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:
Figure 1 - Caption
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:
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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.