A High-Speed Hysteresis Motor Spindle for Machining
Applications
by
Jacob D. Bayless
B.S. University of British Columbia (2011)
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
MA SSACHUSETTS INSTMUT-E
OF TECHNOLOGY
FEBRUARY 2014
MAY 0 8 2014
© Jacob D. Bayless. All rights reserved.
LIBRARIES
The author hereby grants to MIT permission to reproduce
and to distribute publicly paper and electronic
copies of this thesis document in whole or in part
in any medium now known or hereafter created.
Signature of Author:
_
Department of Mechanical Engineering
January 15, 2014
Certified by:
Alexander Slocum
ppalardo Professor of Mechanical Engineering
Thesis supervisor
Accepted by:
David E. Hardt
Chairman, Department Committee on Graduate Theses
1
A High-Speed Hysteresis Motor Spindle for Machining
Applications
by
Jacob D. Bayless
Submitted to the Department of Mechanical Engineering
On January 15, 2014 in Partial Fulfillment of the
Requirements for the Degree of Master of Science in
Mechanical Engineering
ABSTRACT
An analysis of suitable drive technologies for use in a new high-speed machining spindle was performed
to determine critical research areas. The focus is on a hysteresis motor topology using a solid,
inherently-balanced D2 steel shaft. An analytical model of the motor is devised in order to make
performance predictions and optimization, and an experimental apparatus is constructed in order to
verify the predictions of the model and investigate speed limits. The model's limitations due to a stillincomplete understanding of the vector hysteresis properties of magnetic steels are noted, and a
proposal for an experiment to resolve this limitation is presented. The model predicts that the motor
performance is optimized for a very thin ring of hysteretic steel. The experimental apparatus used a
solid rotor. It was run up to a speed of 11,000 RPM and torque-speed curves with various drive
parameters are measured.
Thesis Supervisor: Alexander Slocum
Title: Pappalardo Professor of Mechanical Engineering
2
Table of Contents
1.
2.
3.
4.
5.
Background and M otivation
................................................................................................
1.1 Technology Survey for the Turbotool
..................................................................
1.2 Rotordynam ics
..........................................................................................................
M otor Classifications and First-O rder Estim ates ..................................................................
2.1 Salient Poles
...........................................................................................................
2.2 Stator configuration
................................................................................................
2.3 M otor sizing
................................................................................................................
Hysteresis M otor Design for Turbotool .................................................................................
3.1 Background
................................................................................................................
3.2 Hysteresis m aterial m odels ..............................................................................................
3.3 Com plex perm eability m odel
.................................................................................
4
6
8
9
9
10
10
12
12
13
16
Experim ental Design
25
................................................................................................................
4.1 M echanical Design and Fabrication ..................................................................................
4.2 Electrical Design
.................................................................................................................
4.3 M otor controller .................................................................................................................
Experim ental Results .................................................................................................................
5.1 Determ ining the torque
................................................................................................
5.2 M agnetic fields in the m otor
................................................................................
6. Conclusions
..............................................................................................................................
7. References
...............................................................................................................................
Appendix A: List of Sym bols ................................................................................................................
Appendix B: Derivation of cylinder stresses ................................................................................
B.1 Uniform hollow cylinder
..............................................................................................
B.2 Uniform solid cylinder
...............................................................................................
B.3 Torsional stresses .................................................................................................................
B.4 Surface w rapping .................................................................................................................
3
25
31
35
36
37
41
42
43
44
45
45
47
48
49
1. Background and Motivation
The Turbotool is a concept for a high-speed machining tool intended to increasing the
economical machining speed for roughing aluminum and other metals. The Turbotool is an ambitious
design that combines many novel elements. These include:
- The use of very high-pressure hydrostatic bearings that feed to an internal channel in the
cutting tool, delivering coolant to the flutes
- The use of a direct-drive rotor shrink-fit to the tool shaft, designed to make the entire tool
integrally balanced and interchangeable
- A hybrid electromagnetic and hydrostatic thrust bearing to allow an axial oscillatory motion
- A high-speed spindle developing 100 kW at 100,000 RPM
This design evolved from a previous iteration which was driven directly by a turbine instead of
an electric motor. [1] Variants of this Turbotool design that substitute the hydrostatic bearings for air
bearings may be useful for increasing the accuracy and finish of diesel injector nozzle grinding spindles.
Figure 1 shows a concept sketch of a proposed Turbotool layout. Pressurized water is fed
through a 'stinger' nozzle into the center of the spindle shaft, providing chip-clearing coolant directly to
the cutting flutes. The same fluid supports a hydrostatic thrust bearing, with the clearance between the
spindle and the stinger nozzle serving as a flow restrictor. The thrust bearing is preloaded by an
electromagnet. In this manner, the entire cutting tool and shaft can be easily withdrawn and exchanged.
The radial bearings may be surface self-compensated hydrostatic water bearings, as proposed
by Kotilainen [1], Wasson [2], and Slocum [3], or a more conventional type.
The Turbotool concept combines a number of machine elements into a novel design for highspeed machining. The scope of this thesis will not cover the complete design of the Turbotool, but
covers a preliminary estimate of some important parameters and focuses on a hysteresis motor for the
direct-drive spindle.
4
-Forced coolant
Figure 1. Early Turbotool concept
5
Figure 2: Close-up of the thrust bearing, showing the magnetic flux path (red) and the stinger nozzle
1.1 Technology Survey for the Turbotool
The focus of this research is the development of a high-speed spindle for use in the Turbotool. High
speed motor technology is of interest in many areas, such as machining (including micro-machining and
high-speed drilling, precision grinding, and superfinishing), medical equipment such as dental tools, and
other areas such as gyroscopes, flywheels, and centrifuges.
As a result, there is always an interest in pushing the speed limits of rotating machinery. This section will
review the existing technological limitations on the speed of these machines.
As shown by Borisavljevic, among the high-speed motors documented in published literature,
permanent magnet machines (especially of the slotless type) currently dominate at the highest speeds,
while induction motors are more commonly used for high-power applications. [4]
6
100
induction machines
0 PM machines
0 slotless and very large air-gap PM machines
x
-0
E
-
0
0
%%
CLi Op
(0
0
0
X
00
0
x
10
102
0 0
10
U
00
00
10
P-1/f
036
10
10
10
Rated power [W]
Figure 3: Current achievements in high-speed motor technology (figure due to Borisavljevic [4]).
Permanent magnet machines currently dominate high-speed applications. The Turbotool specification of
10 5 W at 10 5 rpm lies within the range of existing technology.
The limits on the speed of a rotating spindle can come from any of the following:
-
Rotordynamic limitations:
Shaft resonance (critical speeds)
Imbalance loads
Centrifugal stresses
Self-excited vibration instability
Thermodynamic limitations:
Limits on available input power
Limits on operating temperature
Limits on cooling capacity
Voltage limits due to back-EMF
Aerodynamic limitations
Bearing speed limits
Efficiency limitations and speed-dependent losses
Any of these effects can limit the operating speed range of a high-speed machine. Some are 'hard' limits,
set by the bounds of physics and the properties of available materials. For example, driving a flywheel
7
beyond its rated speed limit may cause catastrophic failure and destruction of the machine. Others are
'soft' limits, where increases to the operating speed simply require design modifications that are
technically possible but not economically feasible. For example, driving a commercial electric motor
beyond its rated speed limit may be easily done with a large power supply and an expensive cooling
system.
1.2 Rotordynamics
Detailed rotordynamics calculations are not carried out in this thesis for the Turbotool, as the overall
machine geometry and bearings remain to be finalized. However, it is still important to keep the
rotordynamic limitations in mind when making further design refinements.
As shown in figure 3, the Turbotool design lies on the edge of high-speed rotor technology. Therefore, it
is important to consider the rotordynamic limits. Some of these are summarized in the table below.
Limit
Max. centrifugal
stress in hollow
rotor
Max. stress in solid
rotor
Shaft modal
(bending) stiffness
Shaft modal
(bending) inertia
Equation
Ueer=rin =
Pout - Pin2
Pout + r2 - r? (rin + rout) - pwJ'dut
out
in
=
Ki
Whirl instability
Ensure
0ou
=E (rout
- ri4n)k,
(rUt)
2a,
k =
4
rp(r
_ut2r)
k
=
Shaft modal
(bending)
resonances
Bearing resonances
Rotor imbalance
force
Self-excited
vibration
- 1 a) (rout
~-Esr
2
0-0ro
Implications
For strength, want
rin
«< 1
out
K-
Design for o
Wn
Wn0 i=
Mi
Soft-mounted:
Firn
i
O
-Kb
oJ2 e
sin (wt + q5e)
(n
oVSL<(r
w, = 2'-
Kb i
<K
Design for 4 >> r
and ensure -- < 1
(o <
2
cso
Derivations of many of these limits are presented by Muszynska, and Ishida and Yamamoto. [5] [6] The
top two are derived in Appendix B, along with other results for stresses in rotating cylinders. For the
Turbotool, aiming for operation at 100,000 RPM, the first resonance must be below 200,000 RPM.
Although it is possible to operate above the first critical frequency, whirl-instability in the hydrostatic
bearings above twice the critical frequency is poorly understood and difficult to avoid. Self-excited
vibrations also impose a hard limit on spindle operation, and depend critically on the ratio of the bearing
damping parameters (, and (r, where (n are the damping ratios in the non-rotating (bearing-internal)
and rotating (rotor-internal) frames, respectively. The self-excited vibrations impose a limit even in the
8
case of a perfectly-balanced rotor. [4] [5] [6] Therefore, even the highly-symmetric design of the
Turbotool is not exempt from these limitations.
