Natural frequency intervals for vibrating systems with polytopic uncertainty

We consider the problem of locating the natural frequencies of uncertain systems whose describing matrices are functions of an unknown parameter vector which is included in an assigned bounding set. We face what we call the weak frequency interval detection problem of determining the smallest interval which includes all possible frequencies. We show that if the system matrices depend affinely on the parameter vector, whose bounding set is a compact polyhedron, then this problem requires the solution of a finite number of eigenvalue problems associated with the vertices of such a polyhedron. Unfortunately, detecting the intervals associated with all the natural frequencies (strong frequency interval detection problem), cannot rely on this property, so that one must resort to Monte Carlo methods or numerical optimization to find them. We show that the strong version is solvable “exploring the vertices only” under some stronger assumptions. In the case in which the uncertainty bounding set is not defined by linear inequalities, not even the extremal frequencies can be associated with the vertices of the admissibility domain. Then again, numerical approach is necessary unless we accept to merge the original system in a larger one of an “affine nature”. Finally, we present as an application the study of structures with uncertain mass distribution.