 Free-free bending spectrum

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 $k=0$.