= Solution
A positive-eigenvalue <normal mode> satisfies $KW_n=Ak_n^4W_n$. Solving the constant-coefficient equation gives the <sine>, <cosine>, <hyperbolic sine> and <hyperbolic cosine> family. To avoid confusing modal coefficients with the <filament bending modulus>, write those coefficients as $c_1,c_2,c_3,c_4$.
The free conditions at zero require $c_3=c_1$ and $c_4=c_2$. With $q=kL$, the remaining conditions are
$$
\begin{pmatrix}
\cosh q-\cos q&\sinh q-\sin q\\
\sinh q+\sin q&\cosh q-\cos q
\end{pmatrix}\binom{c_1}{c_2}=0.
$$
Its determinant is $2(1-\cos q\cosh q)$. Therefore the <free-free bending spectrum> is
$$
\boxed{\cos q_n\cosh q_n=1,\qquad k_n=q_n/L\quad(n\geq1).}
$$
A convenient corresponding <eigenfunction> is
$$
W_n(x)=N_n\left[\cos(k_nx)+\cosh(k_nx)
-\gamma_n(\sin(k_nx)+\sinh(k_nx))\right],\qquad
\gamma_n=\frac{\cosh q_n-\cos q_n}{\sinh q_n-\sin q_n},
$$
where $N_n$ gives unit $L^2$ norm. The <stable characteristic equation for free-free bending modes> is $\cos q=\operatorname{sech}q$. Intersections or numerical bracketing give
$$
\boxed{q_1\simeq4.730040745,\quad q_2\simeq7.853204624,\quad
q_3\simeq10.995607838,\quad q_4\simeq14.137165491,\quad q_5\simeq17.278759657,\ldots.}
$$
For the entire positive sequence, let $q_n^0=(n+\tfrac12)\pi$. Expanding $\cos q_n$ near its zero and using $\operatorname{sech}q_n\sim2e^{-q_n}$ gives
$$
\boxed{q_n=(n+\tfrac12)\pi+2(-1)^{n+1}e^{-(n+1/2)\pi}
+O(e^{-2(n+1/2)\pi}).}
$$
There are also two <zero-energy filament modes>, not captured by substituting $k=0$ into the finite-coefficient trigonometric expression. Solving $W^{(4)}=0$ with free endpoints gives the affine kernel. An <orthonormal basis> of that kernel is
$$
W_{\rm tr}(x)=L^{-1/2},\qquad
W_{\rm tilt}(x)=\sqrt{12/L^3}(x-L/2).
$$
The positive <normal modes> are <orthogonal> to both. Including these two rigid-motion modes is essential for completeness; the regular <self-adjoint operator> on a finite interval has a complete discrete eigenbasis.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/iii/paper-74-free-filament-modes.png]
{title=Characteristic roots and bending shapes of a filament with two free ends}
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