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A saturation phenomenon in the forced response of systems with quadratic nonlinearities. (English) Zbl 0574.73075
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 ω 1 ,ω 2 ,...,ω n and let the external excitation of the nth mode be denoted by k n cosΩt. If ω 3 ω 1 +ω 2 , k n =0 for n3 and Ω ω 3 , the solution predicts the existence of a saturation phenomenon. For small values of k 3 , only the third mode is excited. The amplitude of the third mode (a 3 ) increases linearly with k 3 until a critical value, which depends on the damping coefficients and the detuning, is reached. Further increases in k 3 do not cause a further increase in a 3 ; instead all the extra energy goes to the first two modes, which now become strongly excited. A similar phenomenon occurs when ω 3 2ω 1 ; the third mode saturates when it reaches a critical value and all the extra energy goes to the first mode.

MSC:
74H45Vibrations (dynamical problems in solid mechanics)