Dynamic Modeling of High-Speed Impulse Turbine
with Elastomeric Bearing Supports
by
Abraham Schneider
B.S. Mechanical Engineering,
Massachusetts Institute of Technology, 2002
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
Signature of Author............................
Department of Mechanical Engineering
M*9, 2003
C ertified by .......................................
----I
......................
Woodie C. Flowers
Pap palardo Professor of Mechanical Engineering
#',Oesis
Supervisor
Accepted by...................
Ain A. Sonin
Professor of Mechanical Engineering
Chairman, Department Committee on Graduate Students
MASSACHUS ETTS INSTITUTE
OF TEC HNOLOGY
B3PKER
JUL 0 8 2003
LIBR ARIES
DYNAMIC MODELING OF HIGH-SPEED IMPULSE TURBINE WITH
ELASTOMERIC BEARING SUPPORTS
by
ABRAHAM SCHNEIDER
Submitted to the Department of Mechanical Engineering
on May 9, 2003 in partial fulfillment of the
requirements for the degree of Master of Science in
Mechanical Engineering
Abstract
High speed miniature air-driven turbines, operating at rotation rates of up to 500,000 rpm,
are often characterized by their high noise output levels and low bearing life expectancy.
The bearings of high speed air turbines are commonly supported by flexible, elastomeric
O-rings, which provide some level of vibration isolation and damping. In this thesis,
finite-element methods and other dynamic modeling techniques have been used to study
the dynamic characteristics of this high speed rotating machinery. The rotor systems
have been found to traverse a number of critical frequencies during normal operating
conditions. The use of different 0-ring materials has been found to affect the rotor
response and placement of critical frequencies. Rotordynamics have shown that selection
of bearing and support stiffness and damping can have a major effect on the dynamic
behavior of high speed air turbines.
Thesis Supervisor: Woodie C. Flowers
Title: Pappalardo Professor of Mechanical Engineering
2
Acknowledgements
Miniature high-speed turbines are not altogether the easiest device to study, and
the advice, efforts, and support of many people have made this project do-able,
educational, and enjoyable.
Thanks go to the Timken Company for supporting me throughout this project and
others. Specifically, many staff at the Timken Super Precision Company were
instrumental to my efforts. Chancelor Wyatt provided the initial inspiration and
groundwork to kick off this project, as well as continuous commentary. Dick Knepper
and Andy Merrill provided constant support and direction to my work. I am grateful to
Joe Greathouse for the generous allocation of lab space and resources he granted to me.
Keith Gordon was a source of much good engineering advice. I am immensely grateful
to Paul Hubner for his many interesting and useful suggestions, as well as the high
quality machine work he has performed for me. Warren Davis spent countless hours
developing data acquisition methods which, although finally implemented in much
smaller scale than originally envisioned, helped out the project greatly.
My time at MIT has been an intense learning experience. I would like to thank
Professor Woodie Flowers for his advice and counsel. Professor Samir Nayfeh also gave
me useful critiques and suggestions for my work.
I thank my father for some real nuggets of wisdom and innovative design
suggestions. I thank my family for inspiring me to continue working when nothing
seemed to go right
3
Table of Contents
Abstract ...............................................................................................................................
Acknowledgem ents........................................................................................................
Table of Contents ..........................................................................................................
List of Figures .....................................................................................................................
List of Tables ......................................................................................................................
Chapter 1: Background ....................................................................................................
1.1 Introduction...............................................................................................................
1.2 System Com ponents................................................................................................
1.2.1 Bearings ........................................................................................................
1.2.2 Rotors...............................................................................................................
1.2.3 Vibration Isolation M ethods ........................................................................
1.4 Thesis Structure ...................................................................................................
Chapter 2: Theory .............................................................................................................
2.1 Finite Elem ent Analysis......................................................................................
2.2 Axial Vibration...................................................................................................
Chapter 3: Experim ental M ethods .................................................................................
3.1 Sensors ....................................................................................................................
3.2 Flexible bearing supports....................................................................................
3.3 High-speed rotor test-bed....................................................................................
3.2 Spectral analysis with parametric bearing support variation ...............................
Chapter 4: Results and Discussion................................................................................
4.1 Experim ental Results ...........................................................................................
4.1.1 V iton -70 ......................................................................................................
4.1.2 Buna-N .............................................................................................................
4.1.3 Silicone ........................................................................................................
4.2 Finite Elem ent M odel ..........................................................................................
4.2.1 Viton*-70 ......................................................................................................
4.2.2 Buna-N .............................................................................................................
4.2.3 Silicone ........................................................................................................
4.3 Axial Dynam ic Behavior ....................................................................................
Chapter 5: Results .............................................................................................................
W orks Cited ......................................................................................................................
Appendix A : Instrum entation ........................................................................................
Appendix B: Additional Figures....................................................................................
Appendix C: M atlab Script...........................................................................................
2
3
4
5
8
9
9
10
11
11
12
15
15
17
19
21
21
23
26
29
31
31
32
37
42
46
46
61
61
64
66
67
70
73
77
4
List of Figures
Figure 1: Schematic representation of a typical high-speed turbine and its housing.......... 9
Figure 2: A typical high-speed air driven impulse turbine. The aluminum rotor has an
outside diameter of 0.295". The rolling element bearings have an outside diameter
of 0.25". The shaft is 0.0625" diameter stainless steel. Each bearing is supported by
10
an O-ring with a cross-section of 0.030". ..............................................................
Figure 3: Common impulse turbine designs include (a) flat blade (b) double curved (c)
12
simple curved (d) split cup....................................................................................
Figure 4: Schematic of rotor model with reduced degrees of freedom.......................... 16
Figure 5: Modeshapes of simply supported shaft with varying levels of support flexibility
relative to shaft stiffness. With flexible bearing supports, first and second modes are
rigid-body modes. (Source: Handbook of Rotordynamics, Fredrich F. Ehrich) ..... 17
Figure 6: Finite element model of high speed air-driven turbine. 7 shaft stations with 13
substations. Two imbalances 1800 apart on shaft. 4 lumped inertial stations......... 17
Figure 7: Schematic representation of rotor/bearing assembly for axial vibration
19
m o dellin g . .................................................................................................................
Figure 8: Solid model of canister-type high speed air driven turbine assembly........... 23
25
Figure 9: Schematic of O-ring, showing flash dimensions...........................................
Figure 10: Exploded view of testbed setup: 1. Turbine canister 2. Inlet air path 3. Exhaust
26
air path 4. T est block.............................................................................................
Figure 11: Schematic of experimental setup: Front view showing lateral and axial
accelerometers, magnetic pickup, and the low-stiffness open cell foam base..... 28
Figure 12: Waterfall plot showing (a) lateral acceleration and (b) axial acceleration for
rotor with Viton -70 bearing supports. X-axis represents the frequency domain: 0
Hz - 20 kHz. Y-axis represents the changing rotor speed: speed range 9,600 rpm 345,000 rpm. Acceleration is measured on the Z-axis: OG - 8.913G.................. 34
Figure 13: Vibration and phase of testbed, measured in the lateral direction. Phase
35
measured relative to shaft tachometer....................................................................
Figure 14: Vibration and phase of testbed, measured in the axial direction. Phase
36
measured relative to shaft tachometer....................................................................
Figure 15: Buna-N 0-ring after testing and disassembly. Black debris resulted from the
38
disintegration of flash during turbine operation....................................................
Figure 16: Waterfall plot showing (a) lateral acceleration and (b) axial acceleration for
rotor with Buna-N bearing supports. X-axis represents the frequency domain: 0 Hz
- 20 kHz. Y-axis represents the changing rotor speed: speed range 4,800 rpm 480,000 rpm. Acceleration is measured on the Z-axis: OG - 5G.......................... 39
Figure 17: Vibration and phase of testbed, measured in the lateral direction. Phase
40
m easured relative to shaft tachometer....................................................................
Figure 18: Vibration and phase of testbed, measured in the axial direction. Phase
41
m easured relative to shaft tachometer....................................................................
Figure 19 Waterfall plot showing (a) lateral acceleration and (b) axial acceleration for
rotor with silicone bearing supports. X-axis represents the frequency domain: 0 Hz
5
- 20 kHz. Y-axis represents the changing rotor speed: speed range 3,960 rpm 492,000 rpm. Acceleration is measured on the Z-axis: OG - 2G......................... 43
Figure 20: Vibration and phase of testbed, measured in the lateral direction. Phase
44
measured relative to shaft tachom eter....................................................................
Figure 21: Vibration and phase of testbed, measured in the axial direction. Phase
45
measured relative to shaft tachom eter....................................................................
Figure 22: Voigt viscoelastic model with stiffness and damping coefficients. (Atkirk and
47
G oh ar 187) ................................................................................................................
Figure 23: Comparison between two curve-fits for estimation of dynamic properties of
50
Viton*-70 elastomer. (a) Stiffness (b) Loss Coefficient......................................
Figure 24: Modeshapes and modal frequencies for rotor with Viton*-70 bearing supports.
...................................................................................................................................
51
Figure 25: Bode plot of undamped response to imbalance for rotor with Viton®-70
51
bearing supports.....................................................................................................
Figure 26: Bode plot of response to imbalance of rotor with Viton*-70 bearing supports.
Model incorporates damping estimates provided by Atkurk and Gohar.............. 52
Figure 27: X-Y plots and transient motion of shaft center, rotor with Viton* 70 bearing
supports, modeled at 25'C, and incorporating damping estimates developed by
Atkurk and Gohar. (1) 190,000 rpm: Below first rigid-body critical speed. (2)
360,000 rpm: Near first rigid-body critical speed. (3) 500,000 rpm: Above first
rigid-body critical speed. Rotor achieves limit-cycle (stable) motion at all speeds
53
w ithin norm al operating range ...............................................................................
Figure 28: Whirl mode shapes of rotor with Viton*-70 bearing supports. Backward whirl
at 190,000 rpm and 500,000 rpm indicate too high a level of predicted damping in
54
finite elem ent model. .............................................................................................
Figure 29: Rotor system with Viton*-70 bearing supports (a) Stability map (b) Whirl
55
speed map showing damped natural frequencies.................................................
Figure 30: Transmitted force at bearing 1 and 2 for rotor with Viton*-70 bearing
56
sup p orts.....................................................................................................................
Figure 31: Comparison chart of thermal conductivity of Viton*, Buna, and silicone
58
elastom ers versus other m aterials. .........................................................................
Figure 32: Temperature dependence of elastic modulus of Viton*-70 according to
Sm alley, D arrow , and M ehta .................................................................................
