LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTA==In this assignment we examine the flutter phenomenon (at sea level) for the wing that is given in the first home assignment. We find flutter velocity using V-g method.
<Text-field style="Heading 1" layout="Heading 1">Mathematical Model</Text-field>The dynamic system we have (without the unsteady aerodynamics) is: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Adding the unsteady contribution: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And therefore, by neglecting the structure damping and using the relation LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JJW1zdXBHRiQ2JS1GLDYlUSJlRidGL0YyLUYjNiUtRiw2JVEoSSYjOTY5O0YnRi9GMkYvRjIvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjk= we will get the following dynamic system::LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYxLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkMtSSVtc3VwR0YkNiUtSSNtaUdGJDYlUScmIzk2OTtGJy8lJ2l0YWxpY0dGNEYvLUYjNiUtSSNtbkdGJDYkUSIyRidGLy9GTlEldHJ1ZUYnL0YwUSdpdGFsaWNGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictSShtZmVuY2VkR0YkNiYtRiM2JC1GSjYlUSJBRidGVUZXRi9GLy8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJy1GLDYtUSgmbWludXM7RidGL0YyRjVGN0Y5RjtGPUY/RkFGRC1GSjYlUS9JJm9tZWdhOyZyaG87VkYnRlVGVy1GZ242Ji1GIzYkLUZKNiVRIkJGJ0ZVRldGL0YvRl5vRmFvLUYsNi1RIitGJ0YvRjJGNUY3RjlGO0Y9Rj9GQUZELUZHNiUtRko2JVEoJiM5NjE7VkYnRlVGVy1GIzYnRlFGVS8lK2V4ZWN1dGFibGVHRlYvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnRldGWS1GZ242Ji1GIzYlLUZKNiVRIkNGJ0ZVRldGXXFGL0YvRl5vRmFvRmFwLUZnbjYmLUYjNiQtRko2JVEiRUYnRlVGV0YvRi9GXm9GYW8tRiw2LVEifkYnRi9GMkY1RjdGOUY7Rj1GPy9GQlEmMC4wZW1GJy9GRUZici1GLDYtUSI9RidGL0YyRjVGN0Y5RjtGPUY/L0ZCUSwwLjI3Nzc3NzhlbUYnL0ZFRmhyLUZSNiRGZW5GL0Yvwhere:JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= - Generalized inertia matrix.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= - Generalized stiffness matrix.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= - aerodynamic damping matrix.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= - aerodynamic stiffness matrix.Now, we add a friction damping to the structure stiffness matrix (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=) that bring the system to the edge of stability: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 the relation 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 we get: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We got the system from which we will get the flutter velocity.The solution of this system is: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 by finding the eigenvalues of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy we can find the flutter velocity: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JSFHJSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Wing Data</Text-field>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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=The given data of the wing:
PkkiZUc2Ii1JJGFic0clKnByb3RlY3RlZEc2IywmSSZ4X19lYUdGJCIiIkkmeF9fYWNHRiQhIiI= = print(); # input placeholderprint(); # input placeholderPkkiYUc2IiwkLUkiK0clKnByb3RlY3RlZEc2JCQiIiUhIiIkISImRiwiIiM= = print(); # input placeholderQyQ+SSJORzYiIiQrIiEiIg==The length and the mass of each element is:PkklbF9fZUc2IiomSSJsR0YkIiIiSSJOR0YkISIi = print(); # input placeholderPkkibUc2IiomSShtX193aW5nR0YkIiIiSSJOR0YkISIi = print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=The density is:QyQ+SSRyaG9HNiIkIiVENyEiJCEiIg==The number of modes we want to check QyQ+SSNtTkc2IiIiJSEiIg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 2" layout="Heading 2">Global Stiffness and Inertia Matrix</Text-field>The element stiffness Matrix