WOLFRAM|DEMONSTRATIONS PROJECT

Dynamics of Coupled Pendulums

​
frequency ω
5
coupling η
1
θ
1
(0) (degrees)
0
θ
2
(0) (degrees)
0
start motion
Consider a system of two identical pendulums swinging in parallel planes and connected at the top by a flexible string. (A recently published Demonstration considered the related version of two pendulums with their bobs connected by a massless spring.) We describe here a more classic form of the problem, going back to the time of Huygens.
Each pendulum consists of a mass
m
suspended from a fixed support by a massless string of length
L
, with its coordinate described by the angle
θ
from the vertical. Under the action of gravity, the equation of motion is given by
m
2
L
..
θ
+mgLsinθ=0
. Restricting ourselves to small-amplitude oscillations, we can approximate
sinθ≈θ
, leading to the elementary solution
θ(t)=θ(0)cos(ωt)
, where
ω=
g/L
, the natural frequency of the pendulum, independent of
m
. For the coupled pendulum system, to the same level of approximation, the Lagrangian can be written
L
θ
1
,

θ
1
,
θ
2
,

θ
2
=
1
2
m
2

θ
1
+
2

θ
2
-
1
2
m
2
ω

2
θ
1
+
2
θ
2
-m
2
η
θ
1
θ
2
,
where
η
is a measure of the coupling between the two pendulums mediated by the connecting string. The equations of motion are readily solved to give
θ
1,2
(t)=
1
2
[
θ
1
(0)+
θ
2
(0)]cos
2
ω
+
2
η
t±
1
2
[
θ
1
(0)-
θ
2
(0)]cos
2
ω
-
2
η
t
.
The system has two normal modes of vibration. In the in-phase mode, starting with
θ
1
(0)=
θ
2
(0)
, the two pendulums swing in unison at a frequency of
2
ω
+
2
η
. In the out-of-phase mode, starting with
θ
1
(0)=-
θ
2
(0)
, they swing in opposite directions at a frequency of
2
ω
-
2
η
.
A more dramatic phenomenon is resonance. If one pendulum is started at
θ(0)≠0
and another at
θ(0)=0
, then energy will periodically flow back and forth between the two pendulums.