Note as well that damping should be distinguished from drag. Here damping in the bearings and rotor
refers to the damping terms on rotor vibration (radial motion) in the bearing bore, rather than rotational
motion.
Clearly, the speed limit is maximized when non-rotating damping (damping in the bearing frame of
reference) is maximized, and when damping in the rotor is minimized. Damping terms can be difficult to
model deterministically in the machine design process, but in the case of self-excited instability, they
have a very significant effect on the speed limit. One countermeasure is to deliberately include damping
features with predictable behaviour, with the aim of these becoming the dominant contributions to
damping. Layton-Hale documents several such approaches to 'deterministic damping' design (although
not in a rotordynamic context), such as viscoelastic constrained-layer dampers or squeeze-film dampers.
[7] Meanwhile, the rotor should be constructed to minimize any form of damping in the rotating frame
of reference; therefore, use of hard materials with very low internal damping coefficients, and
avoidance of microslip between assembly components, is ideal. This is an advantage of the Turbotool
design, as the simple, one-piece rotor geometry allows a more predictable damping ratio and avoids
many troublesome and unpredictable sources of damping such as epoxy joints and shaft collars. The
rotor can be made of hard, low-damping materials such as hardened tool steel and carbide.
2. Motor Classifications and First-Order Estimates
For machining where speed control is important, synchronous motors have the advantage of a direct
control of speed, but asynchronous motors will not be ruled out as an option, because speed control can
be accomplished by application of a suitable control system later in the design process. Therefore, a
quick analysis is carried to compare the different motor architectures and identify advantages and
disadvantages.
2.1 Salient Poles
Salient-pole motors, such as variable-reluctance motors, are not considered suitable for the Turbotool
spindle for several reasons. First, the salient increase the difficulty of balancing the spindle relative to
that of a round-shaft motor, which can be ground. These features would need to be machined to very
high tolerance. In addition, the salient poles introduce a greater rotordynamic complexity, as the spindle
bending stiffness may become a function of the rotor angle, unless precautions are taken to make the
tooth designs symmetric for bending stiffness. Finally, variable-reluctance motors tend to develop
significant, destabilizing radial forces due to the attraction between the rotor and stator teeth, adding a
significant time-varying negative stiffness which can excite vibrations. Therefore, only round-rotor
machines will be considered for this application.
9
2.2 Stator configuration
Regardless of whether the rotor is configured as an induction motor, a permanent magnet motor, or a
hysteresis motor, the stator design will be essentially unchanged. The stator's purpose is to establish the
rotating magnetic field, and there are many possible ways to arrange the windings in the stator. Iron
slots offer the advantage of a smaller airgap, but displace volume that could be occupied by conducting
coils, and potentially increase high-order harmonics, torque ripple, and negative stiffness.
For high-speed permanent magnet motors, the advantages of slotless motors tend to outweigh their
disadvantages. Optimization of permanent magnet motors for high-speed operation favors large airgap
designs, which in turn favors the use of slotless rotor geometry, as the high airgap reluctance can be
obtained together with a high effective current density. [4, 5]. Furthermore, at high airgap flux densities,
hysteresis motors operate closer to saturation, which may lead to a reduction in torque. However, it is
important to note that the optimization criteria used in these permanent magnet motor studies may not
be the same criteria that would be most relevant to the Turbotool design; for example, the Turbotool
may have water cooling of the rotor, which would make thermal considerations secondary to
mechanical and efficiency concerns.
An important consideration in the design of the motor is the number of poles. High-speed permanent
magnet motors tend to adopt two-pole motor designs. One reason for this is that a two-pole motor
significantly simplifies the manufacturing process and offers structural advantages; the rotor magnet can
be made of a single piece with uniform polarization. Another reason is that the electrical frequency
which is required to drive the motor is proportional to the rotor speed and the number of poles;
therefore, the fewer poles in the rotor, the lower the driving frequency needs to be. This reduces the
hysteresis and eddy-current losses in the stator iron, where the magnetization reverses at the electrical
frequency. A two-pole motor also offers rotordynamic stability advantages: The magnetic attraction
between the rotor and stator behaves as a negative stiffness that reduces the rotordynamic stability of
the rotor. This negative stiffness increases significantly with the number of poles. Because the flux path
of a two-pole motor crosses the airgap at a 180-degree angle, the airgap reluctance is essentially
independent of any radial displacement of the rotor, and therefore the negative stiffness is minimized
when the motor has only two poles.
2.3 Motor sizing
For round-rotor machines, a simple analysis as done by Kirtley [Cit] shows that the magnetic shear stress
on the surface of the rotor is relatively independent of machine size, and therefore the developed
torque goes with the third power of the rotor radius. This torque is also proportional to the current in
10
the windings and on the surface of the rotor (including bound currents, such as in the case of a
permanent magnet motor, or the equivalent currents due to magnetization of a hysteretic rotor).
Kirtley's formula is:
T =
9
KsKr sin (pk)
Where p is the number of pole pairs, 0 is the angle between the rotor and stator surface currents K,
and Kr, g is the airgap thickness, 1 is the rotor length and R is the rotor diameter. This assumes that the
rotor has infinite permeability in addition to the current distributed on its surface, which is not valid for
permanent magnet motors; this can be accounted for by including the magnet thickness in the airgap,
and subtracting the torque from the inner surface of the magnet. Wound-rotor motors (such as brushed
DC motors) are not being considered in this paper. But for an induction motor, Kr would be the induced
surface current in the rotor; for permanent magnet or hysteresis motors, Kr is the magnetization M.
The surface current Ks requires some consideration. Typically the stator current is thermally limited to a
maximum current density in the windings, which depends on how the cooling capacity and acceptable
temperature rise; Js = 3 x 106 A/m 2 is a common estimate for passively-cooled windings. To convert
this into a surface current, an additional linear dimension is needed: either the slot depth, for a slotted
rotor, or the airgap width for a slotless rotor filled entirely with windings. For the latter case, assuming
g <K rm,
Ks :- g Js
This simplifies the equation to:
T =
p
JsKr sin (p#)
For a two-pole motor, accounting for the contribution of the permanent magnet to the airgap
reluctance,
T
Pow(ri - ri)I
r -
1+ "M r'J
IM
For an order-of-magnitude estimate, the rernanent magnetization of NdFeB is about 1 x 10 6 A/m. To
meet the Turbotool specification of 10s W at 10s RPM, a torque of about 10 Nm is necessary. The
maximum value for the sin (po) term is 1, although in the case of a hysteresis motor it may be smaller
as # becomes a material-dependent parameter instead of a controller-dependent parameter.
Then,
T =
puo Jri
-
J5M
r -
In the limit as ri -4 0,
T T IrOm
+
9
11
Or as ri
-+
,x
« 1,
r3il
T -+o~3 JSM
x+ 9
For a motor length of 150 mm, an airgap of 10 mm (filled with conductors), and an inner magnet
diameter of 25 mm, the required outer magnet diameter is about 50 mm.
This order-of-magnitude estimate establishes the scale for the machine being considered; the following
chapters will examine hysteresis motor configurations in more detail. Authors such as Borisavljevic have
discussed the design of permanent-magnet motors for high speed applications in detail, and the reader
is referred to these works for more details on permanent magnet machines. [4] [8]
3. Hysteresis Motor Design for Turbotool
For high-speed motor applications, the most attractive aspect of hysteresis motors is the elimination of
structurally-fragile permanent magnets from the design. A high-speed hysteresis motor's rotor can
potentially be made from a uniform shaft of ground, hardened tool steel. This makes balancing
significantly easier, eliminating an expensive step in high-speed spindle manufacturing. The rotor could
then be made as an integral part of the cutting tool, to be replaced when the tool is worn out. Changing
cutting tools could be done by removing and exchanging the entire rotor as a balanced, one-piece
assembly.
Therefore, the hysteresis motor will be examined to see if it can be made competitive at the speeds and
powers that are currently achieved by permanent magnet and induction motors.
3.1
Background
Hysteresis motors are a type of motor related to permanent-magnet motors, and historically have often
been used in applications that require very constant torque. The operating principle of a hysteresis
motor is that the rotor is composed of, or sheathed by, magnetically-hard steel. The magnetic field
generated by the stator then acts to magnetize the steel, forming it into a semi-permanent magnet. As
the applied magnetic field rotates relative to the poles of the magnetized steel, a torque is developed
proportional to the field strength and the angle between the applied field and the poles (called the 'lag
angle' y). The torque that can be developed is limited by the capability of the hard steel to retain its
polarization against the applied field; if the applied magnetic H-field exceeds the coercive limit of the
steel, the steel will be re-polarized.
Although rarely found in the literature on high-speed motors, hysteresis motors have several
advantages over permanent-magnet types that make them attractive for high-speed applications. The
first is the structural advantage; hysteretic steel has none of the poor mechanical properties that make
permanent magnets a challenge to work with, such as their brittleness. This enables the rotor to be
12
designed with much higher centrifugal stresses, and also enables assembly techniques such as
interference shrink-fits which permanent magnets cannot use. [8] Permanent magnets may be sensitive
to demagnetization at high temperatures or if the motor is over-driven. Some high-performance
materials such as NdFeB may also be prone to corrosion, and others such as SmCo are very expensive.
Hysteretic steels, by comparison, are much more robust and can be magnetized in place by the motor
coils.
