Global dynamics of a flexible asymmetrical rotor
Global dynamics of a flexible asymmetrical rotor resting on vibrating supports is investigated. Hamilton’s principle is used to derive the partial differential governing equations of the rotor system. The equations are then transformed into a discretized nonlinear gyroscopic system via Galerkin’s me...
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Published in | Nonlinear dynamics Vol. 91; no. 2; pp. 1041 - 1060 |
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Main Authors | , , , |
Format | Journal Article |
Language | English |
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Dordrecht
Springer Netherlands
2018
Springer Nature B.V |
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Abstract | Global dynamics of a flexible asymmetrical rotor resting on vibrating supports is investigated. Hamilton’s principle is used to derive the partial differential governing equations of the rotor system. The equations are then transformed into a discretized nonlinear gyroscopic system via Galerkin’s method. The canonical transformation and normal form theory are applied to reduce the system to the near-integrable Hamiltonian standard forms considering zero-to-one internal resonance. The energy-phase method is employed to study the chaotic dynamics by identifying the existence of multi-pulse jumping orbits in the perturbed phase space. In both the Hamiltonian and the dissipative perturbation, the homoclinic trees which describe the repeated bifurcations of multi-pulse solutions are demonstrated. In the case of damping dissipative perturbation, the existence of generalized Šilnikov-type multi-pulse orbits which are homoclinic to fixed points on the slow manifold is examined and the parameter region for which the dynamical system may exhibit chaotic motions in the sense of Smale horseshoes is obtained analytically. The global results are finally interpreted in terms of the physical motion of axis orbit. The present study indicates that the existence of multi-pulse homoclinic orbits provides a mechanism for how energy may flow from the high-frequency mode to the low-frequency mode when the rotor system operates near the first-order critical speed. |
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AbstractList | Global dynamics of a flexible asymmetrical rotor resting on vibrating supports is investigated. Hamilton’s principle is used to derive the partial differential governing equations of the rotor system. The equations are then transformed into a discretized nonlinear gyroscopic system via Galerkin’s method. The canonical transformation and normal form theory are applied to reduce the system to the near-integrable Hamiltonian standard forms considering zero-to-one internal resonance. The energy-phase method is employed to study the chaotic dynamics by identifying the existence of multi-pulse jumping orbits in the perturbed phase space. In both the Hamiltonian and the dissipative perturbation, the homoclinic trees which describe the repeated bifurcations of multi-pulse solutions are demonstrated. In the case of damping dissipative perturbation, the existence of generalized Šilnikov-type multi-pulse orbits which are homoclinic to fixed points on the slow manifold is examined and the parameter region for which the dynamical system may exhibit chaotic motions in the sense of Smale horseshoes is obtained analytically. The global results are finally interpreted in terms of the physical motion of axis orbit. The present study indicates that the existence of multi-pulse homoclinic orbits provides a mechanism for how energy may flow from the high-frequency mode to the low-frequency mode when the rotor system operates near the first-order critical speed. |
Author | Yu, Tian-Jun Yang, Xiao-Dong Zhou, Sha Zhang, Wei |
Author_xml | – sequence: 1 givenname: Tian-Jun surname: Yu fullname: Yu, Tian-Jun organization: Beijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Structures, College of Mechanical Engineering, Beijing University of Technology – sequence: 2 givenname: Sha surname: Zhou fullname: Zhou, Sha organization: Beijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Structures, College of Mechanical Engineering, Beijing University of Technology – sequence: 3 givenname: Xiao-Dong orcidid: 0000-0003-0073-0823 surname: Yang fullname: Yang, Xiao-Dong email: jxdyang@163.com organization: Beijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Structures, College of Mechanical Engineering, Beijing University of Technology – sequence: 4 givenname: Wei surname: Zhang fullname: Zhang, Wei organization: Beijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Structures, College of Mechanical Engineering, Beijing University of Technology |
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Keywords | Chaotic whirl Backward whirl Flexible asymmetrical rotor Gyroscopic system Global dynamics Bidirectional whirl |
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Snippet | Global dynamics of a flexible asymmetrical rotor resting on vibrating supports is investigated. Hamilton’s principle is used to derive the partial differential... |
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SubjectTerms | Automotive Engineering Axis movements Bifurcations Canonical forms Chaos theory Classical Mechanics Control Damping Differential equations Dissipation Dynamical Systems Energy dissipation Engineering Galerkin method Mathematical analysis Mechanical Engineering Orbits Original Paper Perturbation Vibration |
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Title | Global dynamics of a flexible asymmetrical rotor |
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