Ebook: 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.

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This book contains a collection of original papers on nonlinear normal modes and localization in dynamical systems from leading experts in the field. The reader will find new analytical and computational techniques for studying normal modes and localization phenomena in nonlinear discrete and continuous oscillators. In addition, examples are provided of applications of these concepts to diverse problems of engineering and applied mathematics, such as nonlinear control of micro-gyroscopes, dynamics of floating offshore platforms, buckling of imperfect continua, order reduction of nonlinear systems, dynamics of nonlinear vibration absorbers, spatial localization and pattern formation in extended systems, singular asymptotics and nonlinear modal interactions and energy pumping in coupled oscillators.

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This book contains a collection of original papers on nonlinear normal modes and localization in dynamical systems from leading experts in the field. The reader will find new analytical and computational techniques for studying normal modes and localization phenomena in nonlinear discrete and continuous oscillators. In addition, examples are provided of applications of these concepts to diverse problems of engineering and applied mathematics, such as nonlinear control of micro-gyroscopes, dynamics of floating offshore platforms, buckling of imperfect continua, order reduction of nonlinear systems, dynamics of nonlinear vibration absorbers, spatial localization and pattern formation in extended systems, singular asymptotics and nonlinear modal interactions and energy pumping in coupled oscillators.
Content:
Front Matter....Pages I-1
Invariant Manifolds, Nonclassical Normal Modes, and Proper Orthogonal Modes in the Dynamics of the Flexible Spherical Pendulum....Pages 3-31
Normal Vibrations in Near-Conservative Self-Excited and Viscoelastic Nonlinear Systems....Pages 33-48
Nonlinear Normal Modes in a System with Nonholonomic Constraints....Pages 49-64
Nonlinear Normal Modes of a Parametrically Excited Cantilever Beam....Pages 65-77
Normal Modes and Boundary Layers for a Slender Tensioned Beam on a Nonlinear Foundation....Pages 79-93
The Description of Localized Normal Modes in a Chain of Nonlinear Coupled Oscillators Using Complex Variables....Pages 95-109
Spatially Localized Models of Extended Systems....Pages 111-132
Mode Localization in Dynamics and Buckling of Linear Imperfect Continuous Structures....Pages 133-156
Dynamics of Relative Phases: Generalised Multibreathers....Pages 157-182
Nonlinear Modal Analysis of Structural Systems Using Multi-Mode Invariant Manifolds....Pages 183-205
Localization in Nonlinear Mistuned Systems with Cyclic Symmetry....Pages 207-220
Mode Localization Induced by a Nonlinear Control Loop....Pages 221-236
Transition of Energy to a Nonlinear Localized Mode in a Highly Asymmetric System of Two Oscillators....Pages 237-253
Application of Nonlinear Normal Mode Analysis to the Nonlinear and Coupled Dynamics of a Floating Offshore Platform with Damping....Pages 255-274
Performance of Nonlinear Vibration Absorbers for Multi-Degrees-of-Freedom Systems Using Nonlinear Normal Modes....Pages 275-292
Back Matter....Pages 293-293

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This book contains a collection of original papers on nonlinear normal modes and localization in dynamical systems from leading experts in the field. The reader will find new analytical and computational techniques for studying normal modes and localization phenomena in nonlinear discrete and continuous oscillators. In addition, examples are provided of applications of these concepts to diverse problems of engineering and applied mathematics, such as nonlinear control of micro-gyroscopes, dynamics of floating offshore platforms, buckling of imperfect continua, order reduction of nonlinear systems, dynamics of nonlinear vibration absorbers, spatial localization and pattern formation in extended systems, singular asymptotics and nonlinear modal interactions and energy pumping in coupled oscillators.
Content:
Front Matter....Pages I-1
Invariant Manifolds, Nonclassical Normal Modes, and Proper Orthogonal Modes in the Dynamics of the Flexible Spherical Pendulum....Pages 3-31
Normal Vibrations in Near-Conservative Self-Excited and Viscoelastic Nonlinear Systems....Pages 33-48
Nonlinear Normal Modes in a System with Nonholonomic Constraints....Pages 49-64
Nonlinear Normal Modes of a Parametrically Excited Cantilever Beam....Pages 65-77
Normal Modes and Boundary Layers for a Slender Tensioned Beam on a Nonlinear Foundation....Pages 79-93
The Description of Localized Normal Modes in a Chain of Nonlinear Coupled Oscillators Using Complex Variables....Pages 95-109
Spatially Localized Models of Extended Systems....Pages 111-132
Mode Localization in Dynamics and Buckling of Linear Imperfect Continuous Structures....Pages 133-156
Dynamics of Relative Phases: Generalised Multibreathers....Pages 157-182
Nonlinear Modal Analysis of Structural Systems Using Multi-Mode Invariant Manifolds....Pages 183-205
Localization in Nonlinear Mistuned Systems with Cyclic Symmetry....Pages 207-220
Mode Localization Induced by a Nonlinear Control Loop....Pages 221-236
Transition of Energy to a Nonlinear Localized Mode in a Highly Asymmetric System of Two Oscillators....Pages 237-253
Application of Nonlinear Normal Mode Analysis to the Nonlinear and Coupled Dynamics of a Floating Offshore Platform with Damping....Pages 255-274
Performance of Nonlinear Vibration Absorbers for Multi-Degrees-of-Freedom Systems Using Nonlinear Normal Modes....Pages 275-292
Back Matter....Pages 293-293
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