analysis of wing stresses on a prototype aircraft

8/9/2019 Analysis of Wing Stresses on a Prototype Aircraft

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Analysis of Wing Stresses on a PrototypeAircraft

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 onunderstanding the wing stresses during the climbing phases of the flight, where stressesare typically high. The wind tunnel's gust generator induces a controlled gust field tosimulate 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 substantiallydifferent.

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)

Contents Show Configuration of Aircraft Raw Data File Load in the Data Example of Data Management with Datasets Show Plot by Angle Show My Custom Plot Hypothesis Testing of Left/Right Wing Use Hypothesis Testing to Screen Data Add Flag for Sensor Health Analysis of Variance (ANOVA) Clustering (show groups, outliers) Curve Fitting Toolbox Summary Appendix

Show Configuration of Aircraft

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

imshow('TestConfig.jpg')
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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 1denotes the sensor location near the base of the wing and 7 denotes the sensor near the tipof the wing.

imshow('WingStress.jpg')


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Load in the Data

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


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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 datasetstrainUnits = repmat({'kPa = kN/m^2'},1,14);ds.Properties.Units = {'m/s','degrees',strainUnits{:}};

% Show summarysummary(ds)Velocity: [72x1 double, Units = m/s]

min 1st Q median 3rd Q max4.4538 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]


L2: [72x1 double, Units = kPa = kN/m^2]min 1st Q median 3rd Q max60.9627 154.7150 201.6100 252.2235 341.5571






R1: [72x1 double, Units = kPa = kN/m^2]min 1st Q median 3rd Q max44.1058 139.1650 185.5733 239.7453 384.9178

R2: [72x1 double, Units = kPa = kN/m^2]min 1st Q median 3rd Q max6.1899e-009 2.8778e-008 134.2293 213.2698

306.1857

R3: [72x1 double, Units = kPa = kN/m^2]


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You can see how the dataset loaded the data from the spreadsheet file and automaticallyused the column names as variable names within the dataset. The summary commandlists the dataset contents, including the type, size, assigned units, and summary statisticsfor each variable. You can access the variables directly using the "dot" notation.

ds.Velocity(1:5)ans =

5.28485.04414.82924.9279

4.8810

Example of Data Management with Datasets

Datasets make it easy to access my data by name and operate on the data within thedataset 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 R790 159.36 156.15 39.5049 38.81490 160.7978 163.922 43.9022 42.762790 150.5968 137.5926 41.79 51.407990 173.6168 131.1749 45.1426 44.138890 183.4965 160.3539 39.8596 53.091790 151.0959 164.9913 50.1638 43.059390 166.0129 123.9435 48.1589 39.231690 155.5653 137.8742 45.9231 36.5771


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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 103.006129.7064

-90 -90 8 149.3126 26.5415 153.678412.1865

-60 -60 8 230.3514 27.3318 222.677133.259

-30 -30 8 262.2175 44.05 251.462535.3024

0 0 8 275.3241 51.6807 277.386468.5707

30 30 8 260.9625 60.5256 245.388246.0502

60 60 8 216.2059 12.6545 216.382136.0055

90 90 8 162.5677 11.3975 147.000316.1507

120 120 8 102.593 29.6774 99.9800439.1901

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

Show Plot by AngleLet's start exploring my data. I'll use a scatter plot to visualize my data.

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


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This plot makes it easy to see how L1 trends with R1, in a linear fashion. I can also seehow angle affects the stress level. It appears that -120/120 has the lowest stress and 0 isthe 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 45degree 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')


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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 inR2, as can be seen by the zero values in the plot. I know from testing that R2 failedduring the latter part of testing. I need to remove the bad data from R2 and keep the gooddata 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 mycustom 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 thisassumption.

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 methat my assumption L1 = R1 is true, or in other words, the differences I observe in L1 andR1 are due to normal and expected variation.

The hypothesis test (h2) returns a 1 for the assumption that L2 = R2. This tells me L2 isnot equal to R2, which we already know has bad data in it.