Although the operating principle is fairly simple, there is some confusion in the literature about the
working principles of hysteresis motors. As noted by Zaher, some of the past publications on hysteresis
motors contain major analytical errors. [9]
Using the complex-permeability model, Zaher presents an analytical solution for the fields and torque
produced by a simple hysteresis motor configuration. [9] This solution is extended in this thesis for the
case of a slotless motor model with area-distributed windings instead of a stator surface current, and
various rotor geometries.
3.2
Hysteresis material models
The operation of a hysteresis motor depends fundamentally on the magnetic behavior of the hardmagnetic steel material. The traditional approach is to represent the material's magnetization using a
constant complex permeability, p*. This is equivalent to assuming an elliptical B-H magnetization curve.
The real part of the complex permeability determines the angle of the major axis of the ellipse, while the
imaginary part determines the slenderness of the ellipse. The complex permeability is generally fit to an
experimentally-measured B-H curve by matching the enclosed areas, for some the expected maximum B
and H.
0,8
0.8
JOZ5
to
F=3f-U
0.6
F25H
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
-=O~
-0.2
--0.4
-0A
--0.6
-3000
-2000
-1000
0
H(Alm)
1000
2000
3000
-1000
0
H(ANM)
1000
2000
Figure 4: Experimentally-measured B-H curves for D2 steel for various excitation amplitudes (left) and
frequencies (right). Figure is due to Nejad [10]
13
According to the theory developed by Teare and others, the maximum torque production of a hysteresis
motor is proportional to the area enclosed in the B-H curve. [8] This area normally is equal to the cyclic
hysteresis energy loss in a reversing-field application such as a transformer, but in a hysteresis motor it
represents the torque capacity. Therefore, the higher the area enclosed in the B-H hysteresis curve, the
larger the torque production.
This representation does work well for predicting the motor performance, but it is fundamentally very
approximate. A plot of the B-H curve of a sample of magnetic steel indicates the locus of the steel's
magnetization M when the sample is exposed to a particular time-varying field H. Usually the H field is a
uniform, uni-directional, sinusoidal, alternating magnetic field of some particular range of amplitudes
and frequencies. The resulting B-H curve is especially useful for applications such as inductor or
transformer design, where alternating current in a coil will magnetize the steel in exactly the same way.
However, the B-H curve somewhat misleadingly presents the magnetization of the steel as a scalar
quantity, even though it is inherently a vector quantity. [12] [13] [15] When applied reversing-field
applications such as transformers or inductors, the vector nature of the magnetic field is irrelevant, since
the field never changes direction, and so it can be treated as a scalar quantity. But for rotating field
applications, such as in a hysteresis motor the vector nature of the magnetization becomes very difficult
to ignore.
The traditional complex permeability approach models the vector magnetization by assuming linearity:
The x and y components of the B and H fields are assumed to be independent; the magnetization M
likewise has independent x and y components, and therefore the vector magnetization is the sum of the
scalar magnetization components in x and y. For steady-state rotating fields well below saturation, this
approximation seems to hold. [9]
However, if a design is to push the limits of the performance of hysteresis motors, it is worth examining
the literature on vector magnetization, and vector hysteresis. A survey of recent publications suggests
that a proper physical model to represent vector magnetization remains a matter of current
investigation. The starting point is known as the 'Preisach model' for hysteresis, which was originally
developed to model B-H curves in ordinary reversing-field applications. [14] [15]
In the Preisach model, the magnetization state of a sample of steel is represented by a large number of
'hysterons', each of which can assume a discrete state of either positive or negative magnetization. Each
hysteron will flip its state when subjected to a sufficiently large magnetic field equal to some threshold,
but each hysteron has a different threshold; the steel sample contains a characteristic distribution of
hysterons with different thresholds. When the steel is initially unmagnetized, the hysterons have
random values and so the net magnetization is zero. As an external field is gradually applied, the
hysterons with the smallest thresholds flip state to align with the external field, increasing the total
magnetic field within the sample. As the field grows, both due to an increasing external field and the
contributions from the hysterons aligned with it, the more reluctant hysterons begin to realign as well.
Eventually ,all hysterons are uniformly oriented and the magnetic material is considered to be saturated.
14
If, at any point during this magnetization process, the external magnetic field should begin to decrease,
it will have to work against the hysterons that had previously flipped to align with it, which are
collectively generate their own magnetic field. Therefore, the sample holds its magnetization even as
the external field decreases, until eventually the net field is such that hysterons begin to flip back the
other way. This results in the familiar S-shaped hysteresis curve such as seen in figure 4. [15]
But what if the external field can not only increase or decrease, but also change direction? There are a
multitude of approaches in the literature to extending this simple 1-dimensional scalar model to the
case of a 2- or 3-dimensional state of vector magnetization, sometimes even developing the models
specifically for hysteresis motor modeling. [161 [17] [18] [19] [20] As the nature of ferromagnetism is
inherently quantum-mechanical and depends on the coupled interactions of many atoms in a metal, it is
very difficult to solve from first principles, and instead the approaches tend to be empirical models
guided by principles such as conservation of energy. [19]
Recently-proposed vector Preisach models based on fundamental physics suggest that once the applied
field goes beyond saturation before being rotated, the magnetization will align exactly parallel to the
external field, with no energy absorbed as the field reorients. [19] In a hysteresis motor, this would
result somewhat counter-intuitively in the torque decreasing to zero as the magnetic field increases to
the point of saturation. This implies that there is an optimum magnitude of the magnetic field strength
that results in peak torque. Unfortunately, vector hysteresis data for magnetic materials is difficult to
find from manufacturers or in academic literature.
For the further development of hysteresis motors, experimental studies of the vector hysteretic
properties of various magnetic steels will be essential. A vector hysteresis material-property experiment
was designed as a component of this thesis, but unfortunately could not be completed due to time
constraints.
As a last note, from a modelling perspective it is important to notice that hysteretic materials add
additional initial conditions and degrees of freedom to a model. Whereas the magnetization of an ideal,
linearly-permeable 'soft' magnetic material is completely determined by its boundary conditions,
hysteresis significantly complicates this simple scenario. The internal magnetic fields internal to a
hysteretic material are no longer uniquely determined by the boundary conditions alone, but also the
values they assumed in the past.
15
3.3
Complex permeability model
Despite the limitations of the complex permeability model, it makes a good starting point for hysteresis
motor modelling.
0=0
_B
Figure 5: Model regions for the hysteresis motor model.
The motor will be divided into a number of circular regions, shown in figure 5, and solved using
Maxwell's equations. A 2-dimensional solution is assumed, with an infinite-permeability stator. The
derivation is condensed for brevity. The rotor is assumed to be non-conductive, so there are no eddy
currents in the model; this is a very loose approximation which becomes reasonably accurate for
synchronous rotation.
For consistency, the subscripts used for the vector terms below take the following pattern:
Blmn
[r,61 [A,B,C,D]
'[0,1,2,...]
Unit vector Solution region m Spatial harmonic n
A, the magnetic vector potential, is a vector but for the 2D analysis, only the z component is nonzero.
Therefore A is always assumed to refer to (A -2) , and the first subscript Ar indicates that Ar is a
function only of the coordinate r, whereas A0 is a function of 0.
Region
A (conductor)
B (airgap)
C (Hysteretic steel)
D (Center area)
rc <r < rs
rm < r < rc
ri < r < rm
0 < r < ri
Equations
B = poH
B = poH
I
B = ptoIpr*H
B = pOH
16
V 2 A = -poj(r,
V2 A = 0
V2 A = 0
V2 A = 0
)
The magnetic fields B and H are divided into their cylindrical components B(r, 0) = Br(r, 0) + B 6 (r,0)
and H(r, 0) = Hr (r,0) + Ho (r,0), each of which are treated as complex phasor quantities with
various spatial harmonics. The overall approach to identifying the magnetic fields in the rotor will be to
first find the form of the vector potential A, and then in turn derive B from the vector potential, and
then find H from the material property relationships between H and B.
Br =
B0
Brmnei(en-ct),
=
I
n
Bomnei(On-wt)
ni
To begin with, the solution for V 2 A = 0 is found by separation of variables:
A = Ar(r)AO(0)
r2
r dAr
2Ar
2
AAr drrz + Ar or
1
+
-
a 2A 0
2
6
Ae
0
d2 A2
82= --n 2 A 0
n ; 1, n E Z (from the periodic boundary condition in 0)
A0 = AoneitnO-4)
d 2Ar
Ar
dr-
r2 a2
dr 2 +r
Ar=r,
Arn2= 0
a=kn
Therefore,
Amn+rnei(nc-1 'mn+) + Amn-r-nei(n0-0mn-)
A (r,0) =
n
And this solution is valid in all of the regions without free current. (regions B, C, D). Also, note that #Pni
may be a function of time. Furthermore, 4mn can be different for the two coefficients (written as bmn+
and Omn_), because the coefficients Am,+ are allowed to be complex numbers, and a different,
arbitrary phase can be factored out of each to make them real, without loss of generality. This is similar
to the form of the solution used by Zaher, which he argues is critical to satisfying Maxwell's equations
when the boundary conditions are added. [9]
For the case of V2 A = -po](r, 0), in region A, assuming a current density of the form J(r, 0)
Znjn ei(n0-&jt):
V 2A =
1 02 A
1 a /A
rOr
r-
Or
+ -
rz
2
17
-0po2J
LY
71
.
e=(nO t)
r 2 a 2 A.
