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 Ct. 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 Ct. 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.

Invariant Manifolds, Nonclassical Normal Modes, and Proper Orthogonal Modes in the Dynamics of the Flexible Spherical Pendulum.- Normal Vibrations in Near-Conservative Self-Excited and Viscoelastic Nonlinear Systems.- Nonlinear Normal Modes in a System with Nonholonomic Constraints.- Nonlinear Normal Modes of a Parametrically Excited Cantilever Beam.- Normal Modes and Boundary Layers for a Slender Tensioned Beam on a Nonlinear Foundation.- The Description of Localized Normal Modes in a Chain of Nonlinear Coupled Oscillators Using Complex Variables.- Spatially Localized Models of Extended Systems.- Mode Localization in Dynamics and Buckling of Linear Imperfect Continuous Structures.- Dynamics of Relative Phases: Generalised Multibreathers.- Nonlinear Modal Analysis of Structural Systems Using Multi-Mode Invariant Manifolds.- Localization in Nonlinear Mistuned Systems with Cyclic Symmetry.- Mode Localization Induced by a Nonlinear Control Loop.- Transition of Energy to a Nonlinear Localized Mode in a Highly Asymmetric System of Two Oscillators.- Application of Nonlinear Normal Mode Analysis to the Nonlinear and Coupled Dynamics of a Floating Offshore Platform with Damping.- Performance of Nonlinear Vibration Absorbers for Multi-Degrees-of-Freedom Systems Using Nonlinear Normal Modes.