One of the most important features of nonlinear systems with several degrees of freedom is the presence of internal resonances at certain relations between natural frequencies of different modes. This monograph is the first book devoted predominantly to internal resonances in different mechanical systems including those of practical importance.The main purpose is to consider the internal resonances from the general point of view and to elucidate their role in applied nonlinear dynamics by using an efficient approach based on introducing the complex representation of equations of motion (together with the multiple scale method). Considered here are autonomous and nonautonomous discrete two-degree-of-freedom systems, infinite chains of particles, and continuous systems, including circular rings and cylindrical shells. Specific attention is paid to the case of one-to-one internal resonance in systems with cubic nonlinearities. Steady-state and nonstationary regimes of motion, interaction of the internal and external resonances at forced oscillations, and bifurcations of steady-state modes and their stability are systematically studied.
Single-Degree-of-Freedom Systems; Autonomous Two-Degree-of-Freedom Symmetric Cubic Systems with Close Natural Frequencies; Nonautonomus Two-Degree-of-Freedom Cubic Systems with Close Natural Frequencies; Nonlinear Flexural Free and Forced Oscillations of a Circular Ring (with Account of Interaction of Conjugate Modes); Localized Normal Modes in a Chain of Nonlinear Coupled Oscillators; Nonlinear Dynamics of Coupled Oscillatory Chains; Nonlinear Dynamics of Strongly Non-Homogeneous Chains with Symmetric Characteristics; Transversal Dynamics of One-Dimensional Chains on Nonlinear Asymmetric Substrate.