User:Uf.team.aero/HW3
Curved Panels
[edit | edit source]The expression for the finite force applied to a finite unit length of a curved panel is . However, to analyze the curved panel with respects to an x and y component, we can make use of the fact that to render the equation:
For and we can implement the polar coordinate transformations:
(remembering that we are in the yz plane)
and hence:
Integrate to find the total force
For our application we can solve for the magnitude of the force finally arriving to the equation:
Closed Thin Walled Cross Sections
[edit | edit source]We can relate our resultant force, , with the torque, by:
by letting be the perpendicular distance from the point of reference to the corresponding finite force, and by remembering that , we can conclude:
Torsion of Uniform, circular Bars
[edit | edit source]It is first important to define the rate of twist as:
AND
Also, the second polar area moment of Inertia is defined as:
To start, our equation for torque is:
By substituting in that and we get:
Torsion of Uniform, Non-circular Bars
[edit | edit source]Torsion of uniform, non-circular bars typically leads to an occurrence of warping of the cross section. Warping is when a point in the cross section under torsion, subject to deformation, will undergo a axial displacement along the length of bar. A simple example of warping is rolling up a sheet of paper then applying a torque - the rolled sheet will noticeably elongate along its axis.
Figure 4 below shows a typical depiction of displacement in a cross section view of a non-circular bar in torsion.
As can be seen above, the displacement vector, is perpindicular to the original position vector, . This method of describing the displacement vector is chosen because with the assumption that is small the y and z components of the displacement vector can easily be calculated. Figure 4a below shows the y and z components of displacement for the case .
By applying basic trigonometry to the figure above, and can be solved for in terms of :
If the assumption that is made, then and . These equations then become:
For the general case as shown in figure 4, R becomes the initial distance OP and the y-component of the displacement of point P can be expressed as
Similarly the z-component of the displacement of point P is
Recalling the definition of
Generalizing for all points:
- (Eq. 1)
- (Eq. 2)
For warping displacement in the axial direction (i.e. x-direction) it can be shown that
- (Eq. 3)
Road Map for Analysis of Wing Torsion of a Multicell Section
[edit | edit source]- Kinematic Assumptions (Equations (1), (2) and (3) above) [Sun, Sec. 3.2]
- Strain Displacement Relationship [Sun, Sec. 3.2]
- Equilibrium Equation for Stresses [Sun, Ch. 2 , Sec. 3.2]
- Prandtl Stress Function [Sun, Sec. 3.2, Eqn. 3.15]
- Strain Compatibility Equation [Sun, Sec. 3.2, Eqn. 3.17]
- Equation for [Sun, Sec. 3.2, Eqn. 3.19]
- Boundary Conditions for [Sun, Sec. 3.2, Eqn. 3.24]
- Calculation of Torque
- [Sun, Sec. 3.2, Eqn. 3.25]
- 9. Formal Derivation of Torque and Shear Flow Relationship
- [Sun, Sec. 3.5, Eqn. 3.48]
- 10. Calculation of Twist Angle
- [Sun, Sec. 3.5, Eqn. 3.56]
- 11. Solution of Multicell Thin-walled Sections [Sun, Sec. 3.6]
Use of "RoadMap" to Solve a Single Cell Section Airfoil
[edit | edit source]We can simplify the analysis of a complex, multi-cell airfoil by looking at a semi-circle attached to a triangle.
Average Area ()
[edit | edit source]To find we simply look at the area of each shape sepratly.
