Dynamic response of a nonlinear model of a rolling ring

Rick Ian Zadoks, Purdue University

Abstract

This research deals with the dynamic behavior of a rolling ring subject to both an elastic foundation undergoing a prescribed sinusoidal motion and a sinusoidal point load. The nonlinear strain-displacement relations for circular cylinders are derived from first principles using geometric arguments and Taylor series expansions, and are used with the three-demensional Hooke's law expressions to give the nonlinear equations of motion of a rotating ring. Only quadratic nonlinearities are retained, and the resulting equations are seen to include both parametric and external excitation terms. A separable form of the solution is assumed and used to reduce the partial differential equations to a set of coupled nonlinear, nonautonomous ordinary differential equations. These equations are transformed so that their linearized, undamped first-order form is canonical. Since the original equations contain gyroscopic terms the Jordan canonical form is block-diagonal. The equations of the system are reduced to a set of twelve first-order ordinary differential equations. Due to the physical nature of the type of external excitations applied and the system, the eigenvalues and eigenvectors are such that this system can be reduced to only eight equations, which correspond to four modes of motion. Furthermore, when only point loading is applied, it is possible to consider only two modes, or four equations, as long as the forcing frequency $\omega\sb{\rm f}$ is less than the lowest natural frequency $\omega\sb1$. The second major focus of this work is the numerical simulation of the reduced set of the equations of motion. Three different examples are explored, and the results are displayed as functions of the forcing amplitude f, the hub motion amplitude z and the detuning $\sigma$. Through the use of bifurcation diagrams in conjunction with a shooting technique and Poincare maps, it is found that this system is subject to turning point and period-doubling bifurcations, the latter of which lead to regions of chaotic response. Strutt diagrams are used to explore the parametrically excited system and it is found that bifurcations to aperiodic solutions are also possible.

Degree

Ph.D.

Advisors

Krousgrill, Purdue University.

Subject Area

Mechanical engineering

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