lab1 Modeling - Mechanical parameters
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Arizona State University, Tempe *
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Course
324
Subject
Aerospace Engineering
Date
Apr 3, 2024
Type
docx
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Uploaded by ColonelMusic7302
1
st
team member’s Name (Last, First): Lab station #: ____
2
nd
team member’s Name: Date: ___________
3
rd
team member’s Name:
Modeling your plant: mechanical parameters
MAE 318: System Dynamics and Controls lab 1
Before you start designing controllers for your customer, it is important to model your system
mathematically. You will save a lot of time and resources if you try controllers in simulations for this
mathematical model, rather than trying it directly on the setup. In your MAE 318 course you learned how to model an electromechanical plant like a DC motor. For this
lab, your task will be to identify the mechanical parameters of our plant. 1
Pre-lab deliverable
Please write the answers to these questions and submit them on canvas before coming to the lab, as a
pdf file. The answers can be obtained by reading this whole lab handout or some research
. Pre-labs are
to be submitted individually
, they are not worked on as a team. 1.
What is dry friction and what is viscous friction? Research on your own and explain them in a
few sentences each. Also provide formulae to calculate force or torque on a moving mass, due
to dry and viscous friction separately. (2 points)
2.
When you give a voltage V
a
to a dc motor, it moves. Derive step by step (using KVL and
Newton’s laws) the differential equation that relates the voltage V
a
to the motor speed ω
. (2 points)
3.
Write the solution expression (as a function of time) to the following differential equation:
˙
y
+
ay
+
b
=
0
, for initial condition y
(
0
)
=
y
0
. That is, find the solution y
(
t
)
(
a
and b
are constants). (2 points)
4.
Describe in a few sentences what two experiments will be performed in this lab and what three
quantities/parameters they intend to measure? (2 points, EM@FSE(c))
5.
For your set-up you are using a voltage-controlled DC motor+gearbox to model the drive train of
a car. But do the modern electrical cars use DC motors? Research at least two electrical cars
brands you know of and see what type of motors they use. (2 points, EM@FSE(c))
If they do not use the DC motors, then is your set-up with a DC motor still of any value for your
customer? (2 points, EM@FSE(i))
2
The Lab Setup
2. 1 Overview
The experimental setup used throughout the term is shown in Fig. 1. It includes a Personal Computer
(PC) with a USB port that sends and receives information to/from the Arduino board. Arduino interface
(IDE) is the software on the computer that helps facilitate this information exchange. Arduino, using a
digital to Analog converter (DAC) provides low-power control signal to a motor controller that
transforms the low-power control signal to high-power control signal (voltage) to drive the DC motor.
The DC motor is connected through a coupler to a driveshaft that rotates an aluminum disk. Inside the
housing holding the aluminum disk, there are two independent magnets which may be used to add
linear (viscous) damping to the system using a phenomenon known as eddy-current damping. Though,
this need for extra damping will not arise in these set of labs. Along the driveshaft, a rotary encoder is
used to measure the angle θ
of the disk. The encoder signal is read directly by Arduino, and the Arduino
IDE displays that angle data on the computer, via the serial monitor. Figure 1: The lab experimental setup.
Motor Controller
DAC
3
Lab experiments
The mechanical part of the motor-disk setup (shown in Fig. 2) is described by Equation (1) below: T
=
K
t
i
=
J
¨
θ
+
B
˙
θ
+
B
0
,
(1)
where T
is the motor-generated torque, K
t
is the motor torque constant, J
is the moment of inertia of
the rotating disk along with the motor armature and shafts, B
is the damping coefficient for the viscous
friction on the rotating shaft, and B
0
is a constant dry/Coulomb friction on the shaft. The electrical part
of the motor is described by Equation (2) below:
V
a
=
i R
a
+
K
e
˙
θ,
(2)
where V
a
is the voltage applied to the motor from the controller-amplifier, i
is the current flowing
through the motor windings, R
a
is the resistance of the motor windings, K
e
is the motor back-EMF
constant, and ˙
θ
is the motor rotational velocity. Note that the inductance of the motor windings is
considered negligible. You can combine the equations (1) and (2) to get a single relation between the
applied voltage V
a
and the motor speed ˙
θ
as:
V
a
=
R
a
K
t
J
¨
θ
+
(
R
a
K
t
B
+
K
e
)
˙
θ
+
R
a
K
t
B
0
(3)
The purpose of this labs is to identify the
mechanical parameters of the motor in
Equation (1), and the purpose of the next
lab will be to identify the electrical
parameters of the motor in Equation (2). To identify the mechanical parameters, two
experiments will be performed. The first
experiment
identifies
the
friction
parameters B
and B
0
, then the second
experiment identifies the motor torque
constant K
t
.
Note, that inertia J
is the inertia of the
rotating disc plus the inertia of motor’s
rotor and shaft. The inertia of the rotating
disk is given by 1
2
mr
2
, where m
is the disk mass and r
the radius. The disk is made of Aluminum, which has a density of ρ
. The volume of the disk is given by V
=
π r
2
w
where w
is the width of the disk. Therefore, the inertia of the rotating disk is given by: J
=
1
2
mr
2
=
1
2
ρV r
2
=
1
2
ρπ r
4
w
. Where
ρ
=
2800
kg
/
m
3
, w
=
1
∈
¿
0.0254
m
, and r
=
4
∈
¿
0.1016
m
. The inertia of the motor’s rotor and shaft can be experimentally calculated or looked up in the motor specification sheets online, it is 0.015 kgm
2
. But actually
, you don’t need to know J
!! If you divide by J
on both sides of the equation (1), only _
_
+
_
_
Figure 2: The model of the DC motor.
_
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