MATLAB Examples

# Analysis of Wing Stresses on a Prototype Aircraft

Demo file from August 7, 2007 webinar titled Data Analysis with Statistics Toolbox and Curve Fitting Toolbox. View the recorded webinar: http://www.mathworks.com/company/events/archived_webinars.html

We have test results from a prototype aircraft tested in a wind tunnel. The test focused on understanding the wing stresses during the climbing phases of the flight, where stresses are typically high. The wind tunnel's gust generator induces a controlled gust field to simulate severe wind conditions. We measure stress at 7 locations on each wing.

Our primary goals include:

• Identify outliers or bad data
• Compare the stress levels on the 2 wings to make sure they're not substantially different.
• Calculate the shear force the wing experiences as a result of the wind gust
• Determine if the max shear force is above acceptable limits (160 kN)

## Show Configuration of Aircraft

Wind gusts are directed at the aircraft as shown in this image. Sensor locations are shown on the left/right wings.

```imshow('TestConfig.jpg') ```

## Raw Data File

The data is from testing is stored in a Microsoft Excel spreadsheet named 'WingStress.xls'. The data contains measurements of wind velocity (m/s), wind angle (degrees), and stress measurements (kPa) for the left (L) and right (R) wings where 1 denotes the sensor location near the base of the wing and 7 denotes the sensor near the tip of the wing.

```imshow('WingStress.jpg') ```

Take advantage of the dataset array provided in Statistics Toolbox for manipulating and storing my data. Note: You need to have Statistics Toolbox version 6.0 or higher to use dataset arrays.

```clear all, close all % clean up previous work % Load in data to a dataset array (Statistics Toolbox data type) ds = dataset('XLSFile','WingStress.xls'); % Add units to my dataset strainUnits = repmat({'kPa = kN/m^2'},1,14); ds.Properties.Units = {'m/s','degrees',strainUnits{:}}; % Show summary summary(ds) ```
```Velocity: [72x1 double, Units = m/s] min 1st Q median 3rd Q max 4.4538 7.8577 12.0013 15.9402 20.5982 Angle: [72x1 double, Units = degrees] min 1st Q median 3rd Q max -120 -60 0 60 120 L1: [72x1 double, Units = kPa = kN/m^2] min 1st Q median 3rd Q max 65.2751 140.8844 199.7904 252.5293 370.1140 L2: [72x1 double, Units = kPa = kN/m^2] min 1st Q median 3rd Q max 60.9627 154.7150 201.6100 252.2235 341.5571 L3: [72x1 double, Units = kPa = kN/m^2] min 1st Q median 3rd Q max 27.7622 75.4219 95.7420 128.4913 198.7315 L4: [72x1 double, Units = kPa = kN/m^2] min 1st Q median 3rd Q max 61.0577 145.4255 206.3261 239.7442 404.9529 L5: [72x1 double, Units = kPa = kN/m^2] min 1st Q median 3rd Q max 56.0411 134.7558 169.9957 217.6420 358.0392 L6: [72x1 double, Units = kPa = kN/m^2] min 1st Q median 3rd Q max 46.0159 103.9926 128.3909 169.0685 248.7240 L7: [72x1 double, Units = kPa = kN/m^2] min 1st Q median 3rd Q max 13.2258 40.1803 52.8888 61.8663 119.4480 R1: [72x1 double, Units = kPa = kN/m^2] min 1st Q median 3rd Q max 44.1058 139.1650 185.5733 239.7453 384.9178 R2: [72x1 double, Units = kPa = kN/m^2] min 1st Q median 3rd Q max 6.1899e-009 2.8778e-008 134.2293 213.2698 306.1857 R3: [72x1 double, Units = kPa = kN/m^2] min 1st Q median 3rd Q max 67.9553 149.8835 197.8903 248.5895 402.9298 R4: [72x1 double, Units = kPa = kN/m^2] min 1st Q median 3rd Q max 61.3721 149.9545 190.6260 257.7519 358.4477 R5: [72x1 double, Units = kPa = kN/m^2] min 1st Q median 3rd Q max 51.9460 131.8459 168.3736 228.3369 366.4075 R6: [72x1 double, Units = kPa = kN/m^2] min 1st Q median 3rd Q max 33.0364 101.2601 125.0345 170.1178 252.2392 R7: [72x1 double, Units = kPa = kN/m^2] min 1st Q median 3rd Q max 12.8590 39.0228 52.9290 66.3789 105.4329 ```

