Calculation of Eigenvectors with Uniform Accuracy

In nonlinear eigenvalue problems, the standard method for calculating eigenvectors is to first calculate the eigenvalue. The nonlinear governing matrix is then formed using the calculated eigenvalue, a random disturbance is applied, and the response to this gives the eigenvector. In stiffness analyses this is known as the random-force method. It is well established that this approach gives eigenvectors with accuracy of the same order as the eigenvalue, provided the eigenvector is “well represented” by the parameters used in the problem description—the “freedoms.” However, in nonlinear formulations some modes may be poorly represented, or completely unrepresented, by freedom movements—the latter are referred to as u * = 0 modes. The eigenvalues for these modes are found in the normal course of the analysis, but the application of random forces will give modes of lower accuracy, or in the case of the u * = 0 modes, no accuracy at all. A complement to the commonly used random-force method is shown to give eigenmode accuracy that is similar to the eigenvalue accuracy, whether the mode is well represented, poorly represented, or not represented by the freedom movements.