Abstract
Nonlinearities in rotating systems have been seen to cause a wide variety of rich phenomena; however, the understanding of these phenomena has been limited because numerical approaches typically rely on “brute force” time simulation, which is slow due to issues of step size and settling time, cannot locate unstable solution families, and may miss key responses if the correct initial conditions are not used. This work uses numerical continuation to explore the responses of such systems in a more systematic way. A simple isotropic rotor system with a smooth nonlinearity is studied, and the rotating frame is used to obtain periodic solutions. Asynchronous responses with oscillating amplitude are seen to initiate at certain drive speeds due to internal resonance, in a manner similar to that observed for nonsmooth rotor–stator contact systems in the previous literature. These responses are isolated, in the sense that they will only meet the more trivial synchronous responses in the limit of zero damping and out of balance forcing. In addition to increasing our understanding of the responses of these systems, the work establishes the potential of numerical continuation as a tool to systematically explore the responses of nonlinear rotor systems.
Highlights
In nonlinear dynamic systems, the response can change qualitatively with small changes in system parameters, so being able to systematically locate these so-called bifurcations is vital in nonlinear mechanical systems
AUTO, the numerical continuation software employed in this study, uses a method proposed by Fairgrieve and Jepson [11], where the Floquet multipliers are calculated without explicit calculation of the monodromy matrix
From the data sampled at various ζ during this ζ continuation, new solution branches are found with continuation in Ω
Summary
The response can change qualitatively with small changes in system parameters, so being able to systematically locate these so-called bifurcations is vital in nonlinear mechanical systems. The fact that the intermittent contact response can be seen as the synchronisation of resonant frequencies of the rotating system, make the use of the normal forms technique a useful procedure to solve the nonlinear equations [35]. The use of arclength continuation does offer significant benefits in exploring the space around a given solution in the sense that it allows a complete solution family to be found without the interruption of jumping phenomenon, a common phenomenon for time simulations It can pass the turning points, the so-called folds, possibly progressing to the unstable region [30], which make the method feasible for high-resolution bifurcation diagrams [33].
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