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*Modal analysis*, or the mode-superposition method, is a [linear dynamic-response|kb:Nonlinear] procedure where free-vibration mode shapes are evaluated and superimposed to characterize structural displacement. Mode shapes describe the configurations into which a structure will naturally displace. Lateral displacement patterns are generally of primary concern. Mode shapes characterized by low-order mathematical expressions tend to provide the greatest contribution to structural response. As orders increase, mode shapes contribute less. Further, high-order modes are predicted less reliably. Therefore it is reasonable to truncate analysis once sufficiently accurate response is achieved. It is typical for only a few mode shapes to sufficiently describe deflected configuration.

Modal analysis may be conducted for single\- or multi-degree-of-freedom (MDOF) systems. A structure with NDOF will have N corresponding mode shapes. Each mode shape is an independent and normalized displacement pattern which may be amplified and superimposed with the series of mode shapes to indicate structural behavior. The modal superposition of a three-translational-DOF cantilever-column system is shown in Figure 1:

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!Mode shapes.png|align=center,border=1,width=600pxpxpxpx600pxpxpxpxpxpxpxpx!

{center-text}Figure 1 - Resultant displacement and modal components{center-text}

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This visual indicates how modal contributions sum to create a resultant displacement pattern. [Numerical evaluation|Numerical-evaluation summary] proceeds by reducing the equations of motion (_N_ simultaneous differential equations coupled by full mass and stiffness matrices) to a much smaller set of uncoupled second order differential equations (_N_ independent normal-coordinate equations). This reduction is enabled by the orthogonality of mode-shape relations.




{list-of-resources2:label=modal-analysis|drafts-root=Modal analysis drafts}