58
Figure 33: Bode plots for imbalance response of rotor with Viton*-70 bearing supports;
66 'C case. (a) Undamped (b) Damping provided as measured from experimental
59
re su lts. .......................................................................................................................
Figure 34: Finite element analysis of rotor model with Viton*-70 bearing supports. (a)
Whirl speed map indicating damped natural frequencies at 231,000 rpm and 444,000
rpm. (b) Stability map indicating stable rotor behavior........................................
59
Figure 35: Finite element analysis of rotor model with silicone bearing supports. (a)
Whirl speed map indicating damped natural frequencies at 120,000 rpm and 200,000
rpm. (b) Stability map indicating stable rotor behavior up to 480,000 rpm. ........... 63
Figure 36: Shaft center motion of rotor with silicone bearing supports. (a) Stable, limitcycle motion at 200,000 rpm (b) Unstable elliptical motion at 500,000 rpm..... 63
Figure 37: Bode plot of axial vibration of rotor, showing system eigenvalues............. 65
Figure 38: Accelerometer calibration certificate ..........................................................
70
6
71
Figure 39: Accelerometer calibration certificate ..........................................................
72
Figure 40: Magnetic pickup (tachometer) specifications.............................................
to
referenced
elastomer,
moduli
of
Buna
Figure 41: Frequency-dependent elastic and loss
elastic modulus of Viton-70. Predictions based on Smalley, Darrow, and Mehta. . 73
Figure 42: Dynamic force/deformation properties of natural rubber, illustrating different
74
regions of material behavior. (Source: Freakley 68) ............................................
Figure 43: Dynamic Elastic and Loss Modulus for Viton B (durometer 75±5) (Source:
75
Jon es 4 4 ) ...................................................................................................................
Figure 44: Reduced frequency - temperature nomogram for silicone (Source: Jones 46)76
7
List of Tables
Table 1: Stiffness and loss coefficients for power law estimation of Viton-70 dynamic
material properties. (Smalley, Darlow, and Mehta 3-25) ...................................
47
Table 2: Damping values for Viton®-70, based on half-power method applied to
experim ental data...................................................................................................
49
Table 3: Model coefficients for frequency-dependent stiffness of silicone elastomer..... 62
Table 4: Model coefficients for frequency-dependent damping of silicone elastomer. ... 62
8
..
..
..
....
....
......
............
Chapter 1: Background
1.1 Introduction
High speed air-impulse turbines power a multitude of devices, including tools found
in odontology, medicine, and art. The miniature impulse turbines attain speeds exceeding
400,000 rpm. Vibration and noise are common characteristics of these rotors, creating at
the least, an annoyance, and at the worst, a hazardous ergonomic environment (Dyson
219-232).
A typical medical drill is illustrated in Figure 1. The typical air driven drill uses a
high pressure (30-35 psi) air source to drive an impulse turbine, which spins on rolling
element bearings. The rotor/bearing assembly is isolated from the housing by
elastomeric 0-rings.
drive air pipe
(30-35 psi)
pre-load spring
O-ring bearing supports,,,_,.,,,
airflow
handpiece
exhaust pipe
impulse
turbine
ball-
bearings
bit (1/16 in.)
Figure 1: Schematic representation of a typical high-speed turbine and its housing.
Operation of the typical high speed air drill involves a very short startup transient,
followed by a few seconds of work, and finally a short run-down to rest. Typical highspeed rotors spin at speeds between 350,000 - 450,000 rpm. Vibration at steady-state is
9
usually dominated by the once-per-revolution signal between 5.8 kHz and 7.5 kHz. The
most common cause for once-per-revolution vibration spectra is imbalance in the rotor.
In addition to the vibration at rotation rate, several other key frequency multiples are
common, including frequencies typically associated with the rate of ball-bearing retainerpass, as well as misalignment in the bearings.
Regardless of the particular spectral content during the operation of the rotor, the
severity of vibration is largely frequency-dependent. Since the rotor-bearing system is
compliantly supported, the system can be modeled as multiple degree of freedom
mechanical system, possessing fundamental frequencies which amplify the response to an
input disturbance. Understanding the frequency response of the rotor is critical to the
optimization of its dynamic behavior.
1.2 System Components
A typical rotor-bearing and shaft assembly is shown in Fig 2:
Figure 2: A typical high-speed air driven impulse turbine. The aluminum rotor has an outside
diameter of 0.295". The rolling element bearings have an outside diameter of 0.25". The shaft is
0.0625" diameter stainless steel. Each bearing is supported by an O-ring with a cross-section of
0.030".
10
1.2.1 Bearings
High-speed impulse turbines of this type have been historically supported by
either air bearings or ball bearings. However, ball bearings have increasingly replaced air
bearings as the antifriction device of choice because of their ability to supply higher load
capacity, and the resultant resistance to stall. Also, ball bearings enable the use of lower
supply air pressures, and tend to be more stable than air bearings (Dyson 15). Finally, the
high level of precision available in ball bearings, at a low price, has further displaced air
bearings as a choice in high speed turbines.
1.2.2 Rotors
Turbines extract potential energy from a fluid. Turbines can be classified as one
of two types: reaction or impulse (White 742-748). Reaction turbines are low pressure,
large flow devices. The turbine vanes possess a hydrodynamic shape which reacts with a
fluid stream to provide lift, which in turn causes rotation of the turbine around a shaft.
Impulse turbines are momentum-transfer devices, in which a high-velocity jet of fluid, at
atmospheric pressure, impinges upon the turbine blade, causing rotational motion. Both
reaction and impulse designs have been used in high speed air driven machinery, but
according to Dyson, the impulse turbine is the most commonly used design today (16). A
wide range of blade designs have been proposed for use in impulse turbines. Some of the
most common have been illustrated in Figure 3. Despite the variations in blade design,
no reliable evidence has shown significant advantages to any particular design (Dyson
19). The difference appears to be driven mainly by market differentiation between
turbine manufacturers.
11
(a)
(b)
(c)
(d)
Figure 3: Common impulse turbine designs include (a) flat blade (b) double curved (c) simple curved
(d) split cup
Given the high rotational speed of operation, balancing is critical to smooth
operation of these rotors. Thus, many rotor/bearing assemblies are dynamically balanced
as part of the manufacturing process. Dynamic balancing involves the removal of
material from the rotor blades to bring the mass center of the rotor/bearing assembly
close to the axis of rotation of the assembly (Ehrich 3.1-116). The rotor and bearings are
often supplied as a completely assembled "cartridge" to minimize the possibility for an
unbalanced turbine.
1.2.3 Vibration Isolation Methods
Vibration has been a major concern in the operation of high speed air-driven
turbines. If the mass center of the rotor/bearing assembly does not coincide with the
center of rotation, then an oscillatory force will be induced which is proportional to the
square of the speed of operation:
Fnbalance = munbalancerC2
Equation 1
where r is the distance between the mass center and the center of rotation, and (o is the
rotation rate in radians/sec.
12
Dynamic balancing is the method of choice to reduce the vibration level in high
speed air turbines. However, some small level of remaining imbalance is inevitable, so a
means of vibration isolation has been adopted to allow the rotor to rotate about its center
of mass. Elastomeric O-rings, mounted on the outer surface of the bearings, have been
commonly used to provide lateral vibration isolation. Axial vibration isolation has been
provided either by O-rings, or by wavy washers. Common elastomers chosen for this
task include Viton*, Buna-N, and silicone.
Viton® is a fluoroelastomer known for its resistance to heat and for its high
damping properties. Buna-N, or perbunan, is a copolymer of butadiene, natrium
(sodium), and acrylonitrile. It is known for its resistance to oils, but has lower heat
resistance than Viton®. Silicone is known for its extreme temperature range, but it has
lower damping properties than either Viton® or Buna-N (Freakley 15-18).
Elastomers are commonly rated by the Shore A hardness system, which is a
means of classifying the hardness of a material under a point load. Currently, most high
speed turbines are supported by elastomers with a durometer of 65-70.
Powell and Tempest have noted that Viton® and silicone O-rings are effective in
the suppression of whirl in a turbine supported by air bearings with rotation rates of up to
110,000 rpm. The authors noted that in general, increasing temperature and hardness of
the elastomer both tended to reduce the effectiveness of whirl suppression (705-708).
Atktirk and Gohar have also noted that O-rings are effective in vibration isolation.
In a turbine whose maximum rotation rate was 60,000 rpm, Viton* was shown to be
effective in reducing vibration amplitudes. Viton*-70 was shown to be more effective
than Viton®-90, in part because its damping coefficient was larger (187-190).
13
Bearing support stiffness has been shown to be important in the design of smoothrunning rotational machinery. Specifically, the choice of support stiffness can affect the
placement of rotor fundamental frequencies (Gunter 59-69, LaLanne and Ferraris 141,
Ehrich 1.2). As noted by Atkurk and Gohar, an understanding of the dynamic
characteristics of O-rings is critical to their successful use in rotating machinery (189190). Most data on dynamic material properties exists in the 1 - 1,000 Hz frequency
range, largely because most industrial applications of rubber are low-frequency (Freakley
319). In addition, high frequency measurements of rubber are considerably more difficult
to perform than low-frequency measurements (Smalley, Tessarzik, and Badgley 121131). Some attempts to predict the behavior of elastomers in the frequency range of
1,000 Hz - 10,000 Hz, corresponding to shaft speeds of 60,000 rpm - 600,000 rpm, have
been made, but little real-world verification in studies on actual machinery exists (Jones
37-48).
Elastomers exhibit major changes in material properties with changes in
environmental variables such as vibration frequency and temperature (Freakley 56-109,
Payne 25-33). The degree of change in material property varies between elastomers, yet
little literature exists to justify the choice of a certain elastomer for the O-ring bearing
supports in current high speed air turbine designs.
The specification of O-rings as components in precision machinery has been
controversial because of their loose manufacturing tolerances. According to AS568 0ring standards, the width of O-rings with cross sections of 0.030" are held in the ± 0.003"
range, whereas diametrical tolerances on bearings and other steel components are held to
less than 0.0002" (eFunda website). However, as is noted by Powell and Tempest, 0-
14
rings are produced in batches, and dimensional variance within a batch is often less than
0.001"; the larger dimensional tolerance is a cross-batch specification (705). By
choosing O-rings from the same batch, dimensional precision can be improved.
O-rings have been shown to be effective in vibration isolation and damping
applications, but a need exists for better quantification of their performance.
1.4 Thesis Structure
Chapter 2 develops analytical techniques relevant to modeling of dynamics of the
high speed rotor. Chapter 3 describes the experimental setup and outlines the
experimental method for the parametric study of several flexible bearing support
schemes. Chapter 4 presents and discusses experimental and analytical results. Chapter
5 brings the thesis to conclusion, and evaluates the overall success of the project in light
of the hypothesis. In addition, some recommendations for future work are given.