-
Ar Or
2
r OAr
1
+ ++Ar Or
AO
a 2 A1
110
J eitnO-ot)
ArAO In
a02
Substituting the solution for the homogeneous case for Ae, and letting <p = Wt and
r 2 a 2 Ar
Ar
-
r
2
+
r aAr
n
2
rIAOAn ei
r 2Ar
022
=
e-4)
=
Ar
Ar
2
+r--An
Or
r -1
/1oJn
2
Ar
1,
Jn ei(n -t)
PO
Ar Or
AOAfl
AAn+rn + AAn-r-
+
n2
where AAn+,AAnf
are the coefficients for the nth spatial harmonic of the vector potential in region A,
for the terms with increasing and decreasing dependence on r, respectively.
Therefore, in the current-carrying region,
A(r, 0)
AAn+rnei(n6k-PAn+) + AAnr-nei(ne-PAn-) +
=
IoJne i(neO-t)
n
Now the vector potential A has been found in all four regions. Next, the magnetic field B is derived from
A, using the definition of the vector potential:
B =VxA
Br =-
1 aA
_
,
A
Be = -
In regions B, C, D:
Brmn = inAmn+rn
1
e i(n 6O-(mn+) + inAmn-r--
1
6
-mn-)
e i(nO
Bemn = nAmn+rn-le i(ne-*mn+) - nAmn-r n-1ei(nf-lbmn-)
In region A:
6
BrAn = in AAn+rn-1ein6t-An+) + AAnr1eitn
BOAn = n[AAn+rn1e
i(n-An+)
-An-) +
- AAnfr -n-
2O]ei(n6-t)
ei(fO-^PAn-]
Next, H is obtained from B. For regions A, B, D, B = Io H. But for region C, which is the hysteretic steel,
B = popr*H
Where Pr is the complex permeability of the hysteretic rotor steel.
18
Finally, boundary conditions are applied. For four regions and two vector components, there are a total
of eight boundary conditions required.
Boundary equations
Coordinates
Regions
D
C, D
B, C
A, B
r= 0
r = ri
r rm
r=
A
r= r_
B 0 finite
H0 continuous in r
H0 continuous in r
B6 continuous in r
B0 = 0
Br = 0
Br continuous in r
Br continuous in r
Br continuous in r
The finite condition at r = 0 leads to ADn_ = 0 for all n.
For the remaining seven coefficients, rather than plug these equations directly into the solutions, which
would create very large and unwieldly equations, the boundary conditions are expressed as a matrix:
S-i
-
n-1
-r--1
0
-r
-,,n
n2
0
0I
o
-0
rj"
rc~-An
0
r-n-1
_-n
0
rn
rm
-rm
0
0
rm1
0
0
0
0
0
0
Po
-
~
0
0
0
0
r,-n
0
-,-
r-
-rn-i
-r
0
2Jl
0
0
0
0
-i
r
-
1
r
Mr
[r*
-1
1
-r'1
0
-_ n r-
r*
0
0
rn-1
1
AA
0
AAfl.
0
ABn_
ABn-
0
Acn+
B+
r
IPr*
-rm
1
-
n
rn1An
~
n1
[t
r,---1
0
0
A cn +
Dn+
-r-l
The coefficients can then be found by inverting this matrix equation, either symbolically or numerically.
Once the magnetic fields are known, quantities such as the motor torque can be calculated. The torque
can be calculated either by a volume integral of the magnetic field crossed with the current in the
conductor, or a surface integral of Maxwell's stress tensor around the airgap. Using the stress tensor
approach, the traction in the airgap is:
1
TrO =
-Re{BrBnjRefB0Bn}
1
-
+
2
1
sin(2n0 -
2
0Pmn+) n 2 BAn+r
sin(2n0 - 2qNmn-)
2
Anr
2
2
n
2
+
n 2 ABf-ABn+r 2
sin((BBn-
n-2
Averaged over a full circle:
< TrO >=
2f
TrdO
=
n 2ABn-ABn+r 2 sinGPBl
- ''Bn+)
The net torque is therefore:
T
= 2wr2 lrot < Trr
>= 27r n2ABn-ABn+Irot sinG/B?1
19
- ~Bn+)
~PBn+)
Tangential component
Radial component
Magniftude
0.2
0.1
0-2
0.15
0
0.05
01
0.1
0
--
-05
-01
0
0
-0.1
0
200
400
600
800
1000 1200
0-
-0-2
-400-300-200-100 0 100 200 300 400
-1500-1000 -500
0
500
1000 1500
H
H
H
Figure 6: B-H loci for various regions in the rotor showing the ellipsoidal approximation to the measured
B-H curve.
The model serves to guide the design of the hysteresis motor experiment, and in turn, part of the goal of
the experiment is to verify the accuracy of this model. For example, the model shows that the torque
output is maximized for a very thin layer of hysteresis material, and quite sensitive to changes in the
thickness.
The model is evaluated for a motor with the following parameters, to match the experimental test rig:
ri
Inner radius
of rotor
0mm
rc
rm
Outer radius Inner radius
of windings
of rotor
I_
I_
10 mm
9.525 mm
_
_I
Inner radius
of stator
Length
of motor
I
12 mm
40 mm
I
Winding
current
density
5 x 105A
_
2
_2
Permeability
of D2 steel
rotor
129 - 93.5i
The model likewise predicts that for a solid rotor, torque will diminish as the permeability of the rotor
material increases, which although perhaps counterintuitive, agrees with Gavril and Mor. [21] The
reason for this is clear when closely examining the boundary conditions of the hysteretic rotor. A very
high permeability implies that for a given B field in the airgap, the corresponding H field in the rotor will
be very small in magnitude. In order to achieve high torque, the Maxwell stress tensor states that the
product BrB0 should be maximized; in other words, the magnetic field in the airgap must be 'tilted',
neither purely radial nor circumferential. The boundary conditions between the rotor and the airgap are
the continuity of Br and H0, and in the airgap, Bi and H6 are directly proportional. Therefore, high
torque is the result of three conditions:
20
-
-
The rotor reluctance must be sufficiently low that IBI is not small in the airgap;
The hysteresis angle of the complex permeability must be large so that H and B are out of phase
in the rotor;
The rotor reluctance must be sufficiently low so that He is not small in the rotor, and therefore
able to affect the angle of He in the airgap.
Figure 7: Zooming in at the rotor boundary showing B in blue and POH in orange. In the airgap, the
orange and blue vectors are equal, but in the rotor, H is small so the orange vectors are not visible and
the rotor appears blue. The small magnitude
pffof H means that, despite H being at an angle to B inside
the rotor, the field lines through the rotor and airgap are essentially straight.
-----
00
Figure 8: The same simulation run with an artificially-reduced permeability close to the permeability of
air (but with a nonzero lag angle). The field lines are highly skewed, which is necessary for torque
generation. A close look at left shows that, in addition to H and B not being parallel inside the rotor,
they are similar in magnitude, which creates the skewed field in the airgap.
Note that the airgap width is exaggerated in these images in order to show the features more clearly.
21
This third condition is not obvious. So, as an illustrative example, imagine a solid rotor formed from a
very highly permeable material IliI -+ oo, having the maximum possible hysteresis angle, 900, and a
narrow airgap. Continuity of B, will dictate that Br in the airgap is the same as Br in the rotor. The 900
within the
lag angle means that He can be found directly from Br, and is Ho = - -> 0 everywhere
rotor. Continuity of He at the boundary results in Be = 0 in the airgap near the rotor, and therefore the
product BrBe = 0 and there is no torque produced.
Although the permeability is a material-dependent parameter and therefore not easily changed, a lower
permeability could be attained using a bonded powdered metal, for example. But a more practical
approach is using a thin, ring-shaped hysteretic rotor instead of a solid disk.
Optimization
of hysteresis rotor thickness
0.1
0.09
- - - ------.
- ----.
-.
-......
..
-. .......
-.
- ....
-.
.............
0.080.07
- - -. -.
..--
E
..-...-
.
-.
-.-.-.-.-.-.-.-
0.06
-.-.-.-
..... -.0.05 ---- -..-
....
-.--.
0.04 ------- ...-. ...
0.03
0.02
-.
- - -.--.
--.- --.
.....
..
...
' -.
.
...
.-.-......
-..
0.01
ni
10-3
10-1
rotor thickness/rotor radius
10'2
100
Figure 9: Model prediction for torque vs. rotor thickness for a 19.05 mm diameter D2 steel rotor. For
this motor, the optimal thickness is found to be t = 0.038, or t = 0.36 mm. According to this
Other
prediction, a solid hysteretic rotor shaft produces about 1 / 1 0 th of the maximum attainable torque.
solid rotor is
parameters used in the simulation are found in the table above. The predicted torque for a
about 20 mNm.
22
Torque trend with steel permeability, angle =-36 degrees
100
10- 1
z
.....
""
...-...-
1.2..
..
a)
,
.
..
,
...
0 m
- --ri
-..-.-..-.--.-.
- .....
.., ..
m
....
102
)4
101
100
10-1
10'2
102
Rotor permeability / D2 permeability
Figure 10: Torque vs. rotor permeability (at constant lag angle) for a solid rotor and a rotor of optimized
thickness. The measured permeability of the D2 tool steel, for a solid rotor, produces about 1 / 1 0 th of the
maximum predicted attainable torque. The torque drops to zero as the permeability approaches that of
air. For the rotor of optimized thickness, the torque is maximum when the permeability is unchanged.
