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7. Choose reasonable values for each of your model parameters while varying the damping ratio (. Show solutions for the response of the system for the undamped, underdamped, critically damped, and overdamped cases. For the underdamped case choose parame- ters such that ( 0.707 (this is important for resonant behavior in later parts of this project). Plot these solutions using MATLAB. 8. Discuss the characteristic equation and its roots. For example, comments like these show your understanding of the response. Play with numbers in MATLAB to gain more intuition (a) Unstable behavior occurs if any root lies to the right of the imaginary axis. (b) The response oscillates only when a root has a nonzero imaginary part. 1 (c) The greater the imaginary part, the higher the frequency of the oscillation (d) The farther to the left the root lies, the faster the response due to that root decays. You may use additional plots to show the effects the roots have on the response and make qualitative statements. 9. For the underdamped case you have selected, determine important response parameters like wn, S.T, Mg, tp, 4. etc. Derive an expression for them and calculate the value for the constants you chose to plot with. 10. Develop the state-space model of your system. Provide the marriage and vectors A, B, C, and D of the state equation and output equation Yougare free to select whatever output you're interested in Use the MATLAB built-in funct? for state space systems to plot, analyze, and discuss the behavior of your systam 11. Discuss the behavior of your system to a harmoric input of the forment = Asin(wt). Solve for the response using frequency domal techaiques (i.e. Thiw)). Plot some solutions. 12. The log magnitude ratio is defined as m = 10 107 wn Produce a semi-log plan of m in dB vs r forgour system. For example, (=0.1 (=05 (-1 10 to 10 win Note that the resonant frequency Wr is determined to be wr = wnv1-2(2 05650.707 Discuss the importance of resonance to your particular system (e.g. usually a bad thing for mechanical systems, sometimes good for electrical systems). cf. Ogata p617

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%undamped
C=1;
L=1;

sys = tf([0 1],[L*C 10000 1]);

subplot(2,4,1)
step(sys)
subplot(2,4,5)
impulse(sys)

%underdamped
R=1;
C=1;
L=16;

sys = tf([1/(L*C)],[1 (1/(R*C)) 1/(L*C)]);

subplot(2,4,2)
step(sys)
subplot(2,4,6)
impulse(sys)

%critically damped
R=1;
C=1;
L=4;

sys = tf([1/(L*C)],[1 (1/(R*C)) 1/(L*C)]);...

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