Global bifurcations and chaos in nonlinear mechanical systems.

摘要

The local and global bifurcation behavior of various structural and mechanical systems have been examined in detail. The analysis is divided into three main parts. In the first part, global bifurcation analysis is performed for externally excited nonlinear systems with initial imperfections and a semisimple linear operator. Explicit restrictions are obtained on parameters where such systems can exhibit chaotic dynamics. The results are demonstrated on a shallow arch system, which is subjected to a spatio-temporal loading, under various resonance conditions.;In the second part, local dynamics is investigated for a parametrically excited nonlinear system with a non-semisimple linear operator. The global dynamics associated with such systems is also examined by imposing the reversible symmetry on the original system. The results from the general analysis, which are applicable to various physical applications, are used to study the flexural-torsional motion of a rectangular beam.;In the third part, a systematic formulation of nonlinear oscillations of a spinning disc is obtained. These questions of motion include the effects due to inherent bending rigidity, membrane stresses arising from centrifugal forces, non-axisymmetry of the in-plane and transverse displacements, geometric nonlinearities, aerodynamic damping, parametric excitation due to time varying spin rate, etc. For the constant rotation case, the linearized equations of motion are solved by taking both membrane as well as flexural stiffness effects into account. This leads to a power series solution for the radial shape functions and harmonic solutions for the circumferential shape functions. The two-dimensional eigen-functions thus obtained can describe a disc mode with any number of nodal diameters and nodal circles, and the resulting eigen-frequencies match well with the numerical results. The nonlinear and non-axisymmetric in-plane response is also determined. A two-degree-of-freedom system of nonlinear ordinary differential equations, which governs the dynamic evolution of the amplitudes of traveling waves associated with the dominant mode of the transverse motion, is obtained. The local bifurcations are examined in the resulting equations of motion, both in the presence and the absence of imperfections. The existence of chaotic behavior is also proven in the spinning disc system by showing the existence of single and multi-pulse Silnikov type orbits in the presence of dissipation effects. Throughout this research, the relationship between the mathematical results and their physical implications have been interpreted for the engineering applications considered.