Shear flow ()
[edit | edit source]Remembering our derived equation for Torque, , we can solve for the shear flow:
Twist angle ()
[edit | edit source]Our twist angle equation is:
However when looking at each segment of the wall, we have 3 separate sections with constant thickness, so the integral portion becomes a sum:
Proof of Triangle Area Formula
[edit | edit source]The area of a triangle is given as:
To prove this equation, the area of triangle CDE is subtracted from the area of triangle BDE. Notice that area of a right triangle is exactly half the area of a square whose sides are equal to the legs of the triangle. Thus:
and
Subtracting the areas to solve for the triangle of interest:
Second Polar Area Moment of Inertia for a Circle
[edit | edit source]The definition of the Second Polar Area Moment of Inertia is given as:
For a circle of radius a, the second polar moment of inertia can be solved via a double integral as
Second Polar Moment of Inertia of a Thin-walled Cylinder
[edit | edit source]For a thin-walled cylinder of inner radius and outer radius the average radius can be given as:
The square of the average radius is:
If the assumption is made that the wall is thin such that thickness, , is much less than a or b, then
Then,the second polar area moment of inertia for the thin-walled section can be given as
Expanding the last term provides
Recalling the definitions for , and from above
Comparison of a solid, circular cross-section to a hollow, thin-walled, circular cross-section
[edit | edit source]Given
[edit | edit source]A solid, circular cross-section has radius . A hollow, thin-walled, circular cross-section has inner radius and thickness .
Find
[edit | edit source]a) compute and
b) compute and
c) compare
d) find radius of thin-walled section with thickness such that and compare
Solution
[edit | edit source]a)
[edit | edit source]b)
[edit | edit source]c)
[edit | edit source]d)
[edit | edit source]To determine the proper , set the second polar area moment of inertia equations for a thin-walled section and a solid section equal to one another and solve for the unkonwn:
thus,
To compare areas, calculate the area and compare to determined above.
Comparing the two areas,
Analysis of Multi-Cell Sections
[edit | edit source]It is important to understand how to analyze a multi-cell section because of their widespread use in common structures such as wings. We can apply what we already know about a single cells to our analysis of multi-cells by simply adding or integrating all of the single cell sections.
If we look at the torque generated by the single cell, we get the formula:
To apply this equation to a multi-cell structure, we simply add all of the cells. Given that is the number of cells:
As for the rate of twist, , since the structure is a rigid body, such that the rate of twist attributed to the cell is:
NACA Airfoil Matlab Project
[edit | edit source]The purpose of this project is to form the basis for future structural analysis of thin walled bodies, with emphasis on airfoils. The code should be able to plot the body, determine the average area (Ā), and find the location of the centroid. These values are found through splitting the perimeter of the bodies into small discrete segments and then using properties of the cross product to determine the area from a point in space and a quadrature to determine the centroid.
Contribution from Team Aero |
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A code was first developed to determine the average area and centroid location of a circle and NACA 4-digit airfoil based upon user inputs. The Matlab code is posted below and Figures 8 and 9 show examples for a circle of radius 1 m and a NACA 2415 airfoil. % Matlab Script
%*********************************************************************
% Filename: Airfoil.m
%
% PURPOSE:
% Calculates the average area and centroid
% of a NACA airfoil or circle based on user inputs.
% Plots the shape and centroid location.
%
% AUTHOR:
% Team Aero
% EAS 4200C Aerospace Structures, Fall 2008.
%
% Modified on:
% Created on : 1 October 2008
%
% DEPENDENCIES:
% call:
%
% REMARKS:
%
%*********************************************************************
clear % Clear system memory
% Determine what shape to analyze
button = questdlg('Pick which shape you wish to analyze','Shape Selection','NACA Airfoil','Circle','NACA Airfoil');
if strcmp(button, 'NACA Airfoil')
% Input for variables from user
prompt={
' NACA Digit 1 (max camber in percent of chord)' ...
' NACA Digit 2 (position of max camber in tenths of chord)' ...
' NACA Digits 3-4 (maximum thicknes as percentage of chord)' ...
' Chord length in meters'...
' Y coordinate to calculate area around'...
' Z coordinate to calculate area around'...
