## 5 Experimental Validations  
The proposed damage quantification method is applied to quantify the size of the defects in two steel rods. The radius of the two rods at the intact segments is 5 mm and the total length is 960 mm. The experimental setup is illustrated in Figure 10. Instead of applying multiple excitations at different frequencies, a broadband excitation with a pulse signal is used. The frequency-dependent characteristics of the responses are extracted by post signal processing. A pulse-echo circuit developed by our lab is used to generate a 40 V pulse with a duration of 20 \(\upmu\)s. A DPC torsional-wave transducer (ACS(r) S1802 manufactured by Acoustic Control Systems, Saarbrucken, Germany) is applied to excite and capture the torsional waves in the rods with the nominal frequency at 50 kHz. The captured waves are recorded by the oscilloscope with the sampling frequency at 50 MHz. 512 averages are performed to minimize the influence of noise. It should be highlighted the use of the DPC transducer significantly simplifies the measurement system without concerns about its installation. We admit the amplitude of the generated waves by the DPC transducer are vulnerable to the contact force so that it is hard to guarantee the consistency of the generated wave amplitudes in different tests. However, as the proposed method requires only the ratio between the reflections from the defect and end, the influence of the contact force on damage quantification is eliminated. The first rod contains an axial-symmetric defect of 30 mm in width and 1 mm in depth as shown in Figure 10. The center of the defect locates at 645 mm from the left end. Figure 11a depicts the response signal with two highlighted wave packets when the DPC transducer is applied at the left end. By moving the DPC transducer along the length of the rod, the arrival time of the first wave packet varies while that of the second wave packet keeps unchanged. Therefore, it can be deduced that the first wave packet corresponds to the reflection from damage while the second one is induced by the reflection from the end with the total wave propagating distance of 1920 mm. The complex Morlet wavelet transform is applied to extract the wave amplitudes at different frequencies with the result shown in Figure 11b. No obvious dispersion is observed associated with the two wave packets. The velocities of the wave packets are calculated and match well with torsional wave speed in steels. The evidence confirms that the captured wave packets attribute to the torsional waves. Figure 9: (**a**) Variation of the \(SRC\) with respect to the excitation frequency; (**b**) damage image for the 60 mm-wide and 1.05 mm-deep defect. Figure 11b shows that reflection from the defect maximizes at around 30 kHz while the maximum reflection from the end occurs at around 50 kHz, indicating their different variation patterns with respect to the frequency. The ratio between the reflections at different frequencies is then calculated according to the designed damage quantitation method. Slightly different from the numerical simulations, a modified SRC is proposed by taking account of an average effect of the wave amplitude over a time window to reduce the influence of noise as,  
\[\overline{SRC}(f)=\frac{\int_{t_{1}}^{t_{1}+T}W(f,t)\;\mathrm{d}t}{\int_{t_{2}}^ {t_{2}+T}W(f,t)\;\mathrm{d}t} \tag{22}\]  
Figure 11: (**a**) The received signal and (**b**) its complex Morlet wavelet transform result for the first damage case. Figure 10: Experimental set-up. \(W(f,t)\) is the modulus of wavelet coefficients at frequency \(f\). \(t_{1}\) and \(t_{2}\) are the start time of the wave packets associated with reflections from the defect and end, respectively, while \(T\) denotes the length of the wave packets. Considering the effective frequency range of the transducer, the \(\overline{SRC}\) from 30 kHz to 70 kHz is calculated in Figure 12a. It is worth noting the choice of the window parameters does not significantly affect the \(\overline{SRC}\) (not shown here).