Abstract
A method to construct the normal modes for a class of piecewise linear vibratory systems is developed in this study. The approach utilizes the concepts of Poincaré maps and invariant manifolds from the theory of dynamical systems. In contrast to conventional methods for smooth systems, which expand normal modes in a series form around an equilibrium point of interest, the present method expands the normal modes in a series form of polar coordinates in a neighborhood of an invariant disk of the system. It is found that the normal modes, modal dynamics and frequency-amplitude dependence relationship are all of piecewise type. A two degree of freedom example is used to demonstrate the method.
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Atkinson C. and Taskett B., ‘A study of the nonlinearly related modal solutions of coupled nonlinear systems by superposition techniques’, Journal of Applied Mechanics 32, 1965, 359–364.
Benamar R., Bennouna M., and White R., ‘The effects of large vibration amplitudes on the mode shapes and natural frequencies of thin elastic structures’, Journal of Sound and Vibration 149, 1991, 179–195.
Benamar R., Bennouna M., and White R., ‘The effects of large vibration amplitudes on the fundamental mode shape of a clamped-clamped uniform beam’, Journal of Sound and Vibration 96, 1984, 309–331.
Chua L. O., Komuro M., and Matsumoto T., ‘The double scroll family’, IEEE Transactions on Circuits and Systems 33, 1986, 1073–1118.
Cooke C. and Struble R., ‘The existence of periodic solutions and normal mode vibrations in nonlinear systems’, Quarterly of Applied Mathematics 24, 1966, 177–193.
Caughey T. K., ‘Classical normal modes in damped linear dynamics systems’, Journal of Applied Mechanics 32, 1965, 359–364.
Caughey T. K. and Vakakis A. F., ‘A method for examining steady state solutions of forced discrete systems with strong nonlinearities’, International Journal of Non-Linear Mechanics 26, 1991, 89–103.
Greenberg H. and Yang T.-L., ‘Modal subspaces and normal mode vibrations’, International Journal of Non-Linear Mechanics 6, 1971, 311–326.
Kim Y. B. and Noah S. T., ‘Stability and bifurcation analysis of oscillators with piecewise linear characteristics: A general approach’, Journal of Applied Mechanics 58, 1991, 545–553.
Month L. A. and Rand R. H., ‘An application of Poincaré map to the stability of nonlinear normal modes’, Journal of Applied Mechanics 47, 1980, 645–651.
Natsiavas S., ‘On the dynamics of oscillators with bilinear damping and stiffness’, International Journal of Non-Linear Mechanics 25, 1990, 311–326.
Nayfeh A. and Nayfeh S., ‘On nonlinear modes of continuous systems’, Journal of Vibration and Acoustics 116(1), 1994, 129–136.
Nayfeh A. and Nayfeh S., ‘Nonlinear normal modes of a continuous system with quadratic nonlinearitles’, Journal of Vibration and Acoustics 117, 1995, 199–205.
Nayfeh, A. and Chin, C., ‘On nonlinear modes of systems with internal resonance’, Journal of Vibration and Acoustics 118, 1996 (to appear).
Pak C. and Rosenberg R., ‘On the existence of normal mode vibrations in nonlinear systems’, Quarterly of Applied Mathematics 26, 1968, 403–416.
Pak C., Rand R., and Moon F., ‘Free vibrations of a thin elastica by normal modes’, Nonlinear Dynamics 3, 1992, 347–364.
Rand R., Pak C., and Vakakis A., ‘Bifurcation of nonlinear normal modes in a class of 2 degres of freedom systems’, Acta Mechanica 3, 1992, 129–145.
Rand R., ‘Nonlinear normal mode in 2 degrees of freedom systems’, Journal of Applied Mechanics 38, 1971, 561.
Rand R., ‘A direct method for nonlinear normal modes’, International Journal of Non-Linear Mechanics 9, 1974, 363–368.
