Dynamics of Belt-Driven Servomechanisms
Theory and Experiments
by
Dhanushkodi D. Mariappan
Bachelor of Technology (B.Tech), Mechanical Engineering
Indian Institute of Technology, Madras 2001
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
Master of Science in Mechanical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2003
@
Massachusetts Institute of Technology 2003. All rights reserved.
Author .............
Certified by...............
...................................
Department of Mechanical Engineering
May 23, 2003
........
.............
Samir A. Nayfeh
Assistant Professor
sis Supervisor
.............
Ain A. Sonin
Chairman, Department Committee on Graduate Students
Accepted by ...............................
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
Vokoy".101
JUL 0 8 2003
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Dynamics of Belt-Driven Servomechanisms
Theory and Experiments
by
Dhanushkodi D. Mariappan
Submitted to the Department of Mechanical Engineering
on May 23, 2003, in partial fulfillment of the
requirements for the degree of
Master of Science in Mechanical Engineering
Abstract
There is an ever increasing demand for high speed precision positioning systems from a
wide range of industries. These machines typically employ ball-screws, linear motors,
or belt-drives and operate in closed loop to achieve high performance. In this thesis,
we study the dynamics of belt-driven servomechanisms. In these belt-driven machines,
the primary limitation to the performance arises from the belt compliance. The
performance is characterized by parameters which include bandwidth, tracking in the
presence of disturbances, etc. We model the axial dynamics of the belt drives and
discuss collocated and noncollocated feedback strategies. The design and assembly
of a belt-driven linear motion stage is explained in detail. We measure the transfer
functions through sine sweep measurements to verify the theoretical model. Damping
plays a key role in determining the maximum achievable bandwidth of the belt-driven
servomechanism. We present a model for the microslip phenomenon and quantify the
damping that arises out of microslip. In summary, this thesis lays out a dynamic
model of belt-driven servos, a model for microslip, a detailed design process, and
experimental methods for measuring transfer functions.
Thesis Supervisor: Samir A. Nayfeh
Title: Assistant Professor
3
4
Acknowledgment
First of all, I would like to thank Prof. Samir Nayfeh for giving me the opportunity to
work on this exciting project. He continues to amaze me with his vast treasure house
of knowledge and remarkable physical intuition. His deep insights in design, dynamics,
controls and his excellent analytical depth always sets standards I strive to achieve.
He has been very tolerant in admitting all the costly mistakes I did in the course of
completion of this work. I would like to thank Prof. Sanjay Sarma for his encouraging
words and help in moments of trouble. Sanjay's energy is incredible and I cherish the
moments I spent listening to his words of wisdom. My undergraduate advisor Prof.
V. Ramamurti has been a great inspiration in my academic path during and after
my days at IIT, Madras. I would like to thank Kripa for his invaluable guidance and
support. In addition to his lessons on dynamics theory and experiments, he has been
a great mentor . I owe a lot to Kripa for the time he has spent teaching me. Mauricio
always answered my questions patiently and suggested references. I owe a significant
percentage of my design knowledge to him. Justin Verdirame is a very resourceful
person. I always admired his cool and composed approach and I learnt to talk things
with high signal to noise ratio. I consider myself unfortunate not to have worked with
Greg for he is such a vibrant man with lots of design expertise. Andrew Wilson, a
cheerful companion has answered my questions for the nth time without complaining.
I also thank Nader and Lei for their help.
I always worked with machines which
needed atleast two people to handle and I thank Justin, Mauricio, Jonathan, Sup
and others in the lab who took time off from their work and helped me. The LMP
machine shop experience was fun and a lot of learning. I would like to thank Mark and
Jerry for all the hours they spent teaching me patiently and admitting my mistakes.
I acknowledge Jonathan's help in the file conversion issues with Solidworks drawings.
I would like to thank Rick for his help with LVDT and other lessons. Hari, Srini and
Madhu have been excellent companions and motivators. I thank Srini, Ajay and Hari
for patiently proof reading my thesis and giving critical comments. Ajay has been
a great companion who always made me set high standards in research and work
5
towards achieving them. I also thank Anand anna, Carlos, Karen, Vijay, Sriram,
Harsh, Mahadevan, Shorya, Rama, and many others who were directly or indirectly
involved in succesful completion of this work. Above all, I thank my appa, amma,
murugappa, aachi, Juno and Venkatesh who were with me and will continue to be
with me when it matters most. God is great and He has helped me strive, seek, find
and not to yield.
6
Contents
1
1.1
1.2
2
15
Introduction
Servomechanisms: Feedback - Performance Criteria . . . . . . . . . .
16
1.1.1
Parameters of Performance . . . . . . . . . . . . . . . . . . . .
16
1.1.2
Design for Closed-Loop Performance
. . . . . . . . . . . .
17
1.1.3
Limitations
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
[3]
19
Modeling the Axial Dynamics of the Belt Drive
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.2
Modeling the Dynamics
. . . . . . . . . . . . . . . . . . . . . . . . .
20
2.2.1
Model for the Belt Drive . . . . . . . . . . . . . . . . . . . . .
20
2.2.2
Equations with Damping Terms Included . . . . . . . . . . . .
22
Effect of Varying the Stiffness and Damping . . . . . . . . . . . . . .
23
2.3.1
Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.3.2
D am ping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
. . .
24
. . . . . . . . . . . . . . . . . . . . . . .
25
2.3
2.4
2.5
Three D.O.F. Model to Include the Pitch Mode of the Carriage
2.4.1
Equations of Motion
2.4.2
Eigenvalues and Eigenvectors
. . . . . . . . . . . . . . . . . .
25
2.4.3
Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.4.4
Roll and Yaw Modes
. . . . . . . . . . . . . . . . . . . . . . .
27
Bandwidth of the Belt Drive . . . . . . . . . . . . . . . . . . . . . . .
28
2.5.1
Collocated Control . . . . . . . . . . . . . . . . . . . . . . . .
28
2.5.2
Noncollocated control.
30
. . . . . . . . . . . . . . . . . . . . . .
7
2.6
3
Robustness
2.5.4
Crossover of Type 5.
Chapter Summary
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
. . . . . . . . . . . . . . . . . . . . . . .
31
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
Energy Dissipation due to Slip in Belt Drive: Damping and Loss
Factor Estimates
43
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
3.2
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
3.3
M icroslip - Background . . . . . . . . . . . . . . . . . . . . . . . . . .
44
3.3.1
M icroslip and Sliding . . . . . . . . . . . . . . . . . . . . . . .
45
3.3.2
Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . .
45
3.3.3
Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
3.3.4
M indlin's Solution and Results
. . . . . . . . . . . . . . . . .
46
Belt Drive - M icroslip . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
3.4.1
M otivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
3.4.2
Loss Factor - Definition [15] . . . . . . . . . . . . . . . . . . .
50
3.4.3
Origin of M icroslip in Belt Drives . . . . . . . . . . . . . . . .
50
3.4
3.5
3.6
4
2.5.3
M odel: Deforming Control Volume
. . . . . . . . . . . . . . . . . . .
51
3.5.1
Slip Rate: M ass Conservation . . . . . . . . . . . . . . . . . .
51
3.5.2
Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
3.5.3
Energy Loss: Energy Balance
. . . . . . . . . . . . . . . . . .
53
3.5.4
M aximum Potential Energy
. . . . . . . . . . . . . . . . . . .
56
3.5.5
Loss Factor
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
3.5.6
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
Chapter Summary
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
Design of the Belt Drive
61
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
4.2
The Loop
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
4.2.1
Actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
4.2.2
Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
8
Air Bearings . . . . .6
4.2.4
P ulley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
4.2.5
Sizing the Pulley
. . . . . . . . . . . . . . . . . . . . . . . . .
67
4.2.6
Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
4.3.1
Bolted Joints in the Assembly . . . . . . . . . . . . . . . . . .
72
4.3.2
Pulley Assembly
. . . . . . . . . . . . . . . . . . . . . . . . .
74
4.3.3
Air Bearing Assembly
. . . . . . . . . . . . . . . . . . . . . .
75
4.3.4
Motor Assembly . . . . . . . . . . . . . . . . . . . . . . . . . .
76
4.3.5
Carriage Assembly
. . . . . . . . . . . . . . . . . . . . . . . .
76
4.3.6
Belt Assembly and Pre-tension
. . . . . . . . . . . . . . . . .
76
4.3.7
Cleaning and Stoning . . . . . . . . . . . . . . . . . . . . . . .
77
4.4
Feedback Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
4.5
Closed Loop Position Control
. . . . . . . . . . . . . . . . . . . . . .
78
4.6
Chapter Summary
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
4.3
5
Assembly
. .. . ..
89
Experimental Results
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
5.2
Sine Sweep Measurements
. . . . . . . . . . . . . . . . . . . . . . . .
89
Procedure for Transfer Function Measurement . . . . . . . . .
90
5.2.1
6
64
4.2.3
93
Conclusions
95
A Motors
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
A.2 Servomotors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
A.2.1
Motor Principle . . . . . . . . . . . . . . . . . . . . . . . . . .
96
A.2.2
Back E.M.F . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
A.2.3
DC Motor Characteristics
. . . . . . . . . . . . . . . . . . . .
97
A.2.4
Need for Commutation . . . . . . . . . . . . . . . . . . . . . .
98
. . . . . . . . . . . . . . . . . . . . .
98
A.1
A.3
Brushless (BLDC) Servomotors
9
A.4
Classification Based on Commutation Signals
A.5
Voltage Control - Quantitative Picture . . . . . . . . . . . . . . . . .
B Engineering Drawings
. . . . . . . . . . . . . 100
100
103
10
List of Figures
2-1
Two-degree-of-freedom model
. . . . . . . . . . . . . . . . . . . . . .
21
2-2
Collocated transfer function . . . . . . . . . . . . . . . . . . . . . . .
33
2-3
Noncollocated transfer function
. . . . . . . . . . . . . . . . . . . . .
34
2-4
The effect of change in stiffness in the noncollocated transfer function
35
2-5
The effect of damping in the noncollocated transfer function
. . . . .
36
2-6
Three-degrees-of-freedom model . . . . . . . . . . . . . . . . . . . . .
36
2-7
Closed-loop servomechanism . . . . . . . . . . . . . . . . . . . . . . .
37
2-8
Transfer function x .
1
2-9
3 DOF model - collocated transfer function . . . . . . . . . . . . . . .
38
2-10 3 DOF model - noncollocated transfer function . . . . . . . . . . . . .
39
2-11 Nyquist representation of crossover frequencies, Varanasi [3] . . . . .
40
. . .
41
. ...
..
37
....................
2-12 Adding phase at cross over 3 leading to instability, Varanasi [3]
2-13 Nyquist interpretation of robust gain margin (RGM) and phase margin
(PM ), Varanasi [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
3-1
Two spheres in contact under normal and tangential load . . . . . . .
47
3-2
A typical belt drive showing the control volumes on the driven and
driving pulleys
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
. . . . . . . . . . .
53
3-3
Free body diagram to show the forces on the belt
3-4
Variation of loss factor with friction coefficient p for different values of
drive ratio n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
4-1
The m achine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
4-2
Crowned pulleys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
11
4-3
Assembly procedure to align the drive mount on the base . . . . . . .
81
4-4
Air bearing assembly . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
4-5
Measuring the flyheight . . . . . . . . . . . . . . . . . . . . . . . . . .
82
4-6
Assembling the belt between the blocks . . . . . . . . . . . . . . . . .
83
4-7
Pre-tensioning mechanism
83
4-8
Stiffness of the 40 mm flat air bearings [22]
. . . . . . . . . . . . . .
84
4-9
Stiffness of the 50 mm flat air bearings [22]
. . . . . . . . . . . . . .
85
4-10 Figure showing the pitch, roll and yaw axes of the carriage . . . . . .
86
. . . . . . . . . . . . . . . . . . . . . . . .
4-11 Load-deflection characteristics of ball bearings
[24] (Reprinted with
permission from the author) . . . . . . . . . . . . . . . . . . . . . . .
86
4-12 Comparision of preloaded versus non preloaded bearings [24] (Reprinted
with permission from the author)
. . . . . . . . . . . . . . . . . . . .
87
4-13 Locknut and lockwasher mounted on a threaded shaft (Reprinted from
Whittet Higgins catalog with permission) . . . . . . . . . . . . . . . .
4-14 Current mode operation
87
. . . . . . . . . . . . . . . . . . . . . . . . .
88
5-1
Sine sweep experimental setup - Schematic . . . . . . . . . . . . . . .
90
5-2
Measured and predicted collocated transfer function . . . . . . . . . .
91
5-3
Measured and predicted noncollocated transfer function . . . . . . . .
92
A-i
Equivalent circuit of a DC motor
97
. . . . . . . . . . . . . . . . . . . .
A-2 Equivalent-circuit representation of commutation
. . . . . . . . . . .
99
. . . . . . . . . . . . . . . . . . . . . . . . .
101
B-i Drawing of the pulley . . . . . . . . . . . . . . . . . . . . . . . . . . .
104
B-2
Drawing of the carriage plate 1
. . . . . . . . . . . . . . . . . . . . .
105
B-3
Drawing of the carriage plate 2
. . . . . . . . . . . . . . . . . . . . .
106
B-4
Drawing of the carriage plate 3
. . . . . . . . . . . . . . . . . . . . .
107
B-5
Drawing of the carriage plate 4
. . . . . . . . . . . . . . . . . . . . .
108
A-3 Voltage mode operation
B-6 Drawing of the motor mount - Front view
B-7 Drawing of the motor mount - Top view
12
. . . . . . . . . . . . . . .
109
. . . . . . . . . . . . . . . .
110
List of Tables
2.1
Belt parameters . . . . . . . . . . . . . . . .
. . . . . . . . .
23
2.2
Mode shape
eigenvectors . . . . . . . . . .
. . . . . . . . .
26
2.3
The rigid body modes
. . . . . . . . . . . .
. . . . . . . . .
28
4.1
Specifications: BM500E
. . . . . . . . . . .
. . . . . . . . .
63
4.2
Coupling dimensions and specifications . . .
. . . . . . . . .
65
4.3
Bearing nomenclature 7909A5
. . . . . . .
. . . . . . . . .
69
4.4
Loading conditions and fits [23] . . . . . . .
. . . . . . . . .
70
A.1
Inner-rotor versus outer-rotor BLDC motors
. . . . . . . . .
