Combination resonances in the response of the Duffing oscillator to a three-frequency excitation
The authors consider the equation: \(\ddot x+ \omega^2x= \varepsilon(-\alpha x^3- 2\mu\dot x)+ \sum^3_{n=1} K_n\cos(\Omega_nt+ \theta_n)\) which could model a structural element, such a beam or a plate, supporting three rotating machines simultaneously. The response to multifrequency excitation with weak nonlinearity \((\varepsilon= 0,001)\) is studied numerically and analytically with the method of multiple scales. All of the results are in good agreement, and show that when the sum of three excitation frequencies is approximately equal to a natural frequency of the system, the peak amplitude of the system response can be several times larger than that predicted by linear theory. The influence of various parameters is studied, as well as of adding a static load to the three-frequency load. In this case the results show the influence of a nonresonant load on a combination resonance of additive type. The peak at the combination resonance is more than three times the largest peak at the three above excitation frequencies.
scientific article; zbMATH DE number 1270085