Linearsecond-orderdynamicsystemresponse A standard form of a second-order dynamic system, G(s) is G(s) = 2 n s2 + 2ns + 2 n where n is the natural frequency, and is the damping ratio. Inthisexercise, youwillinvestigatethestep and frequencyresponses of theabove system inan open-loopconguration for a variety of natural frequencies and damping ratios. The open loop conguration is shown in Figure 1. The system input is r(t) and the output is y(t). The frequency response is given as a Bode plot, that is a log-log plot of|G(j)|against , and a simultaneous plot ofG(j) against log(). Ifnotyet,youwillcoverthetheoryofBodeplotslaterinthecourse. For a step input function r(t) = 0 for t < 0 1 for t 0 which in the Laplace domain is R(s) = 1 s , the step response is y(t) = g(t) ? r(t) =Z t 0 g(t)dt where g(t) is the impulse response, which in the Laplace domain is Y (s) = G(s)R(s) = G(s) s . The step response is shown as a plot of y(t) against t. 2 1 Stepresponse-PartI(30markstotal) The laboratory exercise makes use of the MATLAB Control Toolbox. You will need to record (write down) in your notebook the values you get during the exercise. Some additional functions which provide unique variables for each group to do this exercise can be found in the following toolbox directory: cd('T:\SEC\ME\ME5112\ME5112lab'); 1.1 Stepresponsewithconstantnaturalfrequency In this section, with a constant natural frequency, n, the effect of different damping ratios, = 0.1,0.3,0.5, 0.707,0.85,1.0,3.0,6.0, on the step response is investigated. To obtain the value of n you will use in this section, type (while you are in the directory given above) [wn]=ME5112ex11('Surname'); where you substitute the groups surname for Surname. ENSURE THAT YOU TYPE IN THE SURNAME CORRECTLY WITHOUT ANY SPACE AND/OR SYMBOL! This places a value into the variable wn in the MATLAB workspace. To check this value, type wn Question1.1.1 Write down this value. (1 mark) Now, change to the following directory: cd('H:\Desktop'); To save the text from your session in a le, type diary filename.txt Now enter into the workspace the rst value of by typing Zt=0.1 Note that MATLAB is case sensitive. Type the command who to check which variables have been dened in the workspace. Now dene the system transfer function G(s) as the ratio of the numerator and denominator polynomials, 2 n and s2 + 2ns + 2 n respectively, by typing G=tf(wn2,[1, 2*Zt*wn, wn2]) Note that in MATLAB, denotes raise to the power of. Also note that in MATLAB, the coefcients of a polynomial in s are dened as a row vector where the coefcients of the polynomial are given in descending powers of s. Record the transfer function. You can check the natural frequency, damping ratio and system poles/eigenvalues by typing damp(G) Alternatively, you can obtain just the system poles/eigenvalues by typing pole(G) Record the values of the poles. To get more information on these or any other MATLAB functions, just type help damp help pole 3 To plot the step response of G(s), type step(G) From the screen output, measure the overshoot Mp and the peak time tp. Do not worry if the values are not very accurate. Record the values. Now repeat the process for the next value of ; Type (or use the up arrow to recall the previous denition of Zt and edit) Zt=0.3 Now redene the new system transfer function G(s) by G=tf(wn2,[1, 2*Zt*wn, wn2]) using the up arrow. Obtain the system poles/eigenvalues by pole(G) Record the values of the poles. Now plot the step response on the same axes as the previous plot. To do this type hold on step(G) Measuretheovershoot Mp andthepeaktime tp. Recordthevalues. Rememberthat tp cannotbemeasuredifthereis no overshoot. Repeat the process for all the values of . From the plot pull-down menu, save the les as an enhanced metale (for Word) or encapsulated Postscript le (for LATEX). Question1.1.2 Produce a table of the results including the transfer functions, poles, overshoots and peak times. Comment on the responses and the results. (10 marks) Plot Mp against and tp against . This has to be done in MATLAB; type help plot to see how. Question1.1.3 Include the plots and explain the curves. The plots should have titles, axes labels and legend. (5 marks) Question1.1.4 What is the signicance of the damping ratio = 0.707? (2 marks) Now clear the plot window by typing hold off; close