A method is presented to determine the regions of dynamic instability of a structural system. The finite element method is employed in problem formulation, and the conjugate gradient method is used to compute the boundaries of dynamic ilzetability by minimizing appropriate Rayleigh quotients. The solution of the governing equation is reduced to a problem of find&g the frequencies which bound the region of dynamic instability. The conjugate gradient method used in the computations has the following advantages: (1) it is not necessary to explicitly invert the stiffness matrix; (2) the computations m,ay be carried out directly from the element stiffness and mass matrices, and therefore, it is not necessary to synthesize the corresponding system matrices; and (3) computer storage required in the computations is less than that required in conventional techniques. An algorithm is presented for computing the boundary frequencies for th,e first mode of dynamic instability. Higher modes of dynamic instability are computed from a modi$ed algorithm based on. revised Rayleigh quotients which sweep out the modes previously calcu,lated. An example, in. which the dynamic stability of a plate is established, illustrates the method.