Study guide: Finite difference methods for vibration problems

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$\newcommand{\uex}{{u_{\small\mbox{e}}}} \newcommand{\half}{\frac{1}{2}} \newcommand{\tp}{\thinspace .} \newcommand{\Oof}[1]{\mathcal{O}(#1)}$

Calculations of an exact solution of the discrete equations

$u^n = IA^n = I\exp{(\tilde\omega \Delta t\, n)}=I\exp{(\tilde\omega t)} = I\cos (\tilde\omega t) + iI\sin(\tilde \omega t) \tp$

$\begin{align*} [D_tD_t u]^n &= \frac{u^{n+1} - 2u^n + u^{n-1}}{\Delta t^2}\\ &= I\frac{A^{n+1} - 2A^n + A^{n-1}}{\Delta t^2}\\ &= I\frac{\exp{(i\tilde\omega(t+\Delta t))} - 2\exp{(i\tilde\omega t)} + \exp{(i\tilde\omega(t-\Delta t))}}{\Delta t^2}\\ &= I\exp{(i\tilde\omega t)}\frac{1}{\Delta t^2}\left(\exp{(i\tilde\omega(\Delta t))} + \exp{(i\tilde\omega(-\Delta t))} - 2\right)\\ &= I\exp{(i\tilde\omega t)}\frac{2}{\Delta t^2}\left(\cosh(i\tilde\omega\Delta t) -1 \right)\\ &= I\exp{(i\tilde\omega t)}\frac{2}{\Delta t^2}\left(\cos(\tilde\omega\Delta t) -1 \right)\\ &= -I\exp{(i\tilde\omega t)}\frac{4}{\Delta t^2}\sin^2(\frac{\tilde\omega\Delta t}{2}) \end{align*}$