is:QyQ+SSVLX19lRzYiLUkwZGVsYXlEb3RQcm9kdWN0RzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJUksVHlwZXNldHRpbmdHNiRGKUkoX3N5c2xpYkdGJTYlKiZJI0VJR0YlIiIiSSVsX19lR0YlISIkLUknTWF0cml4R0YlNiMvSSQlaWRHRiUiNSVcUS5NdVN1WSU9SSV0cnVlR0YpISIiprint(); # input placeholderFrom the Attached file in the course web site, the element Mass Matrix is:QyQ+SSVNX19lRzYiLCQqJkkibUdGJSIiIi1JJ01hdHJpeEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjL0kkJWlkR0YlIjUlNFIuTXVTdVklPUYpI0YpIiIjISIiThe expression of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EnYEpfX3pgRictSSNtb0dGJDYtUSIsRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR1EldHJ1ZUYnLyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiNRJ2BKX194YEYnRjM= was given in home assignment 3:QyY+SSVKX196RzYiLCQqJkkibUdGJSIiIiwmKiRJJWxfX2VHRiUiIiNGKSokSSJ0R0YlRi1GKUYpI0YpIiM3ISIiPkklSl9feEdGJSwkKiZGKEYpLCYtSSJeRyUqcHJvdGVjdGVkRzYkLCRJImNHRiUkIiMmKSEiI0YtRilGLkYpRilGMEYyThe Matrix Assembly procedure:JSFH#Matrix Assembly Procedure
MatAssemb := proc (N::anything, M::Matrix)
local n, j, k, Rng, T;
T := Matrix(3*N, 3*N, 0);
for j to 3 do
for n to 3 do
T[j, n] := T[j, n]+M[j+3, n+3];
end do;
end do;
for k from 0 by 3 to 3*N-6 do
Rng := 1+k .. 6+k;
T[Rng, Rng] := T[Rng, Rng]+M;
end do;
return T;
end proc:The whole wing Stiffness and Inertia matrix:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFHQyY+SSZUX19TTUc2Ii1JKk1hdEFzc2VtYkdGJTYkSSJOR0YlSSVLX19lR0YlISIiPkkmVF9fSU1HRiUtRic2JEYpSSVNX19lR0YlRis=print(); # input placeholderJSFHJSFH
<Text-field style="Heading 1" layout="Heading 1">Theodorsen Function</Text-field>We will build Theodorsen function from a combination of Henkel functions as shown in class:QyQ+SStUaGVvZG9yc2VuRzYiZio2I0kia0dGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiYtSSlIYW5rZWxIMkdGJTYkIiIiOSRGMCwmRi1GMComXiNGMEYwLUYuNiQiIiFGMUYwRjAhIiJGJUYlRiVGOA==JSFHJSFHBefore we start answering the questions, in order to calculate the flutter velocity we need to build the following procedures ( the mathematics model described above )JSFH
<Text-field style="Heading 1" layout="Heading 1">Procedures</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 3" layout="Heading 3">Dynamic's</Text-field>Here we build the oscillatory coefficients as it shown in home assignment 4:QyY+SSJGRzYiZio2I0kia0dGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkmZXZhbGNHRiU2Iy1JI1JlRyUqcHJvdGVjdGVkRzYjLUkrVGhlb2RvcnNlbkdGJTYjOSRGJUYlRiUhIiI+SSJHR0YlZipGJ0YlRilGJS1GLTYjLUkjSW1HRjFGMkYlRiVGJUY3print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=print(); # input 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFH#Total Matrix Assembly Procedure
#This procedure build the whole wing unsteady forces...
#...coefficients matrix (similar to MatAssemb procedure).
TotalMatrix := proc (N::anything, M::Matrix)
local i, T;
T := Matrix(2*N, 2*N, 0);
for i by 2 to 2*N do
T[i .. i+1, i .. i+1] := M[1 .. -1, 1 .. -1];
end do;
return evalc(Re(T));
end proc:JSFHJSFHJSFH#Modes Calculations
#This procedure receive the number of elements, the whole wing...
#...stiffness and inertia matrix and the modes number and return...
#...the whole modes of the wing matrix.
Modes := proc (N, T_I::Matrix, T_S::Matrix, mN)
local lambd, u, i, x, j, phi, phi1;
lambd, u := Eigenvectors(T_I, T_S);
for i to mN do
x[i] := Re(convert(u[1 .. -1, i], 'Vector'));
end do;
phi1 := Matrix(2*N, mN, 0);
phi := Matrix(3*N, mN, 0);
for i to mN do
phi[1 .. -1, i] := x[i][1 .. -1];
for j by 2 to 2*N do
phi1[j, i] := -phi[(3/2)*j-1/2, i];
end do;
for j from 2 by 2 to 2*N do
phi1[j, i] := phi[(3/2)*j, i];
end do;
end do;
return phi, phi1;
end proc:
JSFHJSFH#Omega-V-g Diagram Procedure
#This procedure receives the number of elements, the whole wing stiffness and inertia matrix,...
