Modal matrix

Abstract: We report on a microwave cavity experiment where exceptional points (EPs), which are square root singularities of the eigenvalues as function of a complex interaction parameter, are encircled in the laboratory. The real and imaginary parts of an eigenvalue are given by the frequency and width of a resonance and the eigenvectors by the field distributions. Repulsion of eigenvalues--always associated with EPs--implies frequency anticrossing (crossing) whenever width crossing (anticrossing) is present. The eigenvalues and eigenvectors are interchanged while encircling an EP, but one of the eigenvectors undergoes a sign change which can be discerned in the field patterns.

Abstract: A method has been developed which uses measured normal modes and natural frequencies to improve an analytical mass and stiffness matrix model of a structure. The method directly identifies, without iteration, a set of minimum changes in the analytical matrices which force the eigensolutions to agree with the test measurements. An application is presented in which the analytical model had 508 degrees of freedom and 19 modes were measured at 101 locations on the structure. The resulting changes in the model are judged to be small compared to expectations of error in the analysis. Thus, the improved model is accepted as a reasonable model of the structure with improved dynamic response characteristics. In addition, it is shown that the procedure may be a useful tool in identifying apparent measured modes which are not true normal modes of the structure. Nomenclature - analytical matrix = matrix of changes = identity matrix = full improved stiffness and mass matrices (n x n) = full analytical K, M matrices (n x n) = partitions of KA,MA corresponding to test coordinates = partitions of KA,MA corresponding to coupling elements = partitions of KA,MA corresponding to unmea- sured coordinates = number of measured modes = number of degrees of freedom in the model = measures of changes, Eqs. (15-17) = matrix norm, sum of the squares of all the elements = rectangular modal matrix, normalized (n x m) = /th mode, /th column of $ = measured and unmeasured partitions of ,- = diagonal matrix of measured natural frequencies (m xm) = natural frequency of /th mode = 12 17 = sum of the squares of all elements of matrix ( )