Free--free biharmonic eigenvalue equation
= Free--free biharmonic eigenvalue equation
A filament free of moment and shear at both ends has $h_{xx}=h_{xxx}=0$. Its positive biharmonic eigenvalues $q_n^4$ obey $\cos(q_nL)\cosh(q_nL)=1$, in addition to the two rigid zero modes representing translation and rotation.