99
A.2 PMDC Vs BLDC Motors . . . . . . . . . . .
. . . . . . . . .
100
-
13
14
Chapter 1
Introduction
Precision positioning systems are essential in a wide range of industries. These include the
eiefniconductor,-machine tool, robotics, material handling, packaging, data
storage, and printing industries. Typically these systems use rotary actuators, such
as brushless DC motors and convert the rotary motion to linear motion using mechanical power transmission elements like belts, chains, ball screws, or lead screws.
In addition, linear motors are used in several applications. The choice of the drive
system is often based on the following factors which might vary depending on the
application
1. speed
2. positioning accuracy
3. repeatability
4. range of travel
5. load-carrying capacity
15
1.1
Servomechanisms: Feedback - Performance Cri-
teria
These precision machines may or may not operate in closed loop. When not operated
in closed-loop, they run in open loop using actuators like stepper motors. Open-loop
control is simpler to implement since there is no need for sensors. Feedback control is
more complex and may cause stability problems, but one can achieve significant improvements in the performance of these precision machines using closed-loop control.
When compared to open-loop control, feedback can be used to
1. reduce steady-state error due to disturbances by a factor of 1 + L where L is
the loop gain. L is the product of the controller and plant transfer functions
2. reduce the system's transfer function sensitivity to parameter variations
3. speed up the transient response
4. reduce the sensitivity of the output signal to parameter changes
1.1.1
Parameters of Performance
The two most important issues that concern the designer while designing a machine
that operates in closed loop are the stability and performance. The broad classifications of stability fall into two categories
1. External (OR) Input-Output Stability
2. Asymptotic Stability (OR) Internal Stability
In most cases, these two notions of stability converge as we often work with SISO
sytems that are completely observable and controllable. The most appropriate way of
characterizing stability will be, in the open loop frequency response L(jw), the phase
be greater than -1800 at the cross-over frequency.
Primarily, one has to make sure that the system (in our case, the machine) is
stable under closed-loop control. Once we have a stable system, we can improve the
16
performance of the system by designing the machine and the controller to meet the
following closed loop performance specifications.
1. Trajectory Tracking
2. Disturbance Rejection
3. Noise Rejection
4. Performance Robustness
For detailed descriptions of each of these performance specifications, refer to [1] and
[2]
1.1.2
Design for Closed-Loop Performance
[3]
The closed-loop performance specifications that are mentioned in the previous section
depend on the loop transmission L. The loop transmission encompasses the plant and
the controller dynamics. Therefore, it is important that the controller and the plant
be designed simultaneously to extract the best possible performance out of these
precision machines. This approach to solving the inverse problem of motion control
was addressed by Varanasi and Nayfeh [3]. The inverse problem in motion control can
be stated as: 'Given the performance specifications, design the loop transmission.' In
[3], the authors demonstrate this inverse approach with a case study on a ball-screw
servo system. The ultimate goal in this approach is to be able to obtain closed-form
expressions which serve as strict guidelines for a mechanical designer who sets out to
solve a 'design of precision machines for performance' problem. In this thesis, we lay
out the design of a belt-driven linear motion stage for dynamic performance.
1.1.3
Limitations
The solution to the inverse problem necessitates a good model of the dynamics of the
system. In this thesis, we develop a model of the dynamics and study the maximum
bandwidths attainable with collocated and noncollocated control. The bandwidth
17
of the belt-driven servomechanism is limited by the drive resonance that arises from
the compliance of the belt. In addition, if the stiffness of the components in the
structural loop are not high enough, the compliances add up in series and bring down
the stiffness of the drive. This translates in the form of the axial resonant frequency
of the drive. The lower the resonant frequency, the lower the attainable bandwidth
and hence the larger the time constant of the machine. Hence serious consideration
is to be given to the design of various joints, preloading of bearings, choice of the
coupling, optimization of the structural loop. Damping of the resonant peak also plays
a role in determining the bandwidth. The amount of damping determines the degree
of robustness of the system. In a belt drive, the question of adding deterministic
damping in the load path is yet an unsolved problem. In this thesis, we investigate
the significance of the damping that arises out of microslip in belt drives. We discover
that the damping that arises out of microslip is insignificant and hence one has to
find ways to add damping into the system.
1.2
Contributions
1. A model of the dynamics of the belt-drive.
2. A model for microslip in belt-drives.
3. Development of a design for stiffness approach with details on component selection and the assembly process.
4. Experimental validation of the theoretical results by transfer function measurements.
18
Chapter 2
Modeling the Axial Dynamics of
the Belt Drive
2.1
Introduction
The axial resonance that arises from the compliance of the belt limits the performance
of the belt-driven servomechanism. In this chapter, we derive the equations of motion
for the drive and obtain a closed-form expression for the axial resonance which depends on the inertias in motion: the drive pulley inertia (Ji), the idler pulley inertia
(J 2 ) and the mass of the carriage (M). In the subsequent chapters on the design of
the belt drive, we lay emphasis on the importance of the various compliances in the
dynamic loop. Our model treats the belt compliance as the dominant compliance.
This will not hold true if a bad design of the various components of the machine
leads to one or more compliances at parts of the machine other than belt like joints,
couplings, and so on. The most important assumptions are:
1. The mass of the belt is very small compared with the rest of the inertia.
2. The idler pulley inertia is lumped on to the carriage inertia.
3. The preload in the belt is large enough to avoid slipping.
19
We can check the validity of these assumptions depending on how well the experimental results match with theoretical predictions.
2.2
Modeling the Dynamics
The distributed inertia of the belt is negligible compared to the rest of the inertia
in the system. This allows development of a lumped parameter model consisting of
discrete masses connected by springs and dampers. Thus, the problem formulation
involves a set of ordinary differential equations, the solutions of which propagate in
time; these are commonly referred to as the initial value problems.
of the belt were to be included, it becomes a continuous system.
If the inertia
The motion of
such continuous systems is described by variables depending not only on time but
also on spatial position.
These are governed by partial differential equations.
For
continuous systems, the equations of motion are derived by formulating the problem
using Lagrangian mechanics.
Then it becomes a boundary value problem where
solutions satisfy a differential equation in a given open domain and certain conditions
on the boundaries of the system. A very detailed description of distributed parameter
systems is available in [4]. In addition, the interested reader can refer to [1] and [5]
for an introduction to modeling of dynamical systems and their control.
2.2.1
Model for the Belt Drive
The torque developed by the motor provides the actuation for the system. We represent this torque by
T.
J is the overall inertia on the drive side which comprises of
the inertia of the motor and the drive pulley inertia. We can lump these two inertias
together if the torsional stiffness of the coupling that connects the motor shaft to the
pulley shaft is very high. We will lay more emphasis on this fact in the Chapter 4. In
the Fig. 2-1, F refers to the generalized force which is the torque
T.
The coordinates
X 1 and X 2 represent the displacements of the drive pulley and the carriage respectively. X 1 and X 2 will be referred to as 0 and x 2 through the rest of this chapter. The
generalized mass m 3 refers to the mass of the idler pulley and is given by J 2 /r 2 , where
20
X,
F
X2
M1
n
2
K3
Figure 2-1: Two-degree-of-freedom model
r is the radius of the pulley. Since the pulley m 3 is an idler, it does not transmit
any torque. Provided that the inertia M3 is small, the tensions on either side of the
pulley are equal. Hence, we can consider springs K 2 and K 3 to be in series. Their
equivalent stiffness is given by K' =
Hence, the system reduces to a two-mass
K2K.
system held together by two springs K 1 and K' in parallel.
Writing the equations of motion for this system, we obtain
JiO+ K(rO - X 2 ) = 7
(2.1)
+ K(x 2 - rO) = 0
(2.2)
(M 3 + m 2 )f
2
Here, K represents the overall stiffness. K 1 is the stiffness of the steel belt between
the drive pulley and the carriage given by
1,
where 1 is the length of the belt
between the carriage and the drive pulley. From these equations of motion, we derive
expressions for the transfer functions by taking Laplace transforms, in order to obtain
the behaviour of the system in the frequency domain. This leads to the following set
of equations.
Here, X 2 and
E
J 1 s 2 E + K(rG - X 2 )r = r
(2.3)
(M2 + m 3)s2 X 2 + K(X 2 - r3) = 0
(2.4)
are functions of s. Solving these equations, we obtain the collocated
and non-collocated transfer functions, E(s)/T(s) and X 2 (s)/r(s) respectively.
21
I(S2 +
(s)
T(s)
X 2 (s)
T(s)
2
2
-
s (S + K(m
Kr
1
Ji(m 2 + M3 ) s2 (M2 + K(4
)
2 +M3(2.5)
J,
+
())
+
(2))
The collocated and noncollocated transfer functions are so named because of the
location of the sensors with respect to the actuator. In the former, a rotary encoder
that measures the rotary angle of the drive is mounted on the drive shaft. In the
latter case, a linear encoder provides feedback signal on the position of the carriage.
2.2.2
Equations with Damping Terms Included
In any dynamical system, there are several mechanisms of energy dissipation and it
is important that we characterize these energy dissipations and quantify the damping
in the system. In this section, we derive a model that accounts for damping in the
system. In our dynamic model, for convenience, we use the familiar viscous dashpot
model, where the damping force is given by C
where ± represents the relative speed
of the masses. Assuming that the overall damping is characterized by C, we can write
+ Cr(rse - sX 2 ) + K(re ±S2E
J
X 2) = T
Ms 2 X 2 + C(sX 2 - rsE) + K(X 2 - rE )
=
0
(2.7)
(2.8)
Note that M=m2 + m 3
The transfer functions are given by
E(s)
T(s)
X 2 (s)
T(S)
MS2 + Cs + K
s 2 (JiMs 2 + C(Ji + Mr 2 )s + K(J + Mr2 ))
-
(Cs + K)r
s 2 (J1 Ms 2 + C(J 1 + Mr 2)s + K(J 1 + Mr 2))
Using the values of the parameters listed in Table 2.1, we can plot the above transfer
functions for the case s =
jw, the Bode plots as shown in Figs. 2-2 and 2-3.
The values of K 1, K 2 and K 3 are based on an arbitrary location of the carriage.
The experimental results are compared with the theoretical results for this particular
22
Table 2.1: Belt parameters
Parameter
Symbol
Value
Units
M2
22.0821
Kg
J
2.234 x 10-4
Kg m 2
0.259
Kg
K1
372528
Nm
K2
1358631
Nm
K3
230967
Nm
r
0.0285
mm
Mass of the carriage
Inertia of the drive pulley
m3 =
Lumped Mass of the idler pulley
Stiffness
Radius of the pulleys
location along the length of travel of the stage. As these values of stiffnesses change
with the location, we expect the poles and zeros on the Bode plots to shift accordingly.
2.3
Effect of Varying the Stiffness and Damping
We have derived closed form expressions and we have plotted the Bode plots for
the collocated and noncollocated transfer functions.
The stiffness of the belt and
the damping in the system are the two most important parameters that govern the
resonant frequencies and the magnitudes of the resonant peaks. We present a short
discussion on the effects of varying the stiffness and damping in the Bode plots of
transfer functions.
2.3.1
Stiffness
Stiffness of the belt is a function of the three factors
1. area, A
2. elasticity modulus, E
3. overall length
23
While varying the area of cross section, one has to always compare the stress values
in the belt, which is a constraint on the design. When attempting to increase the
stiffness by increasing the thickness of the belt t, we may cause very high stresses on
the belt as it bends around the pulley. The bending stresses vary as a function of t/R.
This necessitates an increase in the radius of the pulley and hence the inertia of the
pulley. The effect of an increase in the inertia is to lower the axial resonant frequency.
Hence, we have a set of competing constraints. The change in the frequency response
with the change in the stiffness is plotted in Figs. 2-4 and 2-5.
2.3.2
Damping
A very deterministic way of adding damping in belt drives is yet an area that could be
explored more. Damping has a key role to play on changing the dynamical behaviour
of the system and hence impact the performance of the system. We see the effect of
damping on the transfer functions in the Fig. 2-5. Damping affects the maximum
achievable bandwidth and the degree of robustness of the system. We discuss this in
detail in the Section 2.5.
2.4
Three D.O.F. Model to Include the Pitch Mode
of the Carriage
The belt mounts onto the carriage at a height above the center of mass. Hence, in
addition to the force transmitted to the carriage, there is a torque. This torque causes
the pitching motion of the carriage. Depending on the moment stiffness of the air
bearings, the natural frequency of this mode could be very high or of the same order
as that of the axial resonance. This could affect the closed-loop performance. The
carriage can be modeled as a two-degree-of-freedom system with a translation and
rotation. We are not accounting for the yaw and roll modes. A schematic of the
model is shown in Fig. 2-6. We are assuming that the natural frequencies for the roll
and yaw motions of the carriage remain the same even after coupling with the rest of
24
the system. We will derive these values also at the end of this section.
Equations of Motion
2.4.1
= T
(2.11)
+ a6 - R01 ) = 0
(2.12)
Ka(x 2 + a0 - R01)+ C1(_ 22+ aO- RO1) + M,0 = 0
(2.13)
J 1 01 + KR(ROi-
X2-
a
) + CR(ROi -:
mf2 + K(x 2 + aO - R01 ) + C1(-
I
Here,
01,
1+
2
2
- a)
X2 , and 0 represent the angle of rotation of the drive, translation of the
carriage, and the pitching angle of the carriage respectively. Now, we can develop
a state space model for this dynamical system with three degrees of freedom. The
state variables are the three displacements and the three velocities. Hence, we have
the following
Y1 = 01; Y2 = 61; Y 3 = X2; y4 = Y2; Y5 - 0; Y6
2.4.2
(2.14)
Eigenvalues and Eigenvectors
We write the dynamical equations (Eqs. (2.12) - (2.14)) in the following form.
y = Ay + Bu
(2.15)
z
(2.16)
=
Cy + Du
The eigenvalues of matrix A represent the natural frequencies of the system and the
eigenvectors represent the mode shapes. We have not derived closed-form expressions
for the three-degree-of-freedom model as in the two-mass model that we presented in
Section 2.2. Using the values for various quantities from Table 2.1, we use MATLAB
and solve the equations numerically to obtain the eigenvalues and eigenvectors. The
predicted modes are 0 Hz, 226 Hz, 441 Hz. The rigid-body mode of the system is
the 0 Hz mode. Table 2.2 shows the relative phases of the three degrees of freedom.
