UBC Theses and Dissertations
Experimental investigation of nonlinear coupled vibrations of bars and plates Schneider , Bernd C.
The theory presented describes the physical phenomenon of nonlinear coupling of longitudinal and flexural vibrations when a beam is excited transversely at high frequencies. Equations are derived based on the Bernoulli-Euler theory of flexure, by energy methods, to describe the transverse and the longitudinal vibration of a beam of constant cross-section under plane motion. The initial crookedness of the beam and the longitudinal inertia, accounted for in the theory, give rise to the coupled vibrations. No closed form solution is presented. However, a simple analysis of some of the coupling terms suggests the existence of several coupled vibrations. By the method proposed herein, the frequencies of these vibrations can be established. In particular, the theory predicts two longitudinal coupled vibrations with the frequency ratio 1:2. The agreement between the theory and the experimental results is good. The vibrations predicted exist and the frequency ratio for the predicted longitudinal vibrations was 1:2. Further, the experimental results indicate that there are more longitudinal vibrations than indicated by the theory. A longitudinal coupled vibration at three times the frequency of transverse excitation was recorded. There are indications in the data that coupled flexural vibrations at twice the frequency of transverse excitation exist. A circular plate centrally supported and transversely excited was also tested. Two pronounced resonant radial vibrations were recorded. The frequency ratio was 1:2. Coupled flexural vibrations were not identified. The influence of the longitudinal vibration on the flexural vibration of the beam is examined. The limitations of the theory, of the experiment, and the significance of the resonant strains is discussed.
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