# gst: apropos of vibrating pivots, driving a damped coplanar double pendulum.

AA << present results of linear and nonlinear motions of a parametrically driven coplanar double pendulum with velocity-dependent damping. The equations of motion of a damped double pendulum of unequal masses with its pivot vibrated vertically are different from those obtained under gravity modulation. 

Linear stability analysis shows that tongue-shaped marginal stability curves divide the plane of driving parameters into multiple regions of subharmonic and harmonic instabilities. The instability zones for one normal mode overlap with those for the other. 

The double pendulum may oscillate or rotate about its pivot harmonically or subharmonically. The limit cycles corresponding to the normal mode oscillations of a double pendulum of equal masses are squeezed into a line in its configuration space. 

For unequal masses, two marginal curves for subharmonic instabilities merge to form a double-well shaped curve in the presence of damping, which is qualitatively new. The pendulum shows driving amplitude sensitive multi-period complex oscillations for driving parameters near the extrema of the merged instability zones and boundaries of the overlapping zones. 

For larger driving amplitude, the pendulum shows subharmonic, harmonic or chaotic rotations. >>

️

Rebeka Sarkar, Krishna Kumar, Sugata Pratik Khastgir. Parametrically driven damped coplanar double pendulum. arXiv:2208.03292v1 [physics.class-ph]. Aug 2, 2022. 

https://arxiv.org/abs/2208.03292

Also

keyword 'pendulum' in FonT

https://flashontrack.blogspot.com/search?q=pendulum

keyword 'pendolo' | 'pendola' in Notes

(quasi-stochastic poetry)

https://inkpi.blogspot.com/search?q=pendolo

https://inkpi.blogspot.com/search?q=pendola

Keywords: gst, pendulum, double pendulum, instability, chaos, chaotic rotations

# gst: non-trivial behaviors of a rotating (physical) double pendulum

<< The double pendulum, a simple system of classical mechanics, is widely studied as an example of, and testbed for, chaotic dynamics. In 2016, Maiti et al. studied a generalization of the simple double pendulum with equal point-masses at equal lengths, to a rotating double pendulum, fixed to a coordinate system uniformly rotating about the vertical. In this paper, (AA) study a considerable generalization of the double pendulum, constructed from physical pendula, and ask what equilibrium configurations exist for the system across a comparatively large parameter space, as well as what bifurcations occur in those equilibria. >>️

<< the non-trivial bifurcation (AA) have found (..), may actually be rightly understood as three additional bifurcations: there is a narrow region, approximately the ‘crease’ of the surface in Fig. 5, within which a vertical line (..) intersects the surface three times; (..). Thus three non-trival bifurcations would be expected in the corresponding bifurcations plots. >>️

Jonathan Tot, Robert H. Lewis. On the Equilibria and Bifurcations of a Rotating Double Pendulum. arXiv:2204.12437v2 [math.DS]. May 7, 2022. 

https://arxiv.org/abs/2204.12437

Also

keyword 'pendulum' in FonT

https://flashontrack.blogspot.com/search?q=pendulum

keyword 'pendolo' | 'pendola' in Notes

(quasi-stochastic poetry)

https://inkpi.blogspot.com/search?q=pendolo

https://inkpi.blogspot.com/search?q=pendola

Keywords: gst, pendulum, double pendulum, behavior, chaos 

# gst: chaos from a double pendulum

<< Three double pendulums, all starting with near identical initial conditions, all rapidly diverging. >>

Ari Rubinsztejn. Chaos and the Double Pendulum. Nov 19, 2018.

https://gereshes.com/2018/11/19/chaos-and-the-double-pendulum

# gst: intricate transitions in elastoactive structures.

<< The interplay between activity and elasticity often found in active and living systems triggers a plethora of autonomous behaviors ranging from self-assembly and collective motion to actuation. Among these, spontaneous self-oscillations of mechanical structures is perhaps the simplest and most widespread type of nonequilibrium phenomenon. >>️

<< Here, (AA) introduce a centimeter-sized model system for one-dimensional elastoactive structures. >>️

<< such structures exhibit flagellar motion when pinned at one end, self-snapping when pinned at two ends, and synchronization when coupled together with a sufficiently stiff link. (..) these transitions can be described quantitatively by simple models of coupled pendula with follower forces. >>️

Ellen Zheng, Martin Brandenbourger, et al. Self-Oscillation and Synchronization Transitions in Elastoactive Structures. Phys. Rev. Lett. 130, 178202. April 25, 2023. 

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.130.178202

Also:  transition, particle, self-assembly, elastic, pendulum in https://www.inkgmr.net/kwrds.html

Keywords: gst, transition, particle, self-assembly, elastic, pendulum