Merge branch 'master' of https://github.uconn.edu/ltd13002/ME3255_Fin…

…al_Project

Part A/membrane_solution3.m

@@ -19,13 +19,18 @@ w = k\y;
 % Solves for displacement (micrometers)
 % Solution represents a 2D data set w(x,y)
 
-[x,y] = meshgrid(0:10/4:10,0:10/4:10);
-z = zeros(size(x));
-z(2:end-1,2:end-1) = reshape(w,[3 3]);
-surf(x,y,z)
-title('Membrane Displacement')
-zlabel('Displacement (micrometer)')
+
+    [x,y] = meshgrid(0:10/4:10,0:10/4:10);
+    z = zeros(size(x));
+    z(2:end-1,2:end-1) = reshape(w,[3 3]);
+    surf(x,y,z)
+    title('Membrane Displacement (3 x 3)')
+    zlabel('Z Position (micron)')
+    ylabel('Y Position (micron)')
+    xlabel('X Position (micron)')
+    % Membrane displacement is shown on chart
 
 % Membrane displacement is shown on chart
 
-end
+end
+

Part B/membrane_solution3.m

@@ -0,0 +1,25 @@
+function [w] = membrane_solution3(T,P)
+    % T = Tension (microNewton/micrometer)
+    % P = Pressure (MPa)
+
+    od = ones(8,1);
+    od(3:3:end) = 0;
+    k = -4*diag(ones(9,1))+diag(ones(9-3,1),3)+diag(ones(9-3,1),-3)+diag(od,1)+diag(od,-1);
+
+
+    y = -(10/4)^2*(P/T)*ones(9,1);
+    w = k\y; 
+    % Solves for displacement (micrometers)
+    % Solution represents a 2D data set w(x,y)
+
+    [x,y] = meshgrid(0:10/4:10,0:10/4:10);
+    z = zeros(size(x));
+    z(2:end-1,2:end-1) = reshape(w,[3 3]);
+    surf(x,y,z)
+    title('Membrane Displacement (3 x 3)')
+    zlabel('Z Position (micron)')
+    ylabel('Y Position (micron)')
+    xlabel('X Position (micron)')
+    % Membrane displacement is shown on chart
+
+end 

Part C/membrane_solution.m

@@ -22,7 +22,9 @@ w = k\y;
 z = zeros(size(x));
 z(2:end-1,2:end-1) = reshape(w,[n n]);
 surf(x,y,z)
-title('Membrane Displacement')
-zlabel('Displacement (micrometer)')
+title('Membrane Displacement (10 x 10)')
+zlabel('Z Position (micron)')
+ylabel('Y Position (micron)')
+xlabel('X Position (micron)')
 % Membrane displacement is shown on chart
-end
+end

Part D/membrane_solution.m

@@ -0,0 +1,25 @@
+function [w] = membrane_solution(T,P,n)
+    % T = Tension (microNewton/micrometer)
+    % P = Pressure (MPa)
+    % n = # of interior nodes
+
+    od = ones(n^2-1,1);
+    od(n:n:end) = 0;
+    k = -4*diag(ones(n^2,1))+diag(ones((n^2)-n,1),n)+diag(ones((n^2)-n,1),-n)+diag(od,1)+diag(od,-1);
+
+    y = -(10/(n+1))^2*(P/T)*ones(n^2,1);
+    w = k\y;
+    % Solves for displacement (micrometers)
+    % Output w is a vector
+    % Solution represents a 2D data set w(x,y)
+
+    [x,y] = meshgrid(0:10/(n+1):10,0:10/(n+1):10);
+    z = zeros(size(x));
+    z(2:end-1,2:end-1) = reshape(w,[n n]);
+    surf(x,y,z)
+     title('Membrane Displacement (10 x 10)')
+    zlabel('Z Position (micron)')
+    ylabel('Y Position (micron)')
+    xlabel('X Position (micron)')
+    % Membrane displacement is shown on chart
+end

