unsteady aerodynamics flutter analysis
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ME 5301 Term Project Aravind Baskar
03-04-2016 A0136344
1 | P a g e
FLOW SYSTEM ANALYSIS
Part – 1: Unsteady Kernel function for 3D incompressible potential flow
Starting with the definition of acceleration potential, the equation for kernel
function is derived and the steps are detailed as below:
ME 5301 Term Project Aravind Baskar
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Part – 2: Tapered Wing – Analysis
The algorithm of the program sequence is briefly explained above. The grid
points are generated based on the slope of the lines and similarity principle. The
process to determine AIC involve the processes detailed in the referenced paper.
All relevant sub-routines are given by different files to aid debugging process. The
list of files used in the program is as follows:
Project_Final. m – Main File
getKappa.m, get I1.m, getI2.m, getI1pos.m, getI2pos.m – Subroutines
For various cases of reduced frequencies, the lift, moment and coefficients are
computed and the results are as follows:
For M = 0.0
Case
No Lift Moment CL CM
1 --32.7946 +22.5023i
-6.4807e+02 + 8.5323e+02i -0.0547 + 0.0375i -0.0720 + 0.0948i
2 -34.3305 +33.6495i -6.3530e+02 + 1.1559e+03i -0.0572 + 0.0561i -0.0706 + 0.1284i
3 -33.7213 +45.3042i -5.6543e+02 + 1.4620e+03i -0.0562 + 0.0755i -0.0628 + 0.1624i
4 -30.9916 +56.9340i -4.4079e+02 + 1.7575e+03i -0.0517 + 0.0949i -0.0490 + 0.1953i
Pre
-pro
cess
ing
Generate Grid points
Input VariablesD
LM
Doublet at 3 locations
Parabolic Approximation
DLM
Compute Kappa
Compute I
Determine AIC
Post
Pro
cess
ing
Compute Lift, Moment, and its coefficents
ME 5301 Term Project Aravind Baskar
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For M = 0.5
Case
No Lift Moment CL CM
1 -38.2409 + 6.3813i -7.8320e+02 + 4.5654e+02i -0.0736 + 0.0123i -0.1005 + 0.0586i
2 -41.4994 +15.5025i -8.1313e+02 + 7.0952e+02i -0.0799 + 0.0298i -0.1043 + 0.0910i
3 -42.7592 +25.3623i -7.8959e+02 + 9.7169e+02i -0.0823 + 0.0488i -0.1013 + 0.1247i
4 -42.0073 +35.4362i -7.1395e+02 + 1.2293e+03i -0.0808 + 0.0682i -0.0916 + 0.1577i
Code used for the analysis is attached as a part of the appendix. The complete
results set have been added as an attachment to this file
(Case-- 1/2 – a / b / c / d.mat). Plots generated are shown below:
Fig.1 Plot of Pressure & Lift at k=0.086, Mach = 0.0
ME 5301 Term Project Aravind Baskar
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Fig.2 Plot of Pressure & Lift at k=0.010, Mach = 0.0
Fig.3 Plot of Pressure & Lift at k=0.114, Mach = 0.0
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Fig.4 Plot of Pressure & Lift at k=0.128, Mach = 0.0
Fig.5 Plot of Pressure & Lift at k=0.086, Mach = 0.5
ME 5301 Term Project Aravind Baskar
03-04-2016 A0136344
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Fig.6 Plot of Pressure & Lift at k=0.100, Mach = 0.5
Fig.7 Plot of Pressure & Lift at k=0.114, Mach = 0.5
ME 5301 Term Project Aravind Baskar
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Fig.8 Plot of Pressure & Lift at k=0.128, Mach = 0.5
Inferences:
To account for compressibility, the Prandtl-Glauert coefficient has been multiplied
to the coefficients of lift, pressure and moment.
As the Mach number is increased from 0.0 to 0.5 the lift & moment changes as its
dependent upon the compressibility nature. The Coefficient of Lift & Moment
reach their maximum value much faster as the Mach number is increased from 0.0
to 0.5.
