Introduction to Engineering Analysis Assignment 5

ENSC 180: Introduction to Engineering Analysis
Assignment 5

Note: MATLAB codes should include definition of all variables; headings to identify
the program structure plan; and appropriate captions and labels for tables and
figures. Submit a .pdf report documenting your inputs and outputs in addition to a
separate zip-file containing all M-files. Marks will be deducted for solutions that are
unrealistic/impractical (as future engineers student should learn to think practically)
and poorly documented.
1. Write a MATLAB code to find the eigenvalues and eigenvectors of a 3x3 real
symmetric matrix. The program should ask a user to enter the 3x3 matrix and
output each eigenvalue and the corresponding eigenvector. You are not allowed to
use any MATLAB built-in functions except roots. Using [A]= [4 3 1;3 7 -1; 1 -1 9]
check the accuracy of your code by comparing with the eigenvalues and
eigenvectors obtained from the MATLAB eig command. (25 marks)
2.
(a) An eigenvalue problem is defined by the matrix equation, [K]-ꞷ
2
[M]= {0}, where
K is a real symmetric square matrix and [M] is a diagonal square matrix (same
order as [A]) with real positive elements. Given [A]= [4 3 1; 3 7 -1;1 -1 9] and
[M]= [1 0 0;0 2 0;0 0 3], find the eigenvalues ꞷ and the corresponding
eigenvectors using MATLAB. (10 marks).
(b) magic is a MATLAB command that can be used to create a special square matrix.
It has the property that the sums of the columns are the same, as are the sums of
the rows. Create a 6x6 magic matrix and using MATLAB commands find the sum
of each column, sum of each row and the sum of diagonals. Are these equal? (10
marks)
(c) Create a 4x4 magic matrix A. Use MATLAB to assemble a new matrix, B
=[
𝐴 2𝐴
𝐴
2 𝐴 + 2
] and check whether the sum of all rows, columns and diagonals of B
is the same by using MATLAB. (10 marks)
3. At time t=0, the engine of a rocket shuts down and it has reached a height of 500m
and rising at a velocity 125 m/s. Thereafter, the rocket is moving under the action
of gravity and its height, h, is expressed as a function of t.
h(t)=-4.9t2+125t+500
Using MATLAB symbolic calculations, find the velocity and acceleration of the
rocket as a function of time. Plot the height, velocity and acceleration in a single
figure using different colours and find the time to hit the ground using MATLAB
built-in functions. (15 marks).
4. A Fibonacci sequence is composed of elements created by adding the two previous
elements. The simplest Fibonacci sequence is 1,1,2,3,5,8,13,21………. Fibonacci
sequences have many examples in nature (e.g., the shell of the chambered nautilus
grows in accordance with a Fibonacci sequence). Write a MATLAB code to generate
a Fibonacci sequence and prompt a user to enter the first two numbers and the total
number of elements in the sequence. Your code should be able to plot the numbers in
a polar plot using the element number as the angle and the value of the element as the
radius. Show 3 sample polar plots generated by Fibonacci sequences. Check the ratio
of two adjacent numbers of a Fibonacci sequence. Do you see any trend as the
sequence grows? (30 marks).