# Modal Strain Energy-Based Structural Damage Detection Using Convolutional Neural Networks
## 2 Methods  
### Vibration Experiment  
A steel frame was used in this paper for the experiment (Figure 1); this beam consisted of 381 rods, and its length, width, and height were 10.62, 0.354, and 0.354 m, respectively. Each rod had a hollow circular cross section with an external radius of 0.005 m and thickness of 0.002 m. The rods were connected by steel balls. Vibration measurement methods (accelerometers and high-speed camera) were used to obtain the vibration signals. Experimental facilities (Figure 2) included: A FASTCAM SA3 high-speed camera, a JM3840 dynamic data acquisition instrument, 10 accelerometers, an instrumented hammer, and laptops. The weights of the structure, accelerometers and cables were 149.24, 0.52, and 0.84 kg, respectively; hence, the influence of the adding mass on the measurement results were negligible.  
The vibration signals of the structure were collected by accelerometers (YD2150, Jing-Ming Technology Inc., Yangzhou, China) and a high-speed camera (FASTCAM SA3, Photron Inc., Tokyo, Japan), with a sampling frequency of 250 Hz. Vibration signals and force signals were processed by JMTST dynamic signal analysis software, and then modal analysis was carried out to obtain the experimental modal parameters (modal shapes and frequencies). Meanwhile, the high-speed camera  
Figure 1: The steel frame modal with 381 rods.  
Figure 2: Experimental facilities. (**a**) accelerometers; (**b**) hammer; (**c**) JM3840 dynamic data acquisition instrument; (**d**) FASTCAM SA3 high-speed camera; (**e**) laptop.  
recorded the frame pictures (8 bit) in the process of vibration, and the displacement signal of the structure could be obtained by using DIC (digital image correlation) skills. The principle of tracking the movement of points of interest using DIC [36; 37] was expressed in the following formula:  
\[C(\Delta x,\Delta y)=\frac{\iint_{S}I_{0}(x,y)I_{1}(x+\Delta x,y+\Delta y)dxdy}{ \sqrt{\iint_{S}I_{0}{}^{2}(x,y)dxdy}\cdot\sqrt{\iint_{S}I_{1}{}^{2}(x+\Delta x,y +\Delta y)dxdy}} \tag{1}\]  
where \(I_{0}(x,y)\) and \(J(x,y)\) are the gray-scale distributions of two images, and the real displacement \((\Delta x,\Delta y)\) of the reference subset (\(S\)) maximizes the function \(C(\Delta x,\Delta y)\).  
The accelerometers were installed on 10 nodes (Figure 3). The steel frame was excited 5 times with the hammer at the excitation point (the distances between the excitation point and the two supports were 3.894 and 6.726 m, respectively), the force signal was collected by the force sensor, and the accelerometers and the high-speed camera were used to collect the acceleration and displacement of the points of interest. The modal analysis was carried out by using JMTST dynamic signal analysis software, and the first-order frequency and modal shape were obtained. Then the first-order MSE of the structure was calculated (Equation (2)) [12].  
\[MSE_{j}=\phi^{T}K_{j}\phi \tag{2}\]  
where \(\phi\) and \(K_{j}\) were the first-order modal shape and stiffness matrix of the \(j\)-th rod, respectively.  
This paper mainly addressed 4 structural states. State (1): Intact structure; State (2): Damage on Rod 2; State (3): Damage on Rod 11; State (4): Simultaneous damage on Rod 2 and Rod 11. Damage was introduced by cutting off 50% of the cross section (in practical engineering, damage may be caused by corrosion, cracks and other reasons); the intact and damaged rods are shown in Figure 4. The damage index was the change rate of the modal strain energy (MSECR).  
\[MSECR=\frac{\left|MSE_{damage}-MSE_{intact}\right|}{MSE_{intact}} \tag{3}\]  
where the \(MSE_{damage}\) and \(MSE_{intact}\) are the modal strain energy of the damaged structure and intact structures.  
Figure 3: Location of excitation point and accelerometers. N-#: Installation point of accelerometers; R-#: Rods 1–17.