# Dynamics of Coupled Pendulums

Dynamics of Coupled Pendulums

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 suspended from a fixed support by a massless string of length , with its coordinate described by the angle from the vertical. Under the action of gravity, the equation of motion is given by . Restricting ourselves to small-amplitude oscillations, we can approximate , leading to the elementary solution , where , the natural frequency of the pendulum, independent of . For the coupled pendulum system, to the same level of approximation, the Lagrangian can be written where is a measure of the coupling between the two pendulums mediated by the connecting string. The equations of motion are readily solved to give (t)=[(0)+(0)]cos+t±[(0)-(0)]cos-t.

m

L

θ

m+mgLsinθ=0

2

L

..

θ

sinθ≈θ

θ(t)=θ(0)cos(ωt)

ω=

g/L

m

L,,,=m+-m+-m,

θ

1

θ

1

θ

2

θ

2

1

2

2

θ

1

2

θ

2

1

2

2

ω

2

θ

1

2

θ

2

2

η

θ

1

θ

2

η

θ

1,2

1

2

θ

1

θ

2

2

ω

2

η

1

2

θ

1

θ

2

2

ω

2

η

The system has two normal modes of vibration. In the in-phase mode, starting with (0)=(0), the two pendulums swing in unison at a frequency of +. In the out-of-phase mode, starting with (0)=-(0), they swing in opposite directions at a frequency of -.

θ

1

θ

2

2

ω

2

η

θ

1

θ

2

2

ω

2

η

A more dramatic phenomenon is resonance. If one pendulum is started at and another at , then energy will periodically flow back and forth between the two pendulums.

θ(0)≠0

θ(0)=0