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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. Mode shapes characterized by low-order mathematical expressions tend to provide the greatest contribution to structural response. As orders increase, mode shapes contribute less. High-order modes are also 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) structures. A system 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.

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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=600pxpx600pxpxpxpx!

{center-text}Figure 1 - Modal components and the resultant displacement{center-text}

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This visual indicates how modal contributions sum to create a resultant displacement pattern. Numerical evaluation 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 mode-shape relations.

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To briefly summarize the numerical formulation offor modal analysis, the process for a damped structural system is as follows:

* Mode shapes _Φ_ _{~}n{~}_, and their corresponding frequencies _ω_ _{~}n{~}_, are obtained through solution of the following eigenvalue problem:

!eqn for eigenvalue.png|align=center,border=0!

* Modal damping ratio _ξ_ _{~}n{~}_ are typically assumed from empirical data.

* _N_ coupled equations of motion are given as:

!eqn of motion.png|align=center,border=0!

* Their transformation to _N_ uncoupled differential equations is given as:

!eqn for solution.png|align=center,border=0!

{center-text}where{center-text}

!eqn for Mn Pn.png|align=center,border=0!

* Where _Y_ _{~}n{~}_ represents modal amplitude, expressed in the time domain by Duhamel's Integral, given as:

!eqn for Duhamel integral.png|align=center,border=0!

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Total response is then generated by solving each uncoupled modal equation and superposing their displacement. It is advantageous to characterize dynamic response in terms of the displacement time-history vector _v(t)_ because local forces and stresses may then be evaluated directly.


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