Bifurcations in the mean angle of a horizontally shaken pendulum: Analysis and experiment. (English) Zbl 0908.70018

Schmitt, John M.; Bayly, Philip V.

Nonlinear Dyn. 15, No. 1, 1-14 (1998).

Summary: A pendulum excited by high-frequency horizontal displacement of its pivot point will vibrate with small amplitude about a mean position. The mean value is zero for small excitation amplitudes, but if the excitation is large enough the mean angle can take on non-zero values. This behavior is analyzed using the method of multiple time scales. The change in the mean angle is shown to be result of a pitchfork bifurcation, or a saddle-node bifurcation if the system is imperfect. Analytical predictions of the mean angle as a function of frequency and amplitude are confirmed by physical experiment and numerical simulation.

MSC:

70K20	Stability for nonlinear problems in mechanics
70K40	Forced motions for nonlinear problems in mechanics
70-05	Experimental work for problems pertaining to mechanics of particles and systems

Keywords:

method of multiple time scales; pitchfork bifurcation; saddle-node bifurcation