Nonlinear systems have a range of behaviour not seen in linear vibrating systems. In this chapter the phenomena associated with nonlinear vibrating systems are described in detail. In the absence of exact solutions, the analysis of nonlinear systems is usually undertaken using approximate analysis, numerical simulations and geometrical techniques. This form of analysis has become known as dynamical systems theory (or sometimes chaos theory) and is based on using a system state space. In this chapter the basic ideas of dynamical systems are applied to vibrating systems. Finally, the changes in system behaviour as one (or more) of the parameters is varied are discussed. Such changes are known as bifurcations, and they are highly significant for the understanding of nonlinear systems.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Bursi, O. S. and Wagg, D. J., editors (2008). Modern Testing Techniques for Structural Systems. Springer-Verlag. ISBN-10: 3211094440.
Carrella, A., Brennan, M. J., and Waters, T. P. (2007). Static analysis of a passive vibration isolator with quasi-zero-stiffness characteristic. Journal of Sound and Vibration, 301(3–5), 678–689.
Cartmell, M. (1990). Introduction to linear, parametric and nonlinear vibrations. Chapman and Hall.
Coates, R. C., Coutie, M. G., and Kong, F. K. (1972). Structural Analysis. Chapman Hall: London.
Diekmann, O., van Gils, S., Verduyn Lunel, S., and Walther, H. (1995). Delay equations, volume 110. Applied Mathematical Sciences.
Fausett, L. V. (1999). Applied numerical analysis using Matlab. Prentice Hall.
Foale, S. and Thompson, J. M. T. (1991). Geometrical concepts and computational techniques of nonlinear dynamics. Computer Methods For Applications In Mechanical Engineering, 89, 381–394.
Frish-Fay, R. (1962). Flexible Bars. Butterworths: London.
Glendinning, P. (1994). Stability, instability and chaos. Cambridge University Press.
Guckenheimer,J. and Holmes, P. (1983). Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag: New York.
Hsu, C. S. (1987). Cell-to-cell mapping. Springer-Verlag: New York.
Inman, D. J. (2006). Vibration with control. Wiley.
Jordan, D. W. and Smith, P. (1999). Nonlinear ordinary differential equations; an introduction to dynamical systems. Oxford University Press. 3rd Edition.
Khalil, H. K. (1992). Nonlinear Systems. Macmillan: New York.
Krauskopf, B., Osinga, H. M., and Galan-Vioque, J., editors (2007). Numerical Continuation Methods for Dynamical Systems. Springer.
Kuznetsov, Y. A. (2004). Elements of Applied Bifurcation Theory. Springer.
McInnes, C. R., Gorman, D. G., and Cartmell, M. P. (2008). Enhanced vibrational energy harvesting using nonlinear stochastic resonance. Journal of Sound and Vibration, 318, 655–662.
Moon, F. C. (1987). Chaotic vibrations: An introduction for applied scientists and engineers. John Wiley: New York.
Newland, D. E. (1993). An introduction to random vibrations and spectral analysis. Pearson.
Press, W. H., Teukolsky, S. A., Vettering, W. T., and Flannery, B. P. (1994). Numerical recipes in C. Cambridge University Press. 2nd Ed.
Sastry, S. (1999). Nonlinear systems:Analysis, stability and control. SpringerVerlag: New York.
Seyranian, A. P. and Mailybaev, A. A. (2003). Multiparameter stability theory with mechanical applications. World Scientific.
Sontag, E. D. (1998). Mathematical control theory. Springer-Verlag.
Stépan, G. (1989). Retarded Dynamical Systems Stability and Characteristic Functions. Longman Scientific & Technical.
Strogatz, S. H. (2001). Nonlinear Dynamics and Chaos. Perseus Books Group.
Thompson, J. M. T. (1982). Instabilities and catastrophes in science and engineering. John Wiley & Sons.
Thompson, J. M. T. and Hunt, G. W. (1973). A general theory of elastic stability. John Wiley & Sons.
Thompson, J. M. T. and Stewart, H. B. (2002). Nonlinear dynamics and chaos. John Wiley: Chichester.
Virgin, L. N. (2000). An introduction to experimental nonlinear dynamics. Cambridge.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Canopus Academic Publishing Limited
About this chapter
Cite this chapter
(2010). Nonlinear Vibration Phenomena. In: Wagg, D., Neild, S. (eds) Nonlinear Vibration with Control. Solid Mechanics and Its Applications, vol 170. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-2837-2_2
Download citation
DOI: https://doi.org/10.1007/978-90-481-2837-2_2
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-2836-5
Online ISBN: 978-90-481-2837-2
eBook Packages: EngineeringEngineering (R0)