Bifurcations and loss of orbital stability in nonlinear viscoelastic beam arrays subject to parametric actuation
The collective dynamic response of microbeam arrays is governed by nonlinear effects, which have not yet been fully investigated and understood. This work employs a nonlinear continuum-based model in order to investigate the nonlinear dynamic behavior of an array of N nonlinearly coupled micro-elect...
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| Published in | Journal of sound and vibration Vol. 329; no. 18; pp. 3835 - 3855 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
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Elsevier Ltd
30.08.2010
Elsevier |
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| Online Access | Get full text |
| ISSN | 0022-460X 1095-8568 |
| DOI | 10.1016/j.jsv.2010.03.030 |
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| Abstract | The collective dynamic response of microbeam arrays is governed by nonlinear effects, which have not yet been fully investigated and understood. This work employs a nonlinear continuum-based model in order to investigate the nonlinear dynamic behavior of an array of
N nonlinearly coupled micro-electromechanical beams that are parametrically actuated. Investigations focus on the behavior of small size arrays in the one-to-one internal resonance regime, which is generated for low or zero DC voltages. The dynamic equations of motion of a two-element system are solved analytically using the asymptotic multiple-scales method for the weakly nonlinear system. Analytically obtained results are verified numerically and complemented by a numerical analysis of a three-beam array. The dynamic responses of the two- and three-beam systems reveal coexisting periodic and aperiodic solutions. The stability analysis enables construction of a detailed bifurcation structure, which reveals coexisting stable periodic and aperiodic solutions. For zero DC voltage only quasi-periodic and no evidence for the existence of chaotic solutions are observed. This study of small size microbeam arrays yields design criteria, complements the understanding of nonlinear nearest-neighbor interactions, and sheds light on the fundamental understanding of the collective behavior of finite-size arrays. |
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| AbstractList | The collective dynamic response of microbeam arrays is governed by nonlinear effects, which have not yet been fully investigated and understood. This work employs a nonlinear continuum-based model in order to investigate the nonlinear dynamic behavior of an array of N nonlinearly coupled micro-electromechanical beams that are parametrically actuated. Investigations focus on the behavior of small size arrays in the one-to-one internal resonance regime, which is generated for low or zero DC voltages. The dynamic equations of motion of a two-element system are solved analytically using the asymptotic multiple-scales method for the weakly nonlinear system. Analytically obtained results are verified numerically and complemented by a numerical analysis of a three-beam array. The dynamic responses of the two- and three-beam systems reveal coexisting periodic and aperiodic solutions. The stability analysis enables construction of a detailed bifurcation structure, which reveals coexisting stable periodic and aperiodic solutions. For zero DC voltage only quasi-periodic and no evidence for the existence of chaotic solutions are observed. This study of small size microbeam arrays yields design criteria, complements the understanding of nonlinear nearest-neighbor interactions, and sheds light on the fundamental understanding of the collective behavior of finite-size arrays. The collective dynamic response of microbeam arrays is governed by nonlinear effects, which have not yet been fully investigated and understood. This work employs a nonlinear continuum-based model in order to investigate the nonlinear dynamic behavior of an array of N nonlinearly coupled micro-electromechanical beams that are parametrically actuated. Investigations focus on the behavior of small size arrays in the one-to-one internal resonance regime, which is generated for low or zero DC voltages. The dynamic equations of motion of a two-element system are solved analytically using the asymptotic multiple-scales method for the weakly nonlinear system. Analytically obtained results are verified numerically and complemented by a numerical analysis of a three-beam array. The dynamic responses of the two- and three-beam systems reveal coexisting periodic and aperiodic solutions. The stability analysis enables construction of a detailed bifurcation structure, which reveals coexisting stable periodic and aperiodic solutions. For zero DC voltage only quasi-periodic and no evidence for the existence of chaotic solutions are observed. This study of small size microbeam arrays yields design criteria, complements the understanding of nonlinear nearest-neighbor interactions, and sheds light on the fundamental understanding of the collective behavior of finite-size arrays. |
| Author | Gottlieb, O. Gutschmidt, S. |
| Author_xml | – sequence: 1 givenname: S. surname: Gutschmidt fullname: Gutschmidt, S. email: stefanie.gutschmidt@canterbury.ac.nz organization: Department of Mechanical Engineering, University of Canterbury, Christchurch 8140, New Zealand – sequence: 2 givenname: O. surname: Gottlieb fullname: Gottlieb, O. organization: Department of Mechanical Engineering, Technion—Israel Institute of Technology, Haifa 32000, Israel |
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| Keywords | Chaos Dynamic response Viscoelasticity Mode coupling Equations of motion Periodic solution Continuum Vibrations Low voltage Modelling Asymptotic approximation Non linear effect Microelectromechanical device Periodic structures |
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| SubjectTerms | Arrays Beams (radiation) Dynamical systems Electric potential Exact sciences and technology Instruments, apparatus, components and techniques common to several branches of physics and astronomy Mathematical models Mechanical instruments, equipment and techniques Micromechanical devices and systems Microorganisms Nonlinear dynamics Nonlinearity Physics |
| Title | Bifurcations and loss of orbital stability in nonlinear viscoelastic beam arrays subject to parametric actuation |
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