Abstract
Vibro-impact oscillators may undergo zero-velocity impacts, also known as grazing contacts. Near-grazing dynamics refer to low-velocity impacts. The conventional technique of local stability analysis suffers singularity when applied to grazing dynamics. Discontinuity mapping conceived by Nordmark provides a powerful tool to analyze and predict the plethora of complex phenomena due to grazing. This article intends to help beginners in vibro-impact dynamics better understand the concept of discontinuity mapping through a lucid derivation of the discontinuity mapping. The fundamental approach consists of three steps. First, a Poincaré map is introduced for the oscillatory dynamics without impacts. Second, impact dynamics near a grazing contact point are approximated using a series expansion to generate the so-called discontinuity mapping. Finally, the overall oscillations involving low-velocity impacts are analyzed using a combination of the Poincaré map and the discontinuity mapping derived in the previous two steps. We first present a transparent derivation of the discontinuity mapping for a generic one-degree-of-freedom vibro-impact oscillator. Then, the approach is applied to a linear mass spring oscillator whose vibrations are restricted by a rigid wall.
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References
Brogliato, B.: Nonsmooth Mechanics. Springer, New York (1999)
Chin, W., Ott, E., Nusse, H.E., Grebogi, C.: Grazing Bifurcation in Impact Oscillators. Physical Review E 50, 4427–4444 (1994)
Chin, W., Ott, E., Nusse, H.E., Grebogi, C.: Universal Behavior of Impact Oscillators near Grazing Incidence. Physics Letters A 201, 197–204 (1995)
Dankowicz, H., Nordmark, A.B.: On the Origin and Bifurcations of Stick-Slip Oscillations. Physica D 136, 280–302 (1999)
Dankowicz, H., Zhao, X.: Local Analysis of Co-dimension-one and Co-dimension-two grazing bifurcations in Impact Microatuators. Physica D 202, 238–257 (2005)
de Weger, J., Binks, D., Molenaar, J., van de Water, W.: Generic Behavior of Grazing Impact Oscillators. Physical Review Letters 76, 3951–3954 (1996)
de Weger, J., van de Water, W., Molenaar, J.: Grazing Impact Oscillators. Physical Review E 62, 2030–2041 (2000)
di Bernardo, M., Budd, C.J., Champneys, A.R.: Normal form maps for grazing bifurcations in n-dimensional piecewise-smooth dynamical systems. Physica D 160, 222–254 (2000)
di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P.: Piecewise-smooth dynamical systems: theory and applications. Applied Mathematics series no. 163. Springer, Heidelberg (2007)
Leine, R.I., Van Campen, D.H., Van de Vrande, B.L.: Bifurcations in nonsmooth discontinuous systems. Nonlinear Dynamics 23, 105–164 (2000)
Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods. Wiley, New York (1995)
Nordmark, A.B.: Non-periodic motion caused by grazing incidence in an impact oscillator. Journal of Sound and Vibration 145, 279–297 (1991)
Nordmark, A.B.: Universal limit mapping in grazing bifurcations. Physical Review E 55(1), 266–270 (1997)
Molenaar, J., de Weger, J.G., van de Walter, W.: Mappings of Grazing-Impact Oscillators. Nonlinearity 14, 301–321 (2001)
Zhao, X., Dankowicz, H., Reddy, C.K., Nayfeh, A.H.: Modeling and Simulation Methodology for Impact Microactuators. Journal of Micromechanics and Microengineering 14, 775–784 (2004)
Zhao, X.: Modeling and Simulation of MEMS Devices, Ph.D. Thesis, Virginia Tech. (2004)
Zhao, X., Dankowicz, H.: Unfolding Degenerate Grazing Dynamics in Impact Actuators. Nonlinearity 19, 399–418 (2006)
Zhao, X., Dankowicz, H.: Control of Impact Microactuators for Precise Positioning. ASME Journal of Computational and Nonlinear Dynamics 1, 65–70 (2006)
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Zhao, X. (2009). Discontinuity Mapping for Near-Grazing Dynamics in Vibro-Impact Oscillators. In: Ibrahim, R.A., Babitsky, V.I., Okuma, M. (eds) Vibro-Impact Dynamics of Ocean Systems and Related Problems. Lecture Notes in Applied and Computational Mechanics, vol 44. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00629-6_28
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DOI: https://doi.org/10.1007/978-3-642-00629-6_28
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