#...the modes we want to examine, an element unsteady aerodynamic forces matrix, ...
#...the semi chord length, the density and the plot x and y limits. ...
#...The procedure return the V-Omega and V-g plots.
diag := proc (N, T_S::Matrix, T_I::Matrix, mN, B_e::Matrix, C_e::Matrix, b, rho, xlim, ylim)
local GenStiff, GenDamp, StructureInertia, E, K, EignVal, M, F, i, val, j, Pg, Pw, omega, g, v, P, v__D;
P := Modes(N, T_I, T_S, mN);
GenStiff := evalf(P[2]^%T.TotalMatrix(N, B_e).P[2]);
GenDamp := evalf(P[2]^%T.TotalMatrix(N, C_e).P[2]);
StructureInertia := P[1]^%T.T_I.P[1];
E := P[1]^%T.T_S.P[1];
K := [seq(i, i = 1/10000000 .. 1, 0.5e-3)];
EignVal := Matrix(mN, nops(K));
M := Matrix(mN, mN, 0);
F := evalf(StructureInertia+I*rho*b*GenStiff/k+rho*b^2*GenDamp/k^2);
for i to nops(K) do
M := evalf(subs(k = K[i], MatrixInverse(E).F));
val := Eigenvalues(M);
EignVal[1 .. -1, i] := val[1 .. -1];
end do;
for j to mN do
g[j] := [seq(evalc(Im(EignVal[j, i])/Re(EignVal[j, i])), i = 1 .. nops(K))];
omega[j] := [seq(evalc(Re((1/2)/(Pi*sqrt(Re(EignVal[j, i]))))), i = 1 .. nops(K))];
v[j] := [seq(evalc(Re(1/sqrt(Re(EignVal[j, i]))))*b/K[i], i = 1 .. nops(K))];
Pg[j] := plot(v[j], g[j], 0 .. xlim, -2 .. 1, style = point,
symbol = asterisk, legend = typeset(" mod= ", j, ", "),
color = setcolors()[`mod`(j, nops(C))+1], title = "g vs V",
titlefont = ["ROMAN", 15, bold],
axis = [gridlines = [8, color = black]],
labels = ["V [m/sec]", "g"], labelfont = ["HELVETICA", 15, bold]):
Pw[j] := plot(v[j], omega[j], 0 .. xlim, -10 .. ylim, style = point,
symbol = asterisk, legend = typeset(" mod= ", j, ", "),
color = setcolors()[`mod`(j, nops(C))+1], title = "Omega vs V",
titlefont = ["ROMAN", 15, bold],
axis = [gridlines = [8, color = black]],
labels = ["V [m/sec]", "ω [Hz]"], labelfont = ["HELVETICA", 15, bold]):
end do;
return Pg, Pw;
end proc:JSFHJSFHJSFH
<Text-field style="Heading 1" layout="Heading 1">Question 1</Text-field>Show the Omega-V-g diagram which contains the wing's frequencies and flutter (damping) for the first four modes, From the diagram find the flutter velocity and the divergence velocity at sea level. Which of the modes gives us the flutter velocity?Examine the influence of the number of modes we check during the analysis on the results.
<Text-field style="Heading 2" layout="Heading 2">Solution</Text-field>JSFHQyQ+SSRPVkdHNiItSSVkaWFnR0YlNixJIk5HRiVJJlRfX1NNR0YlSSZUX19JTUdGJUkjbU5HRiVJJUJfX2VHRiVJJUNfX2VHRiVJImJHRiVJJHJob0dGJSIkKyMiJF0iISIi%;
LUkoZGlzcGxheUc2IjYjLUkkc2VxRyUqcHJvdGVjdGVkRzYkJiZJJE9WR0dGJDYjIiIiNiNJImlHRiQvRjA7Ri5JI21OR0Yk%;JSFHJSFHLUkoZGlzcGxheUc2IjYjLUkkc2VxRyUqcHJvdGVjdGVkRzYkJiZJJE9WR0dGJDYjIiIjNiNJImlHRiQvRjA7IiIiSSNtTkdGJA==%;JSFHFrom the figure above you can see that the flutter velocity is where the fiction damping LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtaUdGJDYlUSJnRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJ0Y6Rjo= is zero, its happen at 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, we gets the flutter at the second heave mode. The divergence velocity is 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 .We ca also see that the second heave mode that give us the flutter have small values of fiction damping. The larger damping value for this mode is about LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSomdW1pbnVzMDtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GUy1JI21uR0YkNiRRJTAuMTNGJ0Y5Rjk=. From the figure above you can see that the flutter velocity occurs when two modes meet.In our case, the flutter velocity is when the first torque mode meets the second heave mode.The Divergence velocity occurs when the frequency is zero, it's happens in the first heave mode.
