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5 - Dynamics of Systems with Sign-Changing Stiffness

Chaotic Vibration Motion and Stability Conditions

Published online by Cambridge University Press:  29 October 2021

Chang-Myung Lee  and
Vladimir Nicholas Goverdovskiy
Chang-Myung Lee
Affiliation:
University of Ulsan, South Korea
Vladimir Nicholas Goverdovskiy
Affiliation:
University of Ulsan, South Korea
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Summary

Stability in large of the systems with mechanisms of negative and quasi-zero stiffness plays an important role for improvement of the infra-low vibration protection. These mechanisms are predisposed to chaotic vibration motion. Analysis of chaotic vibration and comparative selection of the mechanisms are to be reasonable steps before deciding next steps in designing the vibration protection systems. Their dynamic behavior can be diagnosed and predicted by the qualitative and quantitative methods for analysis of chaotic motion. An algorithm has been developed to study chaotic motion of the mechanisms, and the conditions of dynamic stability of the systems with such mechanisms are formulated. The algorithm is based on the Lyapunov largest exponent and Poincare map of phase trajectory methods and includes (a) formulation of chaotic motion models and criterial experiments for the mechanisms and systems, (b) technique of comparative analysis of the models, (c) computation procedure to estimate their dynamic stability in large, (d) formulation of design and functional parameters for providing stable motion of the systems in the infra-frequency range, including near-zero values. Validity of the algorithm is demonstrated through the development of active pneumatic suspensions supplied with passive mechanisms of variable negative stiffness.

Keywords

System with sigh-changing stiffnesschaotic motiondetecting and measuring chaosstability conditionsselection of the systems
Type
Chapter
Information
Vibration Protection Systems
Negative and Quasi-Zero Stiffness
 , pp. 116 - 144
Publisher: Cambridge University Press
Print publication year: 2021

