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