Free-end bending boundary conditions ($h^{(2)}=h^{(3)}=0$)

Variation of the small-slope bending energy of an elastic filament gives endpoint work $A[h^{(2)}\delta h^{(1)}-h^{(3)}\delta h]$. Arbitrary endpoint displacement and slope variations force the bending moment and transverse force to vanish. These are $h^{(2)}=h^{(3)}=0$. They define a self-adjoint nonnegative fourth-order bending operator on a finite interval; its kernel contains rigid translation and tilt.

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 $k=0$.

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.