I have used free vibration analysis in FEM. After analysis, we can usually use animation to see the motion of each eigenmode (In Abaqus or Comsol, I would  choose either half harmonic or full harmonic). Does anyone know the principle behind it?In my understanding, eigenmode analysis already removes the time factor, i.e., $\exp(i\omega t)$. If so, how can FEM show animation?
How does the animation work in eigenvalue problem of FEM
finite element;eigenvalues
The solution for a particular mode is of the form$$u_i(x,t) = w_i(x)\sin(\omega_i t + \phi)$$where $w_i(x)$ is the $i$th eigenfunction and $\omega_i$ the $i$th eigenvalue, $\phi$ is a phase and, we can assume $\phi=0$. Then, the solution for the mode is $$u_i(x,t) = w_i(x)\sin(\omega_i t) \enspace .$$Thus, you can present your mode as an animation changing the amplitude proportionally to the sine function presented. In commercial software it is common to have the option to vary $t \in [0,\pi]$ and $t \in [0,2\pi]$, and they are termed half cycle or full cycle (or half harmonic and full harmonic).The correct thing to do is to animate over the full range $[0,2\pi]$, but since the amplitude will be the same (but opposite sign) in the other half some people do not consider it necessary.