Mode shape sensitivity of two closely spaced eigenvalues

In this paper the sensitivity of the mode shapes of two closely spaced eigenvalues are studied. It is well known that in case of repeated eigenvalues, the meaningful quantity is not the two individual mode shapes, but rather the subspace defined by the two mode shapes. Following the ideas of a principle that has been released for publishing recently denoted as the local correspondence (LC) principle, it is shown, that in the case of a set of two closely spaced eigenvalues, the mode shapes become highly sensitive to small changes of the system. However, if the two closely spaced eigenvalues have a reasonable frequency distance to all other eigenvalues of the system, then a linear transformation exists between the set of perturbed and unperturbed mode shapes describing the significant changes as a rotation in the initial subspace defined by the two mode shapes. Closed form solutions are given for general combined mass and stiffness perturbations, and it is shown that there is a smooth transition from the case of moderate sensitivity of the mode shapes towards the case of repeated eigenvalues where the sensitivity goes to infinite. In case of “nearly repeated eigenvalues” the perturbed set of mode shapes can be found by solving a special eigenvalue problem for the two closely spaced eigenvalues. The theory is illustrated and compared with the exact solution for a simple 3 dof system.