From the relative phases of the three degrees of freedom, we deduce the mode shapes
for the three modes.
25
Table 2.2: Mode shape - eigenvectors
Frequency
0 Hz
226 Hz
441 Hz
r01 = ryi
X2 = Y3
magnitude
phase
2.8488 x 10-2
90
2.0081x 10-
5
8.0503 x10-
6
aO = ay5
magnitude
phase
magnitude
phase
2
90
8.0219 x 10-6
90
0
1.0211x IO-
7
180
2.2735 x10-
5
-90
2.8488x 10
180
2.5012x 10-7
90
7
1.0027x 10-
-90
The mode at 0 Hz is the rigid-body mode, when everything is moving in phase.
As we go to higher frequencies, we find that the carriage's translation becomes out of
phase with its rotation and drive's rotation at 226 Hz. At 441 Hz, the drive is out of
phase with the carriage's degrees of freedom.
2.4.3
Transfer Function
The transfer function representations can be obtained by taking Laplace transforms
of equations of motion. From the state space model (Eqs.
(2.15) and (2.16)), we
obtain the transfer function H(s) given by
H(s) = C(sI - A)- B + D
(2.17)
This works for most SISO systems. The matrices are
/
0
1
0
0
0
0
J
KR
Ji
CIR
J
KaR
J
CiaR
J
0
0
0
1
0
0
KR
m
CIR
Ka
m
Cia
m
0
0
KR
A -
KaR
7P_
2
m
CaR
K
m
0
Ka
'p
m
0
0
_P1
(2.18)
1
2
(Mp+Ka )
'p
26
Ji
Cia
'p
I
0
1
B
(2.19)
0
0
0
0
For the case where the sensor and the actuator are collocated, the measured output
variable is 01, i.e., z = yi. Therefore,
C
1
=
D =
0 0
0 0
0
(2.20)
(2.21)
0
The Bode plots corresponding to the collocated and noncollocated transfer functions
are as shown in Figs. 2-9 and 2-10.
2.4.4
Roll and Yaw Modes
In the 3 d.o.f. model, we have modeled the pitch mode. But the carriage has yaw
and roll modes also. The roll motion is orthogonal to the direction of travel of the
carriage. Though the yaw motion has a projection in the axial direction (direction
of travel), we have not included it in the model assuming that there is no significant
change in the natural frequency of this mode when added to the rest of the system.
Therefore, we estimate the natural frequency for the yaw and roll motions of the
carriage treating it as a rigid body. The natural frequency for these motions is given
by
1
K(2.22)
2 ir
I!
Km is the net moment stiffness provided by the air bearings for the roll or yaw motion
of the carriage. Km depends on the configuration.
Km
K l 2 + Kil
2
2
27
2
(2.23)
Table 2.3: The rigid body modes
Stiffness
Inertia
Natural Frequency
Yaw
1.5608 X 106 Nm
0.4779 Kg-M 2
288Hz
Pitch
1.5128 X 106 Nm
0.1982Kg-M 2
439Hz
3409420 Nm
0.5241Kg-m 2
406Hz
Roll
There are 2 pairs of air bearings preloaded against each other and they are separated
by a distance of l.
Hence, the summation of two terms
K 1 11.2
The inertia is the
moment of inertia of the carriage about the corresponding axis of rigid-body rotation
of the carriage. The Table 2.3 lists the calculated theoretical values for the rigid body
modes of the carriage.
2.5
Bandwidth of the Belt Drive
The classical definition of bandwidth is the maximum frequency at which the output
of a system will track an input sinusoid in a satisfactory manner. The closed loop
transfer function is given by
Y(s)
G(s)H(s)
R(s)
1 + G(s)H(s)
A plot of this would have a value of 1 at low excitation frequencies and a value
G(s)H(s) at higher excitation frequencies. The frequency which marks this transition is the bandwidth. For systems that have a continuous roll-off (low-pass filter
behaviour), the cross-over frequency is a good approximation for the bandwidth of
the system. In general, the cross-over frequency is defined as the frequency at which
the gain is 0 dB or the magnitude is 1.
2.5.1
Collocated Control
We are interested in precisely positioning the payload (or) the carriage i.e., m 2 . To
achieve this, we can either use
28
1. Collocated control: Feedback from the rotary encoder mounted on the motor
shaft (drive pulley).
2. Noncollocated control: Feedback from the linear encoder reading the position of the carriage in the direction of travel or axial direction.
There are some limitations in using collocated control to precisely position the carriage.
1. Going by the definition of the bandwidth in Section 2.5, we can deduce that
the collocated control can theoretically give an infinite bandwidth precision
machine i.e., the carriage will track the input over all frequencies. But this is
not really true. The collocated transfer function is given by
interested in X 2 or the position of the carriage
(M 2 ).
0/.
But, we are
Therefore, we look at
the transfer function X 2 /X 1 , the ratio of the carriage position X 2 and motor
position X 1. This transfer function is shown in the Fig. 2-8. We see that the
roll-off behaviour starts after the peak in the magnitude plot, which occurs at
the frequency
k/rm2 . This means that the carriage position does not follow
the input signal beyond this frequency. Hence, the frequency range is limited
to this frequency. The frequency
k/m
2
is the frequency of the zero of the
collocated transfer function shown in Fig. 2-2.
2. (Refer to Fig.
2-7) The disturbance rejection transfer function X 2 /D looks
similar to the transfer function in the Fig. 2-8.
The roll-off in the transfer
function means that the disturbances get amplified, and is not desirable.
3. Microslip between the belt and the pulley leads to a cumulative error. Due to
this error, it is difficult to determine the position of the carriage from the rotary
encoder signal.
Therefore, it is difficult to achieve precise positioning of the carriage through
collocated control.
In the next section, we present a discussion on the maximum
achievable bandwidths through noncollocated control.
29
2.5.2
Noncollocated control
Applying the definition of bandwidth in Section 2.5 to the noncollocated transfer
function, we have several possible cross-over frequencies as shown in the Fig. 2-3.
Of these 5 different crossovers possible, crossover of type 2 is the most practical.
We present arguments supporting this optimality of crossover of type 2 in the next
section. Hence, as a rule of thumb, "Draw a line from the resonant peak and locate
the frequency at which it intersects the transferfunction. This is the bandwidth of the
system". But this crossover of type 2 is not realistic due to robustness issues which
is the topic of the next section.
2.5.3
Robustness
Detailed discussions on the detrimental effects of cross overs of type 3 are presented
in
[3].
We will briefly summarize results to emphasize the fact that one cannot
conclusively derive results on stability by just looking at the Bode plots. Nyquist
plots give a more complete picture of stability and the stability margins. These are
important to get insights about rather abstract mathematical definitions of stability
robustness, the small gain theorem, and so on, which are the foundations of robust
control. In Fig. 2-11, unit circle intersects the loop transmission at three points.
These are the three crossover frequencies corresponding to type 2, 3, and 4 crossovers
shown in Fig. 2-3. Going by Bode plots in Fig. 2-3, it appears that at points B and
C, gain is unity and the phase is less than -1800. Hence, we could say that the system
is unstable. But, Nyquist criterion for stability when applied to a crossover of type
3 shows that the system is stable always since the loop transmission L = GHdoes
not encircle the -1 point. Hence, it appears that crossover of type 3 gives us higher
bandwidth.
But, due to uncertainties in a system, crossover frequency of type 3
does not work in reality. We have to design systems with robustness, i.e. systems
that continue to perform satisfactorily even in the presence of uncertainties.
The
stability problem in robust control is about designing a controller that works for a set
of plants rather than a given plant. This is a more realistic description of a physical
30
system as we often do not have an exact description of the plant. Hence, our metrics
for performance and stability should always address robustness issues. To give an
example, our description of damping of the system is not always accurate. If we use
a theoretical value for damping and estimate the bandwidth using our rule of thumb
(cross over of type 2) and it turns out in practice that we overestimated the damping,
we end up making a cross over of type 3. But, cross over of type 3 is detrimental
as it has a very little phase margin and hence not robust. The familiar solution to
this problem is to add a lead compensator to increase the phase margin. Adding
a lead compensator will add phase at the cross over but make the system unstable
at resonance, i.e, loop transmission will encircle the -1 point in the nyquist diagram
(refer to Fig. 2-12). The next best possible cross over would be of type 2. But the
crossover of type 2 is also not very robust, if our plant model had some uncertainties.
For example, how do we accommodate an overestimated value of damping?. Hence we
introduce a gain margin at resonance. The resonance gain margin (RGM) is defined
as the factor by which the loop transmission has to be multiplied without resulting
in multiple cross overs at resonance.
This is shown in Fig. 2-11.
discussion on robust stability, the reader is referred to Dahleh
For a detailed
[2] and Doyle
[6].
In summary, crossover of type 1 works best in reality. The stability margins are
important because our ultimate objective is to be able to derive synthesis rules for
designing a high bandwidth belt-driven servomechanism. In other words, the designer
should be able to size the various components like belt, motor inertia etc., to meet
the performance criteria such as bandwidth. Hence, we have to derive closed-form
expressions for maximum achievable bandwidth for a robust belt-driven system.
2.5.4
Crossover of Type 5
The phase has already dropped to -3600
at the crossover frequency (Type 5 in Fig.
2-3). To keep the system stable, we need to add a phase of atleast 1800 which would
lead to high gains, often leading to actuator saturation. This point can be explained
as follows.
A phase increase of 1800 would require a compensator with two zeros
ahead of the resonance. This compensator would be accompanied with two poles and
31
hence
akes te
form(8+Z) 2 Ti
hence takes the form
,p
.
This means amplifying the input to the amplifier at the
rate of 40 dB/decade, which will lead to actuator saturation. In practice, this method
rarely works.
2.6
Chapter Summary
In this chapter, we have developed a lumped-parameter model for the belt drive.
We have also presented closed-form expressions for the collocated and noncollocated
transfer functions. We have presented a discussion of the maximum achievable bandwidths of the belt-driven system with certain robustness in the presence of uncertainties in the system modeling and other errors.
32
Bode Diagram
100
50
0
-50
-100
150
.45 -
90c
13
0
18
10
10,
10
Frequency (rad/sec)
Figure 2-2: Collocated transfer function
33
0
1
10
Bode Diagram
100
.Typ~e 1
Type 2
Z
CM
Ca
2
RGM
Type 3
A
01
Type 4
Type 5
501
180
135
-a
CD
Ca
.rCL
45
0
10
10
102
10
10
Frequency (rad/sec)
Figure 2-3: Noncollocated transfer function
34
10
Bode Diagram
100
50
CO
2
-50
135
F41fbi i
'
90
- -
-L
r
0~
45
0
10
100
10
104
Frequency (rad/sec)
Figure 2-4: The effect of change in stiffness in the noncollocated transfer function
35
Bode Diagram
50I
100
Drop in resonant peak due to increa 3ed
damping
50
CO
2
0
-Ii
.50
135
90
a
45
10
10
10
10
10
104
Frequency (rad/sec)
Figure 2-5: The effect of damping in the noncollocated transfer function
m
1
a6
X2
Figure 2-6: Three-degrees-of-freedom model
36
carriage
disturbance
D(s)
motor
distu bance
+
X(s)
+
H(s)
+
Y(s)
G(s)
F
Figure 2-7: Closed-loop servomechanism
Bode Diagram
140
10
80,
60
401
2i
-45
-90
135
-
10
103
10
Frequency (Hz)
Figure 2-8: Transfer function xi
x
37
Bode Diagram
50
31
-
3150
270 -
22180
135 10
10
10
10
Frequency (rad/aec)
Figure 2-9: 3 DOE model
-
collocated transfer function
38
Bode Diagram
-50
100
-
Q>
2
-20r250-
-225 -
(L
270
-315
10
10
10
10
10
Frequency (rad/sec)
Figure 2-10: 3 DOF model - noncollocated transfer function
39
10
hn(L(jw))
Unit Circle for
Unit Circlefa
,Cossover (3):
Unit Circle fot
Re(
a~j))
Crossover (4)
Figure 2-11: Nyquist representation of crossover frequencies, Varanasi
40
[3]
Incg2asing
has
Figure 2-12: Adding phase at cross over 3 leading to instability, Varanasi [3]
41
Robustness
CMIL
Margin
Nominal P~lant
PM
R*~(LUw))
2sin
M/2)
unit eircj
D curve
Figure 2-13: Nyquist interpretation of robust gain margin (RGM) and phase margin
(PM), Varanasi [3]
42
Chapter 3
Energy Dissipation due to Slip in
Belt Drive: Damping and Loss
Factor Estimates
3.1
Introduction
This chapter presents a model for microslip in belt drives and estimates for the damping due to microslip. We characterize the damping by the loss factor which is defined
as the ratio of the energy loss and the maximum potential energy during one cycle
of harmonic motion. We are interested in understanding how the slip region varies
under harmonic excitations. We explain the origin of microslip and model the slip
region on the belt-pulley interface as a deformable control volume. Using the mass
conservation, we obtain the rate at which the slip region changes.
The size of the
slip arc is given by the capstan formula. We obtain expressions for the loss factor in
terms of parameters like belt preload To, the drive ratio n, the length of the drive L,
the cross section A, and friction coefficient p. The loss factor estimates show that
the damping one can achieve due to microslip is not very significant. The loss factor
is estimated to be of the order of 10-% for a typical configuration.
43
3.2
Notation
#
Slip are
T
Tangential traction
- Stress
p Poisson's ratio
G Rigidity modulus
q Traction distribution
p Normal pressure distribution
P Normal load
Q
Tangential load
6 Tangential displacement
To Belt preload (or) pre-tension
R 2 Radius of the driving pulley
R 1 Radius of the driven pulley
V2 Peripheral speed of the driving pulley
V1 Peripheral speed of the driving pulley
A Area of cross section of the belt
p Density of the belt material
4 Rate of change of slip arc
p Coefficient of friction
3.3
Microslip - Background
The earliest investigation of slip and the associated energy loss was by Mindlin et
al
[7].
In this section, we will elucidate some of the results from their work. We
also describe the origin of slip and a method used by Mindlin et al for estimating
the energy loss due to slip. They first studied the problem where a pair of elastic
bodies were pressed against each other and a small tangential force is applied across
the elliptic contact surface [8].
