Wolfram Demonstrations Project

A damped spherical spring pendulum consists of a bob suspended by a spring from a fixed pivot. This Demonstration traces the path of the bob.

This system has the following three degrees of freedom: the length of the spring

and the spherical coordinates of the center of the bob,

and

The three equations of motion are:

where

is the mass of the bob and

is the damping coefficient of the system.

Among the many chaotic tracks, the phase curves show many interesting periodic orbits, obtained by changing the parameters of the system.

The equations of motion are similar to those of the damped spherical pendulum with one additional degree of freedom,

, and one more equation:

This equation expresses the longitudinal acceleration of the spring, which consists of four parts:

• gravity,

• spring elasticity,

• radial centrifugal force,

• tangential centrifugal force,

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Damped Spherical Spring Pendulum