You can see how the dataset loaded the data from the spreadsheet file and automatically used the column names as variable names within the dataset. The summary command lists the dataset contents, including the type, size, assigned units, and summary statistics for each variable. You can access the variables directly using the "dot" notation.

```ds.Velocity(1:5) ```
```ans = 5.2848 5.0441 4.8292 4.9279 4.8810 ```

## Example of Data Management with Datasets

Datasets make it easy to access my data by name and operate on the data within the dataset as a group. For example, I can access only 90 degree data for L1/R1 and L7/R7.

```L1R190 = ds(ds.Angle == 90,{'Angle','L1','R1','L7','R7'}) ```
```L1R190 = Angle L1 R1 L7 R7 90 159.36 156.15 39.5049 38.814 90 160.7978 163.922 43.9022 42.7627 90 150.5968 137.5926 41.79 51.4079 90 173.6168 131.1749 45.1426 44.1388 90 183.4965 160.3539 39.8596 53.0917 90 151.0959 164.9913 50.1638 43.0593 90 166.0129 123.9435 48.1589 39.2316 90 155.5653 137.8742 45.9231 36.5771 ```

I can also calculate descriptive statistics by Angle for L1 and R1.

```ByAngle = grpstats(ds,{'Angle'},{@mean,@std},'DataVars',{'L1','R1'}) ```
```ByAngle = Angle GroupCount mean_L1 std_L1 mean_R1 std_R1 -120 -120 8 100.1556 24.1595 103.0061 29.7064 -90 -90 8 149.3126 26.5415 153.6784 12.1865 -60 -60 8 230.3514 27.3318 222.6771 33.259 -30 -30 8 262.2175 44.05 251.4625 35.3024 0 0 8 275.3241 51.6807 277.3864 68.5707 30 30 8 260.9625 60.5256 245.3882 46.0502 60 60 8 216.2059 12.6545 216.3821 36.0055 90 90 8 162.5677 11.3975 147.0003 16.1507 120 120 8 102.593 29.6774 99.98004 39.1901 ```

The dataset array provides much more functionality than I have shown here. I introduced the dataset array so that you would be familiar with it since I use it in my analysis that follows.

## Show Plot by Angle

Let's start exploring my data. I'll use a scatter plot to visualize my data.

```gscatter(ds.L1, ds.R1, ds.Angle) ```

This plot makes it easy to see how L1 trends with R1, in a linear fashion. I can also see how angle affects the stress level. It appears that -120/120 has the lowest stress and 0 is the highest. So flying directly into the wind causes the highest stress on the wings.

## Show My Custom Plot

I created a custom plot using plottool to show left and right sensors plotted with a 45 degree line. I expect my data to follow the 45 degree line if the left and right sensors are behaving the same.

```createfigure(ds.L1,ds.R1,[0 500]), title('L1 vs. R1') createfigure(ds.L2,ds.R2,[0 500]), title('L2 vs. R2') ```

We can see from the plot of L1 vs. R1 that my data on the left and right wings are similar and do follow a 45 degree line. The plot of L2 vs. R2 shows that I have some bad data in R2, as can be seen by the zero values in the plot. I know from testing that R2 failed during the latter part of testing. I need to remove the bad data from R2 and keep the good data that follows the 45 degree line.

## Hypothesis Testing of Left/Right Wing

Let's test the hypothesis that L1 = R1 and L2 = R2. This will perform the same test as my custom plot did, but does not require me to visually inspect the data.

