The dynamics of a spherical pendulum with a vibrating suspension

The motion of a spherical pendulum whose point of suspension performs high- frequency vertical harmonic oscillations of small amplitude is investigated . It is shown that two types of motion of the pendulum exist when it perfor ms high-frequency oscillations close to conical motions, for which the pend ulum makes a constant angle with the vertical and rotates around it with co nstant angular velocity. For the motions of the first and second types the centre of gravity of the pendulum is situated below and above the point of suspension, respectively. A bifurcation curve is obtained, which divides th e plane of the parameters of the problem into two regions. In one of these only the first type of motion can exist, while in the other, in addition to the first type of motion, there are two motions of the second type. The pr oblem of the stability of these motions of the pendulum, close to conical, is solved. It is shown that the first type of motion is stable, while of th e second type of motion, only the motion with the higher position of the ce ntre of gravity is stable. (C) 1999 Elsevier Science Ltd. All rights reserv ed.