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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 $\omega\sb1, \omega\sb2, \ldots, \omega\sb n$ and let the external excitation of the $n$th mode be denoted by $k\sb n\cos \Omega t$. If $\omega\sb 3 {\sim\atop\sim} \omega\sb1 + \omega\sb2$, $k\sb n=0$ for $n\ne3$ and $\Omega {\sim\atop\sim} \omega\sb3$, the solution predicts the existence of a saturation phenomenon. For small values of $k\sb3$, only the third mode is excited. The amplitude of the third mode $(a\sb3)$ increases linearly with $k\sb3$ until a critical value, which depends on the damping coefficients and the detuning, is reached. Further increases in $k\sb3$ do not cause a further increase in $a\sb3$; instead all the extra energy goes to the first two modes, which now become strongly excited. A similar phenomenon occurs when $\omega\sb 3 {\sim\atop\sim} 2\omega\sb1$; the third mode saturates when it reaches a critical value and all the extra energy goes to the first mode.

74H45Vibrations (dynamical problems in solid mechanics)