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*Modal analysis*, or the mode-superposition method, is a linear dynamic-response procedure where the superposition of free-vibration mode shapes characterizes the displacement field of a multi-degree-of-freedom (MDOF) system. 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.

A structural system of NDOF will have _N_ mode shapes. Each mode shape is an independent and normalized displacement pattern. The amplitude of each mode shape then describes a generalized displacement pattern which, when superimposed with the series of patterns, indicates structural behavior.

The modal superposition of a cantilever-column system with three translational DOF is shown in Figure 1:

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

{center-text}Figure 1 - Caption{center-text}

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This visual indicates how modal contributions sum to create a resultant displacement pattern. Only a few shapes are necessary to describe the deflected configuration with sufficient accuracy. Contributions are greatest for lower modes and decrease for higher modes. Analysis may be truncated once sufficiently accurate response is achieved. Another motivation to focus on lower-mode contribution is that higher modes are predicted less reliably.

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.

To briefly summarize the numerical formulation of modal analysis, the process (with damping) 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.

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