Abstract
The responses and codimension-one bifurcations in Masing-type andBouc–Wen hysteretic oscillators are investigated. The pertinent statespace is formulated for each system and the periodic orbits are soughtas the fixed points of an appropriate Poincaré map. The implementedpath-following scheme is a pseudo-arclength algorithm based on arclengthparameterization. The eigenvalues of the Jacobian of the map, calculatedvia a finite-difference scheme, are used to ascertain the stability andbifurcations of the periodic steady-state solutions. Frequency-responsecurves for various excitation levels are constructed consideringrepresentative hysteresis loop shapes generated with the two models inthe primary and superharmonic frequency ranges. In addition to knownbehaviors, a rich class of solutions and bifurcations, mostly unexpectedfor hysteretic oscillators – including jump phenomena,symmetry-breaking, complete period-doubling cascades, fold, andsecondary Hopf – is found. Complex (mode-locked) periodic andnonperiodic responses are also investigated thereby allowing to draw amore comprehensive picture of the dynamical behavior exhibited by thesesystems.
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References
Sauter, D. and Hagedorn, P., 'On the hysteresis of wire cables in stockbridge dampers', International Journal of Non-Linear Mechanics 37, 2002, 1453–1459.
Dyke, S. J., Spencer, B. F. Jr., Sain, M. K., and Carlson, J. D., 'An experimental study of MR dampers for seismic protection', Smart Materials and Structures 7, 1998, 693–703.
Saadat, S., Salichs, J., Noori, M., Hou, Z., Davoodi, H., Bar-on, I., Suzuki, Y., and Masuda, A., 'An overview of vibration and seismic applications of NiTi shape memory alloy', Smart Materials and Structures 11, 2002, 218–229.
Lacarbonara, W. and Vestroni, F., 'Feasibility of a vibration absorber based on hysteresis', in Proceedings of the Third World Congress on Structural Control, Como, Italy, April 7-12, F. Casciati (ed.), 2002, Vol. 3, pp. 421–430.
Bouc, R., 'Forced vibrations of mechanical systems with hysteresis', Abstract in Proceedings of the 4th International Conference on Nonlinear Oscillations, Prague, Czechoslovakia, 1967, p. 315.
Wen, Y. K., 'Method for random vibration of hysteretic systems', ASCE Journal of Engineering Mechanics 120, 1976, 2299–2325.
Masing, G., 'Eigenspannungen und verfestigung beim messing (Self stretching and hardening for brass)', in Proceedings of the Second International Congress of Applied Mechanics, Zurich, Switzerland, 1926, pp. 332–335 [in German].
Capecchi, D. and Masiani, R., 'Reduced phase space analysis for hysteretic oscillators of Masing type', Chaos, Solitons, & Fractals 10, 1996, 1583–1600.
Baber, T. T. and Noori, M. N., 'Random vibration of degrading, pinching systems', Journal of Engineering Mechanics 107, 1985, 1069–1087.
Brokate, M. and Visintin, A., 'Properties of the Preisach model for hysteresis', Zeitschrift für reine und angewandte Mathematik 402, 1989, 1–40.
Bernardini, D. and Vestroni, F., 'Hysteretic modeling of shape-memory alloy vibration reduction devices', Journal of Material Processing and Manufacturing Science 9, 2000, 101–112.
Mayergoyz, I. D., Mathematical Models of Hysteresis, Springer, New York, 1991.
Visintin, A., Differential Models of Hysteresis, Springer-Verlag, Berlin, 1994.
Brokate, M. and Sprekels, J., Hysteresis and Phase Transitions, Springer-Verlag, Berlin, 1996.
Vestroni, F. and Noori, M., 'Hysteresis in mechanical systems - Modeling and dynamic response', International Journal of Non-Linear Mechanics 37, 2002, 1261–1262.
Caughey, T. K., 'Sinusoidal excitation of a system with bilinear hysteresis', Journal of Applied Mechanics 27, 1960, 640–643.
Iwan, W. D., 'The steady-state response of the double bilinear hysteretic oscillator', Journal of Applied Mechanics 32, 1965, 921–925.
Masri, S. F., 'Forced vibration of the damped bilinear hysteretic oscillator', Journal of the Acoustical Society of America 57, 1975, 106–111.
Capecchi, D. and Vestroni, F., 'Steady-state dynamic analysis of hysteretic systems', Journal of Engineering Mechanics 111, 1985, 1515–1531.