Chapter 2: Theory
To completely describe the motions of the single-span rotor, six degrees of
freedom are required: the three translational motions x, y, z and the three rotational
motions of the rotor mass center, which can be interpreted as roll, pitch, and yaw. The
general equations of motion are highly nonlinear and are difficult to solve analytically.
However, these equations may be simplified by assuming constant angular velocity,
small bearing displacements, and zero axial motion. Thus, the total number of degrees of
freedom is reduced from six to four; including the two translational (x, y) and two
rotational (Os, 6,) coordinates (Figure 4).
15
y
x
0X
Figure 4: Schematic of rotor model with reduced degrees of freedom.
We are interested in the forced response of the rotor. Assuming perfect rolling
element bearings, the forcing function for the spinning rotor can come from relative
misalignment between the bearings, aerodynamic cross-coupling between the turbine
blades and the housing, and most commonly, static and/or dynamic imbalance in the
rotor.
Static imbalance occurs when a "heavy spot" on the rotor causes a periodic force
to be exerted perpendicular to the axis of rotation. Dynamic imbalance results from two
or more non-coplanar "heavy spots" interacting to create a wobbling forcing function.
A rotor's response to imbalance will be characterized by a number of critical
frequencies, or mode shapes. The first two critical speeds are rigid-body modes,
especially since the rotor's stiffness is large compared to the support stiffnesses. As is
shown in Figure 5, for a symmetrically suspended rotor on infinitely flexible mounts, the
first and second mode shapes are cylindrical whirl and coning, respectively. The first
flexible rotor critical is the third modeshape.
16
.
N YA1
Mtkdenne flexdi.hty
.
. ......
.. .....
. ........
. . . .........
Infinite fQci jfl
Mod_
Figure 5: Modeshapes of simply supported shaft with varying levels of support flexibility relative to
shaft stiffness. With flexible bearing supports, first and second modes are rigid-body modes.
(Source: Handbook of Rotordynamics, Fredrich F. Ehrich)
2.1 Finite Element Analysis
Implementation of a finite element model provides the most detailed analysis of
the dynamics of the rotor. The rotor can be modeled as a series of shaft elements and
rigid disks (Figure 6). A third-party software package - DyRoBes: Dynamics of Rotor
Bearing Systems - was used to construct the FE model.
6
1
9t
1
Figure 6: Finite element model of high speed air-driven turbine. 7 shaft stations with 13 substations.
Two imbalances 180' apart on shaft. 4 lumped inertial stations.
17
..
....
----
__- -
The rotor model consists of 7 shaft stations with 13 substations. There are 4
degrees of freedom at each substation. The turbine is modeled as a rigid disk centered
between the bearings. The inner rings of both ball-bearings are also modeled as rigid
disks, to capture their contribution of inertial effects at high speeds. A dynamic
imbalance is included in the model following manufacturing tolerances which hold the
rotating imbalance to less than 4.OE-6 oz-in. Thus, an unbalance of 6.477E-10 Lbf-sec 2
is placed at the turbine. A static imbalance of 5.057906E-15 Lbf-sec 2 was added at the
end of the shaft to simulate the imbalanced bit used in testing. The wavy preload washer
is modeled as an axial stiffness of 50 Lbf/in. The rotor supports were modeled as flexibly
supported rolling element antifriction bearings. The bearing stiffness coefficients
account for the nonlinear dynamic behavior of the elastomeric supports.
To simplify the modeling process, a number of assumptions were made. First, all
elements, including the rotor and its flexible supports, are assumed to be axisymmetric.
The outer bearing ring is assumed to be a rigid body, since it is considerably stiffer than
the flexible supports. Moments of inertia of the shaft are calculated at discrete intervals
and lumped at the finite element stations. The shaft is modeled with separate mass and
"stiffness radii" to enable accurate calculation of shaft modeshapes while correctly
representing the inertial behavior of the rotor at very high speeds. The inertia of the balls
and retainer are neglected in the model, but their masses are included in the mass of the
shaft so that the total vibratory mass amount is correct. The O-ring bearing supports are
modeled as speed-dependent bearing elements, while the rolling element bearings are
treated as linearly stiff support elements. This assumption is justified by the high and
relatively linear stiffness behavior of ball bearings compared to elastomers.
18
2.2 Axial Vibration
The rotor/bearing assembly can be modeled as a two degree-of-freedom oscillatory
system (Figure 7).
xl
M
Is
x2
H-*
ring
0. 5 *Kearing, axial
0.5*Kwasher
Kwasher
Kbearing, axial
Figure 7: Schematic representation of rotor/bearing assembly for axial vibration modelling.
Damping is minimal in this system. The equations of motion are:
mouterring 1 + (kwasher± kbearing
Mrotori2+
kbearing
X2
-kbearingXi
)
Equation 2
X2 = 0
1
-kbearing
=
f(t)
Equation 3
Written in matrix notation:
0
Mouter ring
0
m,
1
_
1
2
_
kwasher
bearing
-kbearing
bearing
-
X
kbearing [X
2_
f(t))
Equation 4
or
Mi+ Kx = F
Equation 5
19
The homogenous solution is found by settingf(t) = 0. Assume that Equation 5
has the harmonic solutions:
x(t) =
ek
_jk Xj
ckke
)kt
k =1,2
Equation 6
X2k
where Xk is the mode shape and (Ok is the modal frequency. Substitution of Equation 6
into Equation 5 yields:
M(
cok
2
Mo4k
MK-
ktce' + K(cke')'
=0
Equation 7
Equation 8
=0
Fkahe+
washe, k beainng
otkeg
2i
kbeang
-k
bearing
2
kbearingMoto, W k]
=[G(O)XX)=o0
Ilk
Equation 9
X2k
The matrix [G(wk)] must be singular for a nontrivial solution to exist. In other
words:
G(
iouterring
bearing O(
- (kbearmng
,outerring
+ (kouter ing+
kbearing )lfbearfng )t2+
kouter ring keaing =0
Equation 10
This is known as the characteristic equation of the system. As a quadratic, the
solutions co, and 92 can be found by solving the quadratic formula. These solutions are
the natural frequencies, or modal frequencies, of the system. Knowing the natural
frequencies, the mode shapes can be determined. The mode shapes are defined as the
amplitude and directions of the reactions xj and x2 when the system oscillates at the
natural frequencies. The homogenous solution is:
20
X h=
1"
Je'
(X21 )(X22
where c], c2,
#
, and
+
l
12 C2)e
#2 are constants
= 1
c sin(cwt +
, )+ X 2 c, sin(co2t +$2)
Equation 11
determined by the initial conditions x(O) and i(o).
Thus, the mode shapes and modal frequencies can be calculated for the 2 degree-offreedom model for the axial vibration of the rotor.
Chapter 3: Experimental Methods
Experimental verification of the dynamic behavior of high-speed, miniature
turbines presents a major challenge to laboratory instrumentation. Direct and indirect
measurements of vibration parameters (displacement, velocity, and acceleration) are
possible on the rotor system.
3.1 Sensors
The rotor and bearings are concealed by a housing, and are inaccessible to direct
measurement. The only part of the system accessible to a direct vibration measurement is
the protruding shaft. A capacitive measurement system was available, the ADE
MicroSense 3401, which was capable of measuring the 1/16" shaft. However, because of
limitations imposed by the need to measure the rotor/bearing system in an unconstrained
environment (discussed in section 3.3), the use of the capacitance sensor was prevented.
Vibrations of the housing result from a transfer of force from the rotor;
measurement of vibration on the housing provides an indicator of rotor vibration severity.
In addition, insight into acoustic properties of the rotor is gained by an understanding of
the housing motion. Vibration of the housing is conveniently measured with
accelerometers. Piezoelectric accelerometers are easily obtained with sensitivities from
21
1 Omv/G to 1 00mv/G. High sensitivity accelerometers provide better signal-to-noise
ratios, but at a high price . After preliminary exploration, a small form-factor, IOmV/G
piezoelectric accelerometer was found to provide satisfactory performance, after coupling
with an external amplifier.
Speed measurement of the shaft was accomplished using a magnetic-pickup
sensor, after consideration of several alternatives, including optical sensors and acoustic
methods.
Optical tachometers depend on the ability of a photon emitting-and-receiving pair
to sense changing patterns of light and darkness. Fiber optic sensors are very suitable for
high frequency measurement because of their fast response time of ~0. 1 ps to 1 ps.
However, cost and configurability precluded the use of a fiber optic system.
Acoustic methods have been suggested for speed measurement, since high speed
air impulse micro-turbines exhibit a characteristic "whine", which, at rotation speeds
above 75,000 rpm, is largely due to synchronous (once-per-rotation) spectral content
caused by rotor imbalance. With bandwidth filtering, a prediction of rotation rate can be
inferred by audible frequency content. However, this method has been shown to have
high uncertainty, with accuracy limited to t1,000 rpm (Dyson 95). Also, spurious
acoustic components, due to a variety of causes, may corrupt the prediction. Finally,
acoustic methods of speed prediction eliminate phase information from the data set,
further reducing their usefulness in data analysis.
Magnetic pickups sense fluctuations in magnetic impedance. When a keyway or
a flat is introduced to a shaft, the magnetic pickup will produce a sinusoidally varying
output voltage, which is indicative of the rotational rate. The output voltage of the
22
.
......
............
....
...
. ......
- -...
....
magnetic pickup is inversely proportional to the square of the distance to the target, and
proportional to the rotation rate. With an air-gap of 0.005", the sensor provides 0.6V
peak-to-peak at 30,000 rpm. The magnetic pickup method was chosen as the best
combination of robustness, price, and performance for this project.
3.2 Flexible bearing supports
The turbine chosen for this study is a high-speed air impulse turbine with simple
cup geometry. The turbine is a commercial model, and has the unique characteristic that
the rotor/bearing assembly is packaged within a steel canister (Figure 8). This has the
effect of removing some uncertainty of interaction between the rotor/bearing assembly
and the housing, assuming manufacturing controls on the canister and rotor assemblies
are good. Also, the canister form factor is convenient for experimentation, because it
allows simple test-bed geometry.
7/00
Figure 8: Solid model of canister-type high speed air driven turbine assembly.
23
_--- -----___ ----
--
Each bearing in the test turbine bearing pair is supported by a single elastomeric
O-ring. Thus, to change the support stiffness, O-rings of identical dimensions, but
different material properties, were tested. Three different materials were selected for this
parametric study: Viton*-70, Buna-N, and silicone, each with a durometer rating of 70.
