## 1 Introduction  
Rolling bearings are widely used in the field of engineering machinery. It is one of the parts with the highest failure rate, and causing problems in regard to quality and safety [1]. The fault monitoring and analysis of bearings usually deals with the vibration signals collected by sensors [2; 3; 4]. When the background noise of the vibration data is high and the signal-to-noise ratio (SNR) is low, it is very challenging to obtain the bearings' fault features. In the context of loud noise, the traditional bearing fault analysis methods usually start with noise reduction [5; 6; 7; 8; 9; 10]. Although these methods can reduce the noise, they also can weaken the effective characteristic signal [11; 12; 13; 14; 15; 16; 17; 18]. [PERSON] et al. [19] proposed the theory of stochastic resonance (SR), which can pass noise to weak signals and strengthen the weak characteristic signals submerged in them [20]. This feature has aroused great interest as more and more scholars have begun to study the impact of SR on machine fault diagnosis [21; 22; 23; 24; 25; 26; 27; 28; 29; 30] and its application in other fields [31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41]. In order to quantitatively describe the phenomenon of SR and have a standard for judging the effect of SR, a measurement index is required. In this paper, SNR is used as a measure of SR [42]. When the SNR is larger, it means that the signal has less noise interference and is more pure. In an SR system, the input signal frequency is limited to far less than 1 Hz due to its adiabatic approximation condition [43]. However, in practical engineering application, the bearing has a fault characteristic frequency that is far more than 1 Hz. [PERSON] et al. [44] proposed a standardized scale conversion non-stress test (NST) SR. [PERSON] et al. [45] presented a variable scale SR method to satisfy the small parameter condition of SR, which compresses large frequency signals to small frequency signals (frequency less than 1 Hz) according to a certain scale. [PERSON] et al. [46] studied a method of frequency-shifted and re-scaling SR (FRSR) to achieve small parameter conditions. Large frequency signals are first processed by a high-pass filter, and then combined with frequency-shifted and re-scaling technology. The above two methods play an important role in practical engineering application. In addressing the problem of system parameters selection, [PERSON] et al. [47] found that the selection of classical SR (CSR) system parameters can be achieved under the condition of particle swarm optimization (PSO), which can be easily interfered with by low-frequency information. [PERSON] et al. [48] introduced a method of ant colony optimization algorithm to analyze the signal after FRSR to determine the system parameters, but the operation of this method with the help of a high-pass filter is complex. [PERSON] et al. [49] analyzed the underdamped step-varying second-order SR (USSSR) method. Compared with the first-order SR system, the second-order filtering performance is better, and the dependence on the filter is eliminated. [PERSON] [50] gave specific steps to solve the USSSR parameters by ant colony optimization algorithm. [PERSON] et al. [51] further analyzed the advantages of a second-order multistable SR system. However, the coupling model studies the SR system from another point of view. There is little research on the SR coupling model. [PERSON] et al. [52] studied the coupling bistable system and used it for fault signal detection. [PERSON] et al. [53] further studied a novel adaptive SR method based on coupled bistable systems and its application in rolling bearing fault diagnosis. Although the output of SNR has been improved, its effect is not good. Based on the above analysis, we proposed a second-order coupled step-varying stochastic resonance (SCSSR) model, which is a further evolution of the coupled SR model.