This paper attempts to investigate the behaviors of coupling stochastic resonance (CSR) subject to α-stable noise and a periodic signal by using the residence-time ratio. Then, a nonlinear resonance decomposition is designed to successfully enhance and detect weak unknown multi-frequency signals embedded in strong α-stable noise by decomposing the noisy signal into a series of useful resonant components and a residue, where the residence-time ratio, instead of the output signal-to-noise ratio and other objective functions depending on the prior knowledge of the signals to be detected, can optimize the CSR to enhance weak unknown signals. Finally, the nonlinear resonance decomposition is used to process the raw vibration signal of rotating machinery. It is found that the nonlinear resonance decomposition is able to decompose the weak characteristic signal and its harmonics, identifying the imbalance fault of the rotor. Even the proposed method is superior to the empirical mode decomposition method in this experiment. This research is helpful to design the noise enhanced signal decomposition techniques by harvesting the energy of noise to enhance and decompose the useful resonant components from a nonstationary and nonlinear signal.