3.3.1 Effective Group Velocity Extraction Method using SH Guided Waves
The SH1 mode is dispersive, unlike the SH0 mode, meaning that the group (and phase) velocity of the wave is dependent on the frequency-thickness of the plate in which they are propagating. If an SH1 wave packet is sent between two transducers and a change in thickness occurs, then that change in the thickness of the plate will result in a measurable change of the effective group velocity of the wave packet that is travelling between the two transducers. The setup described, in combination with the algorithm/method below, has been designed to extract a value of the average thickness between two points on a plate or around the circumference of a pipe. As this method determines the average thickness (not remnant thickness) it is possible for different corrosion profiles, along the propagation path, to have the same value of average thickness. One corrosion profile could be uniformly thinned by 1mm over 0.5m propagation path in a 10mm plate, compared to another corrosion profile, which has 2mm of thinning over 0.25m propagation path. Both would have an average thickness of 9mm over the total propagation path of 0.5m. Any timing delays due to transducer design or equipment setup have been assumed to be constant.
55 The group velocity curve of the SH1 (cg) is shown in Figure 3.4 and can be expressed as an
analytical equation, which is dependent only on the frequency (f), thickness (b) and shear velocity (cs):
𝑐𝑔(𝑓𝑏) = 𝑐𝑠√1 −
(𝑐𝑠⁄ )2 2
(𝑓𝑏)2
(3.1)
Figure 3.4. Shows the group velocity curves for the non-dispersive SH0 (blue) and dispersive SH1 (red) modes in steel (cs = 3235m/s) produced in DISPERSE [2]. The effect of a change in the thickness of a plate using a pitch catch SH guided wave setup is shown, as well as the resulting change in the group velocity (at a single frequency- thickness value)
3.3.2 Average Thickness Extraction Algorithm
Group velocity can be extracted from two spatially and temporally separate wave packets as demonstrated by [82] using the zero phase slope method. The zero-phase slope extracts the slope of the phase as a function of frequency of two windowed signals (with a common time frame). The difference between the two phase slopes is inversely proportional to group velocity (see appendix 9.1 for full derivation). In the case of the pipe setup described in this study, an input pulse will be propagated around the circumference of the pipe and detected at the same location, giving the output pulse. In the case of the plate, the input signal is the signal excited by the transmitting transducer and the output signal is the signal received at the receiving transducer. The zero-phase slope of input and output is calculated over the spectra of the pulse and the effective group velocity can be determined using the following equation: 𝑐𝑔(𝜔) = 𝐷 𝑑𝜑(𝜔)𝑖𝑛 𝑑𝜔𝑖𝑛 − 𝑑𝜑(𝜔)𝑜𝑢𝑡 𝑑𝜔𝑜𝑢𝑡 (3.2)
56 where D is the separation distance between the two transducers, ω is the angular frequency of the spectral component, and 𝑑𝜑𝑖𝑛(𝜔)
𝑑𝜔𝑖𝑛 and
𝑑𝜑𝑜𝑢𝑡(𝜔)
𝑑𝜔𝑜𝑢𝑡 are the phase slopes of the input (Figure
3.5 - blue) and output signals (Figure 3.5 - green) respectively. SH1 pulse in the output signal must be windowed to extract the effective group velocity of the SH1 mode (Figure 3.5). In the finite element models, the input signal is known and therefore does not have to be windowed, however, in reality the frequency response of the electronic equipment may alter the input signal and therefore it would have to be detected by another means (e.g. a current sensor – see Figure 3.9).
Figure 3.5. Windowing of the input and output SH1 signal in Figure 3.3.
The analytical equation for SH1 group velocity, equation 3.1, depends on frequency, thickness and shear velocity. As the frequency and shear velocity of the metal are constants (in a stable environment), the analytical group velocity can be fitted to the extracted effective group velocity by varying the thickness, b (see Figure 3.6). This is implemented by minimizing the residuals between the extracted group velocity and the analytical group velocity over the spectra of the wave packet.
min ( ∑ |𝑐𝑔,𝑒𝑥𝑡𝑟𝑎𝑐𝑡𝑒𝑑(𝜔) − 𝑐𝑔,𝑎𝑛𝑎𝑙𝑦𝑡𝑖𝑐𝑎𝑙(𝜔, 𝑏)|𝑐𝑠
𝜔ℎ𝑖𝑔ℎ
𝜔𝑙𝑜𝑤
57
Figure 3.6. Analytical group velocity curve fitted to an experimentally extracted group velocity curve, for a 200kHz 5 cycle Hann toneburst over 0.5m in a 10mm plate with 40dB SNR
3.3.3 Temperature Compensation
The thickness extraction method above depends on the shear velocity of the plate being constant (see equation 3.1); however, the shear velocity of the metal plate is proportional to the temperature of the plate [41]. If not accounted for, this would induce an error in the extracted thickness. Figure 3.7 shows the SH1 group velocity curve plotted using two values of the shear velocity, cs, at -20°C and 20°C. These temperature values were chosen as a
suitable annual variation in a continental climate and the values of cs were found in [83].
Over a 40°C temperature range there is a 24m/s change in group velocity at 2MHz.mm and that equates to a 0.1mm change in extracted average thickness over a 1m propagation distance.
58
Figure 3.7. Two SH1 group velocity curves with cs,-20°C = 3236m/s and cs,20°C = 3260m/s
A mechanism was therefore devised to compensate for changes in temperature, exploiting the fact that the group velocity of the non-dispersive SH0 mode is equal to the shear velocity of the metal. The SH0 mode is non-dispersive and its group and phase velocity is equal to the shear velocity of the elastic medium. By propagating both the SH0 mode and the SH1 mode, the SH0 mode can be windowed (shown in Figure 3.8) and the shear velocity can be extracted using the group velocity extraction described above (in equation 3.2). This value can then be used in equation 3.1 and 3.3 to calculate the average thickness.
Figure 3.8. Windowing of the FEA SH0 and SH1 modes with a 250kHz 5 cycle Hann windowed over a 500mm propagation distance in a 10mm thick plate (cs = 3260m/s)