ENGSCI311 MATHEMATICAL MODELLING 3 ODE ASSIGNMENT - 2022
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ENGSCI311 MATHEMATICAL MODELLING 3
ODE ASSIGNMENT - 2022
The purpose of this assignment is to practice your ordinary differential equation (ODE) knowledge by modelling an RLC circuit (Figure 1). Answer all questions and show all working. Photos of handwritten solutions should be eligible, and any codes must include a header with your UPI, and ID. Submit solutions to each question to the appropriate location on Canvas. Do NOT include your name in any page of the submissions.
Figure 1.
Question 1 [14 marks]
1.1. Write down a first-order ODE describing circuit 1. [1 mark]
1.2. Write down a first-order ODE describing circuit 2. [1 mark]
1.3. Rewrite the system of ODEs of circuits 1 and 2 into the matrix form. [3 marks]
1.4.Given R = 2Q, L = 2H, and C = 0. 5F. Find the complementary function of the system of ODEs, by solving the eigenvalue problem. The solution needs to be in the vector form and in real terms. [9 marks]
Question 2 [5 marks]
2.1 Find and classify the stability of the homogenous part of the system in Task 1.4. [2 marks]
2.2 Plot the phase plane by setting the unknown coefficients to 1, i.e., A = 1, and B = 1, where A and B are the unknowns in the complementary function, over t = 0:0.01:10 s. Remember to add a title and label the axes with units. Submit a copy of your code. [3 marks]
Question 3 [21 marks]
3.1 Solve the particular integral if Vin = sin (t). [10 marks]
3.2 Given initial conditions I1 (0) = 0 and I2 (0) = 0, solve the remaining unknown coefficients. Write down the final solution. [3 marks]
3.3 Plot I1 and I2 over t = 0:0.01:10 s. Remember to add a title, legend, label the axes with units. Do NOT solve numerically using ODE45. Submit a copy of your code. [3 marks]
3.4 Solve and plot the system using numerical method ODE45 in MATLAB. Plot I1 and I2 over t = 0:0.01:10 s. Remember to add a title, legend, label the axes with units. Submit a copy of your code. State the root-mean-square-error (RMSE) relative to the analytical solutions of I1 and I2 .
RSME = lxn(N)=1(N(yn) 一 Yn )2
where yn and Yn are the analytical and numerical solution at time point n, respectively. Note y(1) = initial condition, i.e., time = 0 s. There are total of N solution points. [5 marks]
2022-09-01