Abstract
An experimental and theoretical study of the response of aflexible cantilever beam to an external harmonic excitation nearthe beam's third natural frequency is presented. For a certain range ofthe excitation frequency, we observed experimentally that the responseincludes a large contribution due to the first mode of the beamaccompanied by a slow modulation of the amplitude and phase of the thirdmode. In addition, we noted that the energy transfer between the thirdand first modes is very much dependent upon the closeness of themodulation (or Hopf bifurcation) frequency to the first-mode naturalfrequency. In earlier studies by Nayfeh and coworkers, the modulationfrequency was close to the first-mode natural frequency, and thereforelarge first-mode swaying was observed. But for higher forcingamplitudes, the present experiments show that the modulation frequencytends to shift away from the first-mode natural frequency, andsubsequently very little swaying is observed. We also developed areduced-order analytical model by discretizing the integralpartial-differential equation of motion, derived by Crespo daSilva and Glenn, using the Galerkin procedure with a four-modeapproximation. The reduced-order model demonstrates the energy transferfrom the third mode to the first mode.
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Malatkar, P., Nayfeh, A.H. On the Transfer of Energy between Widely Spaced Modes in Structures. Nonlinear Dynamics 31, 225–242 (2003). https://doi.org/10.1023/A:1022072808880
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DOI: https://doi.org/10.1023/A:1022072808880