JSFHChecking the influence of taking more modes in the calculations:JSFHJSFHQyQ+SSZPVkdxMUc2Ii1JJWRpYWdHRiU2LEkiTkdGJUkmVF9fU01HRiVJJlRfX0lNR0YlLCZJI21OR0YlIiIiIiInRi5JJUJfX2VHRiVJJUNfX2VHRiVJImJHRiVJJHJob0dGJSIkKylGNCEiIg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
LUkoZGlzcGxheUc2IjYjLUkkc2VxRyUqcHJvdGVjdGVkRzYkJiZJJk9WR3ExR0YkNiMiIiI2I0kiaUdGJC9GMDtGLiwmSSNtTkdGJEYuIiInRi4=%;JSFHJSFHLUkoZGlzcGxheUc2IjYjLUkkc2VxRyUqcHJvdGVjdGVkRzYkJiZJJk9WR3ExR0YkNiMiIiM2I0kiaUdGJC9GMDsiIiIsJkkjbU5HRiRGMyIiJ0Yz%;From the figure above you can see that there are higher flutter speeds in higher modes. From here we can assume that if, theoretically, we put the plane in the speed range that is higher then the flutter velocity, but smaller than the second flutter velocity, the plain will be stable. You can also see that the modes that gives us the flutter velocity are the heave modes (3,5,7,...)From the figure above we can see the same Course we saw in the figure on the left.
From V-g figure we can see that the aircraft is not stable when the fictional damping is positive.JSFH
<Text-field style="Heading 1" layout="Heading 1">Question 2</Text-field>Repeat question 1 but now the wing is the wing with the weights at its tip (the wing from home assignment 3.
<Text-field style="Heading 2" layout="Heading 2">Solutions</Text-field>JSFHThe rod influence Only on the last element.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=The changes are in the Inertia moment and in the element mass as shown below:Qyg+SSlKX194LG5ld0c2IiwoSSVKX194R0YlIiIiKiZJJ21fX3JvZEdGJUYoSSdsX19yb2RHRiUiIiMjRigiIzcqJkklbV9fYkdGJUYoLUkiXkclKnByb3RlY3RlZEc2JCwkRisjRihGLEYsRihGLCEiIj5JKUpfX3osbmV3R0YlLCpJJUpfX3pHRiVGKEYpRi0qJkYwRigtRjI2JCwkSSVsX19lR0YlRjZGLEYoRiwqJkYqRihGPUYoRihGNz5JJ21fX25ld0dGJSwoKiZJKG1fX3dpbmdHRiVGKEkiTkdGJUY3RixGKkYoRjBGLEY3print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFHNow we need to switch the last element by the new inertia matrix. We do this in procedure because in question 3 we need to repeat this process twice. #Rod Addition Proc
#This procedure receive the inertia matrix, the new inertia moments, the new mass, the number of elements and...
#...also d and l_e. The procedure Return the new inertia matrix of the whole wing at its new configuration.
