= Central displacement in three freely ended coupled pendulums
Releasing the central bob of three identical spring-coupled <simple pendulums> from displacement $d$ excites only the symmetric <normal modes>. In displacement ratios, $(0,d,0)=(d/3)(1,1,1)-(d/3)(1,-2,1)$. Thus the outer coordinates are $(d/3)(\cos\omega_1t-\cos\omega_3t)$ and the central coordinate is $(d/3)(\cos\omega_1t+2\cos\omega_3t)$, where $\omega_1^2=g/l$ and $\omega_3^2=g/l+3k/m$. The antisymmetric mode is absent by reflection symmetry. The central modulation envelope is $|d|\sqrt{5+4\cos[(\omega_3-\omega_1)t]}/3$, so it never vanishes completely. Slow <beats> require a small frequency difference; the decomposition itself is exact in the linear model.
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