The dimensionless solution is \begin{align}\label{eq:sol:dim} \boxed{U(X,\Omega) = \sum_{n=1}^{\infty} \sin(n\pi X) (a_n \cos n'\Omega + b_n \sin n'\Omega) e^{-\Gamma_n \Omega}} \end{align} where \(n' \equiv \omega_n \ell/\pi c\) is \begin{alignat}{2} n' &= n\sqrt{1 + Bn^2 -\Gamma_n^2/n^2} \,,\hspace{3ex} B&& \equiv EI \pi^2/T\ell^2 \,, \label{eq:freq:dim}\\ &\simeq n + ({B}/{2})n^3 - {\Gamma_n^2}/{2n}\,,\quad n&& \ll 1/\sqrt{B}, \quad n \gg \Gamma_n\,, \label{eq:freq:dim:low}\\ &\simeq \sqrt{B} n^2 \,,\hspace{14ex}\quad\,\,\,\, n &&\gg 1/\sqrt{B}, \quad n \gg \Gamma_n\,.\label{eq:freq:dim:high} \end{alignat}