RodAddition := proc (T_I::Matrix, J_z_new, J_x_new, m_new, N, d, l_e)
local j, n, T_Imass, M_eMass;
M_eMass := (1/2)*m_new*Matrix(1 .. 6, 1 .. 6,
{(1, 1) = 1, (1, 2) = (1/2)*l_e, (1, 3) = -d, (1, 4) = 0, (1, 5) = 0, (1, 6) = 0,
(2, 1) = (1/2)*l_e, (2, 2) = J_z_new/m_new+(1/4)*l_e^2, (2, 3) = -(1/2)*d*l_e, (2, 4) = 0, (2, 5) = 0, (2, 6) = 0,
(3, 1) = -d, (3, 2) = -(1/2)*d*l_e, (3, 3) = J_x_new/m_new+d^2, (3, 4) = 0, (3, 5) = 0, (3, 6) = 0,
(4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = 1, (4, 5) = -(1/2)*l_e, (4, 6) = -d,
(5, 1) = 0, (5, 2) = 0, (5, 3) = 0, (5, 4) = -(1/2)*l_e, (5, 5) = J_z_new/m_new+(1/4)*l_e^2, (5, 6) = (1/2)*d*l_e,
(6, 1) = 0, (6, 2) = 0, (6, 3) = 0, (6, 4) = -d, (6, 5) = (1/2)*d*l_e, (6, 6) = J_x_new/m_new+d^2},
datatype = anything, storage = rectangular, order = Fortran_order, subtype = Matrix);
T_Imass := T_I;
for j from 3*N-2 to 3*N do
for n from 3*N-2 to 3*N do
T_Imass[j, n] := M_eMass[j-3*N+6, n-3*N+6]:
end do:
end do:
return T_Imass:
end proc:QyQ+SShUX19JbmV3RzYiLUksUm9kQWRkaXRpb25HRiU2KUkmVF9fSU1HRiVJKUpfX3osbmV3R0YlSSlKX194LG5ld0dGJUknbV9fbmV3R0YlSSJOR0YlSSJkR0YlSSVsX19lR0YlISIiJSFHprint(); # input placeholderJSFHprint(); # input placeholderQyQ+SSZPVkdxMkc2Ii1JJWRpYWdHRiU2LEkiTkdGJUkmVF9fU01HRiVJKFRfX0luZXdHRiVJI21OR0YlSSVCX19lR0YlSSVDX19lR0YlSSJiR0YlSSRyaG9HRiUiJCsjIiQ/IiEiIg==JSFH%;
LUkoZGlzcGxheUc2IjYjLUkkc2VxRyUqcHJvdGVjdGVkRzYkJiZJJk9WR3EyR0YkNiMiIiI2I0kiaUdGJC9GMDtGLkkjbU5HRiQ=%;JSFHLUkoZGlzcGxheUc2IjYjLUkkc2VxRyUqcHJvdGVjdGVkRzYkJiZJJk9WR3EyR0YkNiMiIiM2I0kiaUdGJC9GMDsiIiJJI21OR0Yk%;%;From the figure above we can see that the flutter velocity for the wing under the given loads is smaller, 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, and the divergence velocity is larger that the wing without loads 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. We can see that the mode that gives us the flutter velocity is the first torque mode. The flutter mode has small damping values 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, this is typical for external loads. We can see that in this wing configuration, the structure self frequencies are smaller than we saw for wing without loads. We can see that now the torque modes slope is positive, where for the wing without loads the slope is negative. The reason for this phenomenon is that the external weights increase the wing torque.
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=When we connect the wing external loads (by rigid connection), the wing self frequencies decrease. Adding more external loads will increase the self frequencies in the low frequency range.The flutter mode is now torque mode. We can assume that adding external load by rigid connection influence on the wing torque more than on the wing heave. JSFHJSFH
<Text-field style="Heading 1" layout="Heading 1">Question 3</Text-field>Repeat question 2 but now, instead of having two weights, examine the influence of fore weight only and the influence of aft weight only.