But as argued above, the key parameter is the rotor reluctance. Assuming the flux travels primarily along
a circumferential path through the rotor (which is true only for reasonably thin rotors), the reluctance
can be approximately expressed as:
wrm
2rotor
=lrm
PI(Tm - ri)
Meanwhile, assuming the flux travels radially through the airgap,
Raap -
In( rs
r
2 -Tnu--
Calculated in this approximate manner, the reluctance calculated for the optimized rotor ring with
ordinary D2 steel is 3.7 times the airgap reluctance. If the inner radius is varied while adjusting the
permeability to maintain a constant ratio of the rotor to airgap reluctance,
'Rrotor
\gap
Tm
.
2I|pr I(rm
23
-
ri) in
= 3.7
Tm
Then under these circumstances, the torque becomes nearly independent of the rotor thickness, as
shown in figure 11. The torque eventually begins to increase when the rotor becomes very thick, but this
is not unexpected, as the assumption that the flux travels circumferentially no longer is valid in this case.
Varying hysteresis rotor thickness, keeping constant reluctance
- .----.-........ ....-.-.-.--
...
---.
..-. -.0.8 ---------.----.
- -.-.-.- ----- -- - - - - -- - --- - --
---.-- --.-.
-.
.-
0.6
C,
0.4
0.2
n
10
10'
10-2
rotor thickness/rotor radius
100
Figure 11: Varying the rotor thickness and permeability together, to maintain a constant 'rotor
reluctance' defined by Rrotor =
7m
p(rm-ri)
Furthermore, the optimal rotor reluctance scales with the airgap reluctance. It appears the torque is
maximized when 0.1 <
-
2jotor
10. As the airgap is made increasingly large by increasing rs, the
2ap
optimal Irotor approaches 1. Therefore, the model predicts that larger-airgap (and thus slotless-type)
R gap
motors will have higher performance using thin rings, whereas motors with a smaller airgap will have
good performance with a thicker ring.
In summary, an optimum design for hysteresis motors is predicted by the complex permeability model,
and it depends strongly on the reluctance of the hysteretic rotor. It should be within the same order of
magnitude as the airgap reluctance. For highly permeable rotor materials, this suggests a thin ring may
be necessary.
Due to time constraints, the only experiment that was conducted was on a solid steel rotor, which lies
significantly below of the predicted optimal value of the torque.
24
4
-
00
_E
U)
CD
C:
1
10
ii
12
3
~~
~ ~~~
. *.
... ....
. ..
...........
.. ....
...
4
.. .
..
..
.. . . .._ ._ _
5
6
7
8
9
10
. . . . . . . . .
. . . .
. . . . .
..
_ . ..
. .. . . .. . .
CL
CU
0
Q
. .. .. ... .. ... ..
1
.
.
3
2
..
4
.
5
.
.1
6
7
8
2
9
10
rS / rM
Figure 12: Optimization study with a varying airgap. The inner radius of the stator r is grown to very
large sizes (with the airgap filled with conductor), and the rotor thickness and rotor reluctance that
maximize torque are calculated for each rs, keeping y, constant.
4. Experimental Design
In order to verify the hysteresis motor model and investigate the speed limits of the basic concept,
an
experimental apparatus was constructed as shown in figure 14.
4.1 Mechanical Design and Fabrication
The general construction of the apparatus is a stack of aluminum plates supported by three 016 mm
steel rods with steel spacing sleeves. The alignment features of all of the plates are line-bored together
to ensure accurate alignment and repeatability. The flexures are also line-bored in the same mounting.
After the precision holes are bored, the surrounding cuts are made on an OMAX waterjet. This approach
helps ensure accurately matched flexures for the torque measurement, as the boring tolerances are
much greater than could be achieved by cutting all of the features on the waterjet directly. The steel
spacing sleeves were turned on one face for perpendicularity, and then match- ground to size on the
25
opposite face for parallelism. The bottom plate is mounted on three match-ground cylindrical steel legs
and the assembly stands on a level granite surface plate.
A shaft of D2 tool steel was heat treated to RC60 hardness and then ground to %" diameter with a g6
tolerance, and 500 mm length. A frameless, slotless motor purchased from Koford Engineering is
bonded to an aluminum heatsink which forms its supporting structure, and this is then mounted
between a pair of of torsionally-compliant flexure bearings. A pair of arms extend from the motor to act
as a lever, with a machined face whose motion is measured by Lion Precision capacitance sensors. This
allows a sensitive measurement of the motor torque by observing the micrometer-scale displacements
of the lever's face. These lever arms have several holes machined in place to reduce their moment of
inertia, in order to increase the dynamic performance of the sensing setup. The machined face on the
lever arm is located to be coincident with the axis of rotation of the spindle. The capacitance gauges are
secured in place using a custom-made split flexure clamp. The faces that control the alignment of the
capacitance probe were milled with the other features, while the flexure arms were cut on the waterjet.
An M3-threaded hole is tapped in the side to allow a setscrew to tighten the clamp.
The torsion flexure plates are manufactured from Y2" (12.7 mm) thick 2024-alloy aluminum plates. 2024
was chosen for its high strength and machinability. The other plates are %" (6.35 mm) thick 6062-alloy
aluminum as it was economical. The plates are 12" square (305 mm).
Flexures are formed by pairs of accurately bored 08 mm holes with an 0.6 mm gap between them,
connected by 8-mm rigid bars. In addition to being easy to machine accurately, the circular-hole flexures
were chosen because they have a much higher ratio of torsional compliance to axial and out-of-plane
stiffness than a comparable leaf flexure design. The geometry is similar to a classical linear-motion fourbar folded flexure to cancel out error motions, but the instant centers of rotation all coincide with the
shaft center to create a torsionally-compliant mount for the motor.
A large cutout is located near the end of the flexure because the original design for torque
measurement was to include a voice coil that would balance the motor torque via a servo loop
maintaining a zero-displacement condition at the capacitance sensor, with the motor torque being
indicated by the voice coil current. This idea was discarded in order to simplify the experiment, but the
mounting area was left in the design to allow for future upgrades.
A hall sensor is bonded with epoxy to the center of each of the three motor coils to measure the
magnetic field, and o2 mm holes are drilled radially in the stator through the center of each coil, in order
to feed the sensor leads through. In addition, a 50kfl thermistor is epoxied to the motor coils in order to
help ensure that the motor does not exceed its maximum rated temperature of 200*C.
The shaft is supported by a pair of radial air bearings and a thrust bearing plate, manufactured by New
Way Air Bearings. A portion of the shaft is colored black around a 180-degree arc, with a reflective
LED/photodiode pair mounted to detect the passing edge and thereby measure the rotational speed.
26
The motor is driven by a Koford Engineering motor controller programmed for open-loop commutation.
The drive current in each of the motor coils is measured by a 100 W-rated 0.050 fl current-sense
resistor.
The original design called for an adjustable eddy-current brake to vary the load characteristics of the
rotor. The eddy-current brake was to be composed of an aluminum disc mounted to the rotor, which
would spin between a pair of permanent magnets on a C-shaped steel yoke, with the yoke position
radially adjustable via a leadscrew. However, this module was omitted from the final assembly due to
time constraints.
Figure 13: Slotless frameless motor from Koford engineering. On the right is a permanent-magnet rotor
supplied with the motor, which is exchanged for the solid D2 steel hysteresis rotor in the experiment.
27
B
i
E
G
Figure 14. Turbotool experimental apparatus cross-section.
A
B
C
D
I Radial air bearing
Eddy current brake
Torsion flexure
Motor
,I
E
F
G
H
Spacer
D2 steel rotor
Tension rod
Air bearing mounting
28
J
I
Thrust air bearing
Feet
I
Figure 15: Flexure setup for torque measurement
A
B
C
D
Match-bored alignment holes
Folded flexure torsion bearing
29
Flexure clamp for capacitance probes
Lever for capacitance probe sensing
Figure 16: Above: Turbotool experimental setup (cover removed from the electronics box for visibility)
Left: Close-up of the
flexure bearing and
capacitance probe
30
4.2 Electrical Design
Host computer
Speed commands
Data
(Data
Digital signal processor
TMS320F28027
Motor controller:
Koford Engineering
S48V20A
Data
Sensors:
Hall x3
Current x3
Temperature
Torque x2
Motor
Speed
Figure 17: Signal flowchart
Each of the measured signals is passed through an amplifier stage to a Texas Instruments
TMS320F28027 digital signal processor, which samples and transmits the data to a host PC along with
timing information, and outputs an analog signal to the motor controller to set the motor speed. The
TMS320F28027 has (among many other modules), sample-and-hold circuits for two-channel
simultaneous analog-to-digital conversion, pulse-width modulation output, a UART for serial
communications, and a high-resolution edge-transition timing module. It runs an interrupt-driven code
that prioritizes high-speed sampling of all of the input channels, and passes data to the serial port to
send to the host computer. The digital signal processor is configured to simultaneously sample the hall
sensor and current-sensor channels.
The rotational speed is measured by an LED-photodiode pair (MTRS6140D) against light/dark colored
patches of the rotating shaft. The photodiode is reverse-biased to reduce its rise time, and the signal is
amplified by a transimpedance amplifier followed by a 10x gain non-inverting amplifier. This amplified
analog voltage signal is transmitted along the shielded ribbon cable to the main board, where it is
compared against a potentiometer-adjustable voltage using a Schmitt trigger. The trigger output
'COUNT' is connected to the ECAP1 (edge capture) input of the digital signal processor. The
potentiometer is adjusted until the encoder reliably reads the edge transitions. The ECAP1 module of
the digital signal processor measures the time between successive edges and calculates the rotational
speed of the rotor.
