Damped Spherical Spring Pendulum

time

0.00

parameters

mass of bob m

0.2

spring constant k

30

damping coefficient μ

0

initial conditions

initial position

θ

0

1.05

angular speed θ

'

0

0.75

angular speed ϕ

'

0

-4

linear speed

L'

0

2

display

pendulum

phase curve

show track

viewpoint

default

front

above

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

L(t)

and the spherical coordinates of the center of the bob,

θ(t)

and

ϕ(t)

.

The three equations of motion are:

mL(t)

′′

θ

(t)=msinθ(t)L(t)cosθ(t)

2

′

θ

(t)

-g-μL(t)

′

θ

(t)

,

mL(t)

′′

ϕ

(t)-L(t)

′

ϕ

(t)(μ+2m

′

θ

(t)cotθ(t))

,

m

′′

L

(t)=gmcosθ(t)+k(L(0)-L(t))+mL(t)

2

′

θ

(t)

sinϕ(t)+mL(t)

2

′

ϕ

(t)

,

where

m

is the mass of the bob and

μ

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.