= Even and odd pendulum-ring modes
{title2=$\omega_j^2=g/L+(4k/m)\sin^2(\pi j/N)$}
For a large ring of equal pendulums with tangential displacements and nearest-neighbour spring coupling, cyclic translation symmetry gives <Fourier modes> $x_s\propto e^{2\pi ijs/N}$. The locally straight-chain approximation gives the displayed spectrum. The lowest mode is uniform. For even $N$, the single highest mode alternates signs and has $\omega_{\max}^2=g/L+4k/m$. For odd $N$, perfect alternation cannot close; the highest pair $j=(N\pm1)/2$ is degenerate and has $\omega_{\max}^2=g/L+(4k/m)\cos^2(\pi/(2N))$. For an exact finite regular polygon with straight chord springs, multiply the spring contribution by $\cos^2(\pi/N)$; it tends to one in the stated large-$N$ limit.
Back to article page
