# Data-Driven Damage Classification Using Guided Waves in Pipe Structures
## 4 Experimental Investigations  
### Test Specimens and Setup  
The specimens used in the experiment were a batch of thin aluminum pipes with dimensions of 102 mm \(\times\) 1.5 mm (outside diameter \(\times\) wall thickness). The damage was manufactured in a properly middle position along the axial direction. The guided wave transducer was a single magnetostrictive one. With the help of the signal generation and reception equipment (MsSRv5, SwRI, San Antonio, TX, US), the transducer can be used to excite and receive the guided wave signals with the time-sharing mode, thus realizing the pulse-echo mode, as shown in Figure 2.  
The excitation signal \(f(t)\), defined as Equation (1), and its frequency spectrum are shown in Figure 3, in which it can be easily seen that the excitation is a sinusoidal tone-burst modulated by the Hanning window to restrain many side lobes at the center frequency of 128 kHz. If the signal frequency is high, the energy attenuation is excessively severe to detect for a long distance, and on the contrary, if the signal frequency is low, it is not sensitive to minor damage. In real application, the excitation signal was controlled by two channels to propagate mainly toward only one direction along the axial direction of the pipe, which can be explained by Equations (2) and (3):  
\[f(t)=a[H(t)-H(t-n/f_{c})][1-\cos(2\pi f_{c}t/n)]\sin(2\pi f_{c}t) \tag{1}\]  
where \(a\) is the amplitude coefficient and equal to 100 here; \(H(\cdot)\) denotes the Heaviside step function; \(n\) is the cycle; and \(f_{c}\) denotes the center frequency of the exciting signal.  
Figure 2: Damage detection setup based on a guided wave using a magnetostrictive sensor.  
Suppose that two channels of transducers were fixed at a pipe, as shown in Figure 4, then signals in the left and right transducers can be expressed as \(s_{1}\) and \(s_{2}\), respectively.  
\[\left\{\begin{array}{l}s_{1}=A_{1}\sin(\omega t+\varphi)\\ s_{2}=A_{2}\sin(\omega t+\varphi+\Delta\varphi)\end{array}\right., \tag{2}\]  
where \(\omega=2\pi f\) is the circular frequency; and \(\Delta\varphi\) is the phase difference, which needed to be designed. In order to produce destructive interference in a certain direction, the amplitude must meet the condition of \(A_{1}=A_{2}=A\). The signal propagating towards the right \(s_{right}\) and left \(s_{left}\) are as follows:  
\[\left\{\begin{array}{l}s_{right}=s_{1}+s^{\prime}{}_{2}=A\sin(\omega t+ \varphi)+A\sin(\omega t+2\pi\frac{d}{\lambda}+\varphi+\Delta\varphi)\\ s_{left}=s^{\prime}_{1}+s_{2}=A\sin(\omega t+2\pi\frac{d}{\lambda}+\varphi)+A \sin(\omega t+\varphi+\Delta\varphi)\end{array}\right. \tag{3}\]  
where \(d\) is the distance of two transducers, and \(\lambda\) is the wavelength of the guided wave at the frequency \(f\). If only the guided wave propagating towards the right is needed, Equation (4) can be obtained by solving Equation (3):  
\[\left\{\begin{array}{l}d=[-2(k_{1}+k_{2})+1]\frac{\lambda}{4}\\ \Delta\varphi=[2(k_{2}-k_{1})+1]\frac{\pi}{2}\end{array}\right. \tag{4}\]  
where \(k_{1}\) and \(k_{2}\) are both integers. For the simplest condition, \(d=\lambda/4\), the phase difference \(\Delta\varphi\) should be \(\pi/2\).  
Figure 4: Sketch map of two channels of transducers in the pipe, where A and B stand for the different pipe ends.  
Figure 3: Excitation signal and its frequency spectrum: (**a**) excitation signal in time domain; and (**b**) frequency spectrum.