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References:
1. DLM Tools from University of Minnesota -
http://gitpaaw.umnaem.webfactional.com/tools.git
2. https://www.grc.nasa.gov/www/k-12/airplane/lifteq.html
3. http://www.grc.nasa.gov/WWW/wind/valid/fplam/fplam01/fplam01.ht
ml
4. Effects of Mach number on the maximum lift coefficient of wings of NACA
– 2300 Series Air Foil Sections
5. Aeroservoelasticty by Ashish Tewari, ISBN 978-1-4939-2367-0
6. The Effect of Wing Damage on Aero elastic Behaviour by Howard J. Conyers
Department of Mechanical Engineering and Materials Science Duke
University.
7. AIAC-2007-109, AEROELASTIC ANALYSIS OF A TAPERED
AIRCRAFT WING S. Durmaz , O. Ozdemir Ozgumus , M. O. Kaya
Istanbul Technical University, Faculty of Aeronautics and Astronautics,
34469, Maslak, Istanbul, Turkey
8. Aero-elastic characteristics of tapered plate wings ARTICLE in FINITE
ELEMENTS IN ANALYSIS AND DESIGN · FEBRUARY 2015 Impact
Factor: 2.02 · DOI: 10.1016/j.finel.2014.09.009
D:\National University Of Singapore...\PROJECT_FINAL.m Page 18 April, 2016 12:09:28 PM
%% Flutter Analysis for Tapered Wing Geometry %%%% References : DLM Tools from University of Minnesota - http://gitpaaw.umnaem.webfactional.com/tools.git %%%% INITIALIZATIONrf = 0.086:0.014:0.128; % Reduced Frequencies for all cases %for num = 1:4 % Loop for Various Reduced Frequencies % rho = 2.38e-3; % Density of Air % M = 0.5; % Mach Number % k = rf(num); % For individual cases % sw = 40; % Span Length % ds = sw/8; % Increment Spanwise % slope_u = 0.1; % Slope dy/dx = 4/40 using median principle % slope_l = - slope_u; % Slope of symmetric opp. line%% GRID COORDINATESi = 0;j = 0;coordinate = zeros(63,4);panel = zeros(48,5); for t = 0:ds:sw;
i = i+1; xl(i,:) = slope_u*t; % Lower Boundary %xu(i,:) = slope_l*t + 19; %#ok<*SAGROW> % Upper Boundary %dx(i,:) = (xu(i,1) - xl(i,1))/6; % Increment %xl_1(i,:) = xl(i,1) - dx(i,1);xu_1(i,:) = xu(i,1) - dx(i,1);for q = xl_1(i,1):dx(i,1):xu_1(i,1);
j = j+1;n(j,:) = j;y(j,:) = t;x_1(j,:) = q + dx(i,1);coordinate(j,1) = n(j,1); % Node Number %coordinate(j,2) = x_1(j,1); % Chordwise Coordinate %coordinate(j,3) = y(j,1); % Spanwise Coordinate %
end endh = 0;%% PANELS for PB = 1:1:48;
if(PB==7)h = 7;
endif (PB==13)
h = 14;endif(PB==19)
h = 21;endif(PB==25)
h = 28;endif(PB==31)
h = 35;
APPENDIX
