ECMM149 Coursework 1. E-BART Deadline 05th November 2021 The linear time invariant second order ordinary differential equation governing the behaviour of a damped oscillator is: For the specific values of a, b, c, U and a value of , you can use your student ID last 4 number to define. For example, xxxxx4258, a=4, b=2, c= 4 2 5 40× × = and U=8. In the meantime, the excitation frequency 1ω is a sum of a, b, c and U. (if your ID number has zero, please use 1 to instead of it). Make sure you note the specific values you used in your report. Develop a MATLAB (or Excel) code to estimate the (zero to peak) amplitude, of the displacement x when a ‘steady state’ response i.e. a clear sinusoidal response has been established. Assume zero initial conditions. Separate the 2nd order ODE into a pair of 1st order ODEs and use Euler (or modified Euler) method. You should use a time state that is 1/30 of the period of the oscillation i.e. ℎ = 2 (301)� 1) Using simple Euler method, what is your estimate of and the ratio for 2) Write down the impulse response function IRF for the system and create a vector of values the same length as the vector of input values u(t) you created for 1). 3) Use the MATLAB ‘conv’ function using the IRF and u vectors from 2) to provide an alternative estimate. 4) Use a larger time step ℎ = 2 (51)� and comment on the effect. 5) Sticking to the correct time step, repeat the numerical solution for nine extra values of i.e. and tabulate values of H. 6) Plot H for ignoring any of such strange values. 7) Now construct the frequency response function H directly using the method from lecture 6 and a result from the tutorial, 8) Plot values of this H for and overlay on the result for 6) above. 9) Use the MATLAB ‘diff’ function or your solution to a numerical calculus tutorial question to differentiate your time series from your calculations for 1) above to obtain velocity (differentiate once) and acceleration (differentiate again). Plot the velocity from the first differentiation and compare that with the values obtained directly from the numerical solution of the ODE. Comment on the comparison.
Report: Create a report where you show the differential equation governing the damped oscillator with your specific parameters (from your student ID number). Then, answer to each of the questions describing your solution and including any figures where asked. Include your Matlab code in your report. See next page for marking scheme: This could be a time-consuming exercise, so it is recommended to devote your major efforts to 1) 3) and 7) which attract most marks, marking scheme is as follows:
Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 25 5 15 10 5 5 15 5 15