Gradients for Large Structural Models with Repeated Frequencies

The physical origin and ambiguity of repeated frequency modes, is discussed, followed by a mathematical derivation of the governing equations for the derivatives of repeated eigenvalues. This is followed by a derivation for the associated eigenvector gradient equations and two small examples to illustrate the solution procedure. Physical interpretations are also included to help explain the analytical results. An efficient computation procedure which preserves the bandwidth for very large matrices is also proposed for systems with repeated or closely spaced eigenvalues.