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References

Clemens, H. and Wauer, J., Free and forced vibrations of a snap-through oscillator, Report of Institute für Technische Mechanik Universität. Karlsruhe, Germany, 1981.Google Scholar
Seydel, R., From Equilibrium to Chaos: Practical Bifurcation and Stability Analysis (New York: Elsevier Science, 1988).Google Scholar
Wiggins, S., Introduction to Applied Nonlinear Dynamic Systems and Chaos (New York: Springer, 1990).Google Scholar
Thompson, J.M.T. and Gray, P., eds., Chaos and Dynamical Complexity in the Physical Sciences (London: McGraw-Hill, 1990).Google Scholar
Moon, F.C., Chaotic and Fractal Dynamics: An Introduction for Applied Scientists (New York: Wiley, 1992).Google Scholar
Kim, J.H. and Stringer, J., eds., Applied Chaos (San Francisco: Wiley, 1992).Google Scholar
Ott, E., Chaos in Dynamical Systems (Cambridge: Cambridge University Press, 1993).Google Scholar
Zakrzhevsky, M.V., Ivanov, Y.M., and Frolov, V.Y., Analysis of nonlinear vibration mechanics and chaos. In Proceedings the Vibration Machines and Technology (Kursk, Russia, 1997), 24‒28. In Russian.Google Scholar
Thomsen, J.J., Vibrations and Stability: Advanced Theory, Analysis, and Tools (Berlin: Springer, 2003).Google Scholar
Kakmeni, F.M.M., Bowong, S., Tchawoua, C., and Kaptouom, E., Strange attractors and chaos control in a Duffing–Van der Pol oscillator with two external periodic forces. Journal of Sound and Vibration, 277 (2004), 783‒799.Google Scholar
Awrejcewicz, J., Papkova, I.V., Krysko, A.V., and Krysko, V.A., Analysis of chaotic parametric vibrations of flexible multi-layer shells. In Dynamical Systems ‒ Applications, ed. Awrejcewicz, J., Kaźmierczak, M., Olejnik, P., and Mrozowski, J. (Lodz: Technical University of Lodz Press, 2013), 687‒694.Google Scholar
Kauderer, H., Nichtlineare Mechanik (Berlin: Springer, 1958). In German.Google Scholar
Volmir, A.S., Stability of Deformed Systems (Moscow: Nauka, 1967). In Russian.Google Scholar
Budianski, B., Theory of buckling and post-buckling behavior of elastic structures. In Advances in Applied Mechanics, ed. Brooke Benjamin, T., Fung, Y. C., German, P. et al. (New York: Academic Press, 1974), 1−65.Google Scholar
Panovko, J.G. and Gubanova, I.I., Stability and Vibration of Elastic Systems, 4th ed. (Moscow: Science, 1987). In Russian.Google Scholar
Karmyshyne, A.V., ed., Methods of Dynamic Analysis and Testing for Thin-Walled Structures (Moscow: Engineering, 1990). In Russian.Google Scholar
Lee, W.K. and Park, H.D., Chaotic dynamics of a harmonically excited spring-pendulum system with internal resonances. Nonlinear Dynamics, 14 (1997), 211−229.Google Scholar
Goverdovskiy, V.N. and Lee, C.-M., A method of stiffness control in a suspension for a vehicle driver compact seat, RU Patent 2214335, 2003.Google Scholar
Goverdovskiy, V.N., Temnikov, A.I., Furin, G.G., and Lee, C.-M., Stiffness control mechanism for a compact seat suspension, RU Patent 2216461, 2003.Google Scholar
Moon, F.C., Chaotic Vibrations: An Introduction for Applied Scientists and Engineers (Chichester: Wiley, 2004).Google Scholar
Syta, A., Litak, G., Lenci, S. and Scheffler, M., Chaotic vibrations of the Duffing system with fractional damping. Chaos, 24 (2014), 013107.Google Scholar
Huynh, B.H., Tjahjowidodo, T., Zhong, Z., Wang, Y. and Srikanth, N., Chaotic responses on vortex induced vibration systems supported by bi-stable springs. In Proceedings the Conference on Noise and Vibration Engineering (Leuven, Belgium), 2016, 1–10.Google Scholar
Liu, X. and Ma, L., Chaotic vibration, bifurcation, stabilization and synchronization control for fractional discrete-time systems. Applied Mathematics and Computation, 385 (2020), 125423.Google Scholar
Awrejcewicz, J., Krysko, A.V., Kuterov, I.E., Zagniborova, N.A., Zhigalov, M.V., and Krysko, V.A., Analysis of chaotic vibrations of flexible plates using fast Fourier transforms and wavelets. International Journal of Structural Stability and Dynamics, 7 (2013). Available at www.worldscientific.com.Google Scholar
Ungar, E.E. and Pirsons, K.S., New constant force spring systems. Product Engineering, 27 (1961), 32‒34.Google Scholar
Alabuzhev, P.M., Kim, L.I., Grytchine, A.A., Migirenko, G.S., Khon, V.F., and Stepanov, P.T., Vibration Protecting and Measuring Systems with Quasi-Zero Stiffness (New York: Taylor and Francis, 1989).Google Scholar
Yurjev, G.S., Vibration Isolation of Precision Instrument (Novosibirsk, Russia: Budker Institute of Nuclear Physics, 1989). In Russian.Google Scholar
Zyuev, A.K. and Lebedev, O.N., Highly Effective Vibration Isolation of Ship Equipment (Novosibirsk, Russia: State Academy of Water Transport, 1997). In Russian.Google Scholar
Lee, C.-M., Goverdovskiy, V.N., and Temnikov, A.I., Design of springs with “negative” stiffness to improve vehicle driver vibration isolation. Journal of Sound and Vibration, 302 (2007), 865−874.CrossRefGoogle Scholar
Lee, C.-M. and Goverdovskiy, V.N., Alternative vibration protecting systems for men-operators of transport machines: Modern level and prospects. Journal of Sound and Vibration, 249 (2002), 635‒647.Google Scholar
Lee, C.-M., Bogachenkov, A.H., Goverdovskiy, V.N., Temnikov, A.I., and Shynkarenko, Y.V., Position control of seat suspension with minimum stiffness. Journal of Sound and Vibration, 292 (2006), 435‒442.Google Scholar
Lee, C.-M., Goverdovskiy, V.N., and Samoilenko, S.B., Prediction of non-chaotic motion of the elastic system with small stiffness. Journal of Sound and Vibration, 272 (2004), 643‒655.Google Scholar
Lee, C.-M. and Goverdovskiy, V.N., Damping control in a spring and suspension with sign-changing stiffness. Journal of Sound and Vibration, 373 (2016), 19‒28.Google Scholar

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