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Title: Combined forced and parametric excitation of a single degree of freedom oscillator
Author: Santos, Daniel
Awarding Body: University of Aberdeen
Current Institution: University of Aberdeen
Date of Award: 1991
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This report looks at two forms of Mathieu's equation, each subjected to an external sinusoidal excitation, and has been appropriately divided into two sections. The first section analyses a nonhomogeneous linear equation considered initially without, and then with damping, The second section investigates a nonlinear nonhomogeneous equation, with damping, and is supplemented by laboratory investigations and analogue computing. The principal mathematical method used throughout has been the perturbation method of multiple scales, with the method of variation of parameters incorporated to investigate the linear case. Some of the results, from both cases, are presented in the form of time trace diagrams and these compared well with results obtained from numerical integration of the original equation using the Runge - Kutta algorithm. All the remaining results are presented in the form of 3D diagrams in an effort to give more information about the frequency content of the response. Experimental results for parametric and forced excitation of a light cantilever with an end mass agreed qualitatively with theoretical predictions and analogue computer results agreed well. Again the results were presented in a similar format to those of the previous sections, using 3D diagrams. The main findings for the linear problem were that the response was dominated by the frequencies ? and ?p - ? , where ? is the forcing frequency and ? is the parametric excitation frequency. Close to the stability boundary the interaction between the forced and parametric responses resulted in a response much larger than that due to either effect on its own. When the nonlinearities considered here were added it was found that, with no external forcing the response in the unstable region was dominated as expected by a component at ? p/2, but as the amplitude of the external forcing was increased this component was suppressed and replaced by dominant components at frequencies ? and ?P- ? In addition there were normally sidebands present, sometimes of relatively quite considerable amplitude and even the appearance, for a limited window, of what might have been chaotic motion, with a continuous range of spectral components. These digital computer findings were confirmed by analogue computer and laboratory observations.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available