unsteady aerodynamics flutter analysis

ME 5301 Term Project Aravind Baskar

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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:


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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


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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


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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

http://gitpaaw.umnaem.webfactional.com/tools.git

https://www.grc.nasa.gov/www/k-12/airplane/lifteq.html

http://www.grc.nasa.gov/WWW/wind/valid/fplam/fplam01/fplam01.html

http://www.grc.nasa.gov/WWW/wind/valid/fplam/fplam01/fplam01.html

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 = 35;

APPENDIX


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;


%% 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 -


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


(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


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 =


-0.0808 + 0.0682i CM = -0.0916 + 0.1577i >>


>> 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


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 =


-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


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


MO = -4.4079e+02 + 1.7575e+03i CL = -0.0517 + 0.0949i CM = -0.0490 + 0.1953i >>


%% 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;


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;


%% 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 -


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


(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

unsteady aerodynamics flutter analysis

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