ROUTES TO CHAOS OF A PENDULUM WITH VERTICALLY ROTATING PIVOT

Each blade of a mechanical shredder may be modeled by a pendulum whose pivot rotates in a vertical circle with angular speed OMEGA. It is well known from field tests that small oscillations about radial lines, which are stable at high operating speed, degenerate into disordered motion when speeds become low enough after the machine is shut off. The transition to chaos of small periodic motion about radial lines as the driving speed of the pivot is reduced is investigated for several values of the system parameters. Numerical methods are used to obtain time histories, Poincare maps, response curves and frequency spectra. The harmonic balance method is used to obtain response curves for oscillatory periodic solutions, whose stability is determined by the Floquet method. In the undamped case, instability associated with pitchfork or tangent bifurcations precedes chaos. When damping is introduced, additional routes involving intermittency and period-doubling cascades may precede chaotic motion characterized by strange attractors.