Homework 5

Problem 1

Calculate the magnitude and phase angle of the following complex numbers:

a. $13 + 2j$

b. $-5 - 3j$

c. $-2.4 + 3.6j$

Problem 2

Given the transfer function $G(s) = \displaystyle\frac{2.4}{0.6s^{2} + 8s + 36}$, do the following:

a. Determine system output, $y(t)$, for input $u(t) = 30.2\cos(20t)$.

b. Calculate the magnitude of the transfer function at $\omega = 20$ rad/s in dB.

c. Calculate the time shift between the input and output waveforms at $\omega = 20$ rad/s.

d. Generate a Bode plot for the transfer function in Matlab.

Problem 3

Given the transfer function $G(s) = \displaystyle\frac{4}{0.2s + 1}$, answer the following:

a. What is the cutoff frequency of the transfer function?

b. What is the bandwidth of the system represented by the transfer function?

Problem 4

Consider the system shown below. Assume $m = 0.6$ kg, $k = 80$ N/m, $b_{1} = 3$ N-s/m, and $b_{2} = 0.4$ N-s/m.

a. Calculate the output of the system if the input is $f_{a}(t) = 2\sin(8t)$ N.

b. Calculate which input frequency, $\omega$, will cause the largest amplitude of mass displacement.