## 2 Establishment of the Mathematical Model  
Figure 2 shows the ultrasonic reflection/transmission model of the flange fastening device in air, which is mainly composed of Q235 and rubber materials and vertically stacked on a horizontal plane. The first and fifth layers of the structure are Q235, the second and fourth layers are air layers, and the third layer is silicone rubber. To reduce the computational complexity of the model, the influence of roughness on the reflection coefficient has not been considered. In the model, the upper and lower layers are solid materials, and there is a gas layer sandwiched between the solid-solid layers. When the sound wave is incident into the medium at an oblique angle, reflection occurs in the upper solid-gas layer and transmission occurs in the lower solid-gas layer. According to the theory of acoustic reflection, the acoustic reflection coefficient is related to the density, sound speed, thickness, and acoustic impedance properties of the intermediate layer and the two-sided media. In the contact separation process of the flange fastening device, the thickness of the air layer changes in the range of 10-110 \(\upmu\)m. Based on the stress and strain continuity at the boundaries of the multilayer media, the first-order resonance model can be used to accurately predict the ultrasonic reflection coefficient between different thicknesses and different material media. When the thickness of the internal air layer is significant in a flange fastening device, its volume and mass cannot be neglected and a first-order resonant model is typically used to measure the air layer thickness [13]. When the sound field is perpendicular to the surface, the reference displacement-stress formula and the continuity boundary conditions of acoustic displacement and stress at the interface can be applied to derive the analytical expression of the ultrasound reflection coefficient \(R\). In a solid medium, the displacement and stress equations for longitudinal waves are given by [14]:  
\[u(x)=Ae^{i\omega x/c}+Be^{-i\omega x/c} \tag{1}\]  
\[\sigma(x)=E\frac{\partial u}{\partial x}=\mathrm{i}\omega z\Big{(}Ae^{i\omega x /c}-Be^{-i\omega x/c}\Big{)} \tag{2}\]  
where \(u\) is the displacement of the sound wave in the x-direction, \(\sigma\) is the stress induced by the sound wave in the x-direction, \(A\) is the amplitude of the incident sound wave, \(B\) is the amplitude of the reflected sound wave, \(\omega\) is the angular frequency, \(c\) is the speed of sound in the medium, and \(c=\sqrt{E/\rho}\), where \(\rho\) is the density of the medium, and \(z\) is the acoustic impedance of the medium, \(z=\rho c\). Neglecting the effect of surface roughness on the metal material, the contact between Q235 and the air layer can be regarded as a perfect interface, so the displacement and stress are continuous when crossing the interface. Therefore, the continuity conditions at \(x=0\) and \(x=h\) for the interface are given by:  
\[ \begin{array}{c}u_{1}|_{x_{3}=0}=\left.u^{\prime}_{1}\right|_{x_{3}=0}\\ \sigma_{33}|_{x_{3}=0}=\left.\sigma^{\prime}_{33}|_{x_{3}=0}\\ u_{3}|_{x_{3}=h}=\left.u^{\prime}_{3}\right|_{x_{3}=h}\\ \sigma_{31}|_{x_{3}=h}=\left.\sigma^{\prime}_{31}|_{x_{3}=h}\end{array}  \tag{3}\]  
where \(h\) is the thickness of the air layer. When assuming the amplitude of the incident wave is 1, Equations (1)-(3) yield [15; 16]:  
\[R=\frac{g^{2}(z_{2}-z_{3})(z_{1}+z_{2})+(z_{1}-z_{2})(z_{2}+z_{3})}{(z_{2}+z_{ 3})(z_{1}+z_{2})+g^{2}(z_{2}-z_{3})(z_{1}-z_{2})} \tag{4}\]  
where \(g\) has no physical meaning in the equation, \(g=e^{i\omega h/c}\), \(z_{i}\) is the acoustic impedance of Q235, air, and silicone, and its calculation formula is \(z_{i}=\rho_{i}c_{i}\), \(\rho_{i}\) is the material density, \(c_{i}\) is the internal sound velocity of the material, and \(i=1,2,3\).