44
3.3.1
Microslip and Sliding
A tangential force whose magnitude is less than the force of limiting friction, when
applied to two bodies pressed into contact, will not give rise to a sliding motion.
But this force will induce tangential surface tractions which arise from a combination
of normal and tangential forces; this does not cause the bodies to slide relative to
each other. When a tangential force
Q is applied to two bodies of non-conformal
geometries (refer to Fig. 3-1) pressed against each other with a normal force P, the
tangential force
Q deforms the bodies in shear. This causes the points on the contact
surface to have tangential displacements relative to the distant points on the bodies.
There will be atleast one point which is at rest as long as there is no gross sliding
motion. But, there are points which slip even though
Q< pP, i.e., there is some slip
even in the absence of gross-sliding. This is referred to as microslip. This slip can be
mathematically expressed as
={ui - 6X1} - {ux 2 slip, s=,
(3.1)
x22
where 6 21 and 6 x2 are displacements of points far away from the contact surface, which
are used to define the tangential compliance.
3.3.2
Boundary Conditions
In order to solve the boundary value problem of two nonconformal spheres in contact,
we need to state the boundary conditions that distinguish the stick and slip regions.
These boundary conditions are
stick region
slip region
sx =0;
=
Ux1 -
Ux2 =
q(x, y) = [p(x, y)
45
6
x1 -
6x2
(3.2)
(3.3)
3.3.3
Assumptions
Effect of the Tangential Force
Q on Hertzian Distribution of Normal Pres-
sure, p(x, y)
A normal force pressing the two bodies together is the Hertzian contact problem
[11]. When a tangential traction exists on the contact surface, we could say that, if
the two solids have the same elastic constants, any tangential traction transmitted
between them gives rise to equal and opposite normal displacements of any point on
the interface and it does not affect the distribution of normal pressure predicted by
Hertz theory. This is because the normal displacements due to these tractions are
proportional to the respective values of
(1-2v)
G
G1
Iv
-
zi (X, y) =
Therefore, we have
G2
2v Uz2(,yY
(3.4)
where, uz refers to displacements in the normal direction. But even between different
materials, the influence of tangential tractions on the distribution of normal pressure
is generally small and it is ignored in all the analysis presented in the previous section.
Amonton's Law
Amonton's Law of static friction is applicable at each elementary area of the interface.
It can be stated as
IQ|
Iq(x, y)I
p(x, y)
3.3.4
(3.5)
-
Mindlin's Solution and Results
Hence the problem of two spheres (refer to Fig. 3-1) solved by Mindlin is a boundary
value problem where the tangential displacement u. and normal pressure p(x, y) are
given over part of the boundary, i.e., the contact region and the three components
of traction (=O) are given over the rest. The solution of this problem assuming 'no
slip' through out the contact region leads to the following distribution of tangential
46
P
-- +Q
Figure 3-1: Two spheres in contact under normal and tangential load
traction over the surface.
T
=
27ra(a 2 - r2 ,
(3.6)
r< a
The tangential traction is everywhere parallel to the direction of the applied force.
The contours of constant tangitial traction are concentric circles. The displacement
is linear and the tangential compliance is given by
1 2- v
8a G 1
2 - v2
C-1-(
where v = Poisson's ratio and G
=
G2
)
(3.7)
rigidity modulus. We see that at the boundary
of the contact area, i.e., at r = a, the tangential traction goes to infinity. But, we
presume that the tangential traction cannot exceed p times the normal traction if
there is no slip. Hence, some portion of the contact region has to slip. Assuming
that there is a slip region and an adherent region, Mindlin solved the boundary value
problem using the second boundary condition given by Eq. (3.3) over a part of the
boundary. The following are some of his results. The inner radius of the annulus of
the slip region is given by
c
=
Q)
a(1 -
(3.8)
pP
From this expression we can see that when the applied tangential force
Q
exceeds PP,
c goes to zero and gross sliding occurs, which we are familiar with. The distribution
of the tangential traction on the contact surface is
T
=
3p-P
3
r
=
23[(a2
2_21
270 (a2
c< r <a
r2),
- r2
-
47
(C
- r2]
(3.9)
r < c
(3.10)
and the displacement of distant points w.r.t the uniform displacement of the adhered
portion is
S= 3
(2 - v)p-P
Q
_
[1 - (I - P)]
16[pa
(1-P_
(3.11)
The tangential compliance for this configuration can be derived as
2 - v
Q 1
)
(1 8pa
pP
(3.12)
Note that in this solution the compliance is a function of
Q, i.e., the Q-6 curve is non-
C8
dJ
dQ
linear. Considering a case of cyclic loading, where the normal force is kept constant
and the tangential force is varied, the expressions for the traction distributions, compliance for loading and unloading and displacements have been derived by Mindlin
[8]. We can see a hysteresis effect and the associated energy loss due to slip over one
cycle is given by
9( 2 - V)p2 p2
{1
1OEa
QmaT
-
(1 --
PP
5Qmax [1+
)3
-
6pQ
[1
+ (1
QMax)
-
pP
)3]}
(3.13)
Experimental results for hard steel spheres pressed against flats are in good agreement with the above results and confirm the energy dissipation due to microslip [9].
In this paper, Johnson has presented the observations from the damping tests conducted to obtain the energy dissipation due to microslip. In the dynamic tests, he has
demonstrated the marked distinctions between the microslip and gross sliding. In the
regime of microslip, the oscillations are harmonic and are about an unvarying datum
position. When
Q
exceeds pP, slide ensues and unsteady non-harmonic motion is
setup. Following this work, there were other researchers who demonstrated the validity of the theory proposed by Mindlin [10]. The experimental studies investigating
the effect of oblique forces and their angles of inclination w.r.t. the plane of contact
were by Johnson [11].
So far, we have discussed the theoretical framework for studying microslip under
static conditions when the bodies are in contact and are at rest, even though the
forces could be oscillating in magnitude.
48
3.4
Belt Drive - Microslip
In this section, we discuss the origin of microslip in the belt drive and derive expressions for energy dissipation when the system is driven by harmonic excitations. This
problem is different from the microslip under static conditions that we have discussed
in the previous sections. In the traction drive under study, the contact surfaces are
moving relative to each other. The boundary condition that defines the slip region
is different in this problem when compared with the one given by Eq. (3.3) and it is
given as A_ = 0 in the stick region. Different components of velocities occur in the
expression for slip velocity
s,
depending on the complexity of the configuration. This
includes rolling, spinning, sliding, and so on. A detailed discussion of the microslip
in rolling elastic bodies in contact is done by Johnson [12].
3.4.1
Motivation
Belt-driven servomechanisms are widely employed in precise positioning applications
which include semiconductor and optical industries. The most important limiting
factors on the performance of these precision machines arise from the inherent dynamics of the system. Hence, in the design of such servomechanisms, a complete
understanding of the dynamics of the system is essential. This would help us derive
synthesis rules for the design of such drives to achieve high bandwidth, accelerations,
and speeds. Damping plays a very important role in the stability and performance
of the belt-driven servos. For example, a well-damped resonance peak would help us
achieve high crossover frequencies and hence high bandwidth
[3].
there is some energy loss when the belt slips on the pulley [12].
In a belt drive,
Researchers have
worked on modeling the slip and obtaining the power loss and efficiency in the context
of power transmission
[13, 14]. These researchers study the mechanics of a steadily
rotating belt drive. Our objective is to understand the mechanics of energy dissipation under harmonic excitations and derive an analytical expression for the loss factor
in the belt drive.
49
3.4.2
Loss Factor - Definition
[15]
Loss factor is a measure of the damping in a system. A vibrating system may have
different types of energy dissipation mechanisms and their mathematical descriptions
in terms of the damping force are quite complicated. Instead, we can characterize
damping by the amount of energy dissipated under steady harmonic motion. The
most common measure of this dissipation is the loss factor q, which is formed by
taking the ratio of the average energy dissipated W per radian to the peak potential
energy U during a cycle. That is
w
(3.14)
2WU
3.4.3
Origin of Microslip in Belt Drives
Due to the compliance of the belt, the belt stretches. The tight side has a higher
tensile force and hence stretches more than the slack side. This explains the origin of
the microslip in the belt drive. To develop a complete picture of how the slip occurs
and locations where the belt slips, we present the following arguments, discussed
in detail by Johnson
[11].
Consider an infinitesimal element of the belt dx.
Let
the tensile strain experienced by that element be c. Using the familiar constitutive
relation
=
dl
Ec
(3.15)
(1 + E)dx
(3.16)
Differentiating the above expression w.r.t. time, we obtain
dl
dx
V=dt
(1 + E) dt
-dx
+ E )(3.17)
dt
where L defines the unstretched velocity of the belt. This clearly indicates that the
tight side of the belt moves faster than the slack side of the belt. Now we obtain
expressions for the speeds of the belt on the tight side and the slack side as V and
50
V2 respectively given by
1+
V2E (+
TO + T1 dx
FA= EA
dt
T - T 1 dx
EA
)
dt
(3.18)
(3.19)
Consider the instant of time when the direction of motion of the driven and driving
pulleys are as shown in the Fig. 3-2 The frictional traction pulls the belt forward on
the driven pulley and it opposes the belt motion on the driving pulley. We also know
that the direction of the frictional traction is such that it opposes the direction of
slip. Therefore,
1. The driving Pulley must be moving faster than the belt in the slip arc.
2. The driven Pulley must be moving slower than the belt in the slip arc.
Hence we deduce that the belt adheres where it runs onto the pulley and it slips as
it leaves the pulley on both driver and driven pulleys.
3.5
Model: Deforming Control Volume
We are interested in estimating the energy loss during one cycle of harmonic motion
of the form eiwt. When the direction of rotation changes, the location of the stick
arcs shift to satisfy the condition stated at the end of the previous section, i.e., the
belt adheres where it runs onto the pulley. The slip arcs are expected to vary
with time as the harmonic input varies from a maximum to a minimum. We propose
a deforming control volume model to accommodate the above variations in slip arcs.
The control volume is as shown in the Fig. 3-2.
3.5.1
Slip Rate: Mass Conservation
Applying the continuity equation for this deformable control volume which is moving
relative to the pulley, we obtain
dt
ddr
10a
t
+ I>pV.VdQ
4a
51
(3.20)
Driven Pulley
Driving Pulley
22
V,
R2
R2
R,
V,
U
-
TO +T1
dotted lines on the driven and the driving pulleys
show the control surface enclosing the control volume
Figure 3-2: A typical belt drive showing the control volumes on the driven and driving
pulleys
where the integration is over the volume represented by Q. The first term in the Eq.
(3.20) goes to zero since 9at = 0. The conservation of mass reduces
ddt =
0
Applying divergence theorem, the second term on the right side of the Eq. (3.20)
reduces to
pjI .ds = Ap(V - V2) - Ap(V 2 - R 2 )
JaQ
=
0
(3.21)
Here, &Q represents the control surface. Thus,
pAR 2 (5
-
02) =
Ap(V 2 - V1 )
(3.22)
Similarly considering a control volume in the slip arc of the driving pulley we obtain,
pAR 1 (
-
01) = Ap(V - V2)
We assume that the rate of change of slip arcs
(3.23)
(4) of the driven and driving pulleys
are equal. Equating these expressions,
R2(0-
2) -
-R
01)
(3.24)
+ R2d2
R1 +R2
(3.25)
R1
1
52
1
(O
-
3.5.2
Forces
Assuming that the preload applied to the belt is 2TO, the tight and the slack sides of
the belt experience tensions To + T, and To - T respectively. To solve for the relation
between the forces, let us consider an infinitesimal element on the driven pulley as
shown in the Fig. 3-3.
T
q *aN
T+dT
Figure 3-3: Free body diagram to show the forces on the belt
This leads to the well known Capstan formula which defines the slip arc
(#)
implicit in the following expression
TO + Ti=
(3.26)
TO - T1
3.5.3
Energy Loss: Energy Balance
We apply the first law of thermodynamics to the deforming control volume.
dQ
dt
= dE
dt
dW
dt
(3.27)
The above expression represents the rate of heat addition as the sum of the rate of
change of internal energy and the rate at which the forces do work. Neglecting the
heat addition, the expression reduces to
(t )ds =dE
Ia.U
dt
(3.28)
The term on the left side of the Eq. (3.28) is the rate at which the forces on the
boundary of the control surface do work. The forces on the boundary are the tension
53
in the belt and the frictional traction. Hence, the term on the left hand side of Eq.
(3.28) can be expanded as
(57)ds = (T. + T 1)(V - R 2 d 2 )
(To - T1)(V2-
-
R 2 5) + j(.)ds
(3.29)
Note that &Q 2 is the region of the control surface on the driven pulley where the
belt and pulley slide against each other. We are interested in evaluating the term
fa 2 (ji.)ds during one cycle of steady harmonic motion of the belt drive. The internal
energy term can be expanded into
dE
dt
where
ddj
dQ+ dJptdQ
dt JQ 2
tQ
(3.30)
is the internal energy per unit volume which depends on the material of the
belt. The integral of the rate of change of internal energy when evaluated over a cycle
goes to zero, i.e.,
S
Idt
.dt = o
(3.31)
Applying the same principle of energy balance for the driving pulley, we obtain
(jQui)ds
=
(To - T 1 )(V 2 - R 1 1 ) - (To + T 1 )(V - Riq) + j
(i)ds
(3.32)
(qii)ds]dt ==0
(3.33)
Summing up the rate of energy loss on both pulleys, we obtain
[T 1 (R1 - R 2 )(V1 + V2) ++ti(R21 + R 2 )(V
R, + R2
-
V2 )
+
+ Ll+-9Q2
Rearranging the terms, we obtain
fr|
fa (q-.-)ds]dt =
Q1 +aQ2
V2 R2
2T1 V22
-
V1 R1
-VR
R1 + R 2
dt
(3.34)
The left hand side of Eq. (3.34) represents the energy dissipation. Performing this
integration gives the energy dissipated over a cycle of steady harmonic motion given
by 01=asin wt of the driver and 02 =bsinwt of the driven pulley. As we explained
earlier, location of the slip regions is directly linked with the direction of the belt drive
motion. When direction of motion reverses, as would happen in a steady harmonic
motion, the location of the slip arcs shifts on both pulleys. But energy is dissipated
54
always since the frictional traction opposes motion. Hence we integrate over quarter
cycle and multiply the result by four to get the energy dissipated during one cycle.
