# zbMATH — the first resource for mathematics

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.

##### MSC:
 74H45 Vibrations in dynamical problems in solid mechanics