Structural Dynamics of Nonlinear Mechanical Systems with Cyclic Symmetry

摘要

The proposer has investigated the dynamics of externally forced, weakly coupled, non-linear oscillators. The objective in this project is to use the constructs of group theory and equivalent bifurcation analysis to produce an in-depth picture of the dynamics of such oscillators. In this report, he reviews the concepts of group theory and equivalent bifurcation theory, derive the mathematical model of a three element oscillator, classify the possible solutions of the system, and discuss the predicted dynamics of the weakly coupled cyclic system. Specifically, the proposer derives the equations of motion for weakly non-linear dynamics and applied the method of averaging to reduce the system of non-linear coupled differential equations to an autonomous dynamical system. The averaged equations of motion prove to be a low order polynomial normal form for the general case of cyclically coupled and harmonically excited mechanical systems. An isotropy sub-group lattice is used to obtain information on fixed point solutions of the averaged system of equations, which correspond to periodic motions in the original system. The author shows that, in the case of the free vibration model, there can be three possibilities of solution for each oscillator depending upon initial conditions.