The circuit diagram for the speed sensor is shown in figure 18.
31
LM358
.LM358
M"1S6 1400
HEHOD
f
I
I
i
+3
PO + .33
.047 JF
LM311N
1
1 uF
F
Figure 18:Speed sensor. The circuit on the left is located adjacent to the sensor and outputs ENCODER
over a shielded cable to the circuit on the right, near the DSP, which digitizes ENCODER into COUNT.
The capacitance probe output voltage ranges between -10 V and +10 V, but the ADC on the digital signal
processor accepts an input between 0 and 3.3 V. An op-amp stage was therefore added to divide the
voltage signal by 6.1 and add a 1.65 V offset, to ensure that the voltage signal remained in the
appropriate range. The circuit diagram for the capacitance probes is shown in figure 19.
+3.3V
20k
4. R64
49 k
0P7
k
Figure 19: Capacitance probe measurement circuit takes the input signal 'CAP' and converts to the signal
'TORQUE' to be read by the ADC.
32
The hall sensors have a ratiometric output that is between the positive and negative supply voltages,
and the minimum operating voltage of 5 V. The hall sensors were therefore connected to GND at the
positive terminal and -5V at the negative signal, and the output was passed through an inverting
amplifier with a gain of 0.66 to produce the desired output signal.
+ V
+33V
4.99 kuF
LMC6482
.1 u F
8250
Figure 20: Circuit for converting hall sensor output ('HALL') to ADC input value ('B')
The thermistor has a 50 kU resistance at room temperature, but at the high expected operating
temperatures near 150 degrees Celsius its resistance decreases to 1.1 kU. To obtain an acceptable
temperature sensitivity and a wide range, a voltage divider is formed by adding a 1.5 kU resistor in
series, and the resulting voltage signal is passed to the ADC. The thermistor triggers an automatic
shutdown of the motor if the temperature exceeds 1400 C.
+3.3V
zi
.iUF
("T E MP
LMC6482
C2
10
nF
Figure 21: Circuit for measuring temperature. The thermistor is part number B57861S0503F040, made
by EPCOS Inc.
33
Although the current sensing resistors are only 0.05 fl and therefore, when a 20 A current (the
maximum possible) is supplied to the motor coils, the voltage across the resistor is only 1 V, each of the
three resistors is subject to a very large common-mode voltage of up to 48 V. That is, neither of the
resistor legs is grounded. Therefore, the voltage across the current sense resistor is first buffered by an
Analog Devices AD629 difference amplifier, which has unity gain and is capable of rejecting very large
common-mode voltages (even larger than the power supply rails) by means of an internal laser-trimmed
voltage divider. The resulting signal is then shifted by a unity-gain inverting amplifier to be within the 03.3V input range of the analog-to-digital converter.
+sv
is
+sv
snseO
LM68
T
kk
- -CURRENT
resistor
!current
(DRIVER
AD629
1
uF
J uF
-5V
Figure 22: Circuit for current measurement in the motor coils, using current sense resistors.
According to the device datasheet, the AD629 can reject a common-mode voltage of up to ±80 V with
the supplied ±5 V power supply voltage, and so the current sense resistor voltages should be well
within the range at which the differential amplifier can isolate the signal.
The current sense resistors are each capable of dissipating up to 100 W, and are fixed to the aluminum
frame of the test structure to act as a heat sink.
The motor controller requires an analog input signal between 0 and 5 V to set the target speed. The
EPWM1A output of the digital signal processor is used to generate this voltage, by passing it through a
low-pass filter to obtain a clean DC signal, and then amplifying with a gain of 1.51 to produce a DC
output between 0 and 5 V.
34
+5V
.1uF
WM21
I
1 . -~
(SPE ED
I
C2
1HC6482
T.47 uF
I
'
I
Figure 23: Filter and output amplifier for the 0-3.3V PWM output to produce a 0-5V analog output signal
for the motor controller.
Figure 24: Signal processing electronics. The DSP board connects via the header pins on this board.
4.3 Motor controller
The motor controller is a Koford engineering S48V20A sensorless motor driver, programmed for openloop operation. The controller accepts a 0-5V analog signal from the DSP, in order to set the drive speed.
This varies both the duty cycle (ie, the amplitude) and the rotation frequency. The drive is programmed
for operation as a permanent magnet motor, with the duty cycle varying in order to maintain constant
torque across the entire range of speeds, and with a maximum speed of 50,000 RPM, with a maximum
current of 20 A at 48V. This maximum current is proportional to the supplied voltage.
35
Because varying the motor controller setpoint changes both drive frequency and amplitude, and also
because the available power supply could not produce the required 48 V, the actual output of the motor
driver is measured via the current sense resistors.
5. Experimental Results
- ---
12000 -------------------
Acceleration curve
- - -
---
10000 - ---- - -- ----.--.--
0Ix
-N
CL~
0
50
100
150
200
Time (s)
250
300
350
400
x 10
...............................
2.5
......... ......... ....... ....... ... ................ ..
2
0~
Ix
.......... .......... .................... ....
C
a,
0)
..I... ... ..
... .I
U)
. . .. ..
....... ..... ......
........ ............... .......... ...
0
0
0.5
-
0
-
50
100
150
200
Time (s)
250
300
350
400
Figure 25: Motor accelerated to top speed using a variety of input profiles to the motor controller. The
top speed attained was about 11,000 RPM. When the controller is switched off, the motor decelerates
due to viscous friction.
36
5.1
Determining the torque
In addition to the capacitance probes, the motor torque can be calculated based on the rotor
acceleration, if the drag on the rotor is known. In order to calculate the rotor drag as a function of
speed, the rotor is spun up to its maximum speed and then the motor is switched off, allowing the rotor
to spin down freely. The speed is recorded as a function of time.
A reasonable first guess for the drag is a viscous damping proportional to speed. This produces
the familiar exponential decay result:
dw
-- = -b&)
dt
w(t) = &joe-bt
The residuals have a clear trend with time, as shown in figure 26, which suggests that the fit could be
improved by a more accurate model. An additional damping term is posited, proportional to the square
of the rotor velocity. This results in a Bernoulli-type differential equation for the spin-down curve, which
has an exact solution:
dw
dt- =-b
dt
6(t) = 0)
-ca2
e-bt
C
1 + woI;(1 - ebt)
Additional higher-order damping terms could be added as necessary, but the Bernoulli drag fit is
considered accurate enough for the purposes of determining the motor torque.
The moment of inertia of the rotor is simply calculated as for a solid cylinder:
p2rtrot
frot =
7700
2
3T(0.00952
m)4 (0.350 m) = 3.477 x 10- 5 kgm 2
The torque is then:
T =Irot d
+ bo + co2)
The numerical derivative is estimated by a first-order central differences model, with high-frequency
noise smoothed by a 100-sample running-average filter.
Note that, as formulated above, w has units of
, b has units of s- 1, and c is dimensionless.
Due to the high accuracy of the damping model, the measurement of torque based on the dynamic
acceleration of the rotor is more accurate than the method using the capacitance probes, and so this
approach is used for the torque measurements in this paper.
37
Fits
1200
data
-- - - - - - linear drag fit
..-------. . ----------------------------..-....... .- - .-- . . ---- ..-- . b e rn o u lli d ra g fit
1000
CO
a)
a)
0
.0
Cl,
1
I
800
. ....
- -- - ...
600
---- --
------.
.-
400
-
.
ni
0
-
--
- -.-- 6---- ---- ---- -- -- - .
*--'---'
- -----.-.-.
-.---.
40
20
-- --
...-.- - -
----.
-
-.-.--.-.-
- -*
----.--
200
+
.-.--
140
120
100
Time (s)
-
Residuals
6
linear drag fit
....- ....
..-----. ..-.---.-...-.-.--.--- --- -.----.--- --- ..-.-- . b e rn o u lli d ra g fit
4
(I)
(D
2
CA)
0
.
76
-
-2
-4
20
0
60
40
8o
140
120
100
Time (s)
Figure 26: Motor is switched off and allowed to decelerate freely under drag. The deceleration is fit to a
linear drag model and a nonlinear (Bernoulli) drag model. The residuals, below, show that the Bernoulli
drag model is a much more accurate description of the rotor drag. This fit was done for each data set;
the one plotted here is just an example.
Fits
1085
- --- - - - -data
- -- -- -- -- -- - - --linear drag fit
...bernoulli drag fit
1080
..-.
.-.
.-.
......
- -. -... ..-.
-.
-- --.-............
.-..
.-.-. .-.-. ..- .- .-.-.-. -.
.... ......
..
..
. .....
......
. .....
..
W1075
CO
1070
CO
1065
0.2
0.1
0.3
0.7
0.6
0.5
Time (s)
Residuals
0.4
0.8
0.9
1
- - - - - - - -linear drag fit
bernoulli drag
4
-
.........
-. -...... ..
.. ........... ......... .......
........... ........... ......
........
.......
-..
-. -.
0
0.2
0.3
0.4
0.5
0.6
Time (s)
0.7
0.8
0.9
1
Figure 27: Close-up view of the fit in the first second of the curve, showing the fine-scale features.
38
0.021
1
1
Bernoulli drag
Linear drag
0.0208
-
-.