Given 01, 02 we obtain expressions for 4, V1 , V2, Ti as follows:
(Ria = R 2 b)w cos wt
R1 + R2
(3.35)
#0 sinwt
(3.36)
R 1a+R 2 b
00 R, + R2
(3.37)
Riaw cos wt
(3.38)
V2 = R 2 b cos wt
(3.39)
Therefore,
0 =
assuming
#
0 at t = 0. where
Also
V1
Substituting the expression for
-
#
given by Eq. (3.36) into the Capstan formula,
we obtain the time evolution of T 1 , which is
1
1 (t)) T(t) = TO(exp(
exp(pL#(t)) + I
(3.40)
Hence, the energy loss is
W = 4 x 2TO
( exp(Po sin wt) - 1 R2 b - R
j
exp(pio sin wt) + 1
R1 + R 2
s
(3.41)
Making the substitution x=sinwt reduces the integral to
8TOf4
0
(
2+ (WOX) - I Rb - Riad
exp
exp (poox) + I R, + R2
(3.42)
Integrating the above expression, we get
- Ra 2 ln( e
W= 8T Rib
R 1 + R 2 POO
55
2
+ e
2
2
(3.43)
3.5.4
Maximum Potential Energy
When a harmonic input drives the pulleys, the tension in the belt also varies harmonically. As a result, the elastic potential energy stored in the belt oscillates between a
maximum and minimum periodically. We are interested in the maximum value of this
potential energy. As we explained earlier, by taking a ratio of the energy dissipated
over a cycle to the maximum potential energy stored in the belt, we obtain a measure
of the damping. We know that energy per unit volume is given by the product of
the stress and strain o-e which when integrated over the volume gives the total strain
energy or the potential energy. Mathematically, this is given by
U = J-e dQ
Potential Energy U
=
1
(3.44)
Uj where Uis are the potential energies stored in the
regions shown in the Fig. 3-2. We write down the individual expressions for Us:
U
=
U2
=
(To-T) 2
( 0 - T1) (L + R2 (27r AE
0
x)2 R, dy
Ti)+
AE
((T
(3.45)
-))
(3.46)
(3.47)
where the integration is over the slip arc of the driver.
U2 =
(To+T1)2
1-
AE
AE
1(
2y
)
(3.48)
)
(3.49)
Similarly
U4 =
(To-T
1)
AE
2
R2(
e 2 '_1
2/p
Using Capstan formula, the expression for U4 can be rearranged as follows
U3
=
U4
=
2
(T + T1 )
AE
(L + R1 (a-
-- T,) 2
R2(1
AE
e
2y
))
(3.50)
(3.51)
)
Summing up the expressions and rearranging them we obtain,
2L
U=AE
+
)+
T
R2(27r - ae - #) (TR,
AE
( - T1)2 +
R 1 (a - )(
AE (TO +T1)2
+ R2 (
2pAE (
56
2.o
+ T1)2(l _ e
(3.52)
The approximation L >> R 1 , R 2 reduces the expression for potential energy to
2L
2
2
U = A(To2 +T)
(3.53)
which has a maximum value of
(3.54)
(T 2 + T2
Umax =2
where Tm is the maximum value of the tension in the belt.
(3.55)
(
- 1
T(t) = To (exp([#(t))
exp(#tc(t)) +
This function is increasing with
#
and its maximum value is
(3.56)
)
P0
exp (p 0 ) +1
TM= TO ( ex
3.5.5
Loss Factor
Recalling Eq. (3.14), we take the ratio of Eq. (3.43) and (3.54)
=
2 A E R 2b - R 2a
A
2 bR
rr To L(R 1 + R 2 )
2In(
2-
e
n (o
(3.57)
1 + {e0 -1 }2
To simplify the above expression, we substitute the drive ratio n
=g
and the ratio
of the amplitudes m =a. Hence,
77 =
2 AE R 1 2(n - m) ln('
ir To L p(n+m) 1+{
)
2
-}2
(3.58)
From the expressions for rate of change of slip arc (#), we can show that
n
m n+ 2
2n + 1
(3.59)
This condition arises due to the assumption that the rate of change of slip arcs of
both driven and driving pulleys are equal. Hence,
#0
3nb
2n
2n + I
57
(3.60)
and
2AE R1
7
To L
2(n-1)
3 /t(n+1)
2 AE R1 2(n - 1)
7r
3.5.6
To
ln(e
2 e
1+{ jj-I}2
n{sinh(
3.6)}
2
L 3p (n + 1) 1 + {(tanh( 2 3 b
1 ))2
-.1
Results
The loss factor varies inversely with the preload To and increases as the drive ratio
and the friction coefficient 1t increase.
Also, 77 decreases as the distance between
the pulleys L increases. These trends are shown in the Fig. 3-4. From the graph
we see that the loss factor due to microslip is very low for a typical configuration.
Hence, we conclude that the microslip between the pulley and the belt in the beltdriven system does not introduce significant damping in the system to achieve good
dynamic performance.
3.6
Chapter Summary
In Chapter 2, we have stressed the importance of damping in the dynamic performance
of high speed precision machines that have high bandwidth. In this chapter, we have
developed a model for microslip phenomenon that has been reported in belt drives.
The analytical expressions that we have developed can serve as tools to add damping
deterministically in the belt-driven system. We understand that the damping that
arises from microslip is negligible and hence it is important to think along different
directions to add damping in belt-driven systems to achieve the best possible dynamic
performance.
58
x 10'
1.81.6-
2 10
E =7.0 x 1 N/ m2
A =5 x 100 mm
I
I
b =1
R1=200
Ty30
N
1.41.2 0
1-
0.8n
0.6
0
0.4
0.2-
0.1
0.15
0.2
0.25
0.4
0.35
0.3
friction coefficient
Figure 3-4: Variation of loss factor with friction coefficient
drive ratio
n
59
0.5
0.45
t
0.55
0.6
for different values of
60
Chapter 4
Design of the Belt Drive
4.1
Introduction
A positioning mechanism is designed to meet several specifications which include
the range of travel, positioning accuracy, maximum velocity and acceleration. When
designing a servomechanism that operates in closed loop, the sensors that provide
feedback signals also have an important role to play. As has been mentioned earlier,
the design of a high-bandwidth belt-driven servomechanism is our objective. The
primary compliance in these machines arises from the belt. The axial resonance associated with this compliance poses a serious limitation on the maximum achievable
bandwidth. The stiffness of the structural loop drops if the parts of the machine like
the coupling, bearings, or bolted joints are not designed and assembled appropriately.
This affects the bandwidth and hence the performance of the machine. Hence the mechanical design and assembly have a significant impact on the dynamic performance
of the machine. This chapter includes a layout of the design and assembly process of
the belt-driven positioning stage.
4.2
The Loop
In this section, we present an outline of the machine with its components. In the
subsequent sections, we present details of the design of individual components and
61
the assembly. The parts of the machine include (refer to Fig. 4-1)
1. Actuator (BM 500E DC Brushless Motor)
2. Coupling
3. Drive Pulley
4. Belt
5. Payload (Carriage)
6. Idler Pulley
7. Air Bearings
8. Angular Contact Bearings
9. Lockwashers
10. Locknut
In addition to these, we have mounted a linear encoder on the machine base to provide
feedback on the linear position of the carriage and the brushless motor has a built-in
rotary encoder.
4.2.1
Actuator
The actuator in the system is a brushless DC servomotor BM500E from Aerotech.
These brushless servo motors have high energy neodymium-iron-boron magnets and
low-inertia rotors. These are suitable for high performance applications. The principle
of operation of a motor is fairly straightforward. When a fixed magnetic field setup by
permanent magnets interacts with the current carrying conductors in a rotor winding,
the rotor is set in motion. The critical issue is to reverse the current vector as the
magnetic field reverses direction. This is possible only if there is a mechanism which
routes the current in the conductor along the appropriate direction depending on the
position of the rotor relative to the magnetic field. This is referred to as commutation.
62
Table 4.1: Specifications: BM500E
Torque Constant
0.19 N-m
Rotor moment of inertia
Maximum acceleration
/
Amp
13.9 x 10-5 Kg-m2
65000 rad/sec 2
8000 rpm
Maximum speed
In DC servo motors, commutation is achieved through mechanical brushes. These
have serious limitations and they need constant maintenance. Also, the construction
of these motors require the commutators to rotate. Hence the inertia in motion is
high and this affects the dynamic performance of the permanent magnet DC(PMDC)
motors which have brushes. There are heat transfer issues since the heat generation
in these motors are high due to mechanical contacts. The solution to these problems
is the brushless motor which uses electronic commutation. These motors have rotor
position sensors which control the commutation signals. The brushless motors have
the following advantages
1. low torque ripple
2. low heat generation and better heat transfer path since the armature windings
are in the stator
3. very high speeds and accelerations due to low inertia
The detailed specifications of the BM500E motor are as listed in the Table. 4.1
The dynamic characteristic of the motor is very critical in motion control applications. So, it is always important to do a sinesweep measurement of the motor to
ensure that the motor behaves as expected.
63
Motor Transfer Function
An ideal servomotor could be modeled as an inertia in motion. Hence its transfer
function is given by
1
J5 2
H(s)
(4.1)
For more details on brushless motors, the reader is referred to [18]. We present
more on this topic in Appendix A
4.2.2
Coupling
The coupling connects the motor shaft to the pulley shaft.
The coupling should
have very high torsional stiffness. The torsional rigidity of the coupling could be a
potential limiting factor on the performance of the system. Referring to Chapter 2,
while modeling the dynamics we have lumped the rotor inertia of the motor and the
pulley inertia together. In this model, the implicit assumption is that the coupling is
several orders of magnitude stiffer and does not affect the dynamics of the machine.
This would be invalid in case the coupling were compliant. If the torsional rigidity of
the coupling is Ct and the inertias of the pulley and the motor are J1 and Jm, then
there is a resonance at the frequency given by
ct( i+
(4.2)
-- )
Hence, while designing for stiffness, the coupling has to be torsionally rigid. In our
design, we have used bellow couplings from R+W
[21].
The specifications of the
coupling are listed in the Table 4.2.
4.2.3
Air Bearings
For very high precision and high speed applications where friction is undesirable,
air bearings could be used. As the name suggests, these bearings utilize a thin film
of pressurized air to provide a zero friction load bearing interface between surfaces,
that would otherwise be in contact with each other. Eliminating the contact using air
64
Table 4.2: Coupling dimensions and specifications
Overall length
59mm
inner diameter
8-28 mm
outer diameter
49mm
20 X 10 3 Nm/rad
Torsional stiffness
1mm
axial misalignment
lateral misalignment
0.15 mm
bearings provides several advantages. The reader is referred to [20, 22] to understand
the physics behind air bearings
Selection of the air bearings
The location of the air bearings on the carriage determines the moment stiffness for
the pitch, roll and yaw motions. The natural frequencies of the pitch, roll and yaw
modes should be very high and should stay outside the bandwidth of the drive. Table
2.3 lists the theoretical values for the rigid body modes.
The stiffness of the air
bearings varies as
K c
P
h3
(4.3)
where P is the preload, i.e., the supply-air pressure and h is the fly height, which is
the height of the bearing above the surface on which it is mounted. Therefore it is
very clear that we need to have as low a fly height as possible to get high moment
stiffness against rigid body motions. For stiffness calculations we have assumed a
fly height of 5 microns and we closed the gap to the order of 5 microns and the air
pressure is 90 psi. The variation of stiffness with pressure and fly height are as shown
in the graphs (Figs. 4-8 and 4-9).
We have used 50 mm diameter air bearings to
provide yaw stiffness and 4 pairs of 40mm diameter air bearings for providing roll
and pitch stiffness.
65
4.2.4
Pulley
The idler and the drive pulley designs are symmetric.
The width of the pulley is
selected primarily taking into account the size of the guideway (steel base) on which
we mount the carriage. The diameter of the pulley is the most critical dimension and
it is constrained by the bending stresses on the belt as given by
Et
E-t
D(1 - v 2 )
(4.4)
where E is the Young's modulus of the belt material, t is the belt thickness, D is the
diameter of the pulley and v is the Poisson's ratio. As we can clearly see from the
above expression, for a belt of given material and thickness, the higher the diameter
of the belt, the lower are the bending stresses. The effect of increasing the diameter
is an increase in the effective inertia of the stage and a drop in the axial resonant
frequency.
Anodizing
The pulley surface is hard anodized after machining to the required diameter. This
provides wear resistance to the pulley surface against the wear due to traction between
steel belt and aluminum pulley.
Belt Tracking
The alignment of the axes of the drive pulley and the idler pulleys is critical. If their
axes are misaligned, the belt will generally work toward the edges of the pulleys which
are nearer together. Hence the belt has a tendency to leave the pulleys. This problem
is commonly referred to as belt tracking (refer to [19]). One common solution is to
use crowned pulleys which are as shown in the Fig. 4-2. If the belt travels in the
direction of the arrow, the point a will, on account of the pull of the belt, tend to
adhere to the cone and will be carried to b, a point nearer to the base of the cone,
than that previously occupied by the edge of the belt. If a pulley is made of two
such cones, the belt tends to climb both the cones and hence runs with its centerline
coinciding with the line on the plane containing the base of the cones.
66
4.2.5
Sizing the Pulley
The pulley width is chosen to be less than that of the width of the guideway (= 100
mm). The pulley width is also greater than the width of the belt. The constraint
on sizing the belt arises from the tensile yield strength of the belt material. Hence,
belt width cannot exceed
T
yield stress x t ,
where t is the thickness of the belt and T is the
maximum tension in the belt.
4.2.6
Bearings
The bearings are the load bearing members in a machine and these can be looked
at as constraints. Depending on the application, these constraints differ and hence
the choice of the bearing. The axial or radial constraints or both have to be added
in rotating shafts depending on the system. In our design, we constrain the pulley
radially and axially using angular contact bearings. Angular contact bearings can
hold a larger number of balls due to their construction and hence offer higher thrust
and radial load capacity. The pulley in our machine is mounted on a pair of angular
contact bearings preloaded against each other and they are mounted in a back to back
configuration. The back to back configuration has some advantages over face to face
mounting in an application where the outer race is fixed and the inner race is rotating.
