Nonlinear Dynamics Of Flexible Beams Undergoing A Large Overall Motion

This dissertation systematically presents a study on the nonlinear dynamics of flexible beams undergoing a large overall motion, including the nonlinear dynamic modeling of these beams, the periodic vibration of a cantilever beam subject to axial movement of base, the dynamic stability analysis of parametrically excited slender beams undergoing a larger linear motion, the nonlinear dynamic behaviors of slender beams under the combination of both parametric excitation and internal excitation, and the largest Lyapunov exponent and the almost certain stability analysis of slender beams under the narrow-band random parametric excitation. The purpose of this dissertation is to gain an insight into the inherent nonlinear dynamics of flexible beams undergoing a large overall motion. The dissertation is organized as following.Chapter 1 surveys the state-of-the-art of dynamics of flexible structures undergoing a large overall motion and the advance in the corresponding theories, and presents the significance, main contents and arrangement of the dissertation.In Chapter 2, a set of nonlinear differential equations is established by using Kane's method for the flexible beams undergoing a large overall motion. Compared with the linear model where the generalized inertial force and generalized active force are linearized, the present model takes these nonlinear terms into consideration, which makes it possible for one to capture and understand their complicated nonlinear dynamics.Chapter 3 focuses on the dynamic behaviors of a cantilever beam under an axial movement of its base. The results show that the nonlinear inertia terms produce a softening effect and play a significant role in the planar response of the second mode and the higher ones. On the other hand, the nonlinear geometric terms produce a hardening effect and dominate the planar response of the first mode. The validity of the present modeling results is clarified by comparing and calculating the corresponding coefficients in the present modeling method and other investigators' modeling methods, which, to a large extent, gives one the necessary theoretical security to continue the further studies.In Chapter 4, the first order approximation to the solution of a set of nonlinear differential equations, which is established in Chapter 2 and governs the planar motion of flexible beams undergoing a large linear motion, are systematically derived via the method of multiple scales. In the case of a simply supported beam with 3:1 internal resonance between the first two modes, the dynamic stability of the trivial state of the system is investigated by using the Cartesian transformation in detail. The equations of approximate stability boundary are derived. Finally, the modulation equations are reduced to a two-dimensional system and the type of the Hopf bifurcations are determined in the corresponding vicinity of the bifurcations via the center manifold theorem and the limit cycles are found.In Chapter 5, the nonlinear planar response of a simply supported flexible beam undergoing a large linear motion, which is seldom dealt with by other investigators, to a principal parametric resonance of either its first or second mode or a combination parametric resonance of the additive type of its first two modes is investigated. The comprehensive periodic motions and periodically and chaotically modulated motions of the beam are determined by investigating the equilibrium and dynamic solutions of the modulation equations and the center manifold theorem are also used to determine the type of the Hopf bifurcations. The numerously complicated nonlinear dynamic behaviors of the beam are revealed.Chapter 6 presents the nonlinear dynamic behaviors of a simply supported flexible beam subject to narrow-band random parametric excitation, in which either the principal parametric resonance of its first mode or a combination parametric resonance of the additive type of its first two modes with or without 3:1 internal resonance between the first two modes is taken in to conside...