Coupled oscillator

Coupled oscillators exchange energy through interactions. Linear equations can be transformed into normal mode coordinates; the spatial pattern of an applied force determines which normal modes are driven.

For equal lengths, end masses $m$, middle mass $M$, and equal springs between adjacent bobs, reflection symmetry gives mode vectors $(1,1,1)$, $(1,0,-1)$ and $(1,-2m/M,1)$. Their squared frequencies are displayed above. These vectors are orthogonal in the inner product weighted by $\mathrm{diag}(m,M,m)$. Equal end masses, rather than equality of all three masses, suffice for this symmetry reduction.

For $x_2=(a/2)[\cos {\omega }_{1}t-\cos {\omega }_{2}t]$, the slowly varying signed factor is $a\sin [({\omega }_2-{\omega }_1)t/2]$. The absolute amplitude envelope repeats in $2\pi /|{\omega }_2-{\omega }_1|$, half the period of that signed factor.

Two equal small-angle pendulums coupled by a spring have displacement patterns $(1,1)$ and $(1,-1)$, with squared angular frequencies $g/l$ and $g/l+2k/m$. An initial displacement of only the first pendulum excites both patterns equally.