SEMM3023 Given that the eigenvalue for matrix [A] in equation (2) is  = βˆ’cos⁑(𝛽𝑛): Applied Numerical Method Report, UTM, Malaysia

School

Universiti Teknologi Malaysia (UTM)

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Assignment Type

Individual Assignment

Subject

SEMM3023 Applied Numerical Method Report

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Date

27/11/2023

Your Task
1. Given that the eigenvalue for matrix [A] in equation (2) is  = βˆ’cos⁑(𝛽𝑛).
i) Derive the characteristic equation of matrix [A] in equation (2). Show all your
work when deriving the characteristic equation.
ii) Is it possible to solve for 𝛽𝑛 analytically? If yes, solve it. If no, suggest another method to use to solve for 𝛽𝑛.

2. Based on the characteristic equation derived from Question 1,
i) Suggest four methods you can use to solve for 𝛽𝑛 numerically. Among the four methods you have suggested, are there any methods that may be less favorable to use? If yes, explain why.
ii) Select a numerical method to solve for 𝛽𝑛 and explain in detail and in your own words, all the steps required to implement the chosen method for this problem. (Calculation is NOT required for this question).

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3. Using ONLY MATLAB or OCTAVE as your programming platform,
i) Explain the algorithm you will apply in your programming platform to implement the chosen numerical method from Question 2. (NO code is required for this question).
ii) Develop a flowchart based on your algorithm and explain the flowchart. (NO code required for this question. You may want to refer to your SEMM/SKMM 1013 notes for this question).
iii) Given that 𝐸 = 69β‘πΊπ‘ƒπ‘Ž, 𝑑 = 0.1β‘π‘π‘š, 𝜌 = 2700 π‘˜π‘”π‘šβˆ’3, 𝐿 = 20β‘π‘π‘š. Develop a
SINGLE script of code in your programming platform to solve for 𝛽𝑛 AND calculate the fundamental natural frequency of the beam,β‘πœ”1, in Hz to four decimal places. For your initial guess use any values between 0 and 2. For your stopping criteria, stop the iteration when πœ€ ≀ 0.01. Also, assume that 𝛽𝑛 = 𝛽1. (Please fully comment your code and display the output of your code in table form as shown in lecture. Paste your code inside the report. DO NOT screenshot your code for the report. Make sure that your code reflects the flowchart, algorithm and steps that you have given).

4. Based on your answer in Question 3,
i) Discuss on a method to verify your result in Question 3. Hence, verify your results by calculating the percentage error and explain the significance of the error. (Hint: Solutions from in-built functions in programming software can sometimes be considered as an β€˜exact’ solution).
ii) Explain what you learnt from the chosen numerical method in Question 2. What improvements can be made in Question 3 to increase the accuracy of your results?

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