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.

Details

The equations of motion are similar to those of the damped spherical pendulum with one additional degree of freedom,
L(t)
, and one more equation:
m
′′
L
(t)=gmcosθ(t)+k(L(0)-L(t))+mL(t)
2
′
θ
(t)
sinϕ(t)+mL(t)
2
′
ϕ
(t)
.
This equation expresses the longitudinal acceleration of the spring, which consists of four parts:
• gravity,
gmcosθ(t)
• spring elasticity,
k((L(0)-L(t)/m
• radial centrifugal force,
L(t)
2
′
θ
(t)
sinϕ(t)
• tangential centrifugal force,
L(t)
2
′
ϕ
(t)

External Links

Spherical Coordinates (Wolfram MathWorld)
Degree of Freedom (Wolfram MathWorld)
Spherical Pendulum
Damped Spherical Pendulum

Permanent Citation

Erik Mahieu
​
​"Damped Spherical Spring Pendulum"​
​http://demonstrations.wolfram.com/DampedSphericalSpringPendulum/​
​Wolfram Demonstrations Project​
​Published: October 19, 2011