In this paper, using the one-dimensional vibrating beam equation with generalized viscous damping as a model of vibration of flexible robot arms, it is shown that for such a system the high eigenmodes decay at a uniform rate. The proof is obtained by perturbation theory of linear operators [Y. H. Luo, Acta Math. Sinica, 32 (1991), pp. 556-563] and asymptotic estimates of eigenvectors based on an earlier work of G. B. Birkhoff and M. H. Stone. Feedback control for this class of system is investigated, and a finite-dimensional controller is presented for an exponentially stable closed-loop system. Our method may be used to study nondissipative systems.