode be denoted by . If , for and , the solution predicts the existence of a saturation phenomenon. For small values of , only the third mode is excited. The amplitude of the third mode increases linearly with until a critical value, which depends on the damping coefficients and the detuning, is reached. Further increases in do not cause a further increase in ; instead all the extra energy goes to the first two modes, which now become strongly excited. A similar phenomenon occurs when ; the third mode saturates when it reaches a critical value and all the extra energy goes to the first mode.