linguistic:voice_change
Coupled oscillations

The harmonic oscillator and the systems it models have a single degree of freedom. More complicated systems have more degrees of freedom, for example, two masses and three springs (each mass being attached to fixed points and to each other). In such cases, the behavior of each variable influences that of the others. This leads to a coupling of the oscillations of the individual degrees of freedom. For example, two pendulum clocks (of identical frequency) mounted on a common wall will tend to synchronise. This phenomenon was first observed by Christiaan Huygens in 1665. The apparent motions of the compound oscillations typically appears very complicated but a more economic, computationally simpler and conceptually deeper description is given by resolving the motion into normal modes.

The simplest form of coupled oscillators is a 3 spring, 2 mass system, where masses and spring constants are the same. This problem begins with deriving Newton's second law for both masses. { m 1 x ¨ 1 = − (k 1 + k 2) x 1 + k 2 x 2 m 2 x ¨ 2 = k 2 x 1 − (k 2 + k 3) x 2 {\displaystyle {\begin{cases}m_{1}{\ddot {x}}_{1}=-(k_{1}+k_{2})x_{1}+k_{2}x_{2}\\m_{2}{\ddot {x}}_{2}=k_{2}x_{1}-(k_{2}+k_{3})x_{2}\end{cases}}}

The equations are then generalized into matrix form. F = M x ¨ = k x, {\displaystyle F=M{\ddot {x}}=kx,} where M = {\displaystyle M={\begin{bmatrix}m_{1}&0\\0&m_{2}\end{bmatrix}}}, x = {\displaystyle x={\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}}, and k = {\displaystyle k={\begin{bmatrix}k_{1}+k_{2}&-k_{2}\\-k_{2}&k_{2}+k_{3}\end{bmatrix}}}

The values of k and m can be substituted into the matrices. m 1 = m 2 = m, k 1 = k 2 = k 3 = k, M =, k = {\displaystyle {\begin{aligned}m_{1}=m_{2}=m,\;\;k_{1}=k_{2}=k_{3}=k,\\M={\begin{bmatrix}m&0\\0&m\end{bmatrix}},\;\;k={\begin{bmatrix}2k&-k\\-k&2k\end{bmatrix}}\end{aligned}}}

These matrices can now be plugged into the general solution. (k − M ω 2) a = 0 = 0 {\displaystyle {\begin{aligned}\left(k-M\omega ^{2}\right)a&=0\\{\begin{bmatrix}2k-m\omega ^{2}&-k\\-k&2k-m\omega ^{2}\end{bmatrix}}&=0\end{aligned}}}

The determinant of this matrix yields a quadratic equation. (3 k − m ω 2) (k − m ω 2) = 0 ω 1 = k m, ω 2 = 3 k m {\displaystyle {\begin{aligned}&\left(3k-m\omega ^{2}\right)\left(k-m\omega ^{2}\right)=0\\&\omega _{1}={\sqrt {\frac {k}{m}}},\;\;\omega _{2}={\sqrt {\frac {3k}{m}}}\end{aligned}}}

Depending on the starting point of the masses, this system has 2 possible frequencies (or a combination of the two). If the masses are started with their displacements in the same direction, the frequency is that of a single mass system, because the middle spring is never extended. If the two masses are started in opposite directions, the second, faster frequency is the frequency of the system.

More special cases are the coupled oscillators where energy alternates between two forms of oscillation. Well-known is the Wilberforce pendulum, where the oscillation alternates between the elongation of a vertical spring and the rotation of an object at the end of that spring.

Coupled oscillators are a common description of two related, but different phenomena. One case is where both oscillations affect each other mutually, which usually leads to the occurrence of a single, entrained oscillation state, where both oscillate with a compromise frequency. Another case is where one external oscillation affects an internal oscillation, but is not affected by this. In this case the regions of synchronization, known as Arnold Tongues, can lead to highly complex phenomena as for instance chaotic dynamics.