' Number of segments to use'};
def={'2','4','15','1','0','0','40'}; % define default values of prompt parameters
text_title='Please Specify Parameters'; lineNo=1;
answer=inputdlg(prompt,text_title,lineNo,def); %store user input into vector
m = str2num(answer{1})/100; % max camber in percent of chord
p = str2num(answer{2})/10; % position of max camber in tenths of chord
t = str2num(answer{3})/100; % maximum thicknes as percentage of chord
cl = str2num(answer{4}); % chord length
xa = str2num(answer{5}); % Y coordinate to calculate abar around
ya = str2num(answer{6}); % Z coordinate to calculate abar around
ns = str2num(answer{7}); % number of segments taken
x1=0:1/ns:1; % mean camber line y-coords
Yc(1)=0; % set initial point
% Determining Points of Airfoil
for i=1:round(ns*p)
Yc(i+1)=m/(p^2)*(2*p*(i/ns)-(i/ns)^2); % mean camber line from y=0 to y=p
theta(i+1)=atan(m/(p^2)*(2*p-2*(i/ns)));
end
for j=round(ns*p)+1:ns
Yc(j+1)=m/((1-p)^2)*((1-2*p)+2*p*(j/ns)-(j/ns)^2); % mean camber line from y=p to y=c
theta(j+1)=atan(m/((1-p)^2)*(2*p-2*(j/ns)));
end
for i=0:ns
Yt(i+1)=t/.2*(.2969*sqrt(i/ns)-.126*(i/ns)-.3516*(i/ns)^2+.2843*(i/ns)^3-.1015*(i/ns)^4); %Airfoil thickness
xu(i+1)=i/ns-Yt(i+1)*sin(theta(i+1));
yu(i+1)=Yc(i+1)+Yt(i+1)*cos(theta(i+1));
xl(i+1)=i/ns+Yt(i+1)*sin(theta(i+1));
yl(i+1)=Yc(i+1)-Yt(i+1)*cos(theta(i+1));
end
% Close up trailing edge
yu(ns+1)=0;
yl(ns+1)=0;
xu(ns+1)=1;
xl(ns+1)=1;
% Scale to chord line
xu = cl*xu;
xl = cl*xl;
yu = cl*yu;
yl = cl*yl;
x1 = cl*x1;
Yc = cl*Yc;
%Average Area Calculation
for i=1:ns
ai=cross([xu(i)-xa yu(i)-ya 0],[xu(i+1)-xu(i) yu(i+1)-yu(i) 0]);
dAu(i)=ai(3)/2;
end
for i=ns+1:-1:2
ai=cross([xl(i)-xa yl(i)-ya 0],[xl(i-1)-xl(i) yl(i-1)-yl(i) 0]);
dAl(i)=ai(3)/2;
end
a_bar=abs(sum(dAu)+sum(dAl));
% Centroid Calculation
for i=1:ns
xbaru(i)=(xu(i+1)-xu(i))*(2*yu(i+1)+yu(i))/(3*(yu(i)+yu(i+1)))+xu(i);
ybaru(i)=(yu(i)^2+yu(i+1)^2+yu(i)*yu(i+1))/(3*(yu(i)+yu(i+1)));
areau(i)=(xu(i+1)-xu(i))*(yu(i)+yu(i+1))/2;
xbarl(i)=(xl(i+1)-xl(i))*(2*yl(i+1)+yl(i))/(3*(yl(i)+yl(i+1)))+xl(i);
ybarl(i)=(yl(i)^2+yl(i+1)^2+yl(i)*yl(i+1))/(3*(yl(i)+yl(i+1)));
areal(i)=-(xl(i+1)-xl(i))*(yl(i)+yl(i+1))/2;
end
abar=sum(areau)+sum(areal);
xareau=areau.*xbaru;
yareau=areau.*ybaru;
xareal=areal.*xbarl;
yareal=areal.*ybarl;
Centroid = [(sum(xareau)+sum(xareal))/abar, (sum(yareau)+sum(yareal))/abar];
% Plot
plot(xu,yu,'k',xl,yl,'k',x1,Yc,'b',Centroid(1), Centroid(2),'+r')
xlabel('Y (m)')
ylabel('Z (m)')
title(['NACA ' num2str(m*100) num2str(p*10) num2str(t*100) ' Airfoil'])
text(.45*cl,.25*cl,['Average area = ' num2str(a_bar) ' m^2'])
text(.45*cl,.2*cl,['Centroid = (' num2str(Centroid(1)) ' m, ' num2str(Centroid(2)) ' m)'])
axis([0 cl -.3*cl .3*cl]);
else
% Input for variables from user
prompt={
' Radius in meters' ...
