= Stable characteristic equation for free-free bending modes
{title2=$\cos q=\operatorname{sech}q$}
Writing the characteristic equation with reciprocal <hyperbolic cosine> avoids exponentially large products. For the positive roots indexed from one, $q_n=(n+1/2)\pi+2(-1)^{n+1}e^{-(n+1/2)\pi}+O(e^{-2(n+1/2)\pi})$. The apparent root at zero represents the separate affine kernel, not another positive bending mode. Evaluating hyperbolic pieces of high modes with decaying exponential forms also avoids cancellation.
Back to article page
