Parametric excitation of an internally resonant double pendulum. II
The second author [ibid. 36, 337-345 (1985; Zbl 0576.70019)] has examined the weakly nonlinear response of a damped, internally resonant double pendulum to parametric excitation at twice the frequency of the dominant (slower) pendulum. He found that, for all values of frequency offset and dissipation, regular motion is obtained in the form of either parametrically excited oscillations or vertical translation of the system as a rigid body. We now suppose that the driving frequency approximates twice the frequency of the satellite pendulum (four times the frequency of the dominant pendulum) and find that, in contrast to I, chaotic motion may occur. We obtain the evolution equations that govern the envelope of the oscillations by postulating the angular displacements, \(\theta_ n\), \(n=1,2\), of the pendulums in the form of slowly modulated sinusoids. We examine the fixed points and bifurcation structure of these evolution equations through the construction of the resonance curves (energy vs frequency). Finally, in Sect. 5, we present numerical results that support the conclusion that chaotic motion occurs in certain parametric neighbourhoods. The corresponding case in which the internally resonant double pendulum is designed so that the satellite has the lower natural frequency is discussed in an appendix.
damped, internally resonant double pendulum
scientific article; zbMATH DE number 4005671