Each of these materials has been previously used commercially as a bearing support in
this application. However, the most common elastomer choice is Viton*-70, most likely
because of its high damping ability and its ability to withstand high temperature medical
autoclaving.
O-rings for the study were purchased from Apple Rubber, Inc. The specified
dimensions exactly matched the stock O-ring dimensions of 0.030"CS x 0.298"OD, with
a 0.003" tolerance on the cross-section. However, one major difference between the
different O-rings concerns the matter of "flash". Flash is an artifact of the O-ring
manufacturing process. It consists of a raised portion of material on the ID and OD, left
by the two halves of the O-ring mold (Figure 9). Two of the elastomeric O-rings in this
study - Silicone and Viton* - were de-flashed; the flash was removed in a cryogenic postmanufacturing operation. To cryogenically de-flash an O-ring, the product is frozen, and
then the flash is broken off of the body, and the surface is ground flush. The flash
dimensions of the Buna-N 0-rings were within the manufacturer's specifications.
24
0.005'" max
A
0.003" max
A
I.D.
0.238" ±0.005"
Cross Section
0.030" ±0.003",
Figure 9: Schematic of 0-ring, showing flash dimensions.
Taking advantage of the convenience of the canister-type design for experimentation,
four canister-type turbines were disassembled and retrofitted with new (different) O-rings
for a different bearing support. The procedure to retrofit a turbine is:
1.
Place canister into turbine press.
2. Gently apply force to axle.
3. After canister cap "pops" off, release force.
4. Separate canister cap from canister body, and remove rotor/bearing assembly.
5. Using needle, pick out old O-rings from canister cap and body.
6. Lubricate new O-rings with Minapore Light Oil.
7. Replace 0-rings.
8. Re-seat rotor/bearing assembly into canister body.
9. Align canister cap with canister body, under press.
10. Gently press canister cap onto body.
25
...
.......
..............
...
....................
...
...
.................
...... . ...................
3.3 High-speed rotor test-bed
The test-bed was designed to study radial and axial vibrations caused by the rotor under
its full operating speed range. Under normal operating conditions, air is forced at up to
35 psi through a converging nozzle to drive the impulse turbine, while exhaust air
escapes out large diameter orifice to the room.
Conditions for the test-bed are:
" high structural stiffness
* appropriate mounting area for radial and axial accelerometers
" appropriate mounting area for magnetic pickup tachometer
3.
Figure 10: Exploded view of testbed setup: 1. Turbine canister 2. Inlet air path 3. Exhaust air path 4.
Test block
A rigid steel block, with a cavity for the turbine canister and appropriate air path
fittings, was chosen as an appropriate form factor for the test-bed (Figure 10). The
natural frequency of the block is calculated with Rayleigh's method on a lumped
parameter model of the block. First, the block is modeled as a simply supported beam of
26
length L. A force, P, equal to the weight of the block, acts on the center of the beam.
The deflection of the beam is:
-Px3L
J(x)=
L
2
2
_4x 2)
48EI
S=P(LxXL2
- 8xL+4x2)
48E1
Equation 12
2 <x < L
2
where I is the area moment inertia of the beam:
bh3
12
Equation 13
The maximum deflection of the beam is:
max
=
4-2
=-
48EI
= -2.09 x 10 7 'in
Equation 14
Rayleigh's method solves for the natural frequency of the beam by equating the
kinetic and potential energies of the system. The potential energy, in the form of strain
energy in the deflected shaft, is maximal at the largest deflection. The potential energy is
defined as:
E
= K(m3a)
2
Equation 15
The beam is assumed to undergo sinusoidal motion, due to an external excitation.
The kinetic energy is maximum when the vibrating shaft passes through the un-deflected
position with maximum velocity. The kinetic energy is defined as:
Ek - Wn" (M2)
2
Equation 16
27
Setting Ep=Ek yields:
Co, =
= -54,270.09Hz
g_:
m
Equation 17
3max
-
The block's resonant frequency is extremely high, so the test-bed dynamics will
not interfere with the rotor. To ensure the free motion of the test-bed, the block is
mounted on a sheet of open cell foam (Figure 11).
Testbed]
-Accelerometers
Magnetic Pick-up
Open Cell Fa
Figure 11: Schematic of experimental setup: Front view showing lateral and axial accelerometers,
magnetic pickup, and the low-stiffness open cell foam base.
The procedure for setup of the test-bed is as follows:
1. Release the back-cap by removing screws.
2. Remove any prior turbine canister from the test-bed
3. If a specific test bit is being used, install it into the new turbine chuck.
4. Place turbine canister to be tested into the cavity; align ball with groove to ensure
proper orientation of airways.
5. Replace back-cap and tighten screws.
28
The block is fitted to accept standard "push-to-connect" plastic airline couplings. The
drive air port is supplied by 6mm plastic tubing, whereas the exhaust is created with a 24inch section of 8mm tubing. This length of tubing is used, instead of directly exhausting
the air to the atmosphere, because it was found that porting the exhaust improved the
stability of the turbine performance. A manual checkvalve regulates airflow to the
turbine, allowing the air pressure to be varied from 0 psi to a line maximum of >60 psi.
3.2 Spectral analysis with parametric bearing support variation
Frequency-dependent rotordynamic characteristics of low mass, high speed
turbines are often masked by the extremely fast start-up time of the impulse turbine,
when operated at the normal operating air pressure of 35 psi. However, variation of input
air pressure can reveal the transient response. Specifically, we are interested in the
synchronous response and its harmonics, spectral content related to ball bearing
frequencies, as well as non-frequency dependent spectral content, such as structural
resonances.
To discover the frequency behavior of the rotor, the input air pressure was varied,
causing the turbine to spin at a series of steady speeds ranging from 0 rpm to the
maximum attainable speed; usually around 500,000 rpm. Approximately 70 discrete
steady state speeds were recorded for each canister, with an average step size of 6,500
rpm. A combination of sensors, digital lab equipment, and computers was used to
analyze the vibration data from the test-bed. Signals from accelerometers in the radial
and axial directions were first passed through a Bruel & Kjaer Model 2525 DeltaTron
amplifier, to boost the signal to noise ratio. The improved signals, as well as the
tachometer signal, were monitored on a Tektronix TDS 1012 oscilloscope. The
29
acceleration signals were then passed to a Hewlett-Packard Model 3561A single-channel
digital spectrum analyzer (DSA) which collected up to sixty sequential samples to create
a cascade plot. Each accelerometer signal was then compared to the tachometer input for
a phase measurement, using a Hewlett-Packard 8562A two-channel digital spectrum
analyzer. An RMS average of 16 samples gave the phase between the tachometer and the
acceleration output, and this value was recorded by hand. In addition, the peak
magnitude of the acceleration was recorded in Excel for each sample.
The procedure to test a canister is:
1. Mount canister inside of test-bed as described above.
2. Adjust air pressure to 35 psi, or to a level that yields a rotational frequency of 6.5
kHz (or 390,000 rpm).
3. Run turbine for 2 minutes at 390,000 rpm to warm up bearings and distribute
lubricant.
4. Reduce air pressure to the minimum needed to stably actuate the turbine. This is
usually 6 psi - 8 psi.
5. Properly scale oscilloscope, DSA's.
6. Record shaft rotational speed as indicated by tachometer signal.
7. For radial accelerometer, record vibration amplitude given by B&K 2525.
8. Record phase for radial accelerometer, from dual-channel DSA.
9. Add a sample to the cascade plot on the DSA 356 IA.
10. Repeat for axial accelerometer.
11. Increase speed by -5,000 rpm and repeat Steps 1-10.
12. Print cascade plots
30
Chapter 4: Results and Discussion
4.1 Experimental Results
This section describes and discusses the response of the rotor to unbalance forces
when operated across its entire speed range from 0 rpm - 400,000 rpm. The effect of
substitution of various elastomers for the bearing supports is presented.
Spectral analysis of waterfall plots of the rotor response reveal frequency
dependent behavior related to the rotor and bearing dynamics. In particular, strong
components of the spectra include vibration at the rotation rate (IX), twice the rotation
rate (2X), and at the rotation rate of the ball bearing retainer. Ball bearings have unique
vibration characteristics, related to geometry and rotation rate. The major characteristic
frequencies are:
RPM
_
cos(#)
Fundamental train frequency: RPM 1 60 2
D
1
Defect on outer race: RPM
60 2
Defect on inner race:
D,
cos()
D
RPM n
1+ Db cos(#)
DP
60 2
Ball defect frequency:
RPM D rD1
60
D- IlDb
D
~ (0.3987)
RPM
60
Equation 18
~(0.3987) RPM
60
Equation 19
RPM
(4.2090) 60
60
Equation 20
cos2(0)
RPM_
= (4.6621) RPM
60
Equation 21
31
where Db is the ball diameter, Dp is the pitch diameter, O is the contact angle in degrees,
and n is the number of balls in the bearing. For the bearings involved in this study, Db 0.03937 in., D = 0.1914 in., #= 10, and n = 7 balls.
4.1.1 Viton*-70
Turbine serial number 2J0213 incorporates Viton® O-rings for its bearing
supports. Since Viton* is the material specified as an "Original Equipment
Manufacturer" (OEM) part, this turbine was not disassembled before testing. The rotor
was operated from a minimum speed of 9,600 rpm to a maximum speed of 345,000 rpm.
Spectral analysis of lateral and axial vibration reveals the existence of rotor
resonances, as well as complex behavior related to ball bearing dynamics (Figure 12).
Prominent vibration is detected at IX rotation rate, which is a result of the imbalance in
the rotor. Vibration at twice the rotation rate (2X) begins to appear at 3,200 rpm and is
prominent for all higher rotation rates. Vibration at the fundamental train frequency
(FTF) appears at 3,250 rpm, and grows somewhat linearly as the rotation rate is
increased, possibly indicating some energy interaction at the cage/ball interface. Other
vibration components exist at non-integer multiples of the rotation rate, but are not
characteristic ball bearing frequencies. Spectral components that are unrelated to rotation
rate also appear, indicating structural resonance or possibly looseness. For instance, a
broad amplification region exists for axial vibration in the range of 14,500 Hz to 18,500
Hz. This amplification effect is not noticed in the lateral vibration measurements.
Instead, some high frequency amplification exists for lateral vibration above 18,500 Hz;
this amplification is not present in the axial vibration measurements.
32
Vibration at the lX frequencies displays non-linear behavior. Lateral acceleration
levels increase from OG through a broad peak to 4.07G at 4,050 Hz, or 243,000 rpm, after
which the vibration level reduces to about 1G. The phase of the lateral acceleration
relative to the rotation of the shaft lags by 90* at 4,050 Hz.