As the shaft expands axially and radially more than the housing, the preload remains
relatively same. Axial expansion decreases the preload and radial expansion increases
the preload and the two effects cancel. In the case of face to face mounting, there is
an opposite effect and both axial and radial thermal expansion tends to increase the
preload and high preloads are detrimental. Hence, back to back mounting provides
high moment load support capacity and it is thermally more stable.
Preload
The stiffness of the bearings is very critical in high precision applications. The balls in
the bearings are preloaded to attain high stiffness for different applications. A typical
ball bearing deflection vs load has a characteristic shown in Fig. 4-11. It can be seen
67
that as the load is increased uniformly, the slope of deflection curve decreases. Hence,
it would be advantageous to operate above the knee of the load-deflection curve from
bearing deflection considerations. This simply summarizes the idea behind preload.
This condition can be realized by axially preloading the angular contact bearings.
The relevant equations and the graphs are derived using Hertzian contact mechanics.
A comparison between the preloaded and non preloaded bearings can be seen in the
Fig. 4-12
The chief advantages of preload are
1. to maintain the bearings in exact position both radially and axially and to
maintain the running accuracy of the shaft
2. to increase the bearing stiffness. The key issue is to get almost all the balls to
bear the load. If not properly preloaded, this may not be the case. If the preload
is not high enough, the compliance of the bearings could become a dominant
compliance and bring down the overall stiffness of the loop. As we have pointed
out earlier, this will lead to a reduced bandwidth and poor performance in terms
of tracking. There are other familiar effects of poorly preloaded bearings in an
assembly such as play and noise.
3. to reduce sliding between rolling elements and raceways. This is very critical
especially when we are talking about high speed applications.
If the preload is larger than necessary, abnormal heat generation, increased frictional
torque, reduced fatigue life etc may occur. The amount of preload must be carefully
determined considering the operating conditions and the purpose of the preload.
Bearing selection
In this section, we will focus on the bearing selection rules, the standards and their
meaning, the preloading mechanism, fits and tolerances, the assembly procedure for
the bearing and lubrication. While choosing the angular contact bearing for a particular application, one has to keep the following criteria in mind in addition to the
constraint picture that we mentioned earlier. These include
68
Table 4.3: Bearing nomenclature 7909A5
7
Angular Contact Bearing
9
diameter series
09
shaft diameter = 09 x 5 = 45
A5
standard contact angle of 25 degrees
1. allowable bearing space
2. shaft size
3. stiffness - very high in our application
4. load capacity - The loads were not very critical in our application. The loads on
the machine are not high enough to cross the permissible limits of the bearings.
5. maximum permissible shaft speeds
6. allowable life in terms of number of cycles
Fits
The fits and tolerances are very important from the assembly point of view. It is
important to specify the nature of fit between the inner race of the bearing and
the shaft on which the bearing is mounted. Also, we have to define the fit between
the outer race of the bearing and the housing or the bearing mount.
discussed in greater detail in
[23].
These are
(Refer A131, A132 of the NSK catalog
[23]).
Some simple rules of thumb are very useful in working out the initial guess for fits
while designing bearing assemblies. These are based on the bearing operation and
the loading conditions as shown in the Table 4.5.
69
Table 4.4: Loading conditions and fits [23]
Bearing Operation
Inner Ring
Load Conditions
Outer Ring
Fitting
Inner Ring
Outer Ring
Tight Fit
Loose Fit
Loose Fit
Tight Fit
Tight Fit
Tight Fit
Rotating
Rotating
Stationary
inner ring load
Stationary
outer ring load
Stationary
Rotating
Rotating
Stationary
Rotating
uter ring load
Stationary
inner ring load
Rotating
Stationary
Rotating
Rotating
Direction
or
or
of load
Stationary
Stationary
indeterminate
70
Preloading mechanism
We assembled the bearing inside the bearing housing as follows
Stepi: The bearings were mounted on the pulley shaft on either side and pressed
inside by using cylindrical rings that pressed on the outer race and inner race simultaneously without disturbing the balls. The lengths of these rings were such that we
have an indication when the bearings were seated at the right lengths.
Step2: We used a pair of rings to press the outer race against walls of the bearing
housing. This was done in a drill press. It is this step that adds a constraint on one
of the critical manufacturing tolerances in the bearing housing design. There is an
annotation in the dimensioning of the drive mount which reads R 0.2 max. This is
to make sure there is a proper mating between the outer race of the bearing and the
bearing housing.
Step3: To preload the balls, we used a lock nut and a lock washer. The arrangement
is as shown in the Fig. 4-13. The pulley has a threaded portion on which the locknut
is mounted and tightened with a wrench. This pushes the inner race against the balls
and preloads them against the outer race.
Lubrication
The main purpose of lubrication is to reduce friction and wear inside the bearings
that may cause premature failure and the preferred practice is to use grease as a
lubricant.
4.3
Assembly
This section is dedicated to details on how to put together different parts of the
machine. Here we discuss bolted joints, preload calculations and graphs, details on
alignments to ensure proper belt tracking, the air bearing assembly, belt-carriage
connection, the mechanism for belt- preload or pre-tension. We list the various sub
assemblies that are mated together to form the overall assembly
71
1. Pulley assembly
2. Carriage assembly
3. Air bearing assembly
4. Motor assembly
4.3.1
Bolted Joints in the Assembly
There are several sub-assemblies in the system as listed above and these are joined
together by bolts. This section is devoted to analyzing these joints, their importance
in the assembly and presenting some rules of thumb that one can use for putting
together a precision machine like the belt drive. The importance of bolted joints can
be listed as follows
1. The bolt is a mechanism for creating and maintaining a force, the clamping
force between joint members.
2. The behaviour and life of the bolted joint depend strongly on the magnitude
and stability of the clamping force.
The preloading of bolted joints is a very important step in a precision assembly. The
effect of preload is to place the bolted member components in compression for better
resistance to the external tensile load and to create a friction force between the parts
to resist the shear load. A good assumption to work with is that the shear load does
not affect the final bolt tension. This leads to an analysis of the effect of the external
tensile load on the compression of the parts and the resultant bolt tension.
Relevant Equations for preload [28]
The idea of preload is to make sure that the joint members take the load when an
external load is applied. When preloaded properly and made sure that the members
are in compression, stiffness of the connection is large compared to the bolt. Hence,
if preloaded properly, the bolt would not fail. The only other possibility of failure
72
of the bolt is during the preload operation when one could torque beyond its proof
strength. There are design tables available for proof strengths of bolts. 90% of the
proof strength is used for putting together aluminum members. The other question
is the number of bolts. There are possibilities of putting too many or too less. Too
less is detrimental for the joint. How do we decide the optimal number of bolts for a
connection? The compressive stress distribution in the members can be determined
using the theories of contact mechanics. In most cases, the rule that works is a 450
pressure cone distribution. Hence, if we have a member that is t units thick, then
the area that is covered is 7rt 2 . If the area of the joint is A, then the number of bolts
should be such that
n7rt 2 > A
(4.5)
The bolt diameter is not taken into account in the above expression.
We use the
following notations to state some results on preload analysis,
P = total external load on a bolted assembly
Fi= Preload on bolt due to tightening and in existence before P is applied
Pbzportion of P taken by bolt
Pm=Portion of the load taken by members
Fb=resultant bolt load
Fm=resultant load on members
E= Young's modulus of the material of the joint member
Eb=Young's modulus of the bolt material
d=diameter of the bolt
t=thickness of the member
D=Diameter of the bolt head or the washer
Km=stiffness of the joint member
Kb=bolt stiffness
d=bolt diameter
F- =
Kb+
Km
73
- Fi
(4.6)
Km
Kn=
7rEd
{2t+D-d)(D+d)
(47)
EbA
(4.8)
(2t+D+d)(D-d)
Kb.
1
Here, I is the effective length of the bolt which is somewhere between the grip length
and the overall length. This is due to the non-uniform stress distribution in the bolt
which is maximum near the inner faces of the head and nut and is zero at the outboard
faces of head and nut. We see from Eq. (4.8), if the external force is large enough to
remove this compression completely, the members will separate and the entire load
will be carried by the bolt. The torque required can be derived and can be shown to
be
T = KFd
(4.9)
where the factor K depends on the friction coefficient and the thread geometry . K
can be found in design handbooks.
4.3.2
Pulley Assembly
The pulley is a part of the stepped shaft on which the angular contact bearings are
mounted and preloaded in a back to back arrangement.
The steps in preloading
the bearings are elaborated in the section on angular contact bearings. The pulley
assembly is referred to as the drive mount in the engineering drawings presented in
Appendix B. While assembling the drive mount on the base, it is aligned such that
the axis of the pulley is perpendicular to the direction of motion of the payload. This
is done using a pair of gauge block sets measuring 70mm, the distance measured
between the face of the base and the drive mount. If this is not done, the pulley
axis could be misaligned to the extent of clearance in the bolt holes. This assembly
procedure is as shown in the Fig. 4-3 Once the drive mount is aligned, it is bolted
down to the base using the four bolts on the drive mount.
74
4.3.3
Air Bearing Assembly
Air bearings support the payload - carriage.
These bearings are mounted to the
carriage by a threaded rod and a pair of locknuts. The threaded rod has a spherical
end which sits in the spherical cup on the top surface of the bearing. A typical air
bearing assembly is as shown in the Fig. 4-4.
We have not used the retainer clips in our assembly. The bearing heights should
be adjusted to make sure that the load is equally shared by all the bearings. If there
are some gross misalignments, the carriage could be at angle and this could affect the
tracking of the belt.
Setting Air Bearing Flyheights
1. Gauge blocks measuring 13mm (= vertical dimension of the bearings) are mounted
on the base and they support the weight of the carriage.
2. The air bearings are placed on the base.
3. The threaded rod is inserted into the hole on the carriage and the spherical end
of the rod sits inside the mating cup on the bearing surface.
4. The locknuts are tightened.
This procedure is repeated on all four faces of the carriage. In this manner, twelve
bearings are assembled. Once the air bearings are set at the right positions, the fine
adjustment to set the flyheight of the air bearings is done by adjusting the threaded
rods. The term flyheight refers to the air film thickness between the bearing surface
and the base when the air is switched on. This height determines the stiffness of the
air bearings. One way of measuring this is as follows.
1. Switch the air supply off. This will make the flyheights to zero
2. Set an LVDT probe (repeatability = 0.1 microns) touching the top of the carriage
(Fig. 4-5).
75
3. Set the reading on the probe to zero.
4. Turn the air supply on.
5. The reading on the probe reads the flyheight
For very small flyheights, this procedure could be adopted by placing the probe tip
on the bearing instead of the carriage. The air supply should be equipped with dryer
and filter to make sure the air supplied to the air bearings is dry and clean. This is
very important to ensure the proper functioning of the air bearings.
4.3.4
Motor Assembly
The motor is mounted to the aluminum block which is bolted to the steel base. Care
is taken to make sure that the centerline of the pulley shaft and the motor shaft are
aligned within the misalignment tolerance limits of the coupling that connects the
two.
A transition fit between the motor flange and the mating hole in the motor
mount is used to generate enough clamping force.
4.3.5
Carriage Assembly
The payload in our system is the carriage. It is made of four aluminum plates bolted
together to form a structure that wraps around the steel base on which it moves.
There are features machined on the carriage plates to seat the locknuts that hold the
air bearing assembly.
4.3.6
Belt Assembly and Pre-tension
The assembly that holds the belt is as shown in the Fig. 4-6. The steps involved in
assembling the belt are as follows
1. The holes are punched in the belt.
2. The belt is sandwiched between the blocks and the belt is inserted.
76
3. The bolts are held finger tight to allow for some adjustability.
4. The belt is aligned at right angles to the blocks. This step is very critical to
ensure proper belt tracking.
5. After ensuring perpendicularity, the blocks are tightened together by the bolts.
A similar procedure is adopted for fastening the belt to the other set of blocks. This
assembly of belt and the two blocks is inserted inside the hollow machine base and the
belt is wrapped around the pulleys. The blocks are bolted to the carriage. To apply
pre-tension, a bolt is used to pull on the block 1.(refer to Fig. 4-7) Once we ensure
that the belt has enough pre-tension, block 1 is bolted down to the carriage and the
pre-tensioning bolt is removed. The belt tracking has to be checked by moving the
carriage along the entire length of travel.
4.3.7
Cleaning and Stoning
The cleaning and stoning operations are performed before mating two parts using
bolted joints. This improves the contact stiffness of the joints.
4.4
Feedback Sensors
The position feedback signals are obtained using the linear and rotary encoder. The
rotary encoder is built in the BM500 motor. This encoder has a resolution of 2000
counts per revolution.
The linear encoder is manufactured by Heidenhain.
The
sinusoidal output of the encoder is converted to square pulses using interpolation and
digitizing electronics.
The square pulses in quadrature are read in dSPACE. The
overall resolution is 0.4 microns at a maximum traversing speed of 2 m/s. The read
head of the linear encoder is mounted to the carriage by bolts. The read head mount
should have high stiffness.
If the read head mount is not designed properly, the
signal to noise ratio will be low; this will affect the performance of the machine. The
distance between the read head and the encoder scale is a critical dimension. This is
77
adjusted while assembling. After assembling the read head, the encoder signals out
of the interpolation box should be 5 V TTL pulses. This is a good check of a proper
assembly of the scale and the read head.
4.5
Closed Loop Position Control
The motor is driven by a PWM (Phase Width Modulation) amplifier which sends
the current input to the stator windings. The torque developed is proportional to the
current. The motor amplifier can be used as a voltage source or the current source.
Depending on what we choose, we have two types of controls that are commonly
encountered. The motor working in current mode can be represented as shown in the
Fig. 4-14. Note that the voltage controlled motor has the additional complications of
the effect of the electric time constant AR and of the induced voltage e=Kw. Hence the
preference for the current controlled motor. Please refer appendix. 1 for a quantitative
treatment of this. In the Fig. 4-14,
T
is the torque in the motor, i is the current, W
the speed of the rotor and v the voltage applied.
4.6
Chapter Summary
The design of various components and their assembly procedure have a significant
impact on the dynamic performance of the machine. In Chapter 2, we developed a
dynamic model in which the belt compliance was treated as the predominant compliance. Our discussions on bandwidth in Chapter 2 also presents limitations that are
posed by the drive resonance that arises out of the belt compliance. This model is
valid only if the other compliances in the machine structure are small compared to
the belt. To be able to have the machine closely resemble the model, there are several
scientific procedures that have to be followed. These details have been the subject of
discussion of this chapter on mechanical design and assembly of belt driven servos. In
the next chapter, we present the details of our experimental setup and the measured
transfer functions.
