An analytical and numerical algorithm for solving the displacement response of fractional oscillator system under cosine excitation is presented. The analytical method means that steady-state response and transient response solutions of the system can be obtained by the average method. The total displacement response solution is the sum of the steady-state solution and transient solutions. In the numerical method, the Grunwald-Letnikov definition of fractional derivative is used to discretize the fractional differential term in the system, so as to reduce the order of the original system. Considering the general periodic excitation, the approximate response solution of the system can be obtained by using the Fourier series expansion method and the linear system superposition principle. Finally, the effectiveness and feasibility of the proposed method are verified by numerical simulation. The effects of fractional order, linear damping coefficient and fractional derivative coefficient on steady-state response amplitude and total displacement response of the system are analyzed.