## Transcribed Text

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)]);...