Nonlinear dynamic response of shallow arches to harmonic forcing
Abstract
Buckled beams and shallow arches exhibit nonlinear force-deflection curves as a result of their geometric shape. These structural components share the common characteristic of nonlinear load-deflection curves, exhibiting both stable and unstable equilibrium states. Previous work has shown that with the application of sufficiently large static or dynamic loads, snap-buckling can occur, in which the structure suddenly jumps from one stable equilibrium configuration to another. The dynamic response due to harmonic forcing has been shown to contain orbits encompassing either of the stable equilibrium positions, or orbits encompassing both the stable and unstable equilibrium positions. The investigation of the dynamic response of a shallow arch is undertaken using a two rigid-link, single degree of freedom (SDOF) model. The method of harmonic balance, coupled with a continuation scheme, is used to find the solutions for an entire range of externally applied loading. Floquet analysis provides the requisite stability information, as well as information about the bifurcation points encountered in the solution. The dynamic response of the arch to harmonic forcing is shown to exhibit both symmetric and asymmetric solutions. Additionally, stable solutions are found that have a fundamental period of an integer multiple of the excitation period. Finally, regions of chaotic motion are found. Results are presented for three cases of varying the magnitude of the applied harmonic loading for a constant execution frequency, and for the three cases of varying the excitation frequency of the applied harmonic loading for constant magnitude. The investigation is extended to the case of a four-link, three degree of freedom model. It is shown that for loading cases away from combination and internal resonances, the above SDOF model provides an accurate representation of the dynamic response. The dynamic response to of the model to antisymmetric loading is investigated. A hardening type response is observed, and the symmetric and asymmetric response curves are traced for three levels of damping. Additionally, the birth of the bifurcation point connecting the symmetric and asymmetric branches of the response is shown.
Degree
Ph.D.
Advisors
Krousgrill, Purdue University.
Subject Area
Aerospace materials|Mechanical engineering|Mechanics
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