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There are two sources of errors that need to be considered carefully in this experiment. Firstly, the errors associated with measuring the absolute velocity of the Lamb waves. Secondly, further errors arise in the accuracy of the graphical fit of equation 7.1; these errors manifest in the accuracy of the determined texture coefficients.

Random errors affecting the accuracy of the absolute velocity measurements include EMAT lift-off variation which intuitively affects the signal received. The increase in stand- off would smear the waveform. Every effort was made to minimise this effect. The design of the apparatus afforded a consistent stand-off distance, particularly for aluminium which had no discernible magnetic interaction with the NeFeB permanent magnets.

100 velocity measurements were taken for each angle to further improve accuracy to reduce the random errors. For example, the texture measurement system has a „fast‟ scan capability, taking 1 velocity measurement at each angle whilst the EMATs are continuously rotated. The data is collected in just a few seconds, though the speed of data acquisition comes at a cost in accuracy. The movement of the EMATs as the Lamb waves are being generated and detected gives larger random errors, with alignment and signal quality being compromised. Continuous movement of the stepper motor increases the presence of electrical noise also. Figure 7.10 illustrates the effect of the noise associated with the „fast‟ experimental setup, here displaying fast data for the 0.2 mm thick aluminium scan (red trace). The mean velocity determined from 100 readings per point is also displayed on the graph (in blue) for comparison.

The main source of systematic error exist in the measured distance between the generation and the two detection EMATs. The Labview program requires the input of the distance between the EMATs to determine the velocity. Values of 95.4 mm and 147.3 mm were used in the program and were checked manually; both were agreeable within ± 0.05 mm, equating to absolute errors in the distances of 0.053% and 0.034% respectively. The texture measurement system had been calibrated previously using samples of known

velocity to confirm the distance accuracy. The effect of the size and design of the coils used was also researched extensively [1].

5530 5520 5510 5500 5490 5480 Ul tr as on ic V el oc ity ( ms -1 ) 6 5 4 3 2 1 0

Angle subtended from RD (radians)

Figure 7.10: Ultrasonic velocity profiles for 0.2 mm thick aluminium sheet AL101325. The red trace is the velocity data taken with just one measurement per point, with the blue trace giving the mean velocity at each point from 100 measurements per point.

Any systematic errors that manifest in the velocity measurement are inherent in all the velocities at each angle of propagation. Therefore the velocity variation will always be detected, indicating that the magnitude of the angular dependent ODCs W420 and W440 will be

unaffected. However, with a 0.034% uncertainty in distance, the absolute variation in velocity detected will have an uncertainty of approximately ± 0.02 ms-1 (figure 7.10 shows a velocity variation of ~ 50 ms-1 for example). This uncertainty is less than the noise in the signal, especially in the case for the „fast‟ data.

Visually comparing the two corresponding sets of data from figure 7.10, the averaged curve is smoother, possessing less noise than the fast scan. This is to be expected as the signal to noise ratio is improved by a factor of 10. Taking the mean will eradicate errors caused by the directionality of the beam, electrical noise and stand-off fluctuations which are more significant with the rotating EMATs.

Applying the mathematical fit to both scans in figure 7.10 to extrapolate the ODCs gave the results displayed in table 7.5.

W400 W420 W440

Slow -0.0107 ± 3.39 10-7 -0.000327 ± 1.63 10-5 -0.00502 ± 2.55 10-5

Fast -0.0109 ± 1.22 10-6 -0.00029 ± 5.79 10-5 -0.00511 ± 9.04 10-5

Table 7.5: ODCs and errors associated with fast and slow texture scans for 0.2 mm thick aluminium sheet.

The errors indicated in table 7.5 are at least  3.5 greater for the fast scan. The ODCs are similar, within 10% with correct orders of magnitude and sign; this suggests a fast scan is a viable option for the estimation of crystallographic texture.

The level of agreement between the mathematical fit and the velocity data needs to be considered however. The graphical fit applied to the ultrasound data was programmed in a proprietary data analysis program (Igor, Wavemetrics), which was incorporated into the Labview data collection program provided by Sonemat Ltd. The fit utilises a least squares algorithm, iteratively determining the optimum solution to equation 7.1, by minimising the sum of the difference between each data point and the line of best fit (root mean squared errors). Error analysis was included in the analysis software, the errors given with the ODCs. Figure 7.11 shows the residual velocity between the measured velocity and the corresponding fit velocity (data given in figure 7.4).

-4 -3 -2 -1 0 1 2 Me as ured v el oc ity - fi t v el oc ity ( ms -1 ) 6 5 4 3 2 1 0

Angle subtended from RD (radians)

Figure 7.11: Residual velocity between EMAT measured and applied fit for 0.2 mm thick aluminium sheet.

standard deviation can be calculated to be ~ 0.76 ms-1. Combining this with the 0.02 ms-1 absolute distance uncertainty leaves a total error in the fit of 0.79 ms-1; this equates to ~ 0.014% error in the velocity for the 0.2 mm thick sheet, as shown in figure 7.12.

5530 5520 5510 5500 5490 5480 Ul tr as on ic V el oc ity ( ms -1 ) 6 5 4 3 2 1 0

Angle subtended from RD (radians)

Figure 7.12: 0.2 mm thick aluminium sheet data including error bars.

The maximum errors for the measured ODCs in both steel and aluminium are given in tables 7.3-7.4. The largest calculated error in the ODCs was ± 3.5 10-5. The orders of magnitude of the ODCs are between 10-2 and 10-4; the smallest ODC value recorded is the

W420 coefficient for the 1 mm thick aluminium sheet (table 7.6)

Sample W400 % error W420 % error W440 % error

Al – 0.2mm thick -0.0032 -4.99 -5.08 Al – 0.5mm thick -0.0022 1.03 1.3 Al – 1.0mm thick 0.0045 2.05 5.79 Al – 1.5mm thick -0.0036 1.53 1.07 DC01 -8.40E-03 -1.18 8.18 DC05 -4.97E-03 -1.11 1.69

Table 7.6: ODC % errors. The largest aluminium and steel errors are highlighted in bold.

The largest percentage error for an ODC in an aluminium sample was 5.79% and for steel it was 8.18%: both were for the W440 ODC. The calculated W440 ODC will in general have