Abstract: In this paper,the optimal problem of the displacement mean square re-sponse for weak nonlinear systems with hardening or softening sprimg subje-cted to a white noise excitation is studied.The relation between the power spectral densities of the displacement and the excitation is obtained by means of the perturbation techniqe.According to the damped least square feasible direction method,the coefficients of the damping and stiffness of the weak nonlinear systems are optimized.The displacement mean square response of the mass m₂ in a certain frequency range is taken as an objective function.The optimal design is accomplished numerically.