' Y coordinate to calculate area around'...
' Z coordinate to calculate area around'...
' Number of segments to use'};
def={'1','0','0','40'};
text_title='Please Specify Parameters'; lineNo=1;
answer=inputdlg(prompt,text_title,lineNo,def);
r = str2num(answer{1}); % radius
xa = str2num(answer{2}); % Y coordinate to calculate abar around
ya = str2num(answer{3}); % Z coordinate to calculate abar around
ns = str2num(answer{4}); % number of segments taken
x1=-r:2*r/ns:r; % y-coords of shape
% Determines points of circle
for i=1:ns+1
yu(i)=sqrt(r^2-x1(i)^2);
yl(i)=-sqrt(r^2-x1(i)^2);
end
% Average area calculation
for i=1:ns
ai=cross([x1(i)-xa yu(i)-ya 0],[x1(i+1)-x1(i) yu(i+1)-yu(i) 0]);
dAu(i)=ai(3)/2;
end
for i=ns+1:-1:2
ai=cross([x1(i)-xa yl(i)-ya 0],[x1(i-1)-x1(i) yl(i-1)-yl(i) 0]);
dAl(i)=ai(3)/2;
end
abar=abs(sum(dAu)+sum(dAl));
% Plot
plot(x1,yu,'k',x1,yl,'k',0, 0,'+r')
xlabel('Y (m)')
ylabel('Z (m)')
title(['Circle with Radius = ' num2str(r) ' m'])
text(.3*r,r,['Average area = ' num2str(abar) ' m^2'])
axis([-1.1*r 1.1*r -1.1*r 1.1*r]);
end
Code was also made to determine the maximum number of segments needed in the calculation of the area to be within 1% accuracy of the true value for a circle and 2 successive values to be within 1% of each other for the NACA airfoil. Figures 10 and 11 shows graphs relating the effects of the number of segments on the average area. Circle code: % Matlab Script
%*********************************************************************
% Filename: Circle_ns_max.m
%
% PURPOSE:
% Calculates the maximum number of segments needed to calculate the
% average area of a circle in order to be within of 1% of the actual value.
% Plots actual versus calculated area for different segment numbers.
%
% AUTHOR:
% Team Aero
% EAS 4200C Aerospace Structures, Fall 2008.
%
% Modified on: 6 October 2008
% Created on : 1 October 2008
%
% DEPENDENCIES:
% call:
%
% REMARKS:
%
%*********************************************************************
clear % Clear memory
% Initial variable values for startup
r = 1; % radius
ns = 1; % number of segments taken
abar = 0; % area initial value
j=1; % series initial value
xa=0; % X coordinate to calculate abar around
ya=0; % Y coordinate to calculate abar around
% Iteration to find max ns
while (abar < 0.99*pi*r^2)
x1=-r:2*r/ns:r; % x-coords of shape
for i=1:ns+1
yu(i)=sqrt(r^2-x1(i)^2);
yl(i)=-sqrt(r^2-x1(i)^2);
end
x(j)=j;
for i=1:ns
ai=cross([x1(i)-xa yu(i)-ya 0],[x1(i+1)-x1(i) yu(i+1)-yu(i) 0]);
dAu(i)=ai(3)/2;
end
for i=ns+1:-1:2
ai=cross([x1(i)-xa yl(i)-ya 0],[x1(i-1)-x1(i) yl(i-1)-yl(i) 0]);
dAl(i)=ai(3)/2;
end
abar(j)=abs(sum(dAu)+sum(dAl));
ns=ns+1;
j=j+1;
end
% Variables for actual area range
x2=0:1:ns;
area=ones(ns+1);
area=area.*0.99*pi*r^2;
area2=ones(ns+1);
area2=area2.*1.01*pi*r^2;
% Plot
plot(x,abar,'k',x2,area,'b',x2,area2,'b',j-1, abar(j-1),'+r')
xlabel('Number of Segments')
ylabel('Computed Area (m^2)')
text(10,1,'Max number of segments = 23')
title('\it{Computed versus Actual Area of Circle of Radius 1 m}')
NACA Airfoil Code: % Matlab Script
%*********************************************************************
% Filename: NACA_ns_max.m
%
% PURPOSE:
% Calculates the maximum number of segments needed to calculate the average
% area of a NACA airfoil in order to be within of 1% of the actual value.
