*(English)*Zbl 0574.73075

[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 ${\omega}_{1},{\omega}_{2},...,{\omega}_{n}$ and let the external excitation of the $n$th mode be denoted by ${k}_{n}cos{\Omega}t$. If ${\omega}_{3}\genfrac{}{}{0pt}{}{\sim}{\sim}{\omega}_{1}+{\omega}_{2}$, ${k}_{n}=0$ for $n\ne 3$ and ${\Omega}\genfrac{}{}{0pt}{}{\sim}{\sim}{\omega}_{3}$, 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 $\left({a}_{3}\right)$ 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 ${\omega}_{3}\genfrac{}{}{0pt}{}{\sim}{\sim}2{\omega}_{1}$; the third mode saturates when it reaches a critical value and all the extra energy goes to the first mode.

##### MSC:

74H45 | Vibrations (dynamical problems in solid mechanics) |