Objectives

• Use Matlab to solve a system of equations.
• Write a function containing MATLAB code to implement part of MP1
• Use the 'diag' and 'length' and 'cumsum' Matlab functions.

References

• Lecture 4 on MATLAB  matrices.
• MATLAB Help: Using MATLAB  Programming and Data Types  M-File Programming
• Matlab by Pratap Chapters 3.1, 3.2, 5.1

 

Instructions

• In this lab, you will work individually. You can ask your group members for help but each person must submit their own answer sheet and write the find_xy.m function.
  
• Read the instructions for each part before filling out the answer sheet.

Part 1: Programming the find_xy function.

In this lab we will be writing MATLAB code to implement the central part of the solution to the suspension bridge problem from MP1. Take time now to read through the first three sections of Part 1 (Section 3. Math shows the equation A*h =  W) of MP1. The first goal of this lab is to get MATLAB to solve a system of equations represented by a matrix equation similar to the one in MP1. To solve the system, we need to solve the matrix equation:
A*h = W

where h is an unknown N x 1 column vector, A is an N x N matrix and W is an N x 1 column vector.

Your code should perform the calculations necessary to solve the matrix equation above to obtain the value of the vector h, which represents the values of the heights of the joint points. The data characterizing the system are provided in the file mp1_data.m. You can download this file here. (Right click and save it to your home directory.)

The file contains the following variables that describe the known quantities of the system.

• N : The number of support cables.
• L : Length of the main cable.
• HL : the height of the left tower. (this is equal to h₀ in the set of equations below)
• HR : the height of the right tower. (this is equal to h_N+1 in the set of equation below)
• d : A row vector of length N+1 containing the horizontal distances between the left tower and the 1^st joint point, between each pair of consecutive joint points and between the right-most(N^th) joint point and the right tower.
•   W : A column vector of length N containing the weights. We will need to modify this vector (just for this function).
  

      K_i+1*(h_i+1 - h_i)+ K_i*(h_i-1 - h_i) = W_i for i=1 .. N
 

These equations define a tri-diagonal system of linear equations for the heights h₁ to h_N corresponding to the weights W₁ to W_N.

For this lab, you will write a function containing MATLAB code called 'find_xy.m' that you can execute from within MATLAB by typing 'find_xy(HF)' at the MATLAB prompt. Of course you must assign a value for HF before you can use 'find_xy(HF)'.

1. Open Matlab and edit a new file called 'find_xy.m' . Add the following lines to your find_xy.m file:

 

function[x,y] = find_xy(HF)
% function[x,y] = find_xy(HF)
% Begin the file by writing a few lines of
% comments (lines that start with '%').
% Your comments should include your names, the date,
% and a brief  description of the program you are about to write.
mp1_data     % mp1_data.m is a script file.
% You can run a script file from inside a function.
% Running a script from inside the function
% find_xy make the values of the
% variables in mp1_data available to the function find_xy.

% Begin typing your  code after this comment.

 

For the next two steps you will go over the math behind the problem for MP1. Don't do any coding for your Matlab function until step 3.



2. Using the equation given above (the one relating h and K variables), write out five equations for i = 1.. 5 (N = 5) on the answer sheet for question #1.
   
   #1. Write down the five equations.
   
   
   We will write the code that works for any value of N but we want to understand how we generate the matrix equation.
   HINTS: h₀ = HL (the left tower), h₆ = HR (the right tower). These aren't part of the vector h that you're solving for in  A*h = W; vector h has N joint points where the weights are attached. HL and HR are given to you as data characterizing the system, so they can be treated as constants. All constants should be moved to the right side of the equation.

 
 
3. Using the equations from the previous step , answer question #2 on the answer sheet by writing out the matrix equation A*h = W (don't use Matlab notation) on the answer sheet. This will make it a lot easier to understand the rest of the lab.  See MP1, at the end of Part 2 where it shows how you should write the matrix equation A*h= W . The example shown in MP1 is for any general N.
   To answer question #2 use N = 5.
   

