Continuation analysis of a nonlinear rotor system

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.

Diversity of ghost notes in tubas, euphoniums and saxhorns

“…The behaviour of an oscillating solution of the system with respect to a control parameter such as p m is then assessed by plotting bifurcation diagrams, which are shown in Section 3. This approach is implemented in several softwares dedicated to advanced numerical bifurcation analysis, such as AUTO [24], which is used in this publication and in [22,25] as well. While AUTO relies on the collocation method, other softwaressuch as Manlab [26][27][28][29][30] are based on the Harmonic Balance Method [31].…”

Diversity of ghost notes in tubas, euphoniums and saxhorns

“…The behaviour of an oscillating solution of the system with respect to a control parameter such as p m is then assessed by plotting bifurcation diagrams, which are shown in section 3. This approach is implemented in several softwares dedicated to advanced numerical bifurcation analysis, such as AUTO [Doedel et al, 1999], which is used in this publication and in [Gilbert et al, 2020, Akay et al, 2021 as well. While AUTO relies on the collocation method, other softwares -such as Manlab , Cochelin and Vergez, 2009, Karkar et al, 2013, Colinot et al, 2019, Fréour et al, 2020 -are based on the Harmonic Balance Method [Nakhla and Vlach, 1976].…”