4 Stereo-DIC Measurements of Thermal Gradient Effects on the Vibratory Response of Metals 39 Fig. 4.5 Comparison of Tchebichef coefficients representing the data obtained from experiment (horizontal axis) and simulation (vertical axis) at (a) room temperature, and (b) elevated temperature. Kernels which have significantly large coefficients are indicated by their kernel number in parentheses of the first kernel, which corresponds to a rigid out-of-plane translation, as shown in Fig. 4.4, and is unrelated to the deformation of the plate, is excluded from the figures. The remaining 54 coefficients are all included, although in reality most of the mode shapes can be reconstructed using relatively few coefficients or kernels. The coefficients with larger magnitudes are farther away from the plot origin and hence represent those kernels that contribute most strongly to the modal shape. The largest coefficients are labeled with the numbers in parentheses corresponding to the kernels in Fig. 4.4. Each plot in Fig. 4.5 also includes a line of gradient one, which is indicative of perfect agreement between the simulation and the experimental results. The closer a data point lies to this line, the better the agreement is for that coefficient between simulation and experiment [11, 13]. In the comparison, it is likely that there will be noise in the experimental data and errors in the simulation which will cause the points to deviate from this line. Therefore an acceptable area can be defined around the ideal line as C/ 2u(SE), defined by the dashed line pairs in the figures. If all of the data points from the comparison of the Tchebichef feature vectors fall within this area then the simulation can be deemed an acceptable representation of the experiment [13]. In general, the uncertainty bands are smaller for the high temperature results (Fig. 4.5b) than for the room temperature results (Fig. 4.5a). This is a result of a lower average residual in the reconstruction of the high temperature experimental results because the larger displacements have less noise. For the majority of the plots, all of the data points fall within the acceptable area, and many of the data points fall on the ideal line, indicating agreement between the simulation and the experiment. For example for Mode 3 in Fig. 4.5, only one Tchebichef kernel is dominant—specifically #6 in Fig. 4.4— which corresponds to a shape produced by bending about the vertical axis. Most other kernels have coefficients clustered at the origin of the plot in Fig. 4.5, i.e., they are near zero. Overall Fig. 4.5a and b show that there is a very good comparison of the spatial distribution of out-of-plane displacements between the predicted and measured resonant modes. One notable exception to this is mode 3 at high temperature (Fig. 4.5b). In this mode there is a kernel, #2, which falls outside the acceptable area defined byC/ 2u(SE). This kernel has a significant non-zero value for the simulation data but is zero-valued for the DIC data. Kernel #2 corresponds to a rigid body out-of-plane rotation about a horizontal axis, and is the dominant kernel in mode 2. The data presented in Fig. 4.5a and b can be summarized as in Fig. 4.6. Each mode is represented by a row in the figure in which those Tchebichef coefficients that were more than 20 % of the value for the largest coefficient for each mode shape have been plotted as bars under the corresponding column for the Tchebichef kernel. Data from both experiment (red) and simulation (blue) have been included as well as at room (dark colors) and high (bright colors) temperature. The exclusion of low value coefficients following the approach of [14] simplifies the figure without excluding any significant information. The shape represented by the kernels is shown along the top of the figure. A comparison of the height of the bars in each chart provides information both about the changes in behavior from room to high temperature (left pair for experiment or right pair for simulation) and about the degree to which the simulation represents the experiment (right pair relative to left pair).
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