To briefly summarize the numerical evaluation of 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:

Technical Knowledge Base > Numerical-evaluation summary > Figure 1.png

Modal damping ratios ξ _n are typically assumed from empirical data.

N coupled equations of motion are given by:

Technical Knowledge Base > Numerical-evaluation summary > Figure 2.png

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

Technical Knowledge Base > Numerical-evaluation summary > Figure 3.png

Technical Knowledge Base > Numerical-evaluation summary > Figure 4.png

From the previous expression, Y _n represents modal amplitude, expressed in the time domain by Duhamel's Integral, which is given as:

Technical Knowledge Base > Numerical-evaluation summary > Figure 5.png

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 vector Y into the geometric coordinate vector 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, v ( t ), because local forces and stresses may then be evaluated directly.

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