33
RANGEt
2'
63
A&
-15
dBV
MAG
STATUSs PAUSED
RMSs 10
d6
/DIV
STARTv
0 Hz
SW* 190.97 Hz
RANGE
-51 dBV
A&MAG
STOP2
20 000 Hz
PAUSED
STATUS
RMSs 10
d8
/D IV
START
0 Hz
SWr
190. 97 Hz
STOPs
20 000 Hz
Figure 12: Waterfall plot showing (a) lateral acceleration and (b) axial acceleration for rotor with
Viton®-70 bearing supports. X-axis represents the frequency domain: 0 Hz - 20 kHz. Y-axis
represents the changing rotor speed: speed range 9,600 rpm - 345,000 rpm. Acceleration is
measured on the Z-axis: OG - 8.913G.
34
Viton-70: Testbed Lateral Vibration
10
0
1
0
0
C)
C)
0.1
1000
0
2000
3000
4000
5000
6000
Rotation rate [cycles/sec]
Viton-70: Phase
0
*
*.4
***44**
*4
4.
-90
444*44444
-180
0
1000
2000
3000
4000
5000
6000
Rotation rate [cycles/sec]
Figure 13: Vibration and phase of testbed, measured in the lateral direction. Phase measured
relative to shaft tachometer.
35
Viton-70: Testbed Axial Vibration
10
4
____________
1
4
0
_______________
_______________
II
_______________
_
4
C)
4
0.1
0.01
1000
0
3000
2000
4000
5000
6000
Rotation rate [cycles/sec]
Viton-70: Phase
90
0
*4
0
4..
4
4
*
.4.
**
4
-e
*
0
4
44
444.4
4
4
-90
-180
0
1000
2000
3000
4000
5000
6000
Rotation rate [cycles/sec]
Figure 14: Vibration and phase of testbed, measured in the axial direction. Phase measured relative
to shaft tachometer.
36
4.1.2 Buna-N
Turbine serial number 2K034 incorporates Buna-N 0-rings for its bearing
supports. This turbine was disassembled before testing in order to replace the original
(Viton -70) O-rings with the Buna-N elastomer. The rotor was operated from a
minimum speed of 48,000 rpm to a maximum speed of 480,000 rpm.
Prominent vibration is detected at IX and 2X rotation rate. The 2X component
begins to appear at 28,000 rpm and is small relative to the IX component until the rotor
speed reaches 432,000 rpm. This rotational rate (7,200 Hz) places the 2X component at
14,400 Hz, where a broad resonant region is excited. This phenomenon is present in both
lateral and axial vibration spectra, but is stronger in the axial data, where a broad
amplification region exists for axial vibration in the range of 14,200 Hz to 17,000 Hz.
Vibration at the FTF appears only in the lateral vibration data, becoming noticeable at
375,000 rpm. The small component of vibration due to the FTF indicates low energy loss
at the cage.
Vibration at the IX frequencies displays non-linear behavior. Lateral acceleration
rise abruptly to 3.3G at 3,300 Hz, drop rapidly to 1.6G at 3,500 Hz, and then gradually
increase through a broad peak to 3.5G at 4,000 Hz, or 240,000 rpm. After this point, the
vibration level reduces to 1.6G at 342,000 rpm, and finally increases to 3G at 480,000
rpm. The phase of the lateral acceleration relative to the rotation of the shaft lags by 630
over the entire operating range.
The rotor on Buna-N bearing supports exhibited very high levels of axial
vibration, both in respect to its own lateral vibration levels, and to the axial vibration
levels from the rotors with Viton-70 and silicone supports. This may be due to the effect
37
of flash remaining on the O-ring, which tended to impart destabilizing forces on the
bearing rings.
The flash on Buna-N bearings is a result of the manufacturing process; it is
actually the parting line on the O-ring, left over from the mold. After running the turbine
with Buna-N rotor supports, the system was disassembled and inspected. A large amount
of black debris was found inside the canister (Figure 15).
Figure 15: Buna-N 0-ring after testing and disassembly. Black debris resulted from the
disintegration of flash during turbine operation.
38
5
dE
ETARTa
Z
~~
zRO
COO Hz
zSOS~
R?4Sv 10
MAG
D/~ V
SToARTs
1) HZ
Y
IW
90. 97 Hz
STOPi
ZO
OC Hz
Figure 16: Waterfall plot showing (a) lateral acceleration and (b) axial acceleration for rotor with
Buna-N bearing supports. X-axis represents the frequency domain: 0 Hz - 20 kHz. Y-axis
represents the changing rotor speed: speed range 4,800 rpm - 480,000 rpm. Acceleration is
measured on the Z-axis: OG - 5G.
39
Buna: Testbed Lateral Vibration
10
4
4
*.
_______
*
*4
4
.'
-44
0
**
4
~-
4
~-4*4
-4
*44
.4
4
q4
4*4
0.1
1000
0
2000
3000
4000
5000
6000
7000
8000
6000
7000
8000
Rotation rate [cycles/sec]
Buna: Phase
0
-90
-180
r--
to) -270
U.
,
-
--......
-360
.4.444*
-450
-540
-630
-720
0
1000
2000
3000
4000
5000
Rotation rate [cycles/sec]
Figure 17: Vibration and phase of testbed, measured in the lateral direction. Phase measured
relative to shaft tachometer.
40
Buna: Testbed Axial Vibration
10
I__
ffi
__-9-
1.
_
9
9.
9*99
99~~~~;
9
___
-9-----
0
1
C.)
C.)
9
9
.9
*9
0.1
1000
0
2000
4000
3000
5000
6000
8000
7000
Rotation rate [cycles/sec]
Phase
90
9
9
0
;9999,***
09
9
9
9
-90
--- 9..9*
9
9
1*
-180
0
1000
2000
3000
4000
5000
99
***
*
9
9
6000
7000
8000
Rotation rate [cycles/sec]
Figure 18: Vibration and phase of testbed, measured in the axial direction. Phase measured relative
to shaft tachometer.
41
4.1.3 Silicone
Turbine serial number 2K074 incorporates silicone O-rings for its bearing
supports. This turbine was disassembled before testing in order to replace the original
(Viton*) O-rings with the silicone elastomer. The rotor was operated from a minimum
speed of 39,600 rpm to a maximum speed of 492,000 rpm.
Prominent vibration is detected at IX and 2X rotation rate as well as many noninteger multiples of the rotation rate. The 2X component begins to appear at 76,000 rpm
and is small relative to the 1X component over the entire operating range. This
phenomenon is present in both lateral and axial vibration spectra, but is stronger in the
axial data, where a broad amplification region exists for axial vibration in the range of
14,200 Hz to 17,000 Hz. Vibration at the FTF appears only in the lateral vibration data,
becoming noticeable at 375,000 rpm. The small component of vibration due to the FTF
indicates low energy loss at the cage.
Vibration at the IX frequencies displays non-linear behavior. Lateral acceleration
levels increase from OG to 1.69G at 1,700 Hz, or 102,000 rpm, after which the vibration
level reduces to 1.1G at 153,600 rpm. A second peak of 2.17G occurs at 3.6 kHz, or
216,000 rpm, before the response falls off sharply and settles at -0.5G. The phase of the
lateral acceleration relative to the rotation of the shaft lags by 900 over the entire
operating range.
42
At
MAG
RMSi 10
2
0
dB
/DIV
START# 0 Hz
A
BW.
190.97 Hz
MAG
STOP.
20 000 Hz
RMSt 10
2
G
5
dB
/DIV
STARTs
0 Hz
BW# 190.97 Hz
STOP.
20 000 Hz
Figure 19 Waterfall plot showing (a) lateral acceleration and (b) axial acceleration for rotor with
silicone bearing supports. X-axis represents the frequency domain: 0 Hz - 20 kHz. Y-axis represents
the changing rotor speed: speed range 3,960 rpm - 492,000 rpm. Acceleration is measured on the Zaxis: OG - 2G.
43
Silicone: Testbed Lateral Vibration
10
0
1
9
04
9*
0.1
1000
0
2000
3000
5000
4000
6000
7000
8000
Rotation rate [cycles/sec]
Silicone: Phase
0
-180
..
r-I
-360
*'..***
49*
-;..-.-.
*.
,
*
-540
0
-720
-900
-1080
0
1000
2000
3000
4000
5000
6000
7000
8000
Rotation rate [cycles/sec]
Figure 20: Vibration and phase of testbed, measured in the lateral direction. Phase measured
relative to shaft tachometer.
44
Silicone: Testbed Axial Vibration
10
1
0
*
0
0
9
0.1
9
-.
9
Th
_
0.01
0
2000
4000
6000
8000
10000
8000
10000
Rotation rate [cycles/sec]
Phase
180 90 -
0-
9 9
99
4140
+
..r
to
4000
6T00
-90
-180
I.9
9
-270
*
9 94
*4
9
9
-360
Rotation rate [cycles/sec]
Figure 21: Vibration and phase of testbed, measured in the axial direction. Phase measured relative
to shaft tachometer.
45
4.2 Finite Element Model
The finite element model verifies the possibility that the rotor traverses critical
speeds within the normal range of operation. In application of the model, undamped
rotor response is first obtained to gain an understanding of the placement of critical
speeds. Then, damping is applied to develop an understanding of the stability and real
behavior of the system.
The accuracy of the finite element model hinges on the accuracy of the estimation
of dynamic properties of the elastomeric bearing mounts. This is because these
properties are much less well understood than the mechanical properties of any other
component in the rotor.
A best choice for model stiffness and damping coefficients was made for each
elastomer, given available data.
4.2.1 Viton-70
Smalley, et al. has suggested a curve fit to determine the elastic properties of
Viton* -70 and Buna-N-70. The elastomer supports are represented using a nonlinear
viscoelastic model. A power-law estimation predicts isothermal stiffness, loss
coefficient, and damping for the elastomer supports at a certain frequency,f, in Hz:
Stiffness : k = A1 (21rf)B1 N/m or Lbf/in
Equation 22
Loss Tangent : q = A 2 (2rf)B2
Equation 23
Damping: c =
2rf
N - s/m or Lbf - s/in
Equation 24
where the coefficients Al, Bl, A2, and B2 can be obtained from Table 1:
46
Table 1: Stiffness and loss coefficients for power law estimation of Viton®-70 dynamic material
properties. (Smalley, Darlow, and Mehta 3-25)
Loss Coefficient
B2
A2
0.1746
0.255
0.2392
0.1103
0.3227
0.0226
Stiffhess Coefficient
B1
Al
0.3747
4.520 x 10"
5
0.4195
1.694 x 10
0.1406
8.850 x 10'
Material: Viton"-70
Temperature
25 C
38 C
66 *C
Using the same core data as Smalley, Darlow, and Mehta, Atkiirk and Gohar
developed a curve fit for Viton®-70, based upon a Voigt viscoelastic model.