We'll use the hypothesis test function kstest2 from the Statistics Toolbox to test this assumption.

```h1 = kstest2(ds.L1,ds.R1,0.1) h2 = kstest2(ds.L2,ds.R2,0.1) ```
```h1 = 0 h2 = 1 ```

Note that the hypothesis test (h1) returns 0 for the assumption that L1 = R1. This tells me that my assumption L1 = R1 is true, or in other words, the differences I observe in L1 and R1 are due to normal and expected variation.

The hypothesis test (h2) returns a 1 for the assumption that L2 = R2. This tells me L2 is not equal to R2, which we already know has bad data in it.

Hypothesis testing allows me to systematically test my data for difference in measured values where I normally would only detect this differences by visual inspection.

## Use Hypothesis Testing to Screen Data

Combine hypothesis testing with my customized plot to automate the process of identifying differences across sensor locations. This next section of code will scan my data and only flag data that may contain bad or suspect data (i.e. any data that fails Lx = Rx will be shown in my custom plot).

```% Left and right wing sensor labels Left = {'L1','L2','L3','L4','L5','L6','L7'}; Right = {'R1','R2','R3','R4','R5','R6','R7'}; for i = 1:length(Left) Lx = double(ds,Left{i}); Rx = double(ds,Right{i}); h(i) = kstest2(Lx,Rx,0.1); % test Lx = Rx if h(i) > 0 % Plot potentially bad data createfigure(Lx,Rx,[0 500]) % modified plot label = [Left{i},' vs. ',Right{i}]; title(label) fprintf('%s failed test, h = %i\n',label,h(i)) end end ```
```L2 vs. R2 failed test, h = 1 L3 vs. R3 failed test, h = 1 ```

We can see that L2 and R2 were flagged, which we expect. We can also see that L3 and R3 were flagged. Looking at the L3 vs. R3 plot, we can see that R3 appears to have a higher stress measurement than L3. But which one is correct?

## Add Flag for Sensor Health

At this point I would like to start removing data that I know is bad from further analysis. I'll use categorical arrays from Statistics Toolbox to categorize each sensor as Alive, Dead, or Suspect. This will allow me to filter my data based upon these categories.

I'll first declare that all sensors are good.

```Health = nominal('Alive'); Health = repmat(Health,length(ds),14); HLeft = {'HL1','HL2','HL3','HL4','HL5','HL6','HL7'}; HRight = {'HR1','HR2','HR3','HR4','HR5','HR6','HR7'};ds = [ds dataset({Health,HLeft{:},HRight{:}})]; HealthUnits = repmat({'[-]'},1,14); ds.Properties.Units= [ds.Properties.Units HealthUnits]; ```

Now I'll flag the data in R2 that I know for sure is bad. Note that the summary command now displays information about the categorical data. Specifically, if we look at HR2, we can see how many data points are good (Alive = 46) vs bad (Dead = 26).

