Damped Spherical Pendulum

This Demonstration traces the path of the bob on a damped spherical pendulum. The pendulum is suspended at the center of an imaginary sphere that marks the outer bounds of the center of the bob.

The equations of motion are

g m sin(θ(t))+L m θ''(t)-L m sin(θ(t)) cos(θ(t)) ((ϕ'(t))^2)+L μ θ'(t)==0,

2 L m θ'(t) sin(θ(t)) cos(θ(t)) ϕ'(t)+L m (sin^2)(θ(t)) ϕ''(t)+L μ (sin^2)(θ(t)) ϕ'(t)==0,

where θ and ϕ are the spherical coordinates of the center of gravity of the bob. The pendulum rod has length L=1 (with no loss of generality) and bob mass m. The damping coefficient of the system is μ. The initial angular positions are (Subscript[θ, 0]) and (Subscript[ϕ, 0]) and the initial angular speeds are (Subscript[θ, 0])' and (Subscript[ϕ, 0])'.

THINGS TO TRY

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DETAILS

Choose a set of parameters with the sliders or click the "randomize" button and start the animation. Run the animation to the end and press the "randomize parameters" button to discover a variety of bob trajectories.

A damping term is added to the equations of motion for the spherical pendulum programmed by Franz Brandhofer.

See also: A. Ertas and S. Garca, "Experimental Investigation of Dynamics and Bifurcations of an Impacting Spherical Pendulum," Experimental Mechanics, 49(5), 2009 pp. 653–662.

RELATED LINKS

Contributed by: Erik Mahieu

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