<Text-field style="Heading 2" layout="Heading 2">Solution</Text-field>JSFH
<Text-field style="Heading 3" layout="Heading 3">Fore mass only</Text-field>Qyo+SSpKX194LGZvcmVHNiIsKEklSl9feEdGJSIiIiomLUkiKkclKnByb3RlY3RlZEc2JCNGKCIjNywkSSdtX19yb2RHRiUjRigiIiNGKC1JIl5HRiw2JCwkSSdsX19yb2RHRiVGMkYzRihGKComSSVtX19iR0YlRihGNEYoRighIiI+SSpKX196LGZvcmVHRiUsKkklSl9fekdGJUYoRilGKComRjpGKC1GNTYkLCRJJWxfX2VHRiVGMkYzRihGKComRjFGKEZBRihGMkY7PkkobV9fZm9yZUdGJSwoKiZJKG1fX3dpbmdHRiVGKEkiTkdGJUY7RjNGMUYyRjpGKEY7PkkoZF9fZm9yZUdGJSwkKiYsJiomRjpGKEY4RihGMi1GKzYkRjAsJEY4I0YoIiIlRihGKCwmRjpGKEYxRjJGO0Y7Rjs=print(); # input placeholderQyQ+SSlUX19JZm9yZUc2Ii1JLFJvZEFkZGl0aW9uR0YlNilJJlRfX0lNR0YlSSpKX196LGZvcmVHRiVJKkpfX3gsZm9yZUdGJUkobV9fZm9yZUdGJUkiTkdGJUkoZF9fZm9yZUdGJUklbF9fZUdGJSEiIg==QyQ+SSpPVkdxM2ZvcmVHNiItSSVkaWFnR0YlNixJIk5HRiVJJlRfX1NNR0YlSSlUX19JZm9yZUdGJUkjbU5HRiVJJUJfX2VHRiVJJUNfX2VHRiVJImJHRiVJJHJob0dGJSIkKyMiJD8iISIi%;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=%;
LUkoZGlzcGxheUc2IjYjLUkkc2VxRyUqcHJvdGVjdGVkRzYkJiZJKk9WR3EzZm9yZUdGJDYjIiIiNiNJImlHRiQvRjA7Ri5JI21OR0Yk%;JSFHLUkoZGlzcGxheUc2IjYjLUkkc2VxRyUqcHJvdGVjdGVkRzYkJiZJKk9WR3EzZm9yZUdGJDYjIiIjNiNJImlHRiQvRjA7IiIiSSNtTkdGJA==%;%;From the figure above we can learn that fore weight gives as larger flutter velocity 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 . We got that for this configuration, the flutter velocity happens in the second heave mode. The divergence velocity is 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.We can see that the damping values for this flutter mode are about LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSomdW1pbnVzMDtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GUy1JI21uR0YkNiRRJTAuMjdGJ0Y5Rjk=. we can assume that this configuration increase the damping of the wing. As we can see from the figure above, the values of the structure self frequencies decrease when comparing to the previous configurations we checked.We can see that the slope of the torque mode is steeper than those in question 2.
LUkoZGlzcGxheUc2IjYjLUkkc2VxRyUqcHJvdGVjdGVkRzYkJiZJKU9WR3EzYWZ0R0YkNiMiIiI2I0kiaUdGJC9GMDtGLkkjbU5HRiQ=%;JSFHLUkoZGlzcGxheUc2IjYjLUkkc2VxRyUqcHJvdGVjdGVkRzYkJiZJKU9WR3EzYWZ0R0YkNiMiIiM2I0kiaUdGJC9GMDsiIiJJI21OR0Yk%;%;From the figure above we can learn that aft weight gives as smaller flutter velocity 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 . The flutter occurs at the first torque mode. The divergence velocity is 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.We can see that the damping values for this flutter mode are about LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSomdW1pbnVzMDtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GUy1JI21uR0YkNiRRJTAuMDdGJ0Y5Rjk=. we can assume that this configuration increase the damping of the wing. Similar to question 2, adding the weight changes the slope of the torque modes.
From the figures of both configurations, we can see that the structure first self frequency does not changes. This is because at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=, there are no aerodynamic forces. The self frequencies depend only be the wing stiffness matrix that does not changes.Adding fore weight moves the elastic axis of the torque mode forward and causes the torque frequency to increase with the velocity increase. Because the frequencies are higher, the meet of two mode delays and that is why we get large flutter velocity. On the contrary, adding aft weight, decrease the torque frequencies and therefore, the flutter occurs in smaller velocity.From this question we can assume that the flutter velocity can be increased by adding fore weight. Note that in rigid connection of the weight, the weight will have only heave. If we want to increase the flutter velocity more, we can connect the weight by more elastic connector. From all the figures above we can see that the divergence velocity is the (approximately) same for every configuration (except for the first question, I think we have there error in our model).Divergence occurs when the moments causing by aerodynamic forces are higher than the moments causes by the structure stiffness. This causes structural failure. According to the physical model, the divergence velocity suppose to stay the same between the all configuration because it's related to the wing stiffness matrix that does not changes. JSFH