78
b belt
couplir
Motor
(D
pre-tension
screw
carriage
encoder
readhead
mount
Motor mount
/
carriag
/Belt
(D
Linear
encoder
(D
Idler Pulley
guiding rail
F~b
belt
Figure 4-2: Crowned pulleys
80
drive mount
gauge block
pulley
Figure 4-3: Assembly procedure to align the drive mount on the base
81
Thraded stt uakeis SIgumeni
Uad puusdnlag assy
fLairsr clip
2
CRO lot WoMM
Pws Wemat igtl Ibids.
emktluAWAOt bassimp.
uside way of WrIutu, as
plulics or wPWins
Figure 4-4: Air bearing assembly
Figure 4-5: Measuring the flyheight
82
Figure 4-6: Assembling the belt between the blocks
pre-tensioning
screw
block 1
&
Figure 4-7: Pre-tensioning mechanism
83
-
33mm
040021
.rarmm
BC
013mm BALL RECESS
x 6m DP
038mm-
LL
-
x SPRjESSUREPORT
4UV,.U
UFT (MICRO-IN)
800,0
400,0
200,0
800,0
-100.0
-
N
-
lrn
aL
____________________
Ih-
-
____________________
____________________
%.... ....- '',---
..
4
4
____________________
4"
I
300.00
z
M3 x. 5
[*- 5 3mm
0.0
500.0450.0
(3)
.
-80.0
-60.0
250.0~
0-J 200,0
-4 0.0
150.0
100.0
2 0.0
50,0_
Sn
0.0
0 .0
5.0
1V
15.0
.0
UFT (MICRONS)
20.0
Figure 4-8: Stiffness of the 40 mm flat air bearings [22]
84
250
a
13mm
021.6mm D.C.
013mm BALL RECESS
x 6mm DP,
048mm
K-5
(3) M3 x,5
x .8 PRESSURE PORT
5.33Em
00
'20010
LIFT (MMAR04N)
4000
6000
00 0
1000
00 0-
00.0
800
4oo 0
o
0
5010
00
0
-j
0
W0 0
-J
0
400
0
010
200
0-
"
0
o
,
5.0
10,0
15
LIFT (MCRONS)
20.0
Figure 4-9: Stiffness of the 50 mm flat air bearings
85
254-
[22]
Yaw
axis
Pit c
axis.
Figure 4-10: Figure showing the pitch, roll and yaw axes of the carriage
6
Figure 4-11: Load-deflection characteristics of ball bearings
permission from the author)
86
[24] (Reprinted with
I.
0
F-
Figure 4-12: Comparision of preloaded versus non preloaded bearings [24] (Reprinted
with permission from the author)
M..I
I
Wars
Ds
F
A
4
Figure 4-13: Locknut and lockwasher mounted on a threaded shaft (Reprinted from
Whittet Higgins catalog with permission)
87
Toad
vomtage
current
Figure 41
Cr
+
torque
K modeAmplifier
Figure 4-14: Current mode operation
88
i
speed
Chapter 5
Experimental Results
5.1
Introduction
In Chapter. 2, we have developed a dynamic model and derived the collocated and
non-collocated transfer functions. We have initiated an experimental study for two
reasons (a) to verify the theoretical model and predictions (b) to determine the complete picture ,i.e to see the unmodeled modes . In this chapter, we summarize the
experimental results of sine sweep measurements and modal tests. In addition we
present a brief note on the key ideas behind setting up these experiments
5.2
Sine Sweep Measurements
This is a traditional method for measuring the frequency response of the structure.
A signal generator is used to provide a sinusoidal command signal to the system
under study. The frequency of the input sinusoid is varied over the frequency range
of interest. A schematic diagram showing the input and output signals, machine and
the signal analyzer is in the Fig. 5-1.
The machine operates in closed loop with feedback signals from the rotary and
the linear encoder. The feedback signal from the encoder is read in the dSPACE controller. The digital to analog converter in dSPACE sends analog signals to the PWM
amplifier which drives the brushless servomotor. The sinusoidal signal generated in
89
Signal
HP35670AI
^1
2
Encoder
+
+
PWMotor-
Machine
Ampl if ie
dSPACE
Controller
Figure 5-1: Sine sweep experimental setup - Schematic
the HP analyzer is the input signal to the system. The signals 1 and 2 are marked in
the Fig. 5-1. By taking the ratio of the signals, we get the loop transmission. From
the loop transmission, we obtain the transfer function of the machine. For measuring
collocated transfer function, we use a constant gain in the controller.
The loop is
inherently unstable in the non-collocated sensor measurements. Hence, we use a lead
compensator H(s)= 0o
5.2.1
Procedure for Transfer Function Measurement
1. The input is a sinusoidal signal and we expect the output to be sinusoidal in
an ideal setting. It is a good practice to set up the oscilloscope to measure the
input and output waveforms
2. The transfer function is measured at different amplitudes of input signals. The
linear regime where the transfer function does not vary with the amplitude of
the input signal is an ideal range to do the measurements
3. The parameters that can be varied in the HP35670A to refine the sine sweep
measurements include integrate time, settle time, resolution, type of sweep (linear or logarithmic).
90
0
-20
-40
-60
W -80
E
-100
--
-120
-140110 1
measured
del
10 2
Freq(Hz)
100
50
00
01
-o -50
a)
CO -100
-150
-200
102
10
Freq(Hz)
Figure 5-2: Measured and predicted collocated transfer function
91
50 --
0 -
E
M-50 --
-
measured
model
102
10
Freq(Hz)
0
-100-
Q- -400 -
-500-
-00
102
10
Freq(Hz)
Figure 5-3: Measured and predicted noncollocated transfer function
92
Chapter 6
Conclusions
We have explained in Chapter 1 the inverse approach of designing a servomechanism.
This approach demands a good model of the dynamics of the system. In this thesis,
we have modeled the dynamics of the belt-driven servomechanism. We have discussed
collocated and noncollocated feedback control methods and the maximum achievable
bandwidths in each of these cases. The primary limitation to the performance of this
system is from the compliance of the system. The drive resonance that arises from the
belt compliance affects the bandwidth and hence the performance of the system. In
addition, we have emphasized the importance of damping in the belt-driven system.
We have presented a model of the microslip phenomenon in belt drives and estimated
the damping from microslip. The design methodology for the belt drive is presented in
detail laying stress on the effects of various compliances that could affect the dynamic
performance of the machine.
Future Work
Our model for microslip shows that the damping achievable by microslip is insignificant.
Hence, other ways of adding damping to this system have to be explored.
We have discussed in detail the limitations posed by drive resonance.
The drive
resonance is explained with a linear model of the dynamic system. Under certain
conditions, the axial excitation can set up excitation in the transverse direction in
the belt. This instability is an interesting topic of research in the area of dynamics
93
of belt-driven servomechanisms. The theoretical model explaining the nonlinear phenomenon of parametric resonance and the experimental evidence can be built to give
a wholesome picture of the design of belt-driven servos which are one of the essential
components in the world of precision motion systems.
94
Appendix A
Motors
A.1
Introduction
This appendix deals with servomotors and their broad classifications. In addition,
we have described some basic motor principles, DC motor characteristics, brushless
motors and their importance in motion control, the velocity control and current control. This appendix is an attempt to give a short introduction to this continuously
evolving area of motors. To get a better understanding, the reader is referred to [25],
[26] and [27]
A.2
Servomotors
Servomotors are the actuators in positioning servosystems which include robot arms,
CNC machines and several other such high speed precision machines. While general
power-use motors are designed to turn basically at one speed, servomotors are designed to carry out operations following a wide range of speed instructions. The word
servo comes from latin servus meaning slave, and a servomotor can be thought of as
a motor that works following the master's orders. Hence, these servomotors should
be able
1. to turn stably over a wide range of speeds
95
2. to change speed swiftly. In other words, the motor should be able to deliver
high torque and should have low inertia.
The most general classification are the DC Motors and the AC motors depending on
whether the power source is a DC source (battery) or an AC source. There are several
different ways in which the DC and AC motors are constructed. In an AC synchronous
motor, there are three phase windings both on the stator and the rotor. These motors
are designed to move at one fixed speed given by L, where
f
is the frequency of the
AC supply and p is the number of magnetic poles. However, in most applications, we
desire to have variable speeds. This is achieved by using devices called inverters. DC
motors and their controls are easier compared to the AC counterparts.
DC motor
characteristics are simple. Here, our primary focus will be on DC motors.
A.2.1
Motor Principle
To explain the motor principle, the best start is the classical current carrying conductor loop that is placed in a magnetic field. The force acting on the conductor is
given by
dV = idt x -9
where
(A. 1)
is the magnetic field, i is the current in the conductor and dl is the in-
finitesimal element of the conductor that is at an angle to the B. The torque is given
by
T= KI
(A.2)
K is the proportionality constant which is just defined by the motor construction.
A.2.2
Back E.M.F
As the rotor is rotating in the magnetic field, the flux lines induce a current, the
direction of which is given by the Fleming's right hand rule. This current direction
is such as to oppose the direction of the current in the conductor, hence trying to
96
reduce the torque output of the motor. The electromotive force that is induced in
this manner is the back e.m.f. and it can be shown that
e
=
(A.3)
Kw
The interesting point to note here is that the constant K in the Eqs. (A.3) and (A.2)
are the same. These can be derived independently and shown to be same.
A.2.3
DC Motor Characteristics
The equivalent circuit is shown in the Fig. A-1. The voltage balance leads to the
following sets of equations. These equations define the ideal DC motor characteristics.
(A.4)
V - E = IaRa
Combining this equation with Eqs.
(A.1), (A.2) and (A.3), we get the following
equation which represents the torque speed relation, given by
T=
(A.5)
( K)(V - Kw)
Ra
Ra
WV
back emf
4-
e=Ko
Figure A-1: Equivalent circuit of a DC motor
We see that the torque-speed relationship is linear.
97
A.2.4
Need for Commutation
If we consider a rotor winding rotating in a magnetic field, as the conductor completes
1800 rotation, it faces a reversed field direction. The developed torque is such that it
makes the rotor rotate in the opposite direction which will affect continuous motor
action. Hence the direction of the current is to be reversed at the appropriate position to keep the rotor in continuous rotation. This necessitates a mechanism which
has come to be known by the term commutation. The first generation motors had
mechanical commutation which had a commutator and a carbon brush which slides
against the commutator disc which is part of the rotor that carries the armature
windings. These have serious limitations posed by mechanical wear and demands
constant maintenance. There are other constraints posed by the temperature limits
and humidity conditions under which wear in the carbon brushes is minimal. Also,
the construction of these motors require that the commutators have to rotate and
hence the inertia in motion is high and this affects the dynamic performance of these
motors with brushes. There are heat transfer issues since the heat generation in these
motors are high due to mechanical contacts. The solution to these problems is the
brushless motor which use electronic commutation. The principle of commutation involves two switches connected as shown in the Fig. A-2. Depending on which switch
is open, the direction of the current is reversed
Hence, mechanically commutated motors are not very useful to act as servomotors.
The commutation can be achieved electronically by using power electronic
circuits consisting of a set of transistors. These electronically commutated motors
with permanent magnet rotors have no brushes. These brushless DC motors are very
convenient for servo applications.
A.3
Brushless (BLDC) Servomotors
BLDC motors are so called because they have a straight line speed-torque curve
like their mechanically commutated counterparts, permanent magnet DC (PMDC)
motors. In PMDC motors, the magnet is stationary and the current-carrying coils ro-
98
V
B
A
armature
If switch A is closed,
current flows in one direction.
When switch B is closed,
current direction reverses
Figure A-2: Equivalent-circuit representation of commutation
Table A.1: Inner-rotor versus outer-rotor BLDC motors
Inner rotor
Outer rotor
Rapid acceleration
Very good
Poor
Heat dissipation
Very good
Poor
satisfactory
Good
Good
Poor to satisfactory
Requirement
Low cogging
Use with speed reducers
tate. The current direction is changed through mechanical commutation as explained
before. In BLDC motors, the magnets rotate and the current-carrying coils are stationary. The current direction is switched by transistors. The timing of the switching
sequence is established by some type of rotary position sensor. The Aerotech motors
in our machine are equipped with hall effect sensors for the commutation position
signal. There are two types of BLDC motors based on construction: outer-rotor motors and inner-rotor motors. As the name suggests, this classification is based on
the construction of the stator and rotor. Table A.1 draws a comparision between the
two types. Brushless servomotors offer several advantages and in the Table A.2, we
compare the BLDC servomotors with the PMDC motors.
99
Table A.2: PMDC Vs BLDC Motors
PMDC Motor
BLDC motor
Electrical
Commutator
Mechanical
Position sensor+Inverter
Large
Small
Maintenance
Periodical
Minimal
Power rating
High
Low
Maximum speed
Low
High
Simple
Complex
High
Low
Size
Speed Control
Moment of Inertia
Poor
Good
(Rotor Winding)
(Stator Winding)
Heat Dissipation
A.4
Classification Based on Commutation Signals
In electronic commutation, there are two types of current signals that are given during
a cycle of the rotor rotation. These are
1. sinusoidal current - AC Synchronous motors
2. square wave current - BLDC motors designed to develop trapezoidal back emf
Getting square wave currents is hard. Due to the inductive effect of the armature,
there is torque loss (worsens at higher speeds) compensated using a technique called
phase advancing. But for this problem, square wave currents are the most preferred
since commutation algorithms are simple.
A.5
Voltage Control - Quantitative Picture
The voltage equation of a motor accounting for its inductance L is given by
di
V- e =L--+iR
dt
100
(A.6)
The equation of motion for the motor is
T
= J
(A.7)
dt
(A.8)
Combining the above equations with Eqs. (A.2) and (A.3) and taking laplace transforms, we obtain
1
U
TeTmS
2
+
TmS +
Te
(A.9)
1
(A.10)
=
R
JR
TM = K 2
(A.11)
K
The voltage controlled amplifier block diagram is as shown in the Fig. A-3
Toad
votage+
1_____
n~
K
q
Figure A-3: Voltage mode operation
101
speed
102
Appendix B
Engineering Drawings
103
8
7
6
I
4
5
3
1
2
METRIC
0.01
SCALE 1:1
Threaded
00.01
D
1
1
THE INFORMATION CONTAINED IN THIS DRAWING IS THE SOLE PROPERTY OF
MIT ANY REPRODUCTION IN PART OR WHOLE WITHOUT
THE WRITTEN PERMISSION OF MIT IS PROHIBITED.