-
-----
44
0.0204-40.0202 4-
0.02-.4
1
12
2
3
I
I
2
3
4
4
5
6
Trial number
7
8
9
10
x 10-
11 (D
E
CU,
10 CD
9 -
7 -6 -
0
4
I
1
I
5
6
Trial number
I
I
I
7
8
9
10
Figure 28: Best-fit parameters for each trial. The parameters in the Bernoulli model are correlated; a
higher b tends to lead to a lower c and vise-versa. The means are shown. The uncertainty on each fit is
very small due to the large number of data points, while the difference between trials is significant. This
may be due to changes in ambient or rotor temperature affecting the bearing airgap, or residual
magnetization causing the eddy current damping.
Parameter
b (Linear drag model)
b (Bernoulli drag model)
c (Bernoulli drag model)
Mean
0.0207 s-I
0.0201 s- 1
1.03 x 10-6
39
Standard deviation
1 x 10-4 s-I
8 x 10~5 s-1
2 x 10-7
Once the load torque is known as a function of the speed, the drive torque can be calculated for each
trial.
--------------- .....
..-.
.-.
.-..-... .
-.
......
0.03
0.025
---- -.
....
-.
-.
..-...
-.
0.02
---...------ .- .-.
- ~~ ~~ ~ ~
E 0.015
....---.--.
..
-
...
-.
...
-.-.-- --.
.--....
~ ~ ...
-
.
Torque-speed curves
-....-....------.
..---.-....- ...-.
.-.
. --.
. . -.
-----..
. -.
.--..---- .
-.-................
- --.--..-- - -- -.--
- - - -.
..---. -.- -.---..--.-.-.-.--- --
..-.
.-.
....
..-.-.-.--. -.-.
.....-..--. ------.-.-.-.-..-----------
-.--......
--.....................
-.....
----.-.-.--.-..---.-
0.01
................
0.005
..-.
-..--
.........
-.. --
- -- -- ----- ----.-.-.- --- --.---.
-.--.--
.-.-.--
.
............ ...-.
. -.-.
-.-
..- .- .-
0
-0.005
-
0
2000
4000
6000
Speed(RPM)
8000
10000
12000
x 10
2.5--
C
-
- -- -- - -
1.5
--
W)
-. .
- -..
0.5
0
2000
40CO
6000
Speed (RPM)
8000
10000
12000
Figure 29: Measured torque-speed curves (top), and drive profiles vs. measured speed (bottom).
As shown in figure 29, there is a distinct peak in the torque-speed curve for the measured motor. At the
maximum drive setting, the torque peaked at 0.027 Nm.
It is important to note that the available power supply was not sufficient to drive the motor to its
maximum performance; the motor driver required 48 V and 20 A, but the driver was capable only of 32
V and 19 A. Therefore, the current density in the windings may have been much lower than intended.
40
5.2 Magnetic fields in the motor
The hall sensors embedded in the motor coils were placed to measure the magnetic field inside the
airgap. The results are shown in figure 30 as a function of the motor controller setpoint. However, the
fields seem to have either saturated the sensor output, or else the motor current reaches a maximum at
a speed setting of 3,000 RPM. This makes it difficult to determine the actual fields with time.
However, the hall sensors are able to measure the residual magnetization in the rotor when the coils are
switched off.
0.9 ------ -------0.8 .............
*..............
.......................................
0.7
0.6
kv.'
4
......... ......... ..........................................
...
............
...... ....................................................
...............
.............................
............................ ........................................................
-~0.5
................
S0.4
.......................................................
..................................
.................... ........................
0.3
..........
.................I ........ ...........................
...........I ............ .............................
..................................................................................
...
0.2
..........................................................
..............................
............................
0.1
0
.........................................................
0
0.5
1
1.5
Motor controller setpoint (RPM)
.............
2
2.5
x 10
Figure 30: Measured magnetic field amplitude vs. motor controller setpoint.
A further problem was that, being embedded in the motor coils and connected to the DSP via long leads,
the hall sensors were susceptible to induced noise from the motor switching. Precautions were taken to
reduce this effect, including shielding the cables, and using hall sensors with on-chip signal amplifiers to
maximize the signal-to-noise ratio. But it is difficult to rule out the motor coils as a source of noise, in
addition to generating the measured magnetic field. A recommended design change would be a filter
below the PWM switching frequency, perhaps even a bandpass filter tuned to the drive frequency of the
coils.
The direct measurement of the current in the motor coils, unfortunately, did not yield the results
anticipated. Although the voltage across the current sense resistor should have been well within the
operating range, the differential amplifier output measured on an oscilloscope saturated to 5V for very
low driving settings, and so unfortunately, no meaningful data about the output current from the motor
driver could be measured.
41
6. Conclusions
An improved Turbotool design would likely result from shrink-fitting a thin hysteretic ring onto a nonmagnetic spindle, such as tungsten carbide or other ceramic spindle. This requires a modification from
the one-piece rotor concept presented in Figure 1, but the overall design can still be maintained. The
torque developed in a solid or thick-rotor hysteresis machine is very low. The most striking conclusion
from the theory is the importance of matching the reluctance of the hysteresis motor to the airgap
reluctance in order to develop maximum torque, as the performance peak is quite sharp.
A platform for high-speed hysteresis motor testing was constructed, using a solid rotor, and detailed
measurements of its performance were taken. The magnetic field inside the rotor was also measured,
but the measurements were susceptible to noise and saturation, so the data does not reveal any
meaningful trends. Unfortunately, the current in the motor coils could not be measured, due to an asyet undetected issue in the current sense resistor isolation circuit. The maximum measured torque
generated by the motor is similar to the model prediction (20 mNm).
Permanent magnet construction is also feasible. The magnet would have to be very large, about 50 mm,
in order to obtain the desired 10 Nm of torque.
Further experiments are necessary to confirm the validity of the hysteresis motor model. However, time
constraints prevented additional experiments from being conducted on thin rings of hysteretic material.
The next research goal should be to verify this prediction, and conduct a design aiming for peak torque
as predicted by the model.
The linear, complex-permeability model was assumed for the rotor, however, it does not adequately
account for the vector nature of the magnetic fields. Vector permeability measurements of hysteretic
steels should be performed in order to assess how this impacts the results, preferably taking into
account large-field effects such as saturation. As well, the current model does not account for eddycurrents and other effects of asynchronous magnetic fields. This is left for future work.
42
7. References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
Kotilainen, Markku S. A. 2000. "Design and Manufacturing of Modular Self-Compensating
Hydrostatic Journal Bearings". PhD thesis, M.I.T., Department of Mechanical Engineering.
Wasson, Kevin L. 1994. "Hydrostatic Radial Bearings for High Speed Precision Machine Tool
Applications". MS thesis, M.I.T., Department of Mechanical Engineering.
Slocum, Alexander. 2007. "Water Hydrostatic Bearings for Precision Machine Tools and
Industrial Machinery", ASPE Annual publications.
Borisavljevic, Aleksandar. 2013. "Limits, Modeling, and Design of High-Speed Permanent Magnet
Machines". PhD thesis, Delft University of Technology.
Muszynska, Agnieszka. 2005. "Rotordynamics". Taylor & Francis Publishing Group.
Ishida, Yukio and Yamamoto, Toshio. 2012. "Linear and Nonlinear Rotordynamics". 2 nd ed.
Wiley-VCH.
Hale, Layton C. 1999. "Principles and Techniques for Designing Precision Machines". PhD thesis,
M.I.T., Department of Mechanical Engineering.
Bianchi, Nicola et al. 2004. "Potentials and Limits of High-Speed PM Motors". IEEE Transactions
on Industry Applications, vol. 40, no. 6.
Zaher, Farouk. 1990. "An Analytical Solution for the Field of a Hysteresis Motor Based on
Complex Permeability". IEEE Transactions on Energy Conversion, vol. 5, no. 1.
Nejad, Mohammed I. 2013. "Self-Bearing Motor Design & Control". PhD thesis, M.I.T.,
Department of Mechanical Engineering.
Teare, B. R. Jr., 1940. "Theory of Hysteresis-Motor Torque". AIEE Transactions.
Enokizono, Masato et al. 1993. "Two-dimensional Magnetic Properties of Silicon Steel Sheet
Subjected to a Rotating Field". IEEE Transactions on Magnetics, vol. 29, no. 6.
Enokizono, Masato et al. 1994. "Magnetic Field Analysis by Finite Element Method taking
Rotational Hysteresis into Account". IEEE Transactions on Magnetics, vol. 30, no. 5.
Mayergoyz, I.D, Friedman, G. 1988. "Generalized Preisach Model of Hysteresis". IEEE
Transactions on Magnetics, vol. 24, no. 1.
Della Torre, E. 1999. "Magnetic Hysteresis". IEEE Press, New York.
Kim, Hong-Kyu et al. 1998. "Finite Element Analysis of Hysteresis Motor Using the Vector
Magnetization-Dependent Model". IEEE Transactions on Magnetics, vol. 34, no. 5.
Bottauscio, Oriano et al. 1998. "Analysis of Isotropic Materials with Vector Hysteresis". IEEE
Transactions on Magnetics, vol. 34, no. 4.
Bergqvist, Anders. 1997. "Magnetic vector hysteresis model with dry friction-like pinning".
Physica B 223, pp. 342-347.
Della Torre, E. et al. 2006. "Vector modeling Part 1: Generalized hysteresis model". Physicaa B
372, pp. 111-114.
Hong, Sunki et al. 1994. "Vector Hysteresis Model for Unoriented Magnetic Materials". IEEE
Transactions on Magnetics, vol. 30, no. 5.
Gavril, S. and Mor, A. 1983. "Asynchronous Performance of Hysteresis Motor Considering EddyCurrent Effects". Electric Machines and Power Systems, vol. 8, pp. 57-67.
