Miami University Electromechanical Systems Lab Report Lab: Electromechanical Systems Due Tuesday by 11:59pm Points 25 Submitting a file upload Topics C

Miami University Electromechanical Systems Lab Report Lab: Electromechanical Systems

Due Tuesday by 11:59pm
Points 25
Submitting a file upload
Topics Covered:
Obtaining the equations of motion of a DC motor based rotary servo.
Creating and validating a system model.
Model validation.
Instructions:

Please complete the following lab assignment tasks associated with this lab:

Complete all steps and questions apart of the Lab Manual
Download and run the MATLAB Program associated with this Lab.
Submit your MATLAB Simulink model to this assignment. Electromechanical System Modeling Lab
Background
The Quanser QUBE-Servo is a direct-drive rotary servo system (Figure 1). Its motor armature circuit
schematic is shown in Figure 2 and the electrical and mechanical parameters are given in Table 1. The DC
motor shaft is connected to the load hub. The hub is a metal disk used to mount the disk or rotary pendulum
and has a moment of inertia of J h . A disk load is attached to the output shaft with a moment of inertia
of J d .
Figure 1: Quanser QUBE-Servo DC motor with inertia disk attachment.
+
–
Figure 2: Schematic of QUBE-Servo DC motor armature circuit and load.
The back-emf (electromotive) voltage, eb ( t ) , depends on the speed of the motor shaft,  , and the backemf constant of the motor, kb . It opposes the current flow and is given by:
eb ( t ) = kb ( t ) → Eb ( s ) = kb  ( s )
(1)
The torque developed by the motor,  ( t ) , depends on the current of the motor circuit, i ( t ) , and the motor
constant, kt . The torque developed by the motor is given by:
Miami University
 ( t ) = kt i ( t ) → T ( s ) = kt I ( s )
(0.2)
ECE/MME 436/436H/536: Control of Dynamic Systems
Page 1
Table 1: QUBE-Servo system parameters
Symbol
Description
Value
R
Terminal resistance
8.4Ω
kt
Torque constant
0.042 N − m/A
kb
Motor back-emf constant
0.042 V/(rad/s)
Jm
Rotor inertia
4.0 × 10−6 kg − m2
L
Rotor inductance
1.16 mH
Mh
Load hub mass
0.0106 kg
rh
Load hub radius
0.0111 m
Jh
Load hub inertia
0.6 × 10−6 kg − m2
md
Mass of disk load
0.053 kg
rd
Radius of disk load
0.0248 m
DC Motor
Load Disk
Lab Assignment
In this lab, we will mathematically model the QUBE-Servo using a first principle approach by referencing
the laws of physics to derive the differential equation of this electromechanical system. From this
differential equation we will derive a transfer function relating output angular speed to input voltage. We
will test the accuracy of this model at predicting the speed of a QUBE-Servo experimentally when driven
at different amounts of input voltage.
Deriving the Differential Equation and Transfer Function
When deriving the governing equations for an electromechanical system, it is often beneficial to examine
the electrical and mechanical components independently. Looking at only the electrical components of
the QUBE-Servo DC motor (as shown in Figure 1):
+
–
Figure 1: Electrical curcuit of the QUBE-Servo DC motor
1. Write the differential equation (in the form of Kirchoff’s voltage law) in the Laplace domain for the
electrical circuit (do not use parameter values given in Table 1). Assume that in this particular motor,
the inductance is much less than the resistance; therefore, the inductance can be neglected. Write the
differential equation in terms of the system variables (NOT component values).
Miami University
ECE/MME 436/436H/536: Control of Dynamic Systems
Page 2
Now take a look only at the mechanical components of the system (as shown in Figure 2). The mechanical
system consists of three inertial components and one viscous damping component. The three inertial
components are a combination of the inertia from the motor, hub, and disk. While there are three inertial
components, they each rotate together and are described by the same rotational displacement variable;
thus, they can be described by a single equivalent inertial component.
(a)
(b)
Figure 2: (a) Mechanical components of the QUBE-Servo DC motor and load, and (b) an equivalent mechanical system.
Q1.
What is the calculated equivalent moment of inertia (with units of kg-mm2) that is
acting on the motor shaft?
2. Write the differential equation in the Laplace domain for the rotational mechanical system. Write the
