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:

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

• 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 _n represents 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 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