Capecchi, D. and Vestroni, F., 'Periodic response of a class of hysteretic oscillators', International Journal of Non-Linear Mechanics 25, 1990, 309–317.
Capecchi, D., 'Periodic response and stability of hysteretic oscillators', Dynamics and Stability of Systems 6, 1991, 89–106.
Wong, C.W., Ni, Y. Q., and Lau, S. L., 'Steady-state oscillation of hysteretic differential model. I: Response analysis', ASCE Journal of Engineering Mechanics 120, 1994, 2271–2298.
Wong, C. W., Ni, Y. Q., and Ko, J. M., 'Steady-state oscillation of hysteretic differential model. II: Performance analysis', ASCE Journal of Engineering Mechanics 120, 1994, 249–263.
Capecchi, D., 'Asymptotic motions and stability of the elastoplastic oscillator studied via maps', International Journal of Solids and Structures 20, 1993, 3303–3314.
Vestroni, F. and Capecchi, D., 'Coupling and resonance phenomena in dynamic systems with hysteresis', in Proceedings of the IUTAM Symposium on New Applications of Nonlinear and Chaotic Dynamics in Mechanics, F. C. Moon (ed.), Kluwer, Dordrecht, 1997, pp. 203–212.
Lacarbonara, W., Vestroni, F., and Capecchi, D., 'Poincaré map-based continuation of periodic orbits in dynamic discontinuous and hysteretic systems', in Proceedings of the 17th Biennial ASME Conference on Mechanical Vibration and Noise, Las Vegas, NV, September 12-15, 1999, Paper No. DETC99/VIB-8088.
Lacarbonara, W., Bernardini, D., and Vestroni, F., 'Periodic and nonperiodic responses of shape-memory oscillators', in Proceedings of the 18th Biennial ASME Conference on Mechanical Vibration and Noise, Pittsburgh, PA, September 9-12, 2001, Paper No. DETC2001/VIB-21458.
Vestroni, F., Bernardini, D., and Lacarbonara, W., 'Nonlinear thermomechanical responses of shape-memory oscillators', in Proceedings of the Ninth Conference on Nonlinear Vibrations, Stability, and Dynamics of Structures, Virginia Tech, Blacksburg, VA, July 28-31, A. H. Nayfeh and D. T. Mook (eds.), 2002.
Awrejcewicz, J., Bifurcations and Chaos in Simple Dynamical Systems, World Scientific, Singapore, 1989.
Awrejcewicz, J., Bifurcations and Chaos in Coupled Oscillators, World Scientific, Singapore, 1991.
Nayfeh, A. H. and Balachandran, B., Applied Nonlinear Dynamics, Wiley Interscience, New York, 1995.
Seydel, R., Practical Bifurcation and Stability Analysis. From Equilibrium to Chaos, Springer, New York, 1994.
Kaas-Peterson, C., 'Computation, continuation, and bifurcation of torus solutions for dissipative maps and ordinary differential equations,' Physica D 25, 1987, 288–306.
Dormand, J. R. and Prince, P. J., 'A family of embedded Runge-Kutta formulae', Journal of Computational Applied Mathematics 6, 1980, 19–26.
Pyke, R., 'Nonlinear soil models for irregular cyclic loadings', Journal of the Geotechnical Engineering Division 105, 1979, 715–726.
Jayakumar, P., 'Modeling and identification in structural dynamics', Report No. EERL 87-01, California Institute of Technology, Pasadena, CA, 1987.
Jennings, P. C., 'Periodic response of a general yielding structure', Journal of Engineering Mechanics 90, 1964, 131–166.
Ramberg, W. and Osgood, W. R., 'Description of stress-strain curves by three parameters', Technical Note, National Advisory Committee on Aeronautics, 1943.
Kunnath, S. K., Mander, J. B., and Fang, L., 'Parameter identification for degrading and pinched hysteretic structural concrete systems', Engineering Structures 19, 1997, 224–232.
Mostaghel, N., 'Analytical description of pinching, degrading hysteretic systems', Journal of Engineering Mechanics 125, 1999, 216–224.
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Lacarbonara, W., Vestroni, F. Nonclassical Responses of Oscillators with Hysteresis. Nonlinear Dynamics 32, 235–258 (2003). https://doi.org/10.1023/A:1024423626386
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DOI: https://doi.org/10.1023/A:1024423626386