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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 \(\omega_1, \omega_2, \ldots, \omega_ n\) and let the external excitation of the \(n\)th mode be denoted by \(k_ n\cos \Omega t\). If \(\omega_ 3 {\sim\atop\sim} \omega_1 + \omega_2\), \(k_ n=0\) for \(n\neq3\) and \(\Omega {\sim\atop\sim} \omega_3\), 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 \(\omega_ 3 {\sim\atop\sim} 2\omega_1\); the third mode saturates when it reaches a critical value and all the extra energy goes to the first mode.

74H45 Vibrations in dynamical problems in solid mechanics