Dynamics of a slender beam with an attached mass under combination parametric and internal resonances, part II: periodic and chaotic responses

摘要

The governing second order temporal differential equation of a slender beam with an attached mass at an arbitrary position under vertical base excitation which retains the cubic non-linearities of geometric and inertial type is reduced to a set of first order differential equations by the method of normal forms for combination parametric and internal resonances of 3:1. These equations are used to find the periodic, quasi-periodic and chaotic responses of the system for various bifurcating parameters, namely, damping, amplitude and frequency of base motion, attached mass and its location. Bifurcation set, mixed-mode oscillation, period-doubling, quasi-periodic orbits and different routes to chaos, namely, alternate periodic-chaotic transition, torus breakdown and intermittency have been studied for the above mentioned bifurcating parameters using phase portrait, Poincare section, time and power spectra.