= Free-free bending spectrum
{title2=$\cos q_n\cosh q_n=1,\quad k_n=q_n/L$}
Nonzero bending modes have the <sine>, <cosine>, <hyperbolic sine> and <hyperbolic cosine> form. Free conditions at the first end identify the <cosine> and <hyperbolic cosine> coefficients, and likewise the <sine> coefficients. The remaining boundary determinant is $2(1-\cos q\cosh q)$. Its first positive roots are about $4.73004,7.85320,10.99561$. The complete basis additionally includes two <zero-energy filament modes>, constant translation and affine tilt; these are absent from the finite-coefficient formula at $k=0$.
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