Bifurcations and dynamical analysis of Coriolis-stabilized spherical lagging pendula

The dynamics of a spherical pendulum with a horizontally rotating support, exhibiting regimes of intriguing lagging behaviour in the presence of aerodynamic drag, is analysed in the present study. The dynamic equilibria of the pendulum are derived through both numerical and analytical means, with excellent experiment agreement across key system parameters. System attractors indicate the critical role played by drag in convergence towards the various equilibria. The stability of the various equilibria is also investigated. Interestingly, one of the folded states is found to be dynamically stabilized by the Coriolis force, with the resulting stability bound by a saddle-node bifurcation from below and a Hamiltonian Hopf bifurcation from above; multiple limit cycles are discovered past the Hopf point. Lastly, a theoretical analogy between the pendulum system and the triangular \(\text {L}_4\)/\(\text {L}_5\) Lagrange points in celestial mechanics is identified. The presented results are of relevance to crane systems and the mechanical modelling of cable or truss-supported structures.