The analysis of systems subjected to periodic excitations can be highly complex in the presence of strong nonlinearities. Nonlinear systems exhibit a variety of dynamic behavior that includes periodic, almost-periodic (quasi-periodic), and chaotic motions. This paper describes a computational algorithm based on the shooting method that calculates the periodic responses of a nonlinear system under periodic excitation. The current algorithm calculates also the stability of periodic solutions and locates system parameter ranges where aperiodic and chaotic responses bifurcate from the periodic response. Once the system response for a parameter is known, the solution for near range of the parameter is calculated efficiently using a pseudo-arc length continuation procedure. Practical procedures for continuation, numerical difficulties and some strategies for overcoming them are also given. The numerical scheme is used to study the imbalance response of a rigid rotor supported on squeeze-film dampers and journal bearings, which have nonlinear stiffness and damping characteristics. Rotor spinning speed is used as the bifurcation parameter, and speed ranges of sub-harmonic, quasi-periodic and chaotic motions are calculated for a set of system parameters of practical interest. The mechanisms of these bifurcations also are explained through Floquet theory, and bifurcation diagrams.
Dynamics of Forced Nonlinear Systems Using Shooting/Arc-Length Continuation Method—Application to Rotor Systems
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Sundararajan, P., and Noah, S. T. (January 1, 1997). "Dynamics of Forced Nonlinear Systems Using Shooting/Arc-Length Continuation Method—Application to Rotor Systems." ASME. J. Vib. Acoust. January 1997; 119(1): 9–20. https://doi.org/10.1115/1.2889694
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