Modal Analysis

The modal dynamic analysis is concerned with the calculation of the natural frequencies and the mode shapes of the structure. As in the static analysis a stiffness matrix can be constructed, but in a modal dynamic analysis a mass matrix is also constructed. The free vibration of the model is then given by

$$\mathbf{M}\ddot{\mathbf{u}} + \mathbf{Ku} = 0$$

The natural frequencies are then given when

$$\left| \mathbf{K} - \lambda\mathbf{M} \right| = 0$$

The eigenvalue problem is then

$$\mathbf{K}\boldsymbol{\phi} - \lambda\mathbf{M}\boldsymbol{\phi} = 0$$

Or across multiple eigenvalues

$$\mathbf{K}\boldsymbol{\varPhi} - \Lambda\mathbf{M}\boldsymbol{\varPhi} = 0$$

where $\left\{ \lambda,\phi \right\}$ are the eigenpairs – eigenvalues (the diagonal terms are the square of the free vibration frequencies) and the eigenvectors (the columns are the mode shapes) respectively.