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end if(PB==37) h = 42; end if(PB==43) h = 49; end if (PB~=7||PB~=13||PB~=19||PB~=25||PB~=31||PB~=37||PB~=43) h = h+1; end panel(PB,1) = PB; % Panel Number % panel(PB,2) = h; % Coordinate 1 % panel(PB,3) = h + 1; % Coordinate 2 % panel(PB,4) = h + 7; % Coordinate 3 % panel(PB,5) = h + 8; % Coordinate 4 % end%% CELL CENTRES OF PANELSi_1 = 0;j_1 = 0;cordinate_c = zeros(48,4); for t1 = ds/2:ds:sw-ds/2; i_1 = i_1 + 1; xl_c(i_1,:) = slope_u*t1; % Upper Boundary % xu_c(i_1,:) = slope_l*t1 + 19; % Lower Boundary % dx_c(i_1,:) = (xu_c(i_1,1) - xl_c(i_1,1))/6; % Increment % xl1_c(i_1,:) = xl_c(i_1,1) - dx_c(i_1,1) + dx_c(i_1,1)/2; xu1_c(i_1,:) = xu_c(i_1,1) - dx_c(i_1,1) - dx_c(i_1,1)/2; for q_c = xl1_c(i_1,1):dx_c(i_1,1):xu1_c(i_1,1); j_1 = j_1+1; n_c(j_1,:)=j_1; y_c(j_1,:) = t1; x_1_c(j_1,:) = q_c + dx_c(i_1,1)/2; cordinate_c(j_1,1) = n_c(j_1,1); % Node Number % cordinate_c(j_1,2) = x_1_c(j_1,1); % Chordwise Coordinate % cordinate_c(j_1,3) = y_c(j_1,1); % Spanwise Coordinate % diff_c(j_1,1) = 6*dx_c(i_1,1); end end%% DOWNWASH POSITIONPO = zeros(PB,3);N = size(PO,1);PO(:,1) = cordinate_c(:,2)+(3/8)*(coordinate(panel(:,3),2) - coordinate(panel(:,2),2) + ... coordinate(panel(:,5),2) - coordinate(panel(:,4),2));PO(:,2) = coordinate(panel(:,2),3) + (coordinate(panel(:,4),3)-coordinate(panel(:,2),3))/2;PO(:,3) = 0;%% DOUBLET - START OF WINGPS = zeros(PB,3);PS(:,1) = coordinate(panel(:,2),2)+(1/4)*(coordinate(panel(:,3),2) - coordinate(panel(:,2),2));PS(:,2) = coordinate(panel(:,2),3);PS(:,3) = 0;
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%% DOUBLET - TIP OF WINGPE = zeros(PB,3);PE(:,1) = coordinate(panel(:,4),2) + ((coordinate(panel(:,5),2) - ... coordinate(panel(:,4),2))/4);PE(:,2) = coordinate(panel(:,4),3);PE(:,3) = 0;%% DOUBLET - HALF SPAN OF WINGPM = zeros(PB,3);PM(:,1) = (PS(:,1) + PE(:,1))/2;PM(:,2) = (PS(:,2) + PE(:,2))/2;PM(:,3) = (PS(:,3) + PE(:,3))/2;%% DEFINING SPAN AND CHORD OF EACH PANELZ_1 = zeros(48,1);SP = (0.5*(coordinate(panel(:,4),3) - coordinate(panel(:,2),3)))';CP = ((abs((coordinate(panel(:,2),2) - coordinate(panel(:,3),2))) + ... abs((coordinate(panel(:,4),2) - coordinate(panel(:,5),2))))/2)';%% DLM NORMALWASH MATRIX - UNNSTEADY CASE %% ROOT DOUBLET LOCATIONx01 = bsxfun(@minus,PO(:,1),PS(:,1)'); y01 = bsxfun(@minus,PO(:,2),PS(:,2)');z01 = repmat(Z_1',48,1);%% SEMI-SPAN DOUBLET LOCATIONx02 = bsxfun(@minus ,PO(:,1),PM(:,1)');y02 = bsxfun(@minus ,PO(:,2),PM(:,2)');z02 = repmat(Z_1',48,1);%% TIP OF DOUBLET LOCATIONx03 = bsxfun(@minus,PO(:,1),PE(:,1)');y03 = bsxfun(@minus,PO(:,2),PE(:,2)');z03 = repmat(Z_1',48,1);cosGamma = ones(48,48);sinGamma = zeros(48,48);%% KAPPAKi_w = getKappa(x01,y01,z01,cosGamma,sinGamma,k,M);Ki_0 = getKappa(x01,y01,z01,cosGamma,sinGamma,0,M);Ki = Ki_w - Ki_0; Km_w = getKappa(x02,y02,z02,cosGamma,sinGamma,k,M);Km_0 = getKappa(x02,y02,z02,cosGamma,sinGamma,0,M);Km = Km_w - Km_0;K0_w = getKappa(x03,y03,z03,cosGamma,sinGamma,k,M);K0_0 = getKappa(x03,y03,z03,cosGamma,sinGamma,0,M);K0 = K0_w - K0_0;%% PARABOLIC APPROXIMATIONe1 = (repmat(SP,N,1));A = (Ki-2*Km+K0)./(2*e1.