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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

Mu¨+Ku=0\mathbf{M}\ddot{\mathbf{u}} + \mathbf{Ku} = 0

The natural frequencies are then given when

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

The eigenvalue problem is then

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

Or across multiple eigenvalues

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

where {λ,ϕ}\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.