KI
Material
Viton-70
KO
3.40 x 101
KI
6.05 x 106
ci
6050
KO
ci
0
Figure 22: Voigt viscoelastic model with stiffness and damping coefficients. (Atkurk and Gohar 187)
Equivalent stiffness and damping factors are given by:
Ke
(K0 + K, )KOK, + K ccij
(K + K1 2+C2
(K
+K
)2+ C2
2
Equation 25
Equation 26
Both curve-fits use the same set of data collected by Smalley, Darrow, and Mehta;
based on an experimental methodology developed in part by Gupta, Tessarzik, and
Czigleni. This data set was collected in a frequency range of 100 Hz - 1,000 Hz
(representing 6,000 rpm - 60,000 rpm). Thus, all values for stiffness and damping
calculated for frequencies above 1,000 Hz are an extrapolation, and subject to increased
47
error. However, the extrapolations can be taken as a rough estimate of true behavior,
when verified against known patterns of behavior of the elastomeric compounds.
Mechanical properties of all elastomers are heavily dependent on environmental
conditions. For instance, most rubbers become less stiff at higher temperatures, and
stiffer at higher excitation frequencies. Elastomeric materials are commonly described by
a complex modulus:
G*, = G + iG
Equation 27
where G, and G,, are the elastic and viscous components, respectively, of the complex
dynamic shear modulus, G, *. G, is also known as the storage modulus, while G"' is
known also as the loss modulus (Freakley 58). The subscript, o, is a reference to the
frequency dependence of the complex moduli. A common expression of the damping
capacity of an elastomer is a relationship between the storage and loss moduli:
ri
=
G"
tan(6)=--"-
Equation 28
If an elastomer is subjected to vibration at an increasing frequency, the material
properties will pass through a series of zones (Figure 42). The known frequencydependent behavior of Viton-70 is shown in (Figure 43).
A comparison between the curve-fits done by Smalley versus Atkirk is shown in
Figure 23. It is clear that the Voigt-element curve-fit performed by Atkfirk provides the
more reasonable prediction of O-ring damping. Whereas the Atkiirk model predicts a
temporary peak in the loss coefficient, the Smalley model predicts a somewhat constant
increase in loss coefficient. This is despite the fact that Smalley, Darrow, and Mehta
have recognized that Viton traverses its transition zone in the frequency range of 100 Hz
48
-
1,000 Hz (3-3). When applied within the finite element model, the Smalley curve-fit
for damping results in a heavily over-damped model, whereas the curve-fit by Atkurk and
Gohar results in a less heavily over-damped model. Observation of actual damping is
accomplished through the half-power method. The amplification factor, Q, of a dynamic
system is defined as:
Q=
f-
Equation 29
-
(f 2 - fl24f
wheref, is the system's observed natural frequency;fi andf 2 are the frequencies
corresponding to an amplitude:
A(fm
A(f2)
A
"
=
Aff)
"
Equation 30
solving for f allows determination of system damping:
(
Equation 31
Br
2Mc>f
Following this methodology, actual damping values have been determined from
the experimental data (Table 2).
Table 2: Damping values for Viton -70, based on half-power method applied to experimental data.
Material
Viton-70
f, [Hz]
4050
f, [Hz]
3475
f 2 [Hz]
Q
4350 4.6286 0.1080
B, [Lbf-s/in.]
0.0093
49
.
.. .
...........
..
.. ....
.......
. ......
_--...
......
......
......
Viton-70 Stiffness Prediction
1.OOE+08
1.OOE+07
1.OOE+06
10000
1000
100
Rotation rate [cycles/sec]
Atkurk & Gohar
extrapolated
-
Smalley, et al.
o extrapo
(a)
Viton-70 Loss Coefficient Prediction
2
r
-
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
_
10000
1000
100
Rotation rate [cycles/sec]
Atkurk & Gohar
-
extrapolated ,
Smalley, et al.
e
extrapolated
(b)
Figure 23: Comparison between two curve-fits for estimation of dynamic properties of Viton -70
elastomer. (a) Stiffness (b) Loss Coefficient.
50
cm
5/
I
WEFLc~u
I
H--L7tL---
7i~j
(a) Mode No. 1
361,007 rpm
-
(c) Mode No. 3
(b) Mode No. 2
654,975 rpm
1,220,675 rpm
Figure 24: Modeshapes and modal frequencies for rotor with Viton®-70 bearing supports.
Bode PtoI
Station: 4, Sub-Station: I - (0-pk)
probe 1 (x) 0 deg - max amp = 0.042536 at 354000 rpm
probe 2 (y) 0 deg - max amp = d04253 at 354000 rpm
......................
.
.
.
.
S
II
0.05
0.04
I
I
- -
--
- -
-
-
- -
-
--
-
-
-
-
-
-
- -
-
--
-
-
---------
-I
-I
I
-
-
-
- -
I
--
-
-
- - - - - - r- - -I -
4------4----------L.I------
I
-
-
I-
-
-
-
-
-
- -- - - - - I
I
--
-----
------------ L..--------------------
-----------------
0.02
0.01
0.001).XEE+00
I 20E+05
2A4E+06
3.80E+05
4O0E+05
&90DE+05
RoatIcnal Speed (rpm)
File: C:\DyRoSeS_ RotoFHHCahcsdvton25.rot
Figure 25: Bode plot of undamped response to imbalance for rotor with Viton®-70 bearing supports.
51
Bode Plot
Station: 4. Sub -Station: I - (0-pk}
pfobe 1 (x) 0 deg - max amp = 000047926 at 386000 rpm
probe 2 (y) 0 deg - max amp = 0.00047928 at 386000 rpm
10
. . .
8I
720
630
- - -
--
-- - -
5
-- - --
-- - - - - -- - - - - -
40
m
-
-
---------------------
-1 -- - -
0
2730
-
-
-
T------r-------I------
------- -- - - -
1I
- -
-- - - - - - -r - - - - - - -- - - - - -
T
I -- -
-
-
-
--
-
-
-
-
-
-
--
-
-
-
-
--
-
--
-
-
20
180-
-- --
I
901
0.00072
Bod
,-
-
-
1.0E0 I.OE0
RoMI~s~n5
Fie
Fiur
---
- - - - - -- -
-
-
--
----
-- -
- L --
3.0+0
-
---
J-
-0E0
-
--
-
O
I.~~oe
oainlSed(~n
0eainsupr.Moe
26 poofrsostoiblneoroowihVo
inopoaesdmpn0stmte6roie
-
I
-
2.0E0
o
--
by Atur
and
L...
52
V& Tenn
Transierd Response
X-Y Plot
X"00
t tp: Mi - -1
i. -. 380M.00
1679E -
MA
m6
x T op Min - -. 6op01E M100 2.8108E-306
I 5154E-006
Ma
.21S3M-006
= I.390E-006
V06010.p
--
------------
50E-06
n-tf
- -----
4;2-000.
-
-- -- 0 -----
- ---
--
-----
-----
22E-00
400E-0.'
-2120E-Ce ------------ -------------------------- ------------
--------
- ---
-.
- --------
00--06
---
F4* C
C3DyRo6.SR0o000HC00aio4020oo00.
,0000
0
-C
OAM
0.0024
ahoo~
F.
---
-
--
-
---
---
-
0o0,06
io.0..20
1. (b)
1.(a)
Tnasdn Response vs. Tkne
X-Y Plot
x
a0
C3 .
. . . ...
0
itp.
dap
000600 3
.000031977
4i=0003072M000.00001010
....
. ...
...
.
x
mi" 400031635,0
di6p;
O
mi' -0 00036, Mexd-M0.0031072
000031977
Max-
0.000319
0.000S 1-----
0000 3
----- ----
Min
Y disp.Min .
4
T0ran.00 Respone vs. Tm
X d-W Mm
Mo - 0.5474E-00
42220.
64E-005
.54330E-0 -,06...
- ---
FY
-
000
400E45
-0-
----
-
Rotmo7CV00A8o0n250rat
4222E0.
- .
: 5.5474E-000
4338E.00&
U-x
M-
S
.984SE-00
P
1
0. .0
O
-
00.0000t.00100
'0e*S
I-: - &Mn
-4~ -4~~
4.000-00
W
3. (a)
00
nsS4
X-Y P101
Xdiq.
C
0010
0
0.001now
=
2. (b)
2.(a)
F0.
0
0
000
0.00 0.00R,010o
F#.*. C.AyRoG*9_Row&deHCVOsdwfto25.not
FiW
CA~yRo.8
0
D15
a MW
0
D25
M+ChedonK
3. (b)
Figure 27: X-Y plots and transient motion of shaft center, rotor with Viton®-70 bearing supports,
modeled at 25'C, and incorporating damping estimates developed by Atkurk and Gohar. (1) 190,000
rpm: Below first rigid-body critical speed. (2) 360,000 rpm: Near first rigid-body critical speed. (3)
500,000 rpm: Above first rigid-body critical speed. Rotor achieves limit-cycle (stable) motion at all
speeds within normal operating range.
53
Precession
Poces&oW ModA Swipe- STABLE BACKWARD
00.8 RAotSopeed =l8000rpm. Mode No.= 2
Wr Speed iDwrped NAtralFroq.) 103 p,, Lo% Do-rol
Pd.e
21W4.64.0
C1Dy~o8.8Roto4Caed1od2SalOt
(a) 190,000 rpm: Stable backward whirl
P1.onoowoao Mod. ShAp. - STABLE FORWARD Piooo. mo
Shed 008000.1 Speed -30800pMm Mod. No. 2
"M80 Speed 1D0,ped NOiuraW Fraq.). 350444 rpm,
Log.
D
rint A083
Fa. ClyRoB SRt*M oohm0ddon25.lot
(b) 360,000 rpm: Stable forward whirl
Poooooosio
1hir0 Speed (Doaped
Fda
Ptoosoo
Speed - SOO 8qom Mod. No.- 2
1075.980
Freq.) 258 rpm. Log Decar*=
Mode Shap. - STABLE BACKWARD
Shoft RooiDnig.
NkrIM
CQyRoB.SRocWMHC80mdvton,25nA
(c) 500,000 rpm: Stable backward whirl
Figure 28: Whirl mode shapes of rotor with Viton*-70 bearing supports. Backward whirl at 190,000
rpm and 500,000 rpm indicate too high a level of predicted damping in finite element model.
54
________
Map
________Stability
__________
S3,OOE*O7
--
--
-------
j---------
------ -----------
I
I
I
IOOE+07
elk tvl=+C*Ml
-.
