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value in 0.57 s, reaches the peak value 2 ⇥ 1.35 = 2.7 at t = 4.6, and settles down within 1% of the
ﬁnal value in 4.6 s. You should sketch it and compare with the matlab plot.
3. For each of the following three systems:
(a) pendulum without friction (b) inverted pendulum (c) mass-spring-dashpot system choose the corresponding transfer function from:
(A) s2 1
+ 2s + 10 (B) s2 1
+4 (C) 1
s2 5 (D) 1
s+1 Solution: A pendulum without friction would oscillate indeﬁnitely, so its transfer function would
have a pair of poles on the imaginary axis. An inverted pendulum would be unstable and diverge
without oscillation, so it should have at least one real positive pole. A mechanical system with mass,
spring, and dashpot would be stable, and of order at least two. Therefore,
(a) B (b) C (c) A 4. Consider the system described by
3¨ + ✓ = u,
z + 2✓ = 10✓.
¨ Find the transfer function from u to ✓.
Solution: Taking the Laplace transform of the differential equations,
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- Winter '09