% Plots actual versus calculated area for different segment numbers.
%
% AUTHOR:
% Team Aero
% EAS 4200C Aerospace Structures, Fall 2008.
%
% Modified on: 6 October 2008
% Created on : 1 October 2008
%
% DEPENDENCIES:
% call:
%
% REMARKS:
%
%*********************************************************************
clear % Clear memory
m = .02; % max camber in percent of chord
p = .4; % position of max camber in tenths of chord
t = .15; % maximum thicknes as percentage of chord
cl = 1; % chord length
ns = 2; % number of segments taken
j=2; % series initial value
xa=0; % X coordinate to calculate abar around
ya=0; % Y coordinate to calculate abar around
x1=0:1/ns:1; % mean camber line x-coords
Yc(1)=0; % set initial point
abar(1)=0; % intial area
abar(2)=.000001; % intial area
% Iteration to find max ns
while (abar(j)*.99 > abar(j-1))
% Determining Points of Airfoil
for i=1:round(ns*p)
Yc(i+1)=m/(p^2)*(2*p*(i/ns)-(i/ns)^2); % mean camber line from x=0 to x=p
theta(i+1)=atan(m/(p^2)*(2*p-2*(i/ns)));
end
for j=round(ns*p)+1:ns
Yc(j+1)=m/((1-p)^2)*((1-2*p)+2*p*(j/ns)-(j/ns)^2); % mean camber line from x=p to x=c
theta(j+1)=atan(m/((1-p)^2)*(2*p-2*(j/ns)));
end
for i=0:ns
Yt(i+1)=t/.2*(.2969*sqrt(i/ns)-.126*(i/ns)-.3516*(i/ns)^2+.2843*(i/ns)^3-.1015*(i/ns)^4); %Airfoil thickness
xu(i+1)=i/ns-Yt(i+1)*sin(theta(i+1));
yu(i+1)=Yc(i+1)+Yt(i+1)*cos(theta(i+1));
xl(i+1)=i/ns+Yt(i+1)*sin(theta(i+1));
yl(i+1)=Yc(i+1)-Yt(i+1)*cos(theta(i+1));
end
% Average Area Calculation
j=j+1;
for i=1:ns
ai=cross([xu(i)-xa yu(i)-ya 0],[xu(i+1)-xu(i) yu(i+1)-yu(i) 0]);
dAu(i)=ai(3)/2;
end
for i=ns+1:-1:2
ai=cross([xl(i)-xa yl(i)-ya 0],[xl(i-1)-xl(i) yl(i-1)-yl(i) 0]);
dAl(i)=ai(3)/2;
end
abar(j)=abs(sum(dAu)+sum(dAl));
ns=ns+1;
end
x=1:j; % Coord. for x-axis
% Plot
plot(x,abar,'k',j, abar(j),'+r')
xlabel('Number of Segments')
ylabel('Computed Average Area (m^2)')
text(5,.03,['Max number of segments = ' num2str(j)])
title('\it{Number of Segments Effect on Average Area of NACA 2415 Airfoil}')
axis([2 j+1 0 .1]);
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Contributing Members
[edit | edit source]The following Team Aero members contributed to this report.
Jared Lee --Eas4200c.f08.aero.lee 03:06, 1 October 2008 (UTC)
Ray Strods --Eas4200c.f08.aero.strods 03:53, 7 October 2008 (UTC)
Oliver Oyama --Eas4200c.fo8.aero.oyama 17:14, 7 October 2008 (UTC)
William Kurth --Eas4200c.f08.aero.kurth 11:53, 8 October 2008 (UTC)
Gonzalo Barcia --Eas4200c.f08.aero.barcia 12:30, 8 October 2008 (UTC)