1. presents a method using a generalized inverse technique to find a solution for eigenvector derivatives withrepeated roots.Also, in Refs. 12-13, the dynamic flexibility method is used to find a solution for eigenvector derivatives. Reference 17 presents an interesting discussion of design sensitivity with a repeated frequency. The sensitivity calculation is obtained with physical interpretations in the vibration mode space.
2. The governingequation for findingeigenvector derivatives withrepeated rootscanbe expressed as1-3
3. ' (4) in whichVisa particular solutionofEq. (1). Eq. (5) is adopted inthe computational processto obtainthe general solutionasshownin Refs. 2-3.
4. Following the procedure used in Refs. 2-3, Eq. (5) is adopted to find the solution of the matrix C. This method needs to compute the second-order derivative matrices K"and M". This approach is not always practical for certain types of engineering problems. Also, derivation of the computational formula of the matrix C is complex and complicated2,3. To avoid the need to find matrices K"and M"and to remove the complicated process of deriving the matrix C analytically, the authors apply the idea of Fox/Hu's method4,5to find eigenvector derivatives with repeated roots. Both Eqs. (1) and (5) are solved simultaneouslyas shown inEq. (6).
5. Eq. (8) is a direct iterative formula and its coefficient matrix is a non-singular matrix10-11. Solving Eq. (8) directly can achieve the required solution for practical engineering problems. There is no need to find the second-order derivative matrices K"and M"during the computation.