## 1 Introduction  
Tiny interference components are widely used in the precision connection of the aerospace and instrumentation fields. Assembly quality will greatly affect the performance and life of mechanical equipment [1; 2; 3; 4]. The stress distribution and connection force are important parameters for evaluating assembly quality [1; 5]. Therefore, studying the stress distribution and connection force on the mating surface of interference components is very important and promising to study. Ultrasonic methods to measure contact stresses have been widely used because ultrasonic waves can be flexibly applied to many practical engineering materials and components [6; 7]. The ultrasonic measurement of contact stress can be achieved based on the mapping relationship of reflection coefficient-contact stiffness-contact stress. In the same contact state, ultrasonic waves with different frequencies have different wavelengths, which affects the amplitude of ultrasonic waves reflected from the contact interface, thereby changing the reflection coefficient. Therefore, the mapping relationship between the reflection coefficient and contact stress measured by ultrasonic transducers with different center frequencies is different. Fortunately, the mapping relationship between contact stiffness and contact stress is not affected by ultrasonic frequency. This is due to the fact that contact stiffness is a mechanical property of the contact interface and is not affected by the frequency of ultrasonic waves [8]. Based on the contact stiffness-contact stress mapping relationship, the contact stress measured by the ultrasonic transducer with different center frequenciesis consistent. Therefore, the calibration of the contact stiffness-contact stress mapping relationship can make the measurement of contact stress independent of frequency, which means the measurement results are unaffected even if transducers with different frequencies are used in the calibration and measurement stages. In practical engineering, the reflection coefficient--contact stress is calibrated first, and then the contact stiffness--contact stress is obtained by the contact acoustic model. As a result, the connection force can be calculated using Coulomb's friction law based on the measured stress distribution. Several acoustic models have been developed over the past few decades [9; 10; 11]. [PERSON] and [PERSON] [10] proposed a spring model, which can be represented by a single spring, to describe the ultrasonic reflection from the stationary and sliding interfaces under the elastic contact assumption to determine the reflection coefficient and contact stiffness mapping relationship. Based on the spring model, [PERSON] et al. measured the contact stiffness of the Al-Al interface at different pressures [8]. Unfortunately, the measured stiffness is much higher than predicted by the theoretical contact model. In contrast, the GW model is in good agreement with the experiments performed using approximation measurements in the elastic deformation range [12]. This model is obtained by integrating the contact states of all asperities using statistical theory. This provides a more reasonable description of the actual contact state of the rough interface. However, this model does not give the relationship between contact stiffness and acoustic parameters. Recently, methods based on virtual material assumptions [13; 14] have been shown to be effective in describing contact properties in engineering applications, such as bolting [15]. However, these previous models were limited to dry, rough interfaces. In practical industrial applications, mechanical systems such as various gears, rolling bearings, and wheel-rail interfaces work in mixed lubricating contacts to reduce friction or wear; that is, lubricants fill concave valleys in rough interfaces. For the lubrication interface, the coupling effect of roughness and lubricant increases the difficulty of constructing the acoustic model.