```R2L2 = ds.R2./ds.L2; % ratio and remove values below 50% (0.5) indx = R2L2 < 0.5; ds.HR2(indx) = 'Dead'; summary(ds) % Plot to verify we can removed R2 bad data createfigure(ds.L2(~indx),ds.R2(~indx),[0 500]) title('L2 vs. R2') ```
```Warning: Categorical level 'Dead' being added. Velocity: [72x1 double, Units = m/s] min 1st Q median 3rd Q max 4.4538 7.8577 12.0013 15.9402 20.5982 Angle: [72x1 double, Units = degrees] min 1st Q median 3rd Q max -120 -60 0 60 120 L1: [72x1 double, Units = kPa = kN/m^2] min 1st Q median 3rd Q max 65.2751 140.8844 199.7904 252.5293 370.1140 L2: [72x1 double, Units = kPa = kN/m^2] min 1st Q median 3rd Q max 60.9627 154.7150 201.6100 252.2235 341.5571 L3: [72x1 double, Units = kPa = kN/m^2] min 1st Q median 3rd Q max 27.7622 75.4219 95.7420 128.4913 198.7315 L4: [72x1 double, Units = kPa = kN/m^2] min 1st Q median 3rd Q max 61.0577 145.4255 206.3261 239.7442 404.9529 L5: [72x1 double, Units = kPa = kN/m^2] min 1st Q median 3rd Q max 56.0411 134.7558 169.9957 217.6420 358.0392 L6: [72x1 double, Units = kPa = kN/m^2] min 1st Q median 3rd Q max 46.0159 103.9926 128.3909 169.0685 248.7240 L7: [72x1 double, Units = kPa = kN/m^2] min 1st Q median 3rd Q max 13.2258 40.1803 52.8888 61.8663 119.4480 R1: [72x1 double, Units = kPa = kN/m^2] min 1st Q median 3rd Q max 44.1058 139.1650 185.5733 239.7453 384.9178 R2: [72x1 double, Units = kPa = kN/m^2] min 1st Q median 3rd Q max 6.1899e-009 2.8778e-008 134.2293 213.2698 306.1857 R3: [72x1 double, Units = kPa = kN/m^2] min 1st Q median 3rd Q max 67.9553 149.8835 197.8903 248.5895 402.9298 R4: [72x1 double, Units = kPa = kN/m^2] min 1st Q median 3rd Q max 61.3721 149.9545 190.6260 257.7519 358.4477 R5: [72x1 double, Units = kPa = kN/m^2] min 1st Q median 3rd Q max 51.9460 131.8459 168.3736 228.3369 366.4075 R6: [72x1 double, Units = kPa = kN/m^2] min 1st Q median 3rd Q max 33.0364 101.2601 125.0345 170.1178 252.2392 R7: [72x1 double, Units = kPa = kN/m^2] min 1st Q median 3rd Q max 12.8590 39.0228 52.9290 66.3789 105.4329 HL1: [72x1 nominal, Units = [-]] Alive 72 HL2: [72x1 nominal, Units = [-]] Alive 72 HL3: [72x1 nominal, Units = [-]] Alive 72 HL4: [72x1 nominal, Units = [-]] Alive 72 HL5: [72x1 nominal, Units = [-]] Alive 72 HL6: [72x1 nominal, Units = [-]] Alive 72 HL7: [72x1 nominal, Units = [-]] Alive 72 HR1: [72x1 nominal, Units = [-]] Alive 72 HR2: [72x1 nominal, Units = [-]] Alive Dead 46 26 HR3: [72x1 nominal, Units = [-]] Alive 72 HR4: [72x1 nominal, Units = [-]] Alive 72 HR5: [72x1 nominal, Units = [-]] Alive 72 HR6: [72x1 nominal, Units = [-]] Alive 72 HR7: [72x1 nominal, Units = [-]] Alive 72 ```

## Analysis of Variance (ANOVA)

I can use ANOVA techniques to test the assumption that all of my sensors have the same mean (i.e. L1 = R1 = R2 = ...L7 = R7). This test will tell me if the differences in stress measurements are different or a result of the normal and expected variation in the data.

```stress = double(ds,[Left,Right]); stress(ds.HR2 == 'Dead',9) = NaN; [p,t,stats] = anova1(stress,[Left,Right]); ```

Let's dig deeper into the differences between L3 and R3. You can see a box plot is produced that shows me the amount of variation each sensor has in my data. The y-axis shows me the median (red bar), the quartile range of the data (end of boxes), the range of the data (dashed bars), and any outliers that might be in my data (red plus symbols). We can see a trend for each wing (looking left to right) that shows decreasing (median) values as we go from the wing base (L1 or R1) to the wing tip (L7 or R7). Also note that L3 and R3 do not behave similarly. We can conclude from this plot that L3 is the sensor that has different behavior.

