Dynamics of multi degree-of-freedom stationary and nonstationary nonlinear systems

In this thesis, the dynamics of weakly nonlinear multi degree-of-freedom systems under resonant excitations is studied. Using a first-order averaging technique, the amplitude equations for damped systems with completely general quadratic nonlinearities are obtained for all the six cases of forcing that are possible. The stability and bifurcation behavior of their solutions is studied for the case when the external excitation is in resonance with the higher frequency mode of the system, which in turn is in 1:2 internal resonance with the lower mode. A second-order averaging technique is used to substantiate the results of the first-order analysis. Using a generalization of the Melnikov method, the parameter regions for chaotic dynamics of the system, as exhibited by the undamped amplitude equations, are found. Numerical investigations show that chaos persists even in the presence of weak damping. It is shown that under conditions of internal resonance, the dynamics of a pendulum attached to an N degree-of-freedom linear structure is essentially governed by the four first-order amplitude equations studied in the main body of this work. The use of the pendulum as a vibration absorber is explored. Finally, numerical investigations are undertaken to study the response of two degree-of-freedom systems with quadratic nonlinearities, also when the excitation frequency evolves slowly in time. An analytical method to predict special motions of the nonstationary system near pitchfork bifurcation points is developed. The technique allows for the study of dependence of solutions on the system parameters and on initial conditions