MECH 412 – System Dynamics and Control
Homework #2
Suggested Completion Date: February 5, 2026
Use Matlab/Simulink as indicated.
1. Determine a state-space model for a system governed by:
x + 3x + 16 x + 25 x = F (t )
Take x as the system output.
2. Obtain the unit-step response curve and unit-impulse response curve of the system in
Question 1 with MATLAB. Your answer should be the figure obtained with Matlab.
3. Find the transfer function of the operational amplifier circuit shown below in figure 2 from
voltage input v1 to voltage output v2. Assume operational amplifiers are ideal.
4. Derive a state-space model for a three-degree-of-freedom mechanical system whose
governing differential equations are stated below. Take the output of the system as the
position and velocity x3 and x3 .
2 0 0 x1 2 − 2 0 x1 3 − 2 0 x1 0
0 4 0 x + − 2 2 0 x + − 2 5 − 3 x = 0
2
2
2
0 0 3 x3 0
0 0 x3 0 − 3 3 x3 F (t )
5. Find the transfer function for each of the two circuits below:
6. Determine (analytically) the transfer function of the single-input single-output system
defined by the following state-space model:
Verify your answer with Matlab.
7. Consider the armature-controlled dc servomotor used to drive one of the joints of the
robotic arm through a gear with the gear ratio n. The load inertia JL represents the effective
inertia of the links ‘down’ the arm from the joint. The torque developed by the motor is T.
The angular displacement of the motor rotor is θm and the angular displacement of the link
is θ such that n = θ / θm . Obtain the transfer function from the input voltage to the link
angle θ.
8. For the system shown in the figure below,
a. Find the closed-loop transfer function 𝑇𝑇(𝑠𝑠) = 𝑌𝑌(𝑠𝑠)/𝑅𝑅(𝑠𝑠).
b. Build a Simulink model of the system for 𝑎𝑎1 = 6, 𝑎𝑎2 = 11, 𝑎𝑎3 = 6, 𝑏𝑏1 = 1, 𝑏𝑏2 =
2, 𝑏𝑏3 = 3 and plot the step response for 0 ≤ 𝑡𝑡 ≤ 12 seconds.
Mech-412: System Dynamics and Control
Solutions to Assignment #2
Problem 1
1
Problem 2
2
Problem 3
3
Problem 4
4
Problem 5
5
6
Problem 6
7
Problem 7
8
Problem 8
Part a)
9
Part b)
10
0
