# Damped Spherical Spring Pendulum

Damped Spherical Spring Pendulum

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 .

L(t)

θ(t)

ϕ(t)

The three equations of motion are:

mL(t)(t)=msinθ(t)L(t)cosθ(t)(t)-g-μL(t)(t)

′′

θ

2

′

θ

′

θ

mL(t)(t)-L(t)(t)(μ+2m(t)cotθ(t))

′′

ϕ

′

ϕ

′

θ

m(t)=gmcosθ(t)+k(L(0)-L(t))+mL(t)(t)sinϕ(t)+mL(t)(t)

′′

L

2

′

θ

2

′

ϕ

where is the mass of the bob and is the damping coefficient of the system.

m

μ

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