-
T. V
tIJZu+tJb
=O-eO
-
-:4-
2,41R4O
P.-RP., ikE q F 3JYt+U5
WIN +05
X~IM-iUo
Rottonal Speed (rpm)
SFile-, C:MDyRoBeSRoo'iHC~hcdvitof25.rot
Whirl Speed Map
Z 3GEtO-------------------
-
5MI
------------ ------------
------------
- -
-
- -
---
I
1 3161
Roaioa
O
-Ayoe
-
-
-
-It
-
- --
-
- -
-
- --
-
- -
- -
04A
I.OE0
File.,
- -
Spe
EI+1356hAj
+0
(rm
IvO~25r~
(b)A
Figure~~~~~~~~
shwn nauareunis
2:RtrssewihVtn70baigspot
a
tblt
a
b
hr
pe
a
dape
55
Transmitted Force (sewi-major axis)
BemhinWSuoport Station : = 2. Station J = 9
Max Forces =4.8752 at 382000 rpm
48-E-TrE030E
4-r++
-------------
3.0----------------:
jr
-----------------
720 -------------
e
240
-r0 0 1
0.00E+00
I
\
I -.
.
...
1.20E+05
I
I
-------
-
077
=
I
I
I
I
240E05
340
V
p
I
I
I
I
I
3.60E+05
I
I
I
I
I
4 80E+05
I
I
I
I
QODE+05
Rotational Speed (rpm)
File. CADyRo8eSRototHHCtmtsdvton25.rot
(a) Bearing 1
Transmitted Force (sent-major axis)
Beauing/Support Station
I 1: = 4, Station J: = 10
Max Forces = 10 717 at 384000 rpmn
12,501
1200
10,00
8
(a
--
-------
7.50---
-
- -
~
-- -
-
I
--
n
-
~
- -
---------
- --- - --- --- -
--
T,
- -
~i-r
- - - - - - - ----\
----
L
I{
rr-r
I------
K--
-- - ----- -1-
I Bern
Figue 3: Tansittd
frceat barig 1and2 fr rtorwithVitn' 70 earng upprts
56
Inspection of the simulation using the extrapolated values for high-frequency
stiffness and damping shows that the experimental results do not match well with the
analytical results for the Viton®-70 elastomer. The predicted first rigid-body critical
speed of 354,000 rpm is considerably higher than the experimentally observed critical
speed of 240,000 rpm. In addition, the predicted damping level appears to be overdamped. A good indication that the system is over-damped is the strange behavior
observed in the whirl speed and stability maps (Figure 28). The first synchronous whirl
mode, mode 2, should be a forward whirl mode. But the model predicts a backward
whirl, a phenomenon that is highly unlikely with a forcing function due to unbalance. In
addition, the stability map predicts extremely high values for the log decrement, on the
order of 107 , indicating a heavily over-damped system.
An improvement in the fit between the model and experimental data can be
achieved by revising the dynamic stiffness predictions of Viton*. It is possible that the
constant 25'C temperature assumption is invalid due to heating of the elastomer through
several mechanisms. First, bearing friction creates heat, which is either dissipated
through a coolant, carried away by the airflow, or is transferred across the outer bearing
ring to the O-ring. Secondly, the elastomer can self-heat through hysteresis.
Hysteresis is the mechanism by which an elastomer dissipates energy, translating
kinetic energy into heat. Viton®is a "lossy" elastomer; its loss coefficient peaks above
1.0. Thus, Viton®tends to dissipate heat internally at a high rate. Air flow through the
turbine should tend to carry away some of the heat, but since Viton®has an extremely
low thermal conductivity, much of the heat will tend to remain in the O-rings
(Goodfellow website, Monachos Mechanical Engineering website).
57
- =
-
.
.. ...
.....
Thermal Conductivity of Various Materials
1000
100
10
1
-1
0.1
--
0.01
0N
.co
0
Figure 31: Comparison chart of thermal conductivity of Viton , Buna, and silicone elastomers versus
other materials.
Temperature Dependence of K
Viton-70, Smalley, et al.
1.OOE+08
____~~A
2 1.OOE+07
____
t_4__1i
____I
_____
1.OOE+06
£iI;;;k10000
1000
100
Rotation rate [cycles/sec]
--
25*C --o- extrapolated
, 38*C
extrapolated
+ 66"C
o extrapolated
Figure 32: Temperature dependence of elastic modulus of Vitone-70 according to Smalley, Darrow,
and Mehta.
58
Bod. Pet
5tt0n. 4, Sub-8*a01
pob. I ix) Og - Sxmp0.01270
Bode
Stem
tO-f)l
@144DOOmOrO
p0b&1
SProe2
(C)
(yU
4.
0
d
g
ISaO ----------- ------ +- --
009
----
20000
-
-
-+-------------
-------
-------
-----
--
Fie Ci0yRoeeS_RoWo.tCcawsntot
4N00
200
2105i06
00
&0t&.06
Speed (rpm
Rawls
t
I - (-oki
0007396 at 240000 poo
oO 240000 Tpm
.00037200
-
-----
--
----------
-------------------------00002
1.200.0
-
1
-
-------
+--
II
ON
ap00
ap.
+
---------
-------
PIg
ub-aton
o-m
d.9
O
Fit-
00E+00
E+200.-
-
-----------
2000+-
24005
----
--
40E0
t
00 00
RS.BtonMlSpood~rpm)
CtDytO0000R0040C00ted.00.SUo
(b)
(a)
Figure 33: Bode plots for imbalance response of rotor with Viton®-70 bearing supports; 66 *C case.
(a) Undamped (b) Damping provided as measured from experimental results.
Whirl
------
Stablity Map
Speed Map
-----------
200E06
400.E-05
--
-
-
O.w12E i r
t0WE+CVCd1W
1.95
L-
2400+0
Ratabonal
-
---
380E.00
Speed (4MM
It.
-
4
E5
0E0506
0. 00E.00
1
0000
2AE.00
RPgl.oCi
20E.06
$peed
(pt
40E+05
0.0.0
0
o
Rio CVyRfooSJ
(a)
(b)
Figure 34: Finite element analysis of rotor model with Viton®-70 bearing supports. (a) Whirl speed
map indicating damped natural frequencies at 231,000 rpm and 444,000 rpm. (b) Stability map
indicating stable rotor behavior.
59
A detailed study of the heat transfer behavior of the O-rings could not be conducted
within the scope of this paper, but prior research has been conducted on the effect of
increasing temperature on the complex dynamic modulus of rubber O-rings. Smalley,
Darrow, and Mehta studied the complex dynamic shear modulus of Viton®-70 O-rings
under three temperature conditions: 25'C, 38'C, and 66'C (4-1). The results of these
studies showed that the elastic and loss moduli both decreased significantly with
increased temperature (Figure 32).
To better model a potential rise in temperature due to hysteretic losses, the
stiffness and damping coefficients from the 66'C case were used. The result of assuming
a higher temperature was a reduction in the undamped first and second rigid-body critical
modes. The first rigid-body critical is predicted to occur at 237,211 rpm, or 3,953 Hz.
The second critical frequency is predicted to occur at 432,424 rpm, or 7,207 Hz. These
speeds correlate very well with the empirically observed natural frequency of 4,050 Hz.
The damping value of ( = 0.108 calculated using the half-power method is
entered into the finite element model, yielding first and second damped natural
frequencies of 231,000 rpm and 444,000 rpm (Figure 34). The rotor undergoes stable
forward whirl. The damped response at the two bearing stations is on the order of 1 Lb.
While the validity of the model coefficients for this model is difficult to assure,
the observed behavior is more similar to the empirical results than the 38'C or 25'C
models. Heating of the Viton®elastomer through hysteresis may be a reasonable
explanation for this phenomenon.
60
4.2.2 Buna-N
The power law estimations created by Smalley, Darrow and Mehta for the
stiffness and loss coefficient of a pair of Buna-N 0-rings at 25'C are (3-24):
0 5 19
k = 2.237x10 6 (2f)
'
Equation 32
77 =.0606(27f)0 .2 32 6
Equation 33
Buna-N is generally less sensitive to frequency than Viton* (Figure 41). Thus,
the undamped critical frequencies for the same rotor with Buna-N 0-ring bearing
supports will be less than for a rotor with Viton® O-ring bearing supports. Based on this
stiffness prediction, the first and second undamped rigid-body critical speeds are 319,137
rpm and 581,797 rpm respectively. The first flexible rotor critical speed is 1,214,216
rpm.
The damping prediction created by Smalley, Darrow, and Mehta cannot be
extrapolated to frequencies >1,000 Hz for the reasons described in Section 4.3.1. The
system amplification factor was determined experimentally to be 6.66. However, the
resolution of the experimental data, combined with a lack of published material
concerning the dynamic behavior of Buna at high frequencies and different temperatures,
reduce the likelihood of accurately modeling the system.
4.2.3 Silicone
The elastic modulus for silicone is considerably lower than Viton*-70 or Buna-N.
In addition, the loss coefficient for silicone is lower than either Buna-N or Viton*-70. A
reduced frequency/temperature nomogram was used to estimate stiffness and damping
61
coefficient for the finite element model (Figure 44). Estimates of the speed-dependent
bearing support stiffness values were:
Table 3: Model coefficients for frequency-dependent stiffness of silicone elastomer.
Frequency [cycles/sec]
1000
2000
4000
6000
Stiffness [Lb-f/in]
1900
2000
2250
2400
With these stiffness coefficients, the undamped rigid-body critical speeds are at
120,000 rpm and 220,000 rpm, respectively. The first flexible rotor critical is at
1,200,000 rpm.
Damping was introduced to the model, based upon the experimentally determined
amplification factor of 6.18.
Table 4: Model coefficients for frequency-dependent damping of silicone elastomer.
Frequency [cycles/sec]
1000
2000
4000
6000
Damping [Lbf-s/in]]
0.01
0.009
0.008
0.006
With damping, the system undergoes stable forward whirl at roughly 120,000 rpm
and 200,000 rpm, but becomes unstable at speeds higher than 480,000 rpm. Both
resonances match well with the experimental values of 102,000 rpm and 216,000 rpm.
The rotor exhibited a sharp increase in measured acceleration at 480,000 rpm, which
could possibly indicate the transition to unstable whirl.
62
.17
~-
-
-
-
-
--
Stability Map
Wirl Speed Map
LogDecorrAint ladcAn M51111044
1409a"
.......