43
Appendix A: List of Symbols
The use of identical symbols for distinct quantities was avoided wherever possible, but unfortunately
because some symbols are used for different purposes in different fields, it could not always be avoided
entirely. The author hopes that all formulas will be unambiguous in their relevant contexts, and with
their relevant dimensions. All calculations are performed using SI units.
Name
Rotor speed (mechanical)
Young's modulus
Density
Magnetizing field
Magnetic field (aka 'flux density')
Permeability of free space
Magnetic vector potential
Angle coordinate
Radial coordinate
Torque
Force
Time
Rotor thickness
Volume-distributed force
Power
Normal pressure
Reluctance
Moment of inertia
Current density
Surface current
Stiffness
Magnetization
Mass
Square root of -1
Wavenumber
Wavelength
Number of poles
Spatial harmonic
Hollow rotor inner radius
Rotor outer radius
Conductor inner radius
Stator inner radius
Airgap width
Motor length
Rotor-stator field lag angle
Relative permeability
Shear stress
Normal stress
Unit
Radian per second
N/M 2
Kg /M 3
A/m
T
N/A 2
T -m
radian
M
N-rm
N
S
M
N/m 3
W
N/M 2
H-1
Kg - M 2
A/m 2
A/m
N/m
A/m
kg
(dimensionless)
m-1
M
(dimensionless)
(dimensionless)
Symbol
w0
E
p
H
B
YO
A
0
r
T
F
t
t
/
P
P
3
IrOt
i
Ks, Kr
K
M, Mr
M
i
k
A
p
n
ri
rm
rr
rs
g
r
r
m
radian
(dimensionless)
N/in 2
Pr
_N/M
_-
44
2
Appendix B: Derivation of cylinder stresses
Under high-speed rotation, centrifugal forces can become significant, and the strength of the rotor is
therefore an important design parameter, especially when fragile materials such as permanent magnets
are to be used. A brief derivation of the stresses on a uniform rotating cylinder is included here.
The equations of elasticity in cylindrical coordinates are:
iaorr Urr ~-U00 1Or
au,()0rz,+
+
+-!
+
+fr=
Or
r
r 50
Cz
189
daO+
2 ar&
--e+-109
OarO
+
Oaez
ao +fo= 0
Or
r
r M
az
auzr azr
auOae 0 azz
+--++
+ fz=o
Or
r
r O + z
z 0
B.1 Uniform hollow cylinder
In the case of a cylindrically-symmetric problem (one-dimensional), this reduces to
a0 rr +rr
+
Or
-
r
For a homogenous, uniform hollow cylinder:
E
1
cGrr
_ V
2
U0+
+fr =0
(Err + VE 6 6 )
E
=
v2
-
+
(e60
VErr)
-v
Ezz
(arr + U00)
=
Expressed in terms of the displacement, u(r),
1 au +
Our
Er
ar
rk
The elastic equation then reduces to
O
U
r
rrO
CZZ
r
Ouz
az
a 2 Ur l Our
1
+-ur + F = 0
Or2 + rOr r2
Where F = fr
2
(1-V )
and
fr = -po
2r
is the centrifugal loading
The general solution is
ur(r) = Cjr + r-+ P
Using this to solve for the stresses,
E_
E
(Yrr =
1+V
(1
C2
-
8E
1
-+p2r2
2
2
E C2 1
-+-p(2r2
+
U00 =
1-v C1 1+vr
2
For a hollow cylinder that is shrink-fit onto a shaft, the boundary conditions are normal stresses due to
the interior (shrink-fit) and exterior (hydrostatic) pressures, respectively
1
-v
E
C1 -_
2
0'rr(rin)
Pin
urr(rout) Pout
Where a positive pressure is compressive, and negative is tensile. Then,
45
1 - V
=
C
(Pout - Pin)r i
+ -r 2 -ri2
in
1+v~ out
Eo
+
E
2
r~t
1
2(r2
2&
r r
+ ru02~
I)
pin) _-P02,
22
rrlt}
irn (pott
Substituting in to the stress equations,
2
rr
Pout+ r2
Tout
+
ri2 (Pout
~
1
Pin)
i
~2
1
2
2
irou
t rin / rz
2
-out
in
out
1
Pin)
12
(r, 2
n + r2ut) +
-P(r
22
p2
2
n
r
oUt
2t
pw 2 r 2
Combining terms,
rr=pout
07rr
+ (pout
-
+
out
r
(pout - Pin)i2n 2(1U00 =Pout +
r2
r2
+
ri22
1
ou t
nrr2outu
r?)
pin)
+
p)2 r2
2n
2r
2
P(
+
2
ri rOUt
2
r
2n
(out r
22
tin-r
2
2
Tin -- rout
The largest stress is often u0,9 (rin)P
(rin) =Pe
- P/
2
+ 1Pout
r2 t)
+
out -n
t
- pnrout
The dependency on w 2 rout is important to consider when centrifugal stress becomes an issue. Also,
attempts to prevent fracture by winding a strong carbon fiber wrap around the outside might be
counterproductive (under centrifugal load, it would create surface pressure similar to an increased
Pout), and so fracture along the inside surface could still occur. This is, of course, only valid for the case
of a uniform hollow cylinder.
It is clear from this analysis that when it comes to centrifugal stresses, an important material property is
not the strength, but the specific strength: u.
P
Some outer surface pressure can be chosen to ensure that Ueo (rin) = 0.
+ ri2
r+o2ut
2
2
p
rout + Pin rt -- ri
Pouto
1+
ru
out
+ rtn
2
out ~tin
Of course, in this situation, the maximum stress would be located elsewhere in the cylinder. The surface
pressure could instead be chosen to minimize the maximum Von Mises stress, or some other criterion.
This sort of optimization would be the goal of constraining the cylinder with some kind of protective
wrapping.
46
B.2 Uniform solid cylinder
A solid cylinder is similar to the hollow cylinder case, but the boundary conditions here are different:
~-yrr (0) = 0
Orrr(rout) = Pout
The first boundary condition requires that C2 = 0. This potentially implies a significant reduction in the
developed stresses, if a solid rotor can be used.
The outer boundary condition implies that the constant C1 is
12
1-V
C1 =
poUt - 2 pwro2
Therefore, both the circumferential and radial stresses become:
1
ae = orr = Pout + _pw(r
2
-
r"zut
Depending on the design, attention would need to be given to ensure P0ut does not become too large.
In a design where the cylinder is constrained by some outer covering, the boundary would be more of a
spring or displacement boundary, relating PoUt to ur(rout). It could be negative (if the expansion of the
cylinder is constricted), or positive (if the outer constraint expands more than the cylinder, pulling it
outward).
47
B.3 Torsional stresses
For a rotor which is subject to an applied torque, there will necessarily be shear stresses TrO. The
governing equation for these would be:
DTre
18T99
Tro
Tez
r +2- ++--+
For a uniform rotor, -=
0 and aT
= 0, so
az
ao0
ar
+2
r 7+ fe =
=
0
It can be seen intuitively that these stresses will be the highest on the inner surface. Any
tangential driving forces exerted on material farther out will propagate to the inner surface, amplified by
the lever arm between the location of the force and the inner surface.
In the case of no distributed forces, but some constant shear pressure on the outer surface,
DTr6
2
+-Tr6 = 0
Dr
r
Guessing that the solution takes the form TrO = ar~" and plugging into the equation,
-anr-f-i + 2ar- '-=
0
It is clear that n = 2, and so Tro =
a
-g. a
is chosen to satisfy the boundary condition of some
shear pressure on the outer surface, as in the case of an electromagnetic shear:
Tr6 r=rot = Tout
And therefore,
r2'
= Tout
For the case where the torsion pressure is distributed throughout the rotor volume, as might be
the case with a hard dipole magnet in a uniform magnetic field,
N
fo = Fe = constant
Tr
[;W]
2
-Tro + F = 0
r
A possible form for the solution for this equation is:
aTre
TrO = ar-n + firm
-anr~- 1 + fimr-"- + 2ar-n-1 + 2#rm-1 + Fe
The solution results if n = 2 and m = 1,
=
0
3f3 + F6 = 0
Fe
# = 3
Therefore, the stress would be
a
Tro = -+
FO
-r
Here a would again be determined by the boundary conditions (a surface drag), such fluid
friction or eddy currents on the surface of the rotor.
48
BA Surface wrapping
The surface of the cylinder has so far been solved assuming some static pressure at the boundary, Put.
The nature of this pressure could be hydrostatic (as would be the case inside the hydrostatic bearing), or
it could be due to a constricting sleeve or casing. The casing could be treated as a thin-walled pressure
vessel, with hoop stresses in the casing proportional to the stresses applied to the cylinder inside, as
well as centrifugal loads due to the casing's own mass.
The sleeve can be described as a thin-walled cylinder with thickness tsleeve, modulus Esteeve, and
density Psieeve.
The interface pressure P0 ur results in a hoop stress and strain on the restrictor
roe~Pout
t
Pout rout
Esieeve t
co
2T
This in turn results in an increase in the diameter of the sleeve of 1
21r Esleeve
1
2
Pt
r
T Esieeve t
rmtt
t
Centrifugal loading in the thin sheath also results in stress:
_2o2ut
U00
-=W2
,
Coo
2
Esleeve
EG op(
Co
1
2
r
22,
2w Esieeve
The ring is fit to the motor with some initial preload, which will subsequently either increase or decrease
as the machine is spun up to full speed.
49