^2);B = (K0-Ki)./(2*e1);C = Km;n0 = y02.*cosGamma;zeta0 = z02.*cosGamma;r2 = sqrt((n0.^2) + (zeta0.^2));%% NORMALWASH MATRIX FACTOR "I"I = (A.*(2*e1)) + ((0.5*B+n0.*A).*log((r2.^2 - 2*n0.*e1 + e1.^2)./(r2.^2 + 2*n0.*e1 + e1.^2)))... +(((n0.^2 - zeta0.^2).*A+n0.*B + C)./abs(zeta0).*atan(2*e1.*abs(zeta0)./(r2.^2 -
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e1.^2)));ind = find(zeta0==0);I2 = ((A.*(2*e1)) + ((0.5*B+n0.*A).*log(((n0-e1)./(n0+e1)).^2))... +((n0.^2).*A+n0.*B + C).*((2*e1)./(n0.^2 - e1.^2)));I(ind) = I2(ind);D = repmat(CP,N,1).*I/(pi*8);%% VLM NORMALWASH MATRIX - QUASI - STEADY STATEbeta = sqrt(1-(M^2));%% AIC MATRIXAIC = inv(D);wbar=ones(48,1);%% PRESSURE DISTRIBUTIONP = AIC\wbar; % delP / Q %P_R = real(P);P_I = imag(P);P_ABS = abs(P);P_PR = vec2mat(P_R,8);P_PI = vec2mat(P_I,8);figure(num)subplot(2,2,1)scatter(y,x_1);xlabel('Span');ylabel('Chord');title('Grid points');subplot(2,2,2)meshz(P_PR);xlabel('Span Divisions');ylabel('Chord Divisions');zlabel('P(R)');title('Pressure - Real Part');subplot(2,2,3)meshz(P_PI);xlabel('Span Divisions');ylabel('Chord Divisions');zlabel('P(I)');title('Pressure - IMG Part');Lift = 0;MO = 0; for i = 1:48 Li(i,:) = P(i)*2*SP(i)*CP(i); % Non dimensional Lift - Lift / Q % Lift = Lift + Li(i,:); MO = MO + Lift*0.4*CP(i); endL_D = vec2mat(abs(Li),8);subplot(2,2,4)meshz(L_D);xlabel('Span Divisions');ylabel('Chord Divisions');zlabel('abs(L)');title('Lift - Absolute');S = 0.5*((coordinate(7,2)-coordinate(1,2)) + (coordinate(63,2)-coordinate(57,2)))*sw; % Surface Area %cavg = 0.5*((coordinate(7,2) - coordinate(1,2)) + (coordinate(63,2) - coordinate
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(57,2))); % Avg. Chord Length %CL = Lift/(beta*S); % Non dimensional Coefficent of Lift %CM = MO/(beta*S*cavg); % Non dimensional Coefficient of Moment %disp(M);disp(k);display(num);display(Lift);display(MO);display(CL);display(CM);end
8/4/16 12:05 PM MATLAB Command Window 1 of 3
>> PROJECT_FINAL 0.5000 0.0860 num = 1 Lift = -38.2409 + 6.3813i MO = -7.8320e+02 + 4.5654e+02i CL = -0.0736 + 0.0123i CM = -0.1005 + 0.0586i 214 end 0.5000 0.1000 num = 2 Lift = -41.4994 +15.5025i MO = -8.1313e+02 + 7.0952e+02i CL = -0.0799 + 0.0298i
8/4/16 12:05 PM MATLAB Command Window 2 of 3
