Stable characteristic equation for free-free bending modes ($\cos q=\mathrm{sech}q$)

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.