Modal dynamic analysis

The modal dynamic procedure provides time history analysis of linear systems.

The following topics are discussed:

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In Other Guides
Transient modal dynamic analysis

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The excitation is given as a function of time: it is assumed that the amplitude curve is specified so that the magnitude of the excitation varies linearly within each increment. When the model is projected onto the eigenmodes used for its dynamic representation, we obtain the following set of equations at time t:

(1)q¨β+Cβαq˙α+ωβ2qβ=(ft)β=ft-Δt+ΔfΔtΔt,

where the α and β indices span the eigenspace; Cβαis the projected viscous damping matix; qβ is the “generalized coordinate” of the mode β (the amplitude of the response in this mode); ωβ=kβ/mβ is the natural frequency of the undamped mode β (obtained as the square root of the eigenvalue in the eigenfrequency step that precedes the modal dynamic time history analysis); (ft)β is the magnitude of the loading projected onto this mode (the “generalized load” for the mode); and Δf is the change in f over the time increment, which is Δt.

If the projected damping matrix is diagonal, Equation 1 becomes the following uncoupled set of equations:

(2)q¨β+2ξβωβq˙β+ωβ2qβ=(ft)β,

where ξβ is the critical damping ratio given by the relation

2ξβωβ=cβmβ,

where cβ is the modal viscous damping coefficient and mβ is the modal mass in mode β.