Normal Modes and Localization in Nonlinear Systems

The nonlinear normal modes of a parametrically excited cantilever beam are constructed by directly applying the method of multiple scales to the governing integral-partial differential equation and associated boundary conditions. The effect of the inertia and curvature nonlin­ earities and the parametric excitation on the spatial distribution of the deflection is examined. The results are compared with those obtained by using a single-mode discretization. In the absence of linear viscous and quadratic damping, it is shown that there are nonlinear normal modes, as defined by Rosenberg, even in the presence of a principal parametric excitation. Furthermore, the nonlinear mode shape obtained with the direct approach is compared with that obtained with the discretization approach for some values of the excitation frequency. In the single-mode discretization, the spatial distribution of the deflection is assumed a priori to be given by the linear mode shape ¢n, which is parametrically excited, as Equation (41). Thus, the mode shape is not influenced by the nonlinear curvature and nonlinear damping. On the other hand, in the direct approach, the mode shape is not assumed a priori; the nonlinear effects modify the linear mode shape ¢n. Therefore, in the case of large-amplitude oscillations, the single-mode discretization may yield inaccurate mode shapes. References 1. Vakakis, A. F., Manevitch, L. I., Mikhlin, Y. v., Pilipchuk, V. N., and Zevin A. A., Nonnal Modes and Localization in Nonlinear Systems, Wiley, New York, 1996.

Vibration analysis of nonlinear systems poses great challenges in both physics and engineering. This innovative book takes a completely new approach to the subject, focusing on nonlinear normal modes (NNMs) and nonlinear mode localization, and demonstrates that these concepts provide an excellent analytical tool for the study of nonlinear phenomena that cannot be analyzed by conventional techniques based on linear or quasi–linear theory.

Written by professor Alexander F. Vakakis and four colleagues from Russia and the Ukraine, the book employs the similarity of NNMs to the normal modes of classical vibration theory to create a new perspective on this highly specialized, yet steadily growing field. Providing a solid foundation in theory, the authors explain, for example, the design of systems with passive or active motion confinement properties and examine applications of essentially nonlinear phenomena to the vibration and shock isolation of flexible, large–scale structures.

Much of the material presented is completely new or appearing here for the first time in Western engineering literature––including numerous mathematical techniques for studying NNMs, their bifurcations, and the localization phenomena associated with them. The authors describe strongly nonlinear analytical methodologies that permit the analytical treatment of oscillators in strongly nonlinear regimes. They then demonstrate the application of these methodologies in numerous practical engineering and physics problems. Also presented is the method of nonsmooth temporal transformations, which enables analytic perturbation studies of strongly nonlinear oscillations, a new asymptotic methodology for analyzing standing solitary waves in some classes of nonlinear partial differential equations, and some new results on localized or nonlocalized oscillations of vibro–impact systems.

Supplemented with an extensive bibliography and numerous illustrations and examples that demonstrate techniques and applications, Normal Modes and Localization in Nonlinear Systems is a useful text and professional guide for physicists studying nonlinear oscillations and waves, for vibration specialists, design engineers, and researchers studying nonlinear dynamics, and for graduate students in applied mechanics and mechanical engineering. It also offers a multitude of new concepts and techniques that can form the basis for future research in nonlinear dynamics and vibrations.

A complete guide to the theory and applications of nonlinear normal modes and nonlinear mode localization

This landmark volume offers a completely new angle on the study of vibrations in discrete or continuous nonlinear oscillators. It describes the use of NNMs to analyze the vibrations of nonlinear systems and design systems with motion confinement properties.

Normal Modes and Localization in Nonlinear Systems features
∗ New and established mathematical techniques that can be used for more refined vibration and shock isolation designs of practical flexible structures
∗ Complete coverage of free and forced motions in systems with weak or strong nonlinearities, including results that cannot be captured with existing linear or quasi–linear techniques
∗ A new method for analyzing strongly nonlinear systems that permits perturbation analysis of systems with essential nonlinearities
∗ A theoretical link between nonlinear normal modes and standing solitary waves
∗ The first experimental verifications of nonlinear mode localization and nonlinear motion confinement in flexible engineering structures