Hypothesis testing allows me to systematically test my data for difference in measuredvalues 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 mydata 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 labelsLeft = {'L1','L2','L3','L4','L5','L6','L7'};Right = {'R1','R2','R3','R4','R5','R6','R7'};

for i = 1:length(Left)


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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 datacreatefigure(Lx,Rx,[0 500]) % modified plotlabel = [Left{i},' vs. ',Right{i}];title(label)fprintf('%s failed test, h = %i\n',label,h(i))

endendL2 vs. R2 failed test, h = 1L3 vs. R3 failed test, h = 1


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We can see that L2 and R2 were flagged, which we expect. We can also see that L3 andR3 were flagged. Looking at the L3 vs. R3 plot, we can see that R3 appears to have ahigher 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 = [dsdataset({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 commandnow displays information about the categorical data. Specifically, if we look at HR2, wecan 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 datacreatefigure(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 max4.4538 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




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R2: [72x1 double, Units = kPa = kN/m^2]min 1st Q median 3rd Q max6.1899e-009 2.8778e-008 134.2293 213.2698

306.1857






HL1: [72x1 nominal, Units = [-]]Alive

72


72


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72


72


72


72


72

HR1: [72x1 nominal, Units = [-]]Alive

72

HR2: [72x1 nominal, Units = [-]]Alive Dead

46 26


72


72


72


72


72


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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-axisshows 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). Wecan 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 thatL3 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 aresignificant and not due to random variation. If the p-value was at or above 0.05, this testwould tell me that my different sensors are really measuring the same thing. Now theANOVA 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. Ican 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 differentthan all of the other sensors, 12 to be precise. So L3 must be the sensor that is wrong andR3 is good. Now I don't konw the casue of L3's difference. It could be bad dataacquisition hardware, improper installation or placement of the L3 sensor, or some other


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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 alsofound L3 to be suspect, and should remove it from analysis based upon it near behavingas expected. Now let's take a look at angle effects on our data. I'll use cluster analysishere instead of hypothesis testing or ANOVA, to show a different method from StatisticsToolbox 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 eachother. 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.


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60 is interesting in that it is more like -90 and 90 than it is like -60. This is surprising inthat we should have a -60/60 grouping, based upon symmetry of the airplane. I need toinvestigate 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 toduring 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 ithere to calculate the shear force.

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

M = I/t * stress

where I is the cross-sectional moment of inertia for the wing and t is the location of thesensor 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^4t = [0.1105 0.1085 0.1065 0.1045 0.1001 0.0962 0.920]; % mspan = [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 loadedcftool([span span],[ML MR])


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Show cftool analysis.

cubicfit([span span],[ML MR]) % generated from cftoolopen('analysis.fig') % show analysis figure (saved from Analysis incftool)


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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 forceappears 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 wingmeasurements to be similar after we removed bad and suspect data.

Appendix


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Custom plot createfigure

type createfigurefunction 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 figurefigure1 = figure;

% Create axesaxes('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 plotplot(X1,Y1,'Marker','o','LineStyle','none');

% Create xlabelxlabel('Left');

% Create ylabelylabel('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 cubicfitfunction 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":


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% X = x:% Y = y:% Unweighted%% This function was automatically generated on 06-Aug-2007 16:03:00

% Set up figure to receive datasets and fitsf_ = clf;figure(f_);set(f_,'Units','Pixels','Position',[852 350 680 477]);legh_ = []; legt_ = {}; % handles and text for legendxlim_ = [Inf -Inf]; % limits of x axisax_ = 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 limitsif 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 datacf_ = 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 fith_ = plot(cf_,'fit',0.95);legend off; % turn off legend from plot method callset(h_(1),'Color',[0 0 1],...

'LineStyle','-', 'LineWidth',2,...'Marker','none', 'MarkerSize',6);

legh_(end+1) = h_(1);


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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 legendset(h_,'Interpreter','none');xlabel(ax_,''); % remove x labelylabel(ax_,''); % remove y label

Published with MATLAB 7.4

analysis of wing stresses on a prototype aircraft

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