CM = -0.1043 + 0.0910i 0.5000 0.1140 num = 3 Lift = -42.7592 +25.3623i MO = -7.8959e+02 + 9.7169e+02i CL = -0.0823 + 0.0488i CM = -0.1013 + 0.1247i 0.5000 0.1280 num = 4 Lift = -42.0073 +35.4362i MO = -7.1395e+02 + 1.2293e+03i CL =
8/4/16 12:05 PM MATLAB Command Window 3 of 3
-0.0808 + 0.0682i CM = -0.0916 + 0.1577i >>
8/4/16 12:04 PM MATLAB Command Window 1 of 5
>> PROJECT_FINAL 0 0.0860 num = 1 Lift = -32.7946 +22.5023i MO = -6.4807e+02 + 8.5323e+02i CL = -0.0547 + 0.0375i CM = -0.0720 + 0.0948i 0 0.1000 num = 2 Lift = -34.3305 +33.6495i MO = -6.3530e+02 + 1.1559e+03i CL = -0.0572 + 0.0561i
8/4/16 12:04 PM MATLAB Command Window 2 of 5
CM = -0.0706 + 0.1284i 0 0.1140 num = 3 Lift = -33.7213 +45.3042i MO = -5.6543e+02 + 1.4620e+03i CL = -0.0562 + 0.0755i CM = -0.0628 + 0.1624i 0 0.1280 num = 4 Lift = -30.9916 +56.9340i MO = -4.4079e+02 + 1.7575e+03i CL =
8/4/16 12:04 PM MATLAB Command Window 3 of 5
-0.0517 + 0.0949i CM = -0.0490 + 0.1953i >> PROJECT_FINAL 0 0.0860 num = 1 Lift = -32.7946 +22.5023i MO = -6.4807e+02 + 8.5323e+02i CL = -0.0547 + 0.0375i CM = -0.0720 + 0.0948i 0 0.1000 num = 2 Lift = -34.3305 +33.6495i MO = -6.3530e+02 + 1.1559e+03i
8/4/16 12:04 PM MATLAB Command Window 4 of 5
CL = -0.0572 + 0.0561i CM = -0.0706 + 0.1284i 0 0.1140 num = 3 Lift = -33.7213 +45.3042i MO = -5.6543e+02 + 1.4620e+03i CL = -0.0562 + 0.0755i CM = -0.0628 + 0.1624i 0 0.1280 num = 4 Lift = -30.9916 +56.9340i
8/4/16 12:04 PM MATLAB Command Window 5 of 5
MO = -4.4079e+02 + 1.7575e+03i CL = -0.0517 + 0.0949i CM = -0.0490 + 0.1953i >>
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%% Flutter Analysis for Tapered Wing Geometry %%%% References : DLM Tools from University of Minnesota - http://gitpaaw.umnaem.webfactional.com/tools.git %%%% INITIALIZATIONrf = 0.086:0.014:0.128; % Reduced Frequencies for all cases %for num = 1:4 % Loop for Various Reduced Frequencies % rho = 2.38e-3; % Density of Air % M = 0.5; % Mach Number % k = rf(num); % For individual cases % sw = 40; % Span Length % ds = sw/8; % Increment Spanwise % slope_u = 0.1; % Slope dy/dx = 4/40 using median principle % slope_l = - slope_u; % Slope of symmetric opp. line%% GRID COORDINATESi = 0;j = 0;coordinate = zeros(63,4);panel = zeros(48,5); for t = 0:ds:sw; i = i+1; xl(i,:) = slope_u*t; % Lower Boundary % xu(i,:) = slope_l*t + 19; %#ok<*SAGROW> % Upper Boundary % dx(i,:) = (xu(i,1) - xl(i,1))/6; % Increment % xl_1(i,:) = xl(i,1) - dx(i,1); xu_1(i,:) = xu(i,1) - dx(i,1); for q = xl_1(i,1):dx(i,1):xu_1(i,1); j = j+1; n(j,:) = j; y(j,:) = t; x_1(j,:) = q + dx(i,1); coordinate(j,1) = n(j,1); % Node Number % coordinate(j,2) = x_1(j,1); % Chordwise Coordinate % coordinate(j,3) = y(j,1); % Spanwise Coordinate % end endh = 0;%% PANELS for PB = 1:1:48; if(PB==7) h = 7; end if (PB==13) h = 14; end if(PB==19) h = 21; end if(PB==25) h = 28; end if(PB==31) h = 35;
