## 2 Theoretical Background
### Contact Acoustic Nonlinearity  
The nonlinear acoustic responses generated by CAN are always stronger than those resulting from the other nonlinearities [18], and thus, the nonlinear coefficients derived from the CAN model are widely used as the damage indices of SHM and NDT&E.  
In a bolted joint, the preload \(F\) induced by the applied torque \(T\) can be described by  
\[F=T/(\tau d), \tag{1}\]  
where \(\tau\) is the friction coefficient between the bolt and nut, and \(d\) is the diameter of the bolt.  
In the VAM-based method, the loose joint is subjected to mixed excitation from a low-frequency pumping vibration (with an equivalent force \(A_{1}cos\omega_{1}t\)) and a high-frequency probing wave (with an equivalent force \(A_{2}cos\omega_{2}t\)), which are independent of each other, as exhibited in Figure 1.  
When the motion of the bolted interfaces excited by the two waves conforms to the \"opening-closing\" behavior in the CAN model, the relationship between the contact stiffness and the applied load is suggested to be [21]:  
\[K_{1}=Cp^{m}\propto F^{m}\propto T^{m}, \tag{2}\]  
\[K_{2}=0.5 mC^{2}p^{2m-1}\propto F^{2m-1}\propto T^{2m-1}, \tag{3}\]  
where \(p\) is the contact pressure, \(K_{1}\) is the linear stiffness, and \(K_{2}\) is the nonlinear stiffness of the contact interface. \(C\) and \(m\) are associated with the surface properties of the material in contact.  
The equation of the motion of the joint can be described as:  
\[M\ddot{x}+K_{1}x-\phi K_{2}x^{2}=A_{1}\cos\omega_{1}t+A_{2}\cos\omega_{2}t, \tag{4}\]  
where \(\omega_{1}\) and \(\omega_{2}\) are the frequencies of the pumping vibration and probing wave, respectively, and \(\phi\) is a small quantity to scale the perturbation to be minute.  
To solve Equation (4), the analogous method based on perturbation theory is used, and the second-order nonlinearity of the solution is considered. The details of the derivation of this solution are presented in [4; 22]. As a result, we can have x in the form of a combined wave, including the fundamental waves at \(\omega_{1}\) and \(\omega_{2}\), the SOHs at \(2\omega_{1}\) and \(2\omega_{2}\), the three-order harmonics (TOHs) at \(3\omega_{1}\) and \(3\omega_{2}\), and the SBs, viz., the right sideband (RS) at \(\omega_{1}+\omega_{2}\) and the left sideband (LS) at \(\omega_{1}-\omega_{2}\). The amplitudes of the signal components satisfy the relations  
\[A_{SOH}=\frac{0.5K_{2}}{K_{1}-4M\omega_{1}{}^{2}}A_{LF}^{2}, \tag{5}\]  
\[A_{TOH}=\frac{0.5K_{2}^{2}}{(K_{1}-9M\omega_{1}{}^{2})(K_{1}-4M\omega_{1}{}^{ 2})}A_{LF}^{3}, \tag{6}\]  
\[A_{SB}=\frac{K_{2}}{K_{1}-M(\omega_{1}\pm\omega_{2})^{2}}A_{LF}A_{HF}. \tag{7}\]  
Figure 1: Simplified model of a joint subjected to two harmonic forces.  
Then, the nonlinear coefficients \(\beta\) can be defined in the unit of dB as  
\[\beta_{SOH}=A_{SOH}-2A_{LF}, \tag{8}\]  
\[\beta_{TOH}=A_{TOH}-3A_{LF}, \tag{9}\]  
\[\beta_{VAM}^{\rm M}=(A_{LS}+A_{RS})/2-A_{LF}-A_{HF}, \tag{10}\]  
\[\beta_{VAM}=(A_{LS}+A_{RS})/2-A_{HF}. \tag{11}\]  
By eliminating the linear terms (i.e., \(LF\) and \(HF\)), the nonlinear coefficients \(\beta\) only depend on the contact stiffness \(K_{1}\) and \(K_{2}\) and are related to the preload \(F\) and torque \(T\). According to CAN theory, the nonlinear coefficients decrease with the increase in the preload (i.e., the applied torque).