The ANOVA table (p-value, prob>F) tells me that the differences across sensors are significant and not due to random variation. If the p-value was at or above 0.05, this test would tell me that my different sensors are really measuring the same thing. Now the ANOVA test is not all interesting in this case, and I used ANOVA to generate the box plot for visualization. But I also want to test for differences across sensors individually. I can do this using the multcompare function and the output stats structure from ANOVA.

```multcompare(stats); ```

The multcompare function provides an interactive plot that I can use to test for differences across groups. For example, if you click on L3, you can see that it is different than all of the other sensors, 12 to be precise. So L3 must be the sensor that is wrong and R3 is good. Now I don't konw the casue of L3's difference. It could be bad data acquisition hardware, improper installation or placement of the L3 sensor, or some other reason. The point is that I do not trust L3's values and will remove it from further analysis.

```ds.HL3(:) = 'Suspect'; % Label L3 as Suspect (I should look into why) stress(:,3) = NaN; ```
```Warning: Categorical level 'Suspect' being added. ```

## Clustering (show groups, outliers)

We looked at our data by sensor location and found R2 to contain bad data. We also found L3 to be suspect, and should remove it from analysis based upon it near behaving as expected. Now let's take a look at angle effects on our data. I'll use cluster analysis here instead of hypothesis testing or ANOVA, to show a different method from Statistics Toolbox that can be used to identify difference in your data.

```rmv = isnan(stress(1,:)); [d,p,stats] = manova1(stress(:,~rmv),ds.Angle); manovacluster(stats) xlabel('Wind Gust Angle (deg)') ```

You can see the dendrogram plot show that -120/120 is a group. You can see the same for -90/90. The height of the bars indicate how closely related the different cluster are to each other. For example, if I cut the bars at around a value of 6, I would get the grouping of -120/120, -90/90, and -60 for the left half of the plot.

60 is interesting in that it is more like -90 and 90 than it is like -60. This is surprising in that we should have a -60/60 grouping, based upon symmetry of the airplane. I need to investigate why this might be so.

Notice that 0 is in a cluster all by itself. This is expected, since it is the only measurement that does not have a symmetric counterpart.

My ultimate goal is to determine the maximum shear force that the wing was exposed to during testing. And from the gscatter plot shown earlier, I know the 0 degree condition produces the maximum stress on the wings. Let's focus attention on this test condition.

## Curve Fitting Toolbox

I want to estimate the maximum shear force the wing was exposed to during testing. Shear force is simply the integration of the bending moment over the wing length, or span. The Curve Fitting Toolbox provides integration capability of curves, so let's use it here to calculate the shear force.

Bending moment (M) as a function of wing span can be calculated from my stress measurements:

` M = I/t * stress`

where I is the cross-sectional moment of inertia for the wing and t is the location of the sensor from the neutral axis of the wing. So let's fit a curve to M, extrapolate down to the base of the wing (span = 0), and then integrate the curve to get Shear Force. I'll do this for the 0 degree wind gust data for the highest wind gust velocity tested (this is the maximum stress condition).

```I = [0.0127 0.0120 0.0113 0.0107 0.0095 0.0084 0.0073]; % m^4 t = [0.1105 0.1085 0.1065 0.1045 0.1001 0.0962 0.920]; % m span = [0.5, 1, 1.5, 2, 3, 4, 5]; % m % Moment for max wind gust velocity at 0 degrees (head on) maxV = max( ds.Velocity(ds.Angle == 0) ); s = stress(ds.Angle == 0 & ds.Velocity == maxV, :); ML = I./t .* s(1:7); MR = I./t .* s(8:end); % Open curve fitting tool with left and right wing data loaded cftool([span span],[ML MR]) ```

Show cftool analysis.

```cubicfit([span span],[ML MR]) % generated from cftool open('analysis.fig') % show analysis figure (saved from Analysis in cftool) ```

We can see in the upper plot that the extrapolation of the cubic fit down to 0 appears to be reasonable. Also note that our confidence in this area has been reduced (wider confidence intervals).

