Analysis And Experiments On Nonlinear Vibrations Of Typical Structure

Theoretical and experimental study of the nonlinear vibration of typical structures is an important part of nonlinear dynamics. This thesis explores the coupled vibration of cantilever, the effect of high order terms and the nonlinear resonance. Three main aspects are listed as follow:Part I: Models for the bending-torsional vibration and the transverse-longitudinal vibration of a cantilever. High order Taylor expansion of bending angle is used. Then, Hamilton principle is applied to acquire the nonlinear coupled equation that governs the motion of cantilever structure. In the bending-torsional coupled model the wrapping effect is also taken into consideration.Part II: Effects of the fifth order terms on the nonlinear resonances of Duffing equation. Multi-scale method is used to obtain the averaging equation and bifurcation equation. Singularity theory is then applied to investigate the bifurcation equation, transitional sets and their corresponding bifurcation diagram. Theoretical results and numerical simulation are in good match. The fifth order nonlinearity has significant effect on the system bifurcation. When the fifth order nonlinearity has the same sign as the cubic term, neglecting the fifth order nonlinearity will not bring much difference and there are two types of frequency response curves. However, when two nonlinearities are in different signs, the fifth order nonlinearity will cause another three types of frequency response and thus its effect must be considered.Part III: Experiments on internal and combinational resonances of a three-layer structure. Through modal design the frequency ratio between the first and the second modal is set near1:3. The internal resonance behaviors are found when the excitation frequency is near that of the first or second modal. Meanwhile, several types of combinational resonances are found in this structure. These results made a foundation for future modification of the nonlinear vibration equipment.