## 2 Proposed Method and Principle  
The new smart washer structure is shown in Figure 1. It is composed of two annular disks, the contact surfaces of which are machined into convex and concave, respectively. Two piezoelectric patches are bonded onto two non-contact surfaces of the proposed smart washer sensor. One piezoelectric patch serves as an actuator to generate an ultrasonic wave transmitting through the contact surface, and the other one as a sensor to detect the response signals. Furthermore, the time reversal method is employed, which can ensure a high signal to noise ratio. The first response signal is reversed in the time domain and then reemitted. Then, the reverse focused signal is detected by the sensor piezoelectric patch. In addition, based on the fact that the waves are the energy transmission and the transmitted energy is proportional to the true contact area of the bolted joint interface, the bolted loosening status can be determined by extracting the focused signal peak that has a certain relationship with the transmitted energy. In order to describe the relationship between the contact area and loading-force, an analysis of the contact is necessary. According to the Classical Hertz contact theory, two elasticaxis-parallel cylinders\" internal contact is shown in Figure 2; the contact surface shape of the cylinders is rectangular [34]. Furthermore, the unit's pressure on the contact surface is distributed in an elliptic cylindrical pattern as shown in Figure 3. [PERSON] theory relates loading force \(F\) and contact area \(S\) as follows:  
\[a=\sqrt{\frac{4 FR}{\pi LE^{*}}} \tag{1}\]  
\[R=\frac{R_{1}R_{2}}{R_{2}-R_{1}} \tag{2}\]  
\[\frac{1}{E^{*}}=\frac{1-\upmu_{1}^{2}}{E_{1}}+\frac{1-\upmu_{2}^{2}}{E_{2}} \tag{3}\]  
\[S=2 aL=2\sqrt{\frac{4 FLR}{\pi E^{*}}}=4\sqrt{\frac{FLR}{\pi E^{*}}} \tag{4}\]  
Figure 1: Structure sketch. where \(a\) is half the width of the rectangle, \(L\) is the length of cylinders, \(E_{1}\) and \(E_{2}\) are the elasticity modulus of the two cylinders, \(R_{1}\) and \(R_{2}\) are the curvatures of the cylinders, and \(\mu_{1}\) and \(\mu_{2}\) are Poisson's ratios. For the smart washer in this paper, the contact area is an annulus; the existing contact formula is not applicable to this model; after the differential, the annulus can be regarded as being composed of numerous small rectangles, made up of countless small rectangles; the area of each rectangle is d\(S\); and the contact area, \(S\), can be obtained by the equation as follows:  
\[\mathrm{d}S=2a\mathrm{d}L \tag{5}\]  
\[S=\int\mathrm{d}S=\int_{0}^{2\pi,r}2a\mathrm{d}L=4\sqrt{\frac{2 FrR}{E^{*}}} \tag{6}\]  
where \(r\) is the distance between the center of the washer and the center of the convex and concave, as shown in Figure 1, and \(2\pi r\) is the perimeter of the annulus center. It can be seen that the contact area is decided by the acting force when \(r\) and \(R\) are given, more specifically, the area will grow with an increasing load. Therefore, the contact area can be an indicator of the loading force. The essence of wave propagation is energy transmission, and the wave leakage and dissipation occur at the contact interface. Thus, the energy will be reduced when the wave transmits through the interface. For the washer, the energy received by the PZT1 and PZT2 is the transmitted wave energy that is transmitted through the convex and concave contact interfaces. It is found that the transmitted energy, \(E\), is proportional to the true contact area, \(S\)[35], as follows:  
\[E\propto C_{1}S \tag{7}\]  
The relationship implies that the energy through the contact area increases with the enlargement of the contact area, when the force increases according to Equation (6). As a result, more energy is  
Figure 3.0 Contact surface status. Figure 3.1 Axis-parallel cylinders’ internal contact. transmitted to the sensor. After considering the relationship in Equation (6), the transmitted energy, \(E\), can be expressed as follows:  
Figure 3.2  
In Equations (7) and (8), \(C_{1}\) and \(C_{2}\) are constants.