2500M
-T ------------ r ------------ ------------ I ------------
- -------------------
T
-.2
---------
- '
'
' '-' '--'
510000
------------
-'-'-'---
-------------~~~~~~~
------ ------------
U
1050505
U
-----------I ------------ -----------
-------------------
-0301
a
A
------------ T------------ r------------ ------------------------------------ ------------ ----------- I ----------- I ------------
000E+00
2.0E0a5
520E0
40&E-
30E.0a
fts
Fikr CAyR0~S1_R0Wftom4CW4MKAVdn2r
&0Dt00E
Pi.
0 +0
~~
1W I E+-W
snas(rPMo
booo~
CiVyR0o&8R..r04C5thcfSk.Z5
1ADE+-M
~MOW
360E+0
ako-4000
...........
6.00+05
440E+05
rd
(b)
(a)
Figure 35: Finite element analysis of rotor model with silicone bearing supports. (a) Whirl speed
map indicating damped natural frequencies at 120,000 rpm and 200,000 rpm. (b) Stability map
indicating stable rotor behavior up to 480,000 rpm.
.0..
xdp
fa 1
O.0DE-05
--------
e.ns.us..
a
.
i. - D.00001204..
0M..
Y diY0 .p 4.000
. . .
. 7 .6---
.
X
.00012903
Ydip
00001207
. .
. .
.
. .
.
. .
isp.
-401E0.
M. -8.5011E-005.
45128E-00
9 32740-
Mu
-
0.1%.......___
-- -----6t OOE-05
.
Mw
N
-----------
000DE.
400E-05
40OE-05
Figure
(y6:
Saftcenter otonp
FO. CSDA0
(a)
rot40
k..20
F",
C 0yRB.sRo
-------
0S"Cth.d
---
-
--------
&ik.25 k
(b)
Figure 36: Shaft center motion of rotor with silicone bearing supports. (a) Stable, limit-cycle motion
at 200,000 rpm (b) Unstable elliptical motion at 500,000 rpm.
63
-
4.3 Axial Dynamic Behavior
The bearing pair used in the turbines in this study was designed to operate under a
0.75 lb pre-load force. This force is provided by a set of wavy-washers, with an effective
spring force of Kwasher = 50 Lb-f/in. Under this loading condition, the bearing pair
develops an axial stiffness of Kbearing, radial= 12,152 Lb-f/in. Thus, a 2 degree-of-freedom
model can be built for axial vibration, which is excited by bearing forces.
In this model, the mass of the ball-bearing balls and retainers are lumped into the
rotor mass, so that Mrotor = 1.3122 x 10-3 kg. The two bearing outer rings are lumped as
an equivalent mass,
Mouter ring
= 2.91 x
10 A
kg.
There are two modes of oscillation of this system. The eigenvalue of the first
axial mode is at 526.3 Hz. This mode is passed through very quickly as the rotor is
actuated; it is below the measured speed range for the experiments in this study.
The second axial mode of oscillation occurs at a frequency of 15,313.25 Hz. This
frequency is above the highest attainable synchronous rotor speed, but it is well within
the audible range of 0-20,000 Hz. In fact, every turbine measured in this study displayed
a resonance phenomenon in the region between approximately 14,000 and 18,000 Hz.
This resonance was only observed in the axial behavior of the turbines; no such force
amplification was observed in the lateral vibration measurements. It is highly probable
that the second axial mode of oscillation of the rotor is excited by synchronous multiples
of the rotation rate.
Damping of the second axial mode is small; mostly attributed to the interface
between the outer ring and the 0-ring bearing supports. Damping in the rolling-element
bearing is usually set to zero, but with a grease lubricant, and given the small size of the
64
system, the bearing may contribute some small amount of damping (Norton 654, Slocum
456).
The excitation of the second axial vibration mode is of concern because of the
acoustical performance properties of the turbine. 15 kHz is perceived by humans as a
piercing, annoying whistle. Damping of this mode could reduce the amount of audible
noise emitted by turbines of this type.
Bode Diagram
100
-
C
-- - . -
-------
-0
..--.---
++-
w-200
~ ~
I
------ -I
-t -
--,.-
t-------+-
S
- --- --- I
I ------
I
I -------
-3100
L- .
90 -- - - ----
..
a> 45
. - - - ..
-.
T
> 90
+-i
10
3
------ t
J
.. . - - - - -J - . J .
T
-
te---
10 4
-
. . - .. -
-,
,
-
+
64
A ingsyste Leienles
s
Fgr37 Bode po o ial virtino r
.- .. . .. . -. .. L
--
,r
+
+ +
105
-
-
---
+
- :-.. L.
- '- - ..' - 'L..
-
-
+
9
M
10 6
Frequjency (rad/sec)
Figure 37: Bode plot of axial vibration of rotor, showing system eigenvalues.
65
Chapter 5: Results
The purpose of this research effort was to clarify the dynamic behavior of a lowmass, high speed rotor with elastomeric O-ring bearing supports. A testing apparatus was
designed, constructed, and instrumented to capture relevant data to give an indication of
the imbalance response of the rotor across its entire speed range. The effects of O-ring
supports made from three elastomeric materials: Viton* 70, Buna-N, and silicone, on the
rotor response were observed. A finite element model was built to numerically simulate
the rotor and support structure dynamics. A Matlab model was constructed to analyze the
dynamics of axial vibration of the rotor/bearing assembly.
The rotor with Viton*-70 bearing supports was found to have a critical speed at
243,000 rpm, with peak acceleration value of 4.07 G and an amplification factor of 4.63.
The rotor with Buna-N bearing supports peaked at 198,000 rpm and at 200,000 rpm, with
peak acceleration values of 3.3G and 3.5G respectively. The rotor with silicone bearing
supports has a critical speed at 102,000 rpm, where the vibration levels peak at 1.69G. A
second peak of 2.17G occurs at 3.6 kHz, or 216,000 rpm, before the response falls off
sharply and settles at -0.5G.
The rotor/bearing assembly, when modeled as a 2 degree-of-freedom oscillatory
system, has an important resonant frequency at 15.3 kHz, which caused strong
amplification of vibration between 13 kHz and 17 kHz in all the turbines in this study.
To improve the reliability of the observations noted in this paper, and to aid in the
design of higher performance high speed air-driven turbomachinery, a more robust data
gathering study should be conducted for more in-depth analysis.
66
Works Cited
1.
Atk rk, N. and R. Gohar. "Damping the Vibrations of a Rigid Shaft Supported by
Ball Bearings by Means of External Elastomeric O-Ring Dampers." Proceedings
of the Institution of Mechanical Engineers 208 (1994): 183-190.
2. Dyson, John Edwin. Aspects of the Behavior and Design of Dental High Speed
Ball Bearing Air Turbine Handpieces. Ph.D. Thesis, University of London. 1993
3. eFunda: Engineering Fundamentals. "AS568 A Standard O-ring Sizes"
<http://www.efunda.com/designstandards/oring/oring
as568.cfm> (January 10,
2003).
4. Ehrich, Fredric F. Handbook of Rotordynamics. New York: McGraw-Hill, Inc.
1992.
5. Freakley, P.K. Theory and Practice of Engineering with Rubber. Barking, Essex,
England: Applied Science Publishers Ltd. 1978.
6. Goodfellow. "Material Information: Hexafluoropropylenevinylidenefluoride
Copolymer." <http://www.goodfellow.com/csp/active/static/A/FV31 .HTML>
(April 25, 2003).
7. Gunter, E.J. "Influence of Flexibly Mounted Rolling Element Bearings on Rotor
Response Part I - Linear Analysis." Journal of Lubrication Technology 92
ASME, January 1970: 59-75.
8. Gunter, E.J. and W. J. Chen. Dynamics of Rotor Bearing Systems Version 5.0:
User's Manual. Charlottesville, VA: RODYN Vibration Analysis, Inc. 2000.
67
9. Gupta, Pradeep K., Juergen M. Tessarzik, and Loretta Cziglenyi. "Development
of Procedures for Calculating Stiffness and Damping Properties of Elastomers in
Engineering Applications." NASA Report CR-134704, prepared for NASA-Lewis
Research Center under Contract NAS3-15334. 1974.
10. Jones, D.I.G. "Viscoelastic Materials for Damping Applications." Damping
Applications for Vibration Control. Ed. Peter J. Torvik. New York: American
Society of Mechanical Engineers. 1980. 27-5 1.
11. Lalanne, Michel, and Guy Ferraris. Rotordynamics Prediction in Engineering.
Chichester, West Sussex, England: John Wiley & Sons Ltd. 1990.
12. Monachos Mechanical Engineering. "Conductivity Table for Various Materials."
<http://www.monachos.gr/eng/resources/thermo/conductivity.htm> (April 21,
2003).
13. Norton, Robert L. Machine Design: An Integrated Approach. Upper Saddle River,
N.J.: Prentice-Hall Inc, 2000.
14. Payne, A.R. "Dynamic Properties of Rubber" Use of Rubber in Engineering. Ed.
P.W. Allen, P.B. Lindley, and A.R. Payne. London: Maclaren and Sons Ltd.
1967.
15. Powell, J.W. and M.C. Tempest. "A Study of High Speed Machines With Rubber
Stabilized Air Bearings." Journal of Lubrication Technology 90 ASME, October
1968: 701-708.
16. Slocum, Alexander H. Precision Machine Design. Dearborn, MI: Society of
Manufacturing Engineers, 1992.
68
17. Smalley, A. J., J.M. Tessarzik, and R.H. Badgley. "Testing for Material Dynamic
Properties." ASME Publication. Vibration Testing - Instrumentation and Data
Analysis. (12): 117-41
18. Smalley, A.J., M.S. Darrow, and R.K. Mehta. "Stiffness and Damping of
Elastomeric O-Ring Bearing Mounts." NASA Report CR-135328 , prepared for
NASA-Lewis Research Center under Contract NAS 3-19751. 1997.
19. Snowdon, J.C. Vibration and Shock in Damped Mechanical Systems. New York:
John Wiley & Sons. 1968.
20. White, Frank M. Fluid Mechanics. Boston, MA: McGraw-Hill, 1999.
21. Winn, L.W., and F.D. Jordan. "Dynamic Behavior of a 140,000 rpm 3 kW TurboAlternator Simulator on Resiliently Mounted Ball Bearings." Proceedings of
International Automotive Engineering Congress and Exposition. Detroit, MI:
Society of Automotive Engineers, 1977. 1-21.
69
Appendix A: Instrumentation
CalibrationCertificate
IWW166-21
P4Wr
Model Number:
352C22
Serial Number:
3948S
Method:
[CP* Accelerometer
Description:
Manufacturer:
Back-t-fack Comparison Cnlibration
PCB
CalibrationData
sensitivity ( 100.0 Hz
mV/g
9.25
(0.943 mV/m/s')
3.0
