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An iterative method can help in the calculation of the eigenvector derivatives of symmetric, real-valued systems, as it is numerically stable and converges to an accurate solution for any initial value. To maintain high convergence rates for eigenvalue ratios close to one, the method has a second generalized stiffness matrix inverse. The method can be applied to zero and repeated eigenvalue systems. Two examples illustrate the method's efficiency.

A.M. Kabe suggested a technique that employs, in addition to the measured modes, the structural connectivities to optimally adjust the stiffness matrix. The weaknesses of this procedure are the large size of the governing equation and the complicated solution algorithm. The Baruch/Berman solution was generalized to allow the structural connectivities to be utilized as explicit constraints in the optimization process, and the dimension of the governing equation became identical to the number of zero elements in the upper-side triangular part of the stiffness matrix. The above method can be applied for the case of limited number of zero stiffness elements.

A modified modal method that incorporates an iterative process accurately computes the eigenvetor derivatives of structures. The iterative method is restricted to one design variable and specific eigenvalues. The iterative process is stable and numerically converges to the correct solution for the derivative. It removes the problems, such as the requirement for a large number of eigenvalues, associated with the modal method. A numerical example demonstrates the accuracy of the method.

A new method performs optimal orthogonalization of incomplete measured modes on the assumption that the exact analytical mass matrix is known. The method does not fix the type of the transformation matrix and this results in adjusted modes closest to the measured ones. A closed-form solution and an iterative algorithm of solution are analyzed. A numerical example demonstrates the fast convergence of the iterative algorithm.

A mathematical method is proposed for measuring the mode shapes and frequencies of free-free or unconstrained structural systems. The method uses a dynamic equation based on stiffness matrices to evaluate the different parameters that affect the systems' mode shapes and frequency stability. A numerical example involving a uniform straight beam validates the computational efficiency of the proposed method.