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end if(PB==37) h = 42; end if(PB==43) h = 49; end if (PB~=7||PB~=13||PB~=19||PB~=25||PB~=31||PB~=37||PB~=43) h = h+1; end panel(PB,1) = PB; % Panel Number % panel(PB,2) = h; % Coordinate 1 % panel(PB,3) = h + 1; % Coordinate 2 % panel(PB,4) = h + 7; % Coordinate 3 % panel(PB,5) = h + 8; % Coordinate 4 % end%% CELL CENTRES OF PANELSi_1 = 0;j_1 = 0;cordinate_c = zeros(48,4); for t1 = ds/2:ds:sw-ds/2; i_1 = i_1 + 1; xl_c(i_1,:) = slope_u*t1; % Upper Boundary % xu_c(i_1,:) = slope_l*t1 + 19; % Lower Boundary % dx_c(i_1,:) = (xu_c(i_1,1) - xl_c(i_1,1))/6; % Increment % xl1_c(i_1,:) = xl_c(i_1,1) - dx_c(i_1,1) + dx_c(i_1,1)/2; xu1_c(i_1,:) = xu_c(i_1,1) - dx_c(i_1,1) - dx_c(i_1,1)/2; for q_c = xl1_c(i_1,1):dx_c(i_1,1):xu1_c(i_1,1); j_1 = j_1+1; n_c(j_1,:)=j_1; y_c(j_1,:) = t1; x_1_c(j_1,:) = q_c + dx_c(i_1,1)/2; cordinate_c(j_1,1) = n_c(j_1,1); % Node Number % cordinate_c(j_1,2) = x_1_c(j_1,1); % Chordwise Coordinate % cordinate_c(j_1,3) = y_c(j_1,1); % Spanwise Coordinate % diff_c(j_1,1) = 6*dx_c(i_1,1); end end%% DOWNWASH POSITIONPO = zeros(PB,3);N = size(PO,1);PO(:,1) = cordinate_c(:,2)+(3/8)*(coordinate(panel(:,3),2) - coordinate(panel(:,2),2) + ... coordinate(panel(:,5),2) - coordinate(panel(:,4),2));PO(:,2) = coordinate(panel(:,2),3) + (coordinate(panel(:,4),3)-coordinate(panel(:,2),3))/2;PO(:,3) = 0;%% DOUBLET - START OF WINGPS = zeros(PB,3);PS(:,1) = coordinate(panel(:,2),2)+(1/4)*(coordinate(panel(:,3),2) - coordinate(panel(:,2),2));PS(:,2) = coordinate(panel(:,2),3);PS(:,3) = 0;
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%% DOUBLET - TIP OF WINGPE = zeros(PB,3);PE(:,1) = coordinate(panel(:,4),2) + ((coordinate(panel(:,5),2) - ... coordinate(panel(:,4),2))/4);PE(:,2) = coordinate(panel(:,4),3);PE(:,3) = 0;%% DOUBLET - HALF SPAN OF WINGPM = zeros(PB,3);PM(:,1) = (PS(:,1) + PE(:,1))/2;PM(:,2) = (PS(:,2) + PE(:,2))/2;PM(:,3) = (PS(:,3) + PE(:,3))/2;%% DEFINING SPAN AND CHORD OF EACH PANELZ_1 = zeros(48,1);SP = (0.5*(coordinate(panel(:,4),3) - coordinate(panel(:,2),3)))';CP = ((abs((coordinate(panel(:,2),2) - coordinate(panel(:,3),2))) + ... abs((coordinate(panel(:,4),2) - coordinate(panel(:,5),2))))/2)';%% DLM NORMALWASH MATRIX - UNNSTEADY CASE %% ROOT DOUBLET LOCATIONx01 = bsxfun(@minus,PO(:,1),PS(:,1)'); y01 = bsxfun(@minus,PO(:,2),PS(:,2)');z01 = repmat(Z_1',48,1);%% SEMI-SPAN DOUBLET LOCATIONx02 = bsxfun(@minus ,PO(:,1),PM(:,1)');y02 = bsxfun(@minus ,PO(:,2),PM(:,2)');z02 = repmat(Z_1',48,1);%% TIP OF DOUBLET LOCATIONx03 = bsxfun(@minus,PO(:,1),PE(:,1)');y03 = bsxfun(@minus,PO(:,2),PE(:,2)');z03 = repmat(Z_1',48,1);cosGamma = ones(48,48);sinGamma = zeros(48,48);%% KAPPAKi_w = getKappa(x01,y01,z01,cosGamma,sinGamma,k,M);Ki_0 = getKappa(x01,y01,z01,cosGamma,sinGamma,0,M);Ki = Ki_w - Ki_0; Km_w = getKappa(x02,y02,z02,cosGamma,sinGamma,k,M);Km_0 = getKappa(x02,y02,z02,cosGamma,sinGamma,0,M);Km = Km_w - Km_0;K0_w = getKappa(x03,y03,z03,cosGamma,sinGamma,k,M);K0_0 = getKappa(x03,y03,z03,cosGamma,sinGamma,0,M);K0 = K0_w - K0_0;%% PARABOLIC APPROXIMATIONe1 = (repmat(SP,N,1));A = (Ki-2*Km+K0)./(2*e1.^2);B = (K0-Ki)./(2*e1);C = Km;n0 = y02.*cosGamma;zeta0 = z02.*cosGamma;r2 = sqrt((n0.^2) + (zeta0.^2));%% NORMALWASH MATRIX FACTOR "I"I = (A.*(2*e1)) + ((0.5*B+n0.*A).*log((r2.^2 - 2*n0.*e1 + e1.^2)./(r2.^2 + 2*n0.*e1 + e1.^2)))... +(((n0.^2 - zeta0.^2).*A+n0.*B + C)./abs(zeta0).*atan(2*e1.*abs(zeta0)./(r2.^2 -
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e1.^2)));ind = find(zeta0==0);I2 = ((A.*(2*e1)) + ((0.5*B+n0.*A).*log(((n0-e1)./(n0+e1)).^2))... +((n0.^2).*A+n0.*B + C).*((2*e1)./(n0.^2 - e1.^2)));I(ind) = I2(ind);D = repmat(CP,N,1).*I/(pi*8);%% VLM NORMALWASH MATRIX - QUASI - STEADY STATEbeta = sqrt(1-(M^2));%% AIC MATRIXAIC = inv(D);wbar=ones(48,1);%% PRESSURE DISTRIBUTIONP = AIC\wbar; % delP / Q %P_R = real(P);P_I = imag(P);P_ABS = abs(P);P_PR = vec2mat(P_R,8);P_PI = vec2mat(P_I,8);figure(num)subplot(2,2,1)scatter(y,x_1);xlabel('Span');ylabel('Chord');title('Grid points');subplot(2,2,2)meshz(P_PR);xlabel('Span Divisions');ylabel('Chord Divisions');zlabel('P(R)');title('Pressure - Real Part');subplot(2,2,3)meshz(P_PI);xlabel('Span Divisions');ylabel('Chord Divisions');zlabel('P(I)');title('Pressure - IMG Part');Lift = 0;MO = 0; for i = 1:48 Li(i,:) = P(i)*2*SP(i)*CP(i); % Non dimensional Lift - Lift / Q % Lift = Lift + Li(i,:); MO = MO + Lift*0.4*CP(i); endL_D = vec2mat(abs(Li),8);subplot(2,2,4)meshz(L_D);xlabel('Span Divisions');ylabel('Chord Divisions');zlabel('abs(L)');title('Lift - Absolute');S = 0.5*((coordinate(7,2)-coordinate(1,2)) + (coordinate(63,2)-coordinate(57,2)))*sw; % Surface Area %cavg = 0.5*((coordinate(7,2) - coordinate(1,2)) + (coordinate(63,2) - coordinate
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(57,2))); % Avg. Chord Length %CL = Lift/(beta*S); % Non dimensional Coefficent of Lift %CM = MO/(beta*S*cavg); % Non dimensional Coefficient of Moment %disp(M);disp(k);display(num);display(Lift);display(MO);display(CL);display(CM);end
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