## 2 Principles
### Elastic Contact Theory between a Rigid Cylinder and a Plane  
To establish the relationship between the contact area and the force \(F\), it is essential to analyze the contact model. According to classical [PERSON] contact theory, the contact area between a rigid cylinder and an elastic half space can be simplified as a rectangle with a length of \(L\) and a width of \(2a\)[48], as illustrated in Figure 3.  
In the case of contact between a rigid cylinder and an elastic half-space (Figure 3), the contact stress, \(p\), distributed in a semi-elliptic cylinder is  
\[p(x)=p_{0}\sqrt{1-\frac{x^{2}}{a^{2}}},x\leq a \tag{1}\]  
where \(p_{0}\) represents the maximum contact stress, \(x\) denotes the radial distance between the contact point and the contact center, and \(a\) represents half of the contact width.  
Half of the contact width is determined by the following equation:  
\[a=\sqrt{Rd} \tag{2}\]  
where \(R\) represents the radius of the cylinder and \(d\) represents the penetration depth.  
The total force is calculated as follows:  
\[F=\int\limits_{0}^{a}p(x)2 Ldx=\frac{\pi p_{0}aL}{2} \tag{3}\]  
where \(F\) represents the external force and \(L\) represents the contact length.  
The order of magnitude of elastic deformation in the contact region is \(\varepsilon=d/2a\). The maximum contact stress is equal to  
\[p_{0}=\frac{E^{*}d}{2a} \tag{4}\]  
where \(E^{*}\) represents the effective elastic modulus, \(E^{*}=E/(1-\
u^{2})\); \(E\) represents the elasticity modulus, and \(\
u\) denotes the [PERSON]'s ratio.  
Substituting Equation (4) into Equation (3) gives  
\[F=\frac{\pi}{4}E^{*}Ld \tag{5}\]  
The force is linearly proportional to the penetration depth \(d\).  
Substituting Equation (2) into Equation (5) yields  
\[a=\sqrt{\frac{4 FR}{\pi E^{*}L}} \tag{6}\]  
Figure 3: Schematic diagram of Hertz contact between cylinder and plane.  
Given that the contact area is \(A=2 aL\), the relationship between the contact area and the force is as follows:  
\[A=4\sqrt{\frac{RL}{\pi E^{*}}}\cdot\sqrt{F} \tag{7}\]  
It can be observed that the contact area, \(A\), is directly proportional to the square root of the external force, \(F\). Therefore, the contact area can serve as an indicator of the magnitude of the external force.  
When an elastic wave propagates through an interface, a portion of the wave is dissipated due to interface friction (\(\Omega_{dissipated}\)), another portion of the wave passes through the contact interface (\(\Omega_{pass}\)), and the remaining portion is reflected (\(\Omega_{reflectal}\)). The incident elastic wave can then be expressed as  
\[\Omega=\Omega_{dissipated}+\Omega_{pass}+\Omega_{reflected} \tag{8}\]  
The relationship between the passing wave \(\Omega_{pass}\), the contact area, and the external force is as follows [41]:  
\[\Omega_{pass}\propto A\propto\sqrt{F} \tag{9}\]  
As shown in Figure 3, the wave energy that passes through the interface is received by the PZT patch on the lower annular cylinder. As can be seen from Equation (9), the energy of the incident wave remains unchanged with the increase in external force, and the energy of the transmitted wave will increase with the increase in external force. This is because the contact area also increases. Thus, the wave energy received by the lower annular cylinder can be utilized to accurately measure the level of prestress tightness.