M 45 1.5 pitch
15 mm long
Key way
width 7 mm
depth 3.6 mm ±0.1mm
length 20 mm
D
Fillet
R3
057±0.025
I'll
CD
C
C
I
I
'I
Oq
0
B
R
8
Pulley surface
Hard Anodized to 1/1000 in
and ground to the tolerance levels on
045_0020
OR
IDENIYING NO
NO.
0
-l
1X
.X ±0.05L
X+.
.X±5
.XX ±0.01
A
LIST
Massachusets Insitute of Technology
Dept of Mechanical Engineering
CAD GENERATED DRAWING,
DO NOT MANUALLY UPDATE
APRDVALS
DRo-kASI S .
DATE
A
7/12/02
Pulley QJY 2)
CHECKQED
RE$P ENG
©
ON
SCIPIOEI
PARTS
UNLESS OTHERWISE SPECIFIED
DIMENSIONS ARE IN MM
TOLERANCES ARE[
and
SIZE
DWG. NO.
REV
IAI
FINISH
QUAL ENG
0
1
U0
I
T
0
14
'
JCAD FIL
SHEET
2
I
1
8
7
6
4
5
1
SOLE
1
3
2
v
8
d
I, q
1
1
THE INFORMATION CONTAINED IN THIS DRAWING IS THE
PROPERTY OF
MIT ANY REPRODUCTION IN PART OR WHOLE WITHOUT
THE WRITTEN PERMISSION OF MIT IS PROHIBITED.
METRIC
SCALE 1:2
All fillets R6.35 Unless
specified
ISTN
IL~-4Drill and Tap
D
D
1 mm deep
38.1
31.75
50
50
635
203.2
C
C
P-I
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33
0 10 THRU
L- 024~6.35 WP. 2 PLS.
-0
c-TI
16
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63.00
c+I
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33
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0.00
|
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NO
NO
C5
0
8 8
8
UNLESS OTHERWISE SPECIFIED
DIMENSIONS ARE IN MM
TOLERANCES ARE:
APPROVALS
-X 0.j
D.M.
Dept of
8
1
7
1
6
1
5
Wnd Weak
4
sharp edgfs
A
Carriage bottom plate
SIZE IDAG.
WUAL ENG
'
'
Mechanical Engineering
7/14/02
CHECKED
RESP ENGT-
8
R
DATE
MARI AL
FINISH
DebUff
O
LIST
CAD GENERATED DRAWING,
DO NOT MANUALLY UPDATE
DhanushkodI
xxO.25
SPE
ESCRIPTON
PARTS
3
'
NO.
ICAD
'
2
'L
REV.
L:
1
1-EET
OF
'
1
8
1
3
1
4
l
5
1
1
2
THE INFORMATION CONTAINED IN THIS DRAWING IS THE SOLE PROPERTY OF
MIT ANY REPRODUCTION IN PART OR WHOLE WITHOUT
THE WRITTEN PERMISSION OF MIT IS PROHIBITED.
METRIC
SCALE 1:2
All fillets are R6.35
unless
I
D
I
-:
pifd
I
I
I
-
0-.
-
C
8
g
8
Drill and Tap 8 x M 4 X 0.7
11.5 mm deep
8
S
5.00
II
4-
40.00 -
+ -+
.1
0.00
I
-~
40.00
I-f
101.00
---
76.00
-
4-
50.00
---
25.00 1B
~
4-~
--
63.0
8 x M 6 X 1.0 Drill and Tap
17 mm deep
NO
N
O
GNO.
SPECIFIED
UNLESS OTHERWISE
DIMENSIONS ARE IN MM
TOLERANCES ARE:
A
l
50
Dhmnushkodi DM.
Massachusetts
ESP ENG
A
Carriage top plate
Same as carriage bottom plate
except for the features mention )d
7/14/02
Debuff ond Breok Sh=r edgms
8
1
7
6
5
t
11 1 G.
VFus
LN
4
.)UAL
l
Insitute of Technology
Dept of Mechanical Engineering
DAIE
CHECKED
MA TEIAL
REOR
UST
CAD GENERATED DRAWING,
DO NOT MANUALLY UPDATE
AFPROVALS
5
SPECICATON
ESCIPTON
PARTS
0.00
NO.
RE
A
ENG
ICAD FILE:
3
2
1.E
OF
,
8
THE
7
6
1
5
IN
T.HIWMTTFN
PRHIRT
PFRMISSjNF
D
4
4
.1.
INFORMATION CONTAINED IN THIS DRAWING IS THE SOLE PROPERTY OF
MT AYREPRUCTION
PART OR WHOLE WITHOUT
3
I
1
I
3
I
I
SCALE 1:3
All fillets R6.35
Unless specified
203.2
D
I
METRIC
0.0
6.35 _
D
31.75
53.60
1--
I 33.60
tz
C
C
41
L
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(D
-q
0 1d THRU LJ 0 24w6.35 FROM
FAR SIDE TYP. 4 PLS.
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298.60
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323.60
20.32
348.60
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ECRPTTON
lZN
ba
UNLESS OTHERWISE SPECIFIED
DIMENSIONS ARE IN MM
TOLERANCES ATE:
DRAWING-
CAD GENERATED
DO NOT MANUALLY UPDATE
APPROVALS
85
X ±0.501
X I.
38.1
[hanushkodi
D.M.
Massachusetts Insitute of Technology
Dept of Mechanical Engineering
DATE-
A
7/14/02
Carriage side plate
CHECKCED
RESP ENG
8 X M6 Clearance Thru
FINISH
Debufr Wd Brek Sharp edges
8
I
7
V
IREQD
PARTS UST
I
6
'IS
4
WUAL ENG
ICAD 1LE:
3
2
1
IS-11 -
8
,
7
I
5
3
44
THE INFORMATION CONTAINED IN THIS DRAWING IS THE SOLE PROPERTY OF
MIT ANY REPRODUCTION IN PART OR WHOLE WITHOUT
1
22
METRIC
THE WRITTEN PERMISSION OF MIT IS PROHIBITED.
SCALE 1:3
All fillets R6.35 Unless specified
38.10
D
n
~-~-tttt
0-.
-L
1~
C
25 1.1
10.00
00
0
H
40
L
-6
36.00
26
EDO
00
cE
40
'-1
___50.0
50.00
30)
6:0
Req
-(-H-
0T
-(-H-
5 X M6X 1.0
Drill and Tap 17mm deep
NO
DENFYING
SPEC IFCAIN
OR DSCRIPTIO
NO
RE
PARTS UST
UNLESS OTHERWISE SPECIFIED
DIMENSIONS ATE IN MM
TOLERANCES ARE:
A
203.2
.x ±0.50
XX ±0.25
X l.5
.X
CAD GENERAE DRAWIG
APPROVALS
Ch-nushk.l D.M.
CHECKED
RESP
MATERIAL
FINISH
8 1
tt
I
Deburr
//
I
00
I
1
00
1
Ori
4
Dept of Mechanical Engineering
Carriage side plate(encoder slot)
Same as Carriage side plate
except for the features mentioned
with dimensions
SWZ
A
and Break sharp eages
4
ENG
DATE
7114102
[REG.
NO.
SHEET
I
i
I
2
i
I
I
G
' THE INFORMATION
CONTAINED IN THIS DRAWING IS THE SOLE PROPERTY OF
MIT
Front View
D
CN
0
<c5
c5
c
NO
N
0
5
1
ANY REPRODUCTION IN PART OR WHOLE WITHOUT
THE WRITTEN PERMISSION OF MIT IS PROH IBITED.
1
4
I~
1
C0
o
o
0
00
N
0
0
3
0
0
0
11
2
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000 0
--; clL>
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0
0
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MET RIC
ISCALE 1:2
52.0
34.80
10 4 .0
0~
)
50
c
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(®)
73±0.02f
e ,
I
35.50
-'~
, --
,,8.50
0.00
Ii
34.80
I-
I
I
-I
0
20%
-4xM5x0.8
Drill
and Tap
(1-
A_
K-
135.50
(®)
SPECIFIED
S OTHERWISE
DIM'ENSIONS
ARE IN MM
TOLERANCES ARE:
*xx
xx71
-±025
/
I
6
1
DATE
A
8/1502
Motor Mount
CHECKED
?ESP ENG
0 12 THRU
Li 0 22 T 15 TYP. 5 PLS.
I
j 717
-6
lEAR)
Massochusefts Insitute of Technology
Dept of Mechanical Engineering
CAD GNERAE DRAWIG
APPROVALS
Dha"''k-",D.M
ICATION
UST
PARTS
All fillets are R6.35
Unless specified
SPEC
OS DESCIPTON
TDENTIFYNG NO
NO
SECTION B-B
SCALE 1: 2
A
L -B
254.0
76.2-
E
b
T
HRr
and Reak hrM Ndgfss
4
IS-::
AU
UALEtNG
I
S
I
2
I
''5
8
1
4
1
42
THE INFORMATION CONTAINED IN THIS DRAWING IS THE SOLE PROPERTY OF
MIT ANY REPRODUCTION IN PART OR WHOLE WITHOUT
THE WRITTEN PERMISSION OF MIT IS PROHIBITED.
3
21
METRIC
SCALE 1:2
D
D
Top View
2-1
76.2
'I!
F~i
56.2
I
I
I
CD
I
I
g
I
I
I
I
I
I
I
I
I
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I
I
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10.0
0.0
-0
ILr~#
I
I
0
I
I
I
I
I
I
I
I
I
I
_________
_________
-'J
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L~1 ]
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
C
I
I
I
I
I
I
I
CD
B
-54.0
All fillets are R6.35
Unless specified
I .
D
0
j
L6)
0
0
04
l~
IEM~O
PAR
IDENTINGNO
NO
OR DESCRIPION
PART
IscxsT
PARTS
UNLESS OTHERWISE SPECIFIED
DIMENSIONS ARE IN MM
TOLERANCES ARE:
X["l'
50TF
A
lX ±0.21
~
j
UPDATE
D-AnTERdi
SM.
A
7/14/02
Motor Mount
CHECKED
RESP ENG
SIZE IDNG.
NO.
REV.
FINISH
QUAL ENG
I1
//
I
00
I
11
05
'~
7
LI
4
RE
DATE
MATERIAL
85
IRCUa
AI
REPD
of Techno1ogy
SPECPICATCRR
Massachusetts Insillute of Technology
Dept of Mechanical Engineering
CAD GENERATED DRAWING,
DO NOT MANUALLY
APPROVALS
I
LIST
FILE:
.5
13
I
1
L
z
ICAD
I
I
SHEE
I
I
Bibliography
[1] G.F. Franklin , J.D. Powell and A. Emami-Naeini 1994 Feedback Control of Dynamic Systems. Reading, Massachusetts: Addison-Wesley.
[2] M.Dahleh, M.A.Dahleh and G.Verghese Dynamic Systems and Control Lecture Notes,
Department of Electrical Engineering and Computer Science, MIT, Fall 2002.
[3] Kripa K. Varanasi 2003 S.M. Thesis, Mechanical Engineering Department, MIT,
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Performance.
[41 L.Meirovitch 1980 Computational Methods in Structural Dynamics. Rockwille,
MD: Sijthoff & Noordhoff.
[5]
Andre Preumont 1997 Vibration Control of Active Structures An Introduction.
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[6] J.Doyle, B. Francis, A. Tannenbaum 1990 Feedback Control Theory. Macmillan
Publishing Company.
[7] Mindlin R. D. and Deresiewicz H. 1953 J. Appl. Mech. Trans.ASME 75, 327-344.
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[8] Mindlin R. D. 1949 J. Appl. Mech., Trans. ASME 71, 259-268. Compliance of
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[9] Johnson K. L. 1955 Proc. Roy. Soc. A 230, 531-549. Surface interaction between
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[10] Goodman L. E. and Brown C. B. March 1962 J. Appl. Mech., 17-22. Energy
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[11] Johnson K. L. 1961 J. Mech. Eng. Sci. 3(4), 362-368. Energy dissipation at
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[12] Johnson K. L. 1985 Contact Mechanics Cambridge University Press, London,
Chap. 8.
[13] Leamy M. J. and Washy T. M. 2001 ASME J. Appl. Mech., 69 763-771. Analysis
of Belt-Drive Mechanics Using a Creep-Rate-Dependent Friction Law.
[14] Betchel S. E., Vohra S., Jacob K. I. and Carlson C. D. 2000 ASME J. Appl.
Mech., 67 197-206. The Stretching and Slipping of Belts and Fibers on Pulleys.
[15] Nayfeh S. A. 1998 Ph.D Thesis, Mechanical Engineering Department, MIT,
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[16] Frank M. White 1994 Fluid Mechanics, McGraw Hill Inc.
[17] Fung Y. C. 1969 A First Course in Continuum Mechanics Prentice Hall Inc.
[18] Electrocraft standard servo product catalog
[19] Peter Schwamb, Allyne L. Merrill, Walter H. James 1921 Elements of Mechanism.
John Wiley sons, Inc.
[20] Alexander H. Slocum 1992 Precison Machine Design. Michigan: Society of Manufacturing Engineers.
[21] R+W, website url: http://www.rw-america.com/
[22] Newway, website url: http://www.newwaybearings.com/productpages/airbearings.
html
[23] NSK Corporation, NSK-MOTION AND CONTROL Rolling Bearing Catalog
[24] Tedric A. Harris 1991 Rolling Bearing Analysis. John Wiley and sons Inc.
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[25] J. C. Compter 2000 Mechatronics Introduction to Electromechanics. Mass Products and Technologies Philips Centre for Technology.
[26] William H. Yeadon, Alan W. Yeadon 2002
Handbook of small electric motors.
McGraw-Hill
[27] Tak Kenjo 1991 Electric Motors and their Controls. OXFORD UNIVERSITY
PRESS
[28] J.E.Shigley 1986 Mechanical Engineering Design Metric Editions, Mechanical
Engineering Series, McGraw-Hill Book Company.
113