The bottom plot shows the integration of the upper plot. The maximum shear force appears to be 151 kN.

## Summary

We can se from the integration curve that the maximum shear force is around 151 kN, which is below our requirement of 160 kN. We also found the left and right wing measurements to be similar after we removed bad and suspect data.

## Appendix

Custom plot createfigure

```type createfigure ```
```function createfigure(X1, Y1, X2) %CREATEFIGURE(X1,Y1,X2) % X1: vector of x data % Y1: vector of y data % X2: vector of x data % Auto-generated by MATLAB on 27-Jul-2007 12:20:13 % Create figure figure1 = figure; % Create axes axes('Parent',figure1,'YGrid','on','XGrid','on'); % Uncomment the following line to preserve the X-limits of the axes % xlim([0 5e-005]); % Uncomment the following line to preserve the Y-limits of the axes % ylim([0 5e-005]); box('on'); hold('all'); % Create plot plot(X1,Y1,'Marker','o','LineStyle','none'); % Create xlabel xlabel('Left'); % Create ylabel ylabel('Right'); % Create plot plot(X2,X2,'DisplayName','[0 4e-5] vs [0 4e-5]','LineWidth',2,... 'Color',[0 0 0]); axis equal tight ```

Generated code from cftool

```type cubicfit ```
```function cubicfit(x,y) %CUBICFIT Create plot of datasets and fits % CUBICFIT(X,Y) % Creates a plot, similar to the plot in the main curve fitting % window, using the data that you provide as input. You can % apply this function to the same data you used with cftool % or with different data. You may want to edit the function to % customize the code and this help message. % % Number of datasets: 1 % Number of fits: 1 % Data from dataset "y vs. x": % X = x: % Y = y: % Unweighted % % This function was automatically generated on 06-Aug-2007 16:03:00 % Set up figure to receive datasets and fits f_ = clf; figure(f_); set(f_,'Units','Pixels','Position',[852 350 680 477]); legh_ = []; legt_ = {}; % handles and text for legend xlim_ = [Inf -Inf]; % limits of x axis ax_ = axes; set(ax_,'Units','normalized','OuterPosition',[0 0 1 1]); set(ax_,'Box','on'); axes(ax_); hold on; % --- Plot data originally in dataset "y vs. x" x = x(:); y = y(:); h_ = line(x,y,'Parent',ax_,'Color',[0.333333 0 0.666667],... 'LineStyle','none', 'LineWidth',1,... 'Marker','.', 'MarkerSize',12); xlim_(1) = min(xlim_(1),min(x)); xlim_(2) = max(xlim_(2),max(x)); legh_(end+1) = h_; legt_{end+1} = 'y vs. x'; % Nudge axis limits beyond data limits if all(isfinite(xlim_)) xlim_ = xlim_ + [-1 1] * 0.01 * diff(xlim_); set(ax_,'XLim',xlim_) end % --- Create fit "Cubic" ok_ = isfinite(x) & isfinite(y); ft_ = fittype('poly3'); % Fit this model using new data cf_ = fit(x(ok_),y(ok_),ft_); % Or use coefficients from the original fit: if 0 cv_ = { -0.5107375651142, 1.784383271041, -4.397432434511, 42.28260479588}; cf_ = cfit(ft_,cv_{:}); end % Plot this fit h_ = plot(cf_,'fit',0.95); legend off; % turn off legend from plot method call set(h_(1),'Color',[0 0 1],... 'LineStyle','-', 'LineWidth',2,... 'Marker','none', 'MarkerSize',6); legh_(end+1) = h_(1); legt_{end+1} = 'Cubic'; % Done plotting data and fits. Now finish up loose ends. hold off; leginfo_ = {'Orientation', 'vertical', 'Location', 'NorthEast'}; h_ = legend(ax_,legh_,legt_,leginfo_{:}); % create legend set(h_,'Interpreter','none'); xlabel(ax_,''); % remove x label ylabel(ax_,''); % remove y label ```