This paper proposes the stiffness nonlinearities and asymmetric SD (smooth and discontinuous) oscillator under time-delayed feedback control with the fractional derivative damping. With the effect of displacement and velocity feedback, the oscillator can be adjusted to the desired vibration state, and then the stochastic resonance (SR) is achieved. This article discusses the contribution of various system parameters and time-delayed feedback to SR, especially which induced by fractional order. It should be noted that this paper provides effective guidance for fault diagnosis and weak signal detection, energy harvesting, vibration isolation and vibration reduction.