Abstract: A superelastic shape memory alloy (SMA) damper is designed. Based on the Graesser’s constitutive model of SMA wire, the conventional mechanical model of the SMA damper is set up. Further, a simplified mechanical model of the SMA damper is put forward. Using a stochastic equivalent linearization method, the analyses of the stationary random vibration of a single-degree-of freedom structure with the SMA damper subjected to Gaussian white noise excitation are formulated. Numerical simulations are carried out to compare the vibration responses of the structure between the stochastic equivalent linearization method and the Monte Carlo simulation method, and the results testify that the stochastic equivalent linearization method is valid. The dynamic properties and the vibration responses of the structure without and with the SMA damper are computed, and the results show that the SMA damper can decrease the vibration of the structure effectively due to its capability of increasing the stiffness and the damping ratio of the structure.