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.