Double pendulum dynamics with non-vertical parametric excitation

Experimental and theoretical investigations of the dynamics of a double pendulum, with the periodic excitation making an angle of 22.5 with the vertical direction are presented. Such non verticality of the excitation induces a horizontal base-excitation component which acts as a direct excitation and as such can generate primary resonances. The equations of motion are given by two differential equations with time-varying coefficients: For vertical excitation there are four regions in the parameter space (fp,a), in which there are no angular oscillations, Three of them exhibit one or both arms in the upward position. Outside these regions there are angular oscillations or spinning of one or both arm. These regions are delimited the by the so called transition lines that are loci of bifurcations. For non vertical excitations, the stabilized angular positions no longer exist and the new four regions appears in which stable periodic oscillations takes place with frequency equal to the excitation frequency (f=fp). The four transition lines were obtained by direct integration of the equations of motion and also numerical continuation technique was applied to classify the bifurcations, so period doubling, saddle node, etc. bifurcations were identified due to the vertical symmetry breaking of the oscillations.