Abstract
In this paper, the nonlinear behavior of a motion amplifier used to obtain large rotations from small linear displacements produced by a piezoelectric stack is studied. The motion amplifier uses elastic (buckling) and dynamic instabilities of an axially driven buckling beam. Since the amplifier is driving a large rotary inertia at the pinned end and the operational frequency is low compared to the resonant frequencies of the beam, the mass of the buckling beam and the dynamics of the PZT stack are neglected and the system is modeled as a single-degree-of-freedom, nonlinear system. The beam simply behaves as a nonlinear rotational spring having a prescribed displacement on the input end and a moment produced by the inertial mass acting on the output end. The moment applied to the mass is then a function of the beam end displacement and the mass rotation. The system can, thus, be modeled simply as a base-excited, spring–mass oscillator. Results of the response for an ideal beam using this reduced-order model agree with the experimental data to a high degree. Inclusion of loading and geometric imperfections show that the response is not particularly sensitive to these imperfections. Parameter studies for the ideal buckling beam amplifier were conducted to provide guidance for improving the design of the motion amplifier and finding the optimal operating conditions for different applications.
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References
Burgreen, D. and Brooklyn, N., ‘Free vibration of a pin-ended column with constant distance between pin ends’, Journal of Applied Mechanics 18, 1951, 135–139.
Eisley, J., ‘Nonlinear vibration of beams and rectangular plates’, ZAMP 15, 1964, 167–175.
Eisley, J., ‘Large amplitude vibration of buckled beams and rectangular plates’, AIAA Journal 2, 1964, 2207–2209.
Tseng, W. and Dugundji, J., ‘Nonlinear vibrations of a buckled beam under harmonic excitation’, Journal of Applied Mechanics 38, 1971, 467–476.
Afaneh, A. and Ibrahim, R., ‘Nonlinear response of an initially buckled beam with 1:1 internal resonance to sinusoidal excitation’, Nonlinear Dynamics 4(6), 1993, 547–571.
Abou-Rayan, A., Nayfeh, A., Mook, D., and Nayfeh, M., ‘Nonlinear response of a parametrically excited buckled beam’, Nonlinear Dynamics 4(5), 1993, 499–525.
Nayfeh, A., Kreider, W., and Anderson, T., ‘Investigation of natural frequencies and mode shapes of buckled beams’, AIAA Journal 33(6), 1995, 1121–1126.
Lestari, W. and Hanagud, S., ‘Nonlinear vibration of buckled beams: Some exact solutions’, International Journal of Solids and Structures 38(26–27), 2001, 4741–4757.
Kreider, W. and Nayfeh, A., ‘Experimental investigation of single-mode responses in a fixed–fixed buckled beam’, Nonlinear Dynamics 15(2), 1998, 155–177.
Lacarbonara, W., Nayfeh, A., and Kreider, W., ‘Experimental validation of reduction methods for nonlinear vibrations of distributed-parameter systems: Analysis of a buckled beam’, Nonlinear Dynamics 17(2), 1998, 95–117.
Ji, J.-C. and Hansen, C., ‘Non-linear response of a post-buckled beam subjected to a harmonic axial excitation’, Journal of Sound and Vibration 237(2), 2000, 303–318.
Perkins, N., ‘Planar vibration of an elastica arch: Theory and experiment’, Journal of Vibration, Acoustics, Stress, and Reliability in Design 112(3), 1990, 374–379.
Levitas, J. and Weller, T., ‘Dynamic global postbuckling behavior of beams by cell-to-cell mapping’, International Journal of Non-Linear Mechanics 28(6), 1993, 651–662.
Nayfeh, A., Lacarbonara, W., and Chin, C.-M., ‘Nonlinear normal modes of buckled beams: Three-to-one and one-to-one internal resonances’, Nonlinear Dynamics 18(3), 1999, 253–273.
Min, G. and Eisley, J., ‘Nonlinear vibrations of buckled beams’, Journal of Engineering for Industry, 1972, 637–646.
Yamaki, N. and Mori, A., ‘Non-linear vibrations of a clamped beam with initial deflection and initial axial displacement. Part I: Theory’, Journal of Sound and Vibration 71(3), 1980, 333–346.
Abhyankar, N., Hall, E., and Hanagud, S., ‘Chaotic vibrations of beams: Numerical solution of partial differential equations’, Journal of Applied Mechanics 60(1), 1993, 167–176.
Nayfeh, A. and Lacarbonara, W., ‘On the discretization of distributed-parameter systems with quadratic and cubic nonlinearities’, Nonlinear Dynamics 13(3), 1997, 203–220.
Benamar, R., Bennouna, M., and White, R., ‘Effects of large vibration amplitudes on the mode shapes and natural frequencies of thin elastic structures: Part I. Simply supported and clamped–clamped beams’, Journal of Sound and Vibration 149(2), 1991, 179–195.
Emam, S. and Nayfeh, A., ‘Nonlinear dynamics of a clamped–clamped buckled beam’, in Proceedings of the AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 2002.
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An erratum to this article can be found at http://dx.doi.org/10.1007/s11071-006-9159-0
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Jiang, J., Mockensturm, E. A Motion Amplifier Using an Axially Driven Buckling Beam: II. Modeling and Analysis. Nonlinear Dyn 45, 1–14 (2006). https://doi.org/10.1007/s11071-005-0780-0
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DOI: https://doi.org/10.1007/s11071-005-0780-0