In this study, a new, deterministic method is discussed for estimating the maximum, or least favorable frequency, and the minimum, or best favorable frequency, of structures with uncertain but non-random parameters. The favorable bound estimate is actually a set in eigenvalue space rather than a single vector. The obtained optimum estimate is the smallest calculable set which contains the uncertain system eigenvaluse. This kind of engenvalue problem involves uncertain but non-random interval stiffness and mass matrices. If one views the deviation amplitude of the interval matrix as a perturbation around the nominal value of the interval matrix, one can solve the generalized eigenvalue problem of the uncertain but non-random interval matrices. By applying the interval extension matrix perturbation formulation, the interval perturbation approximating formula is presented for evaluating interval eigenvalues of uncertain but non-random interval stiffness and mass matrices. A perturbation method is developed which allows one to calculate eigenvalues of an uncertain but non-random interval matrix pair that always contains the system's true stiffness and mass matrices. Inextensive computational effort is a characteristic of the presented method. A numerical example illustrates the application of the proposed method. Copyright.