Nonlinear vibration of beam and multibeam systems

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Virginia Tech

In this dissertation, an experimental and theoretical investigation into the nonlinear vibration of beam and beam-like structures with rectangular cross sections is presented. Two structures, a cantilever beam subject to a harmonic external excitation and a portal frame subject to a harmonic base motion, are the objects of study.

For the cantilever beam, we present experimental results regarding multimode behavior. The beam was tested in both a vertical and horizontal configuration. Our experiments show that. for a forcing frequency near the fourth natural frequency of the beam, a low-frequency mode can be activated through a nonlinear mechanism. The nonlinear mechanism responsible for the transfer of energy to a low-frequency mode of the beam in the horizontal configuration was a subcombination internal resonance. However, for the same beam in the vertical configuration, both a subcombination internal resonance and a nonresonant modal interaction were observed to transfer energy to a low-frequency mode. The subcombination internal resonance consisted of contributions from the directly excited fourth mode, the fifth mode, and the low-frequency second mode. The response due to the nonresonant modal interaction consisted of contributions from the directly excited fourth mode and the indirectly excited low-frequency first mode. Both of these interactions are the result of a system with a dominant cubic nonlinearity.

The single-mode response of the cantilever beam in the horizontal configuration was the subject of study. A comparison between the theoretically and experimentally obtained frequency-response curves revealed a discrepancy for an assumed ideal clamp. The model was brought into agreement by incorporating a quadratic damping term modeling the effect of air damping and a nonlinear rotational spring to model the flexibility of the clamp.

The portal frame is a structure with a dominant quadratic nonlinearity. Experimental results are presented for the cases of a single combination resonance and multiple combination resonances. For the multiple combination resonances, excitation of a single mode was found to eventually activate contributions from six other modes, most of them possessing lower frequencies. The amplitudes of these lower-frequency modes were at times larger than that of the directly excited mode.

The final topic is parameter identification for nonlinear systems. A scheme of experiments is designed that in conjunction with a multiple scales analysis can accurately estimate the nonlinear coefficients of a single-degree-of-freedom model. Parameters for a portal frame were ascertained by activating a subharmonic resonance of order one-half.