# Bol Looseness Detection Based on Piezoelectric Impedance Frequency Shift
## 2 The Proposed Method and Principle  
According to the piezoelectric effect, piezoelectric patches can be used as sensors and actuators. In the piezoelectric impedance method, a piezoelectric patch is bonded on the monitored structure surface or integrated into the structure, and the piezoelectric patch can be excited by a sine sweep voltage signal generated by an impedance analyzer. According to the inverse piezoelectric effect, the sinusoidal voltage input causes the vibration of the piezoelectric patch, and the vibration is transferred to the host structure. Then, the vibration of the structure reacts to the piezoelectric patch simultaneously, which causes electrical signal changes in the circuit due to the direct piezoelectric effect. The coupled admittance, the reciprocal of the impedance, which is closely related to the substrate structural impedance, can be captured by the impedance analyzer. Based on the 1-D coupling model of piezoelectric patch and the host structure, and considering the force equilibrium and the piezoelectric constitutive equations, [PERSON] [20] derived the following electro-mechanical admittance expression:  
\[\overline{Y}=2\omega\frac{w_{a}l_{a}}{h_{a}}\left[\overline{\epsilon_{33}^{T} }+\left(\frac{Z_{a}}{Z+Z_{a}}\right)d_{31}^{2}\overline{Y_{11}^{E}}\left(\frac {\tan\kappa l_{a}}{\kappa l_{a}}\right)-d_{31}^{2}\overline{Y_{11}^{E}}\right] \tag{1}\]  
where \(\omega\) is the angular frequency of the excitation voltage, and \(i\) is the imaginary unit. \(w_{a}\), \(l_{a}\) and \(h_{a}\) are respectively width, length and thickness of the PZT patch. \(\overline{\epsilon_{33}^{T}}\) represents the complex electric permittivity at zero mechanical stress, \(\overline{Y_{11}^{E}}\) is the complex Young's modulus at zero electric field. \(Z_{a}\) is the mechanical impedance of the PZT patch and \(Z\) is the mechanical impedance of the host structure. \(d_{31}\) and \(\kappa\) are the piezoelectric strain coefficient and the wave number, respectively.  
Equation (1) shows that the output admittance signature is affected by both the structural and sensor dynamics, since the admittance is the response of the structure to the harmonic actuation of the piezoelectric transducer and the applied load will influence the response of the structure in the presence of mechanical impedance. The stress changes at a specific location on the structure will cause resonant frequency shift, and vibration mode change of the structure, which will be reflected in changes in the admittance signature [37; 38].  
For the bolted joints, the bolted connection status affects only the local dynamic characteristics of the structure and the bolt itself. When the bolt preload is changed, the stress distribution on the bolt and the bolt joint will change. Since the piezoceramic patch is bonded on the bolt head, the sensing region will be concentrated in a small range of the bolt head and the bolt head stress status change will be reflected in the admittance signature as the resonance frequency shift. By analyzing the interrelation of a bolt head stress-caused frequency shift and the bolt preload, a new bolt joint health monitoring method may be formed based on the piezoelectric impedance frequency shift. The local structure has high frequency dynamic characteristics, and the piezoelectric impedance method is appropriate to study the local dynamics of the bolt joint with appropriate sensing frequency.