# The Effects of Stress on Second Harmonics in Plate-Like Structures
## 4 Experimental Verifications  
The validation experiments are designed for a steel plate with a thickness of 15 mm, length of 2 m, and width of 20 cm. The material properties of the steel plate are listed in Table 1. In order to make a comparison, one mode pair from mode pairs with increasing second harmonic amplitude (mode pair 10 and 11), another mode pair from mode pairs with decreasing second harmonic amplitude (mode pair 7-9, 12-15), and another mode pair from mode pairs with no second harmonic (mode pair 1-6) are selected. It can be seen from Figure 5 that the amplitude coefficients, \(\overline{A}_{n}(x)\), of mode pairs 10 and 15 are the largest in the same type of mode pairs, so the amplitudes of the second harmonics generated by mode pairs 10 and 15 are also the largest. Therefore, the mode pairs 6, 10, and 15 in Table 2 are selected as excitation and reception modes, respectively, and the amplitudes of second harmonics of different mode pairs are compared. In the experiment, the longitudinal wave angle probes with adjustable angle are used as the excitation and receiving transducers, and the distance between the excitation transducer and the receiving transducer is 95 mm. The excitation and receiving angles of transducers meet [PERSON]'s theorem, as follows:  
\[\frac{\sin\theta}{C_{\text{glass}}}=\frac{\sin 90^{\circ}}{C_{p}} \tag{27}\]  
where \(C_{\text{glass}}\) is the speed of ultrasonic waves propagating in plexiglass, \(C_{\text{glass}}=2740\ m/s\). \(C_{p}\) is the phase velocity of ultrasonic waves propagating in steel plate; the \(C_{p}\) corresponding to mode pairs 6, 10, and 15 is \(C_{p1}=5904\ m/s\), \(C_{p2}=3933\ m/s\), and \(C_{p3}=3376\ m/s\), respectively. Substitute the above values into Equation (27) and get \(\theta_{1}=28^{\circ}\), \(\theta_{2}=44^{\circ}\), and \(\theta_{3}=55^{\circ}\). The excitation and receiving transducers are placed on the steel plate, as shown in Figure 8a. In the experiment, the surface of the place where the transducers are placed is flat, which ensures there is no micro crack and bulge. At the same time, a coupling agent with good viscosity is added between the transducer and the steel plate, to increase the bonding strength of the interface and further improve the signal transmission efficiency. The Ritec Ram-5000 high-energy ultrasonic test system is used to excite and receive the signal of the ultrasonic transducers, as shown in Figure 8b. The test system can amplify the maximum excitation signal by 100 times. Because the signal excitation, amplification, and reception are completed by the same test system, the loss of signal strength in the process of transmission is greatly reduced. The experimental system diagram is shown in Figure 9.  
Figure 7: Amplitudes of the second harmonic mode generated by mode combinations vary with the stress: (**a**) mode pair 7; (**b**) mode pair 8; (**c**) mode pair 9; (**d**) mode pair 10; (**e**) mode pair 11; (**f**) mode pair 12; (**g**) mode pair 13; (**h**) mode pair 14; and (**i**) mode pair 15. In the experiment, the excitation signal is a 20-period sine wave modulated by the Hanning window, with a frequency of 2.5 MHz. At the same excitation amplitude, by adjusting the inclination angle of the transducers, the selection of mode pair is changed. The experimental received signal from mode pair 10 is shown in Figure 10 (the received signals of mode pairs 6 and 15 are similar). Fast Fourier transform (FFT) is performed on the inner part of the circle of the received signal, under each mode pair, to extract the basic frequency and second harmonic signals. The experimental spectrum analysis results are shown in Figure 11.  
Figure 11 illustrates that, when mode pairs 10 and 15 are selected as the excitation and reception modes, the received signals have obvious second harmonic, while mode pair 6 is selected as the excitation and reception mode, without obvious second harmonic. Under the same excitation amplitude, the second harmonic amplitude of mode pair 10 is larger than that of mode pair 15.