Homework 4

Problem 1

For each of the systems shown below, find (1) $c(t)$, (2) the time constant, (3) rise time, and (4) settling time.

Problem 2

For each of the transfer functions shown below, do the following:

1. Plot the poles and zeros on the s-plane (complex plane)
2. Write an expression for the general form of the step response
3. State the type of each response (overdamped, underdamped, etc.)

a. $T(s) = \displaystyle\frac{2}{s + 2}$

b. $T(s) = \displaystyle\frac{5}{(s + 3)(s + 6)}$

c. $T(s) = \displaystyle\frac{10(s + 7)}{(s + 10)(s + 20)}$

d. $T(s) = \displaystyle\frac{20}{s^2 + 6s + 144}$

e. $T(s) = \displaystyle\frac{s + 2}{s^2 + 9}$

f. $T(s) = \displaystyle\frac{s + 5}{(s + 10)^2}$

Problem 3

Find the poles of the transfer function

$$G(s) = \frac{s^2 + 2s + 2}{s^4 + 6s^3 + 4s^2 + 7s + 2}$$

Problem 4

For each transfer function shown below find $\zeta$, $\omega_n$, $T_s$, $T_p$, $T_r$, and %$OS$.

a. $T(s) = \displaystyle\frac{16}{s^2 + 3s + 16}$

b. $T(s) = \displaystyle\frac{0.04}{s^2 + 0.02s + 0.04}$

c. $T(s) = \displaystyle\frac{1.05 \times 10^7}{s^2 + 1.6 \times 10^3 s + 1.05 \times 10^7}$

Problem 5

Find the transfer function of a second-order system that yields a 12.3% overshoot and a settling time of 1 second.

Problem 6

For the system shown below, a step torque is applied at $\theta_1(t)$. Find:

A. The transfer function $G(s) = \Theta_2(s)/T(s)$.

B. The percent overshoot, settling time, and peak time for $\theta_2(t)$.

Problem 7

Anesthesia induces muscle relaxation (paralysis) and unconsciousness in the patient. Muscle relaxation can be monitored using electromyogram signals from nerves in the hand; unconsciousness can be monitored using the cardiovascular system’s mean arterial pressure. The anesthetic drug is a mixture of isoflurane and atracurium. An approximate model relating muscle relaxation to the percent isoflurane in the mixture is

$$\frac{P(s)}{U(s)} = \frac{7.63 \times 10^{-2}}{s^2 + 1.15s + 0.28}$$

where $P(s)$ is muscle relaxation measured as a fraction of total paralysis (i.e., it has a value between 0 and 1) and $U(s)$ is the percent mixture of isoflurane.

A. Find the damping ratio and the natural frequency of the paralysis transient response.

B. Find the maximum possible percent paralysis if a 2% mixture of isoflurane is used.

C. Plot the step response of paralysis if a 1% mixture of isoflurane is used.

D. What percent isoflurane would have to be used for 100% paralysis.

Problem 8

Find $J$ and $K$ in the rotational system shown below to yield a 30% overshoot and a settling time of 3 seconds for a step input in torque.