From this point on, you will be writing MATLAB code to work with vectors and matrices. Remember that all the code you write should go into your find_xy.m file. Although the mp1_data.m datafile sets N=9 and we wrote the equations for N=5 above, the Matlab code you now write should work for any N.

a) Now you should start putting together the pieces of the matrices in the equation above. We'll start with the A matrix on the left-hand side of the equation mentioned above. We will do this by creating three matrices, A_main, A_upper and A_lower so that
                A = A_main + A_upper + A_lower ;


  Don't type the above in your Matlab function yet as we need to first create the three matrices  A_main, A_upper and A_lower.
  A_main is a diagonal matrix whose only non-zero values lie on the main diagonal and are   -(K_i + K_i+1)  i = 1,...N  .
 
Answer questions #3 and #4 on the answer sheet.

#3. Fill in the blank to create a row vector K from d and HF. See the MP 1 instructions, equation (4) in section 3. Math.

     K = _______________________________

#4. Fill in the blanks to create the matrix A_main. Assume that the variable N has already been created.
     A_main = diag( - (K(______________) + K(_____________) ) , ________ ) ;


A_upper is the upper diagonal matrix whose only non-zero values lie above the main diagonal and are  K_i  i = 2,...,N  .
 
Answer question #5 on the answer sheet.
 #5. Fill in the blanks to create the matrix A_upper. Assume that the variable N has already been created.
       A_upper = diag( K( ____________ ) , _________ );

A_lower is the lower diagonal matrix whose only non-zero values lie below the main diagonal and are  K_i  i = 2,...N  .
 
Answer question #6 on the answer sheet.

#6. Fill in the blanks to create the matrix A_lower. Assume that the variable N has already been created.
      A_lower = diag( K(_____________) , ________ ) ;
      Now add the line of Matlab code,

        A = A_main + A_upper + A_lower;


in you find_xy.m file.



b) We need to modify the right-hand vector W.  This vector is created using the weight vector W, the HF value and the values HL and HR, all of which are given to you.

The first element is W(1). The last element is W(N).

Answer question #7.
#7. Fill in the blanks to modify the first and the last elements of the vector W:

                W(1) = W( ______ ) - ______________ ;

c) Since we know the left-hand side matrix A and the right-hand side vector W, we can solve the system of equations A*h = W.
Store the solution in a column vector called h. The vector h shall be amended by first making it a row vector (by transposing it) and then inserting the values HL to its left and HR to its right. Then we shall assign h to the output variable y, which is supposed to be a row vector of length N+2 holding the vertical coordinates of the 2 towers and the N joint points.


Answer questions #8,#9 and #10.
 
#8. Fill in the blank to solve the system of equations using the backslash (\) operator, and store this result in a vector h.

        h = ___________________________ ;
 
#9. Fill in the blank to change h from a column vector into a row vector.

         h = _____________________ ;

#10. Fill in the blank to prepend h with HL and append h with HR. (Hint: example, if h = [1 2 3 4 5]  and HL = 0 , HR = 6 then we need to create
        [ 0 1 2 3 4 5 6])

        h = __________________________________________ ;
 
        %   assign h to y , that's one of our output variables

        y = h;




d) Create a vector x to hold the corresponding x-coordinates of the towers and the joint points, so that you can plot all of these points. The distances are measured from the left tower. They are derived from the d vector that you are given (in mp1_data.m). Your first value in this x vector should be 0 (corresponding to the left tower), and the rest of them should cumulatively add up the distances between the left tower and the first joint point(weight), each pair of successive joint points(weights), and the last joint point(weight) and the right tower (think cumsum).


Answer question #11. We’re finished with the function at this point, so you can save it.
#11. Fill in the blank to create a row vector x of distances from the left tower. You should have N+2 (7 for our example) elements in this vector.     
        Assume the variable d (with N+1 elements) has already been created. (Hint: use 'cumsum' and prepend 0 )

           x = ___________________________________________ ;

Part 2 Testing the find_xy function

Congratulations, you are done!