README.md

@@ -3,31 +3,46 @@
 # Part A
 ```matlab
 function [w] = membrane_solution3(T,P)
-    % T = Tension (microNewton/micrometer)
-    % P = Pressure (MPa)
-    
-    od = ones(8,1);
-    od(3:3:end) = 0;
-    k = -4*diag(ones(9,1))+diag(ones(9-3,1),3)+diag(ones(9-3,1),-3)+diag(od,1)+diag(od,-1);
-   
-    
-    y = -(10/4)^2*(P/T)*ones(9,1);
-    w = k\y; 
-    % Solves for displacement (micrometers)
-    % Output w is a vector
-    % Solution represents a 2D data set w(x,y)
+% membrane_solution3: dispalacement of node for membrane with 3x3 interior
+% nodes
+% [w] = membrane_solution3(T,P)
+% input:
+% T = Tension (microNewton/micrometer)
+% P = Pressure (MPa)
+% output:
+% w = vector of displacement of interior nodes
+
+od = ones(8,1);
+od(3:3:end) = 0;
+k = -4*diag(ones(9,1))+diag(ones(9-3,1),3)+diag(ones(9-3,1),-3)+diag(od,1)+diag(od,-1);
+
+
+y = -(10/4)^2*(P/T)*ones(9,1);
+w = k\y;
+
+% Solves for displacement (micrometers)
+% Solution represents a 2D data set w(x,y)
+
 
     [x,y] = meshgrid(0:10/4:10,0:10/4:10);
     z = zeros(size(x));
     z(2:end-1,2:end-1) = reshape(w,[3 3]);
     surf(x,y,z)
-    title('Membrane Displacement')
-    zlabel('Displacement (micrometer)')
-    % Membrane displacement is shown on chart
-    
-end 
-```
+    title('Membrane Displacement (3 x 3)')
+    zlabel('Z Position (micron)')
+    ylabel('Y Position (micron)')
+    xlabel('X Position (micron)')
 
+% Membrane displacement is shown on chart
+
+end
+   
+```
+### Approach
+- A matrix is set up using finite element analysis of the interior nodes of a 5x5 matrix
+- A vector for the y direction is then set up for the membrane
+- Using linear algebra, a vector for the displacement of the nodes is created
+- The vector can then be transformed to represent the actual surface as a matrix
 
 # Part B
 ```matlab
@@ -40,9 +55,14 @@ end
 # Part C
 ```matlab
 function [w] = membrane_solution(T,P,n)
-    % T = Tension (microNewton/micrometer)
-    % P = Pressure (MPa)
-    % n = # of interior nodes
+% membrane_solution: dispalacement of node for membrane with nxn interior nodes
+% [w] = membrane_solution(T,P,n)
+% input:
+% T = Tension (microNewton/micrometer)
+% P = Pressure (MPa)
+% n = number of rows and columns of interior nodes
+% output:
+% w = vector of displacement of interior nodes
 
     od = ones(n^2-1,1);
     od(n:n:end) = 0;
@@ -58,12 +78,16 @@ function [w] = membrane_solution(T,P,n)
     z = zeros(size(x));
     z(2:end-1,2:end-1) = reshape(w,[n n]);
     surf(x,y,z)
-    title('Membrane Displacement')
-    zlabel('Displacement (micrometer)')
+    title('Membrane Displacement (10 x 10)')
+    zlabel('Z Position (micron)')
+    ylabel('Y Position (micron)')
+    xlabel('X Position (micron)')
     % Membrane displacement is shown on chart
 end
 ```
-
+### Approach
+- This problem uses the same steps as with the membrane_solution3 function, except it allows for the user to change the number of interior nodes
+- This required alterations of the all the indexing and sizing in the function along with a few calculations
 
 # Part D
 ```matlab
@@ -76,6 +100,18 @@ end
 # Part E
 ```matlab
 function [pw_se,w]=SE_diff(T,P,n)
+% SE_diff: calculates difference between strain energy and work done by pressure in
+% membrane
+% [pw_se,w]=SE_diff(T,P,n)
+% input:
+% T = Tension (microNewton/micrometer)
+% P = Pressure (MPa)
+% n = number of rows and columns of interior nodes
+% output:
+% pw_se = difference between strain energy and work done by pressure in
+% membrane
+% w = vector of displacement of interior nodes
+
 E = 1; %TPa Units may need to be changed
 v = .31; %Poissons ratio
 t = .3; %nm
@@ -102,6 +138,12 @@ end
 se = E*t*h^2/(2*(1-v^2))*sum(sum(0.25.*dwdx.^4+.25.*dwdy.^4+0.5.*(dwdx.*dwdy).^2));
 pw_se = pw-se;
 ```
+### Approach
+- Using the membrane_solution function, a vector of the displacements is formed
+- Next, the average displacement for each element is calculated. For each elements, the dispalcement at all four courners is taken and then averaged
+- Using these values, the work done by pressure can be calculated
+- For the change in dispalcement over the x and y coordinate system, the values of the change on the left and right (y-axis) or top and bottom (x-axis) are taken and averaged
+- Using these values, the strain energy can be calculated
 
 # Part F
 ```matlab
@@ -115,7 +157,17 @@ pw_se = pw-se;
  end
  ```
  ```matlab
- function [T,ea] = tension_sol(P,n)
+function [T,ea] = tension_sol(P,n)
+% tension_sol: outputs tension of a membrane given the pressure and number
+% of nodes
+% [T,ea] = tension_sol(P,n)
+% input:
+% P = Pressure (MPa)
+% n = number of rows and columns of interior nodes
+% output:
+% T = Tension (microNewton/micrometer)
+% ea = approximate relative error (%)
+ 
 y =@(T) SE_diff(T,P,n);
 [T,fx,ea,iter]=bisect(y,.01,1);
 ```
@@ -160,6 +212,13 @@ root = xr; fx = func(xr, varargin{:});
 ```
 ```matlab
 function re = Rel_error (T)
+ Rel_error: calculates relative error of a vector
+% re = Rel_error (T)
+% input:
+% T = vector of numbers
+% output:
+% re = relative error of vector
+
 re = zeros(1,length(T)-1);
 for i = 2:length(T)
     re(i-1)= abs(T(i)-T(i-1))/T(i-1);
@@ -174,7 +233,9 @@ end
 |35|0.0602|0.09%|
 |40|0.0603|0.06%|
 
-
+### Approach
+- This problem uses the bisect method for locating roots and the SE_diff function for calculating the difference in work and strain energy of the membrane
+- The script runs through all iterations of different amounts of nodes, zeroing the SE_diff function output and saving the values for tension in the T variable as a vector
 # Part G
 ```matlab
 P = linspace(.001,.01,10);
@@ -200,3 +261,9 @@ ylabel('Pressure (MPa)')
 print('Part g','-dpng')
 ```
 ![](https://github.uconn.edu/ltd13002/ME3255_Final_Project/blob/master/Part%20G/Part%20g.png)
+
+### Approach
+- This script uses the tension_sol function as described in the part above, running though all iterations of pressure
+- Additionally, the max deflection for each pressure and tension is calculated at each iteration
+- From there, a general linear regression is calculated with the formula P(x) = A*w^3
+- The results are plotted