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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