# Cable Force Health Monitoring of Tongwamen Bridge Based on Fiber Bragg Grating
## 2 Monitoring Method and FBG Vibration Sensor
### Design of FBG Vibration Sensor  
Through an earlier field test on the bridge suspension cables, it was found that the cable's vibration frequencies are generally within 6 Hz. Thus, the resonant frequency of the FBG vibration sensor should be far higher than 6 Hz so as to avoid interference caused by superimposition of the resonant frequency of the sensor with measured frequencies. Principles of fiber grating vibration sensors vary widely, and have been a hot research topic in the field of optical fiber sensing [14; 15; 16; 17; 18; 19; 20; 21; 22]. As a classic elastic element, cantilever beam shows broad applications in the design of FBG vibration sensors. Sensors based on this principle are simple and reliable with strong engineering applicability. Due these favorable reasons, FBG vibration sensor based on an equal-strength cantilever beam was designed. Compared with other cantilever-based vibration sensors, this FBG vibration sensor possesses a simpler structure of just an original equal-strength cantilever beam, which avoids complex assembly and has higher reliability. In our previous work [20], a high frequency FBG vibration sensor was made through metallization packaging. The sensor possesses a high resonance frequency of 3.6 kHz but a low sensitivity of 0.17 pm/(m/s\({}^{2}\)), which is not suitable for measuring cable vibration with low frequency and high measurement sensitivity. As shown in Figure 2, a FBG was attached on the beam surface, and a block  
Figure 1: Arrangement of fiber Bragg grating (FBG) vibration sensors on Tongwamen bridge cables: (**a**) Side view; (**b**) Overhead view. mass was fixed on the free end of the beam. Under the influence of external vibration acceleration, the beam vibrated with inertial force on the mass, generating alternating bending strain on the surface. This bending strain was detected and converted into wavelength shift by the FBG sensor (Fenglan Tech, Co, Ltd, Anshan, China). Thus, external vibration acceleration was retrieved using wavelength shift information detected by the demodulator. Based on the principle from mechanics of materials, for a cantilever beam of equal strength, relationship between its deflection \(w\) and force acting on its end can be written as:  
\[w=\frac{6L^{3}}{Ebh^{3}}\times F \tag{2}\]  
where \(L\) is length of the beam, \(b\) is width of the beam's fixed end, \(h\) is thickness, and \(E\) is Young's modulus of the beam. The force on the free-end of the beam is inertia force \(F\) generated by the block mass under external acceleration \(a\) can be calculated by:  
\[F=ma \tag{3}\]  
where \(m\) is mass of mass block. For an equal-strength beam, the relationship between strain \(\varepsilon\) of its surface, which can be sensed and measured by the FBG, and its deflection \(w\), can be described as:  
\[\varepsilon=hw/L^{2} \tag{4}\]  
For a FBG with initial central wavelength \(\lambda\), its wavelength shift \(\Delta\lambda\) has the following relationship with its longitudinal strain change \(\Delta\varepsilon\) and its environmental temperature change \(\Delta T\):  
\[\frac{\Delta\lambda}{\lambda}=(1-P_{e})\Delta\varepsilon+(\alpha_{f}+ \xi)\Delta T \tag{5}\]  
where \(\alpha_{f}\), \(\xi\), and \(P_{e}\) are optical fiber's thermal expansion coefficient, thermo-optic coefficient and elasto-optic coefficient, respectively. \(P_{e}\) is approximately 0.22 at normal temperature. For the vibration acceleration sensor, which only measures frequency, wavelength shift caused by temperature change has no impact on the frequency. Therefore, only effect of the strain change \(\Delta\varepsilon\) remained on grating wavelength, and Equation (5) can be rewritten as:  
\[\frac{\Delta\lambda}{\lambda}=(1-P_{e})\Delta\varepsilon \tag{6}\]  
Figure 2: Structural schematic diagram of the designed FBG sensor.