Modal analysis, or the mode-superposition method, is a linear dynamic-response 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) 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.
The modal superposition of a three-translational-DOF cantilever-column system is shown in Figure 1:
Figure 1 - Modal components and the resultant displacement
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.
To briefly summarize the numerical formulation for 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:
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- Modal damping ratio ξ n are typically assumed from empirical data.
- N coupled equations of motion are given as:
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- Their transformation to N uncoupled differential equations is given as:
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where
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- Where Y n represents modal amplitude, expressed in the time domain by Duhamel's Integral, given as:
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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.