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

**Mode shapes***Φ*, and their corresponding_{n}**frequencies***ω*, are obtained through solution of the following eigenvalue problem:_{n}

**Modal damping ratios***ξ*are typically assumed from empirical data._{n}

*N*coupled**equations of motion**are given by:

- Their
**transformation**to*N*uncoupled differential equations is given through the following expression:

- From the previous expression,
*Y*represents_{n}**modal amplitude**, expressed in the time domain by Duhamel's Integral, which is given as:

- Solution then yields the relationship for
**total displacement**, given as:

where**Φ**is the*N*x*N*mode-shape matrix which transforms the generalized coordinate vectorinto the geometric coordinate vector*Y*.*v*

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

# References

- Clough, R., Penzien, J. (2010).
*Dynamics of Structures*(2nd ed.). Berkeley, CA: Computers and Structures, Inc.

Available for purchase on the CSI Products > Books page

- Wilson, E. L. (2004).
*Static and Dynamic Analysis of Structures*(4th ed.). Berkeley, CA: Computers and Structures, Inc.

Available for purchase on the CSI Products > Books page