Rand R., ‘A higher order approximation for nonlinear normal modes in two degrees of freedom systems’, International Journal of Non-Linear Mechanics 6, 1971, 545–547.
Rand R., ‘The geometrical stability of nonlinear normal modes in 2 degrees of freedom systems’, International Journal of Non-Linear Mechanics 8, 1973, 161–168.
Rosenberg R. M., ‘The normal modes of nonlinear n degrees of freedom systems’, Journal of Applied Mechanics 30, 1962, 7–14.
Rosenberg R. M., ‘On nonlinear vibrations of systems with many degrees of freedom’, Advance in Applied Mechanics 9, 1966, 155–242.
Rosenberg R. M. and Kuo J. K., ‘Nonsimilar normal mode vibrations of nonlinear systems having two degrees of freedom’, Journal of Applied Mechanics 31, 1964, 283–290.
Shaw S. W., ‘On the dynamic response of a system with dry friction’, Journal of Sound and Vibration 108, 1986, 305–325.
Shaw S. W., ‘An invariant manifold approach to nonlinear normal modes of oscillation’, Journal of Nonlinear Science 4, 1994, 419–448.
Shaw S. W. and Holmes P. J., ‘A periodically forced piecewise linear oscillator’, Journal of Sound and Vibration 90, 1983, 305–325.
Shaw S. W. and Pierre C., ‘Normal modes for nonlinear vibratory systems’, Journal of Sound and Vibration 164, 1993, 85–124.
Shaw S. W. and Pierre C., ‘Nonlinear normal modes and invariant manifolds’, Journal of Sound and Vibration 150, 1991, 170–173.
Shaw S. W. and Pierre C., ‘Normal modes of vibration for nonlinear continuous systems’, Journal of Sound and Vibration 169, 1994, 319–347.
Sparrow C. T., ‘Chaos in a three-dimensional single loop feedback system with a piecewise linear feedback function’, Journal of Mathematical Analysis and Applications 83, 1981, 275–291.
Szemplinska-Stupnicka W., ‘The resonant vibration of homogeneous nonlinear systems’, International Journal of Non-Linear Mechanics 15, 1980, 407–415.
Szemplinska W., ‘The Behavior of Nonlinear Vibrating Systems, Kluwer, Dordrecht, 1990.
Thompson J. M. T., Bokaian A. R., and Ghaffari R., ‘Subharmonic resonance and chaotic motions of a bilinear oscillator’, IMA Journal of Applied Mathematics 37, 1983, 207–234.
Vakakis, A., ‘Analysis and identification of linear and nonlinear normal modes in vibrating systems’, Ph.D. Dissertation, California Institute of Technology, 1990.
Vakakis A. F. and Caughey T. K., ‘A theorem on the exact nonsimilar steady state motions of a nonlinear oscillator’, Journal of Applied Mechanics 5D, 1992, 418–424.
Vakakis A. and Rand R., ‘Normal modes and global dynamics of a 2 degrees of freedom system—I. Low energies’, International Journal of Non-Linear Mechanics 27, 1992, 861–874.
Vakakis A. and Rand R., ‘Normal modes and global dynamics of a 2 degrees of freedom system—II. High energies’, International Journal of Non-Linear Mechanics 27, 1992, 875–888.
Van der Varst, P., ‘On normal mode vibrations of nonlinear conservative systems’, Doctoral Thesis, Germany, 1982.
Yen D., ‘On the normal modes of nonlinear dual-mass systems’, International Journal of Non-Linear Mechanics 9, 1974, 45–53.
Zuo L. and Curnier A., ‘Nonlinear real and complex modes of conewise linear systems’, Journal of Sound and Vibration 174, 1994, 289–313.
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Chen, SL., Shaw, S.W. Normal modes for piecewise linear vibratory systems. Nonlinear Dyn 10, 135–164 (1996). https://doi.org/10.1007/BF00045454
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DOI: https://doi.org/10.1007/BF00045454