differential equation in terms of the system variables (NOT component values).
3. Noting that these two systems are coupled together, mainly the relationships between eb and  as
well as  and i , combined the two differential equations derived for the electrical and mechanical
systems independently, such that the result is a single differential equation in the Laplace domain
relating input voltage to output angular speed. Write the differential equation in terms of the system
variables (NOT component values).
Q2.
Using the differential equation of the electromechanical system, solve for the transfer
function relating input voltage to output angular speed. Write the transfer function
in terms of the system variables (NOT component values).
Testing the Model Predictions
Now that we have a mathematical model of this QUBE-Servo DC motor, we can test the accuracy of the
model at predicting dynamics of this electromechanical system. We will do so by measuring the step
response of the QUBE-Servo DC motor experimentally and using the model to predict the speed of the
motor.
4. Use the parameter values provided in Table 1 and the provided MATLAB program to measure and
predict the speed of the QUBE-Servo DC motor. Assume that in this particular motor that friction is
negligible; thus, viscous damping can be neglected ( D  0 ). Run the program for such that the
motor has an input of 2 V. Does the mathematical model predict the motor speed well?
5. Re-run the program for different input voltages between -5 V and +5V.
Q3.
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How does changing the input voltage affect the performance of the model in
predicting the measured disk rotation?
ECE/MME 436/436H/536: Control of Dynamic Systems
Page 3
6. Re-run the program for different values of the model until the model best predicts the motor speed.
That is, change the value of the back EMF constant, the motor constant, and the motor resistance
until the curve of the model response matches the curve of the measured response. What are the final
values of the back EMF constant, the motor constant, and the motor resistance?
Q4.
What effect does adjusting the motor resistance, R , have on the simulated response?
Q5.
What effect does adjusting the back EFM constant, kb , and the motor constant, kt ,
have on the simulated response?
Q6.
What is the estimated value of the motor resistance, R ? This is the value that resulted
in the best prediction of the QUBE-Servo angular speed.
Q7.
What are the estimated values of the back EFM constant, kb , and the motor constant,
kt ? This is the value that resulted in the best prediction of the QUBE-Servo angular
speed. Recall that kb = kt ?
Modeling the Open-Loop System
We now have a mathematical model that accurately predicts the dynamics of the QUBE-Servo DC motor.
In future labs, we will use this model to predict the response of the QUBE-Servo DC motor when exposed
to different input voltages. The ability to predict the response of a system based on its input is the basis
for developing automatic control. It is from this information, that we can develop control laws (or
algorithms) that determine the required inputs to produce the desired system output. To prepare for future
labs, and controlling the QUBE-Servo DC motor system, we will develop a block diagram of the openloop system (Figure 3).
Plant
Figure 3: General block diagram of the open-loop system for the QUBE-Servo DC motor.
7. Use MATLAB Simulink to create and simulate the open-loop system of the QUBE-Servo DC motor.
Your Simulink model should include two inputs (the voltage and the disturbance), a plant transfer
function, and an output. To complete the development of this Simulink model you should use: (1)
the Step Input block for the input voltage, (2) the Band-Limited White Noise block for the
disturbance, (3) the Sum block for the summing junction, (4) the Transfer Fct block for the plant
transfer function, and (5) the Scope block (or To Workspace block) for the output. Set the parameters
of the system to those that you found in 6 and simulate the Simulink model. Submit a copy of the
response for a 3 V step input. Save this model for future use.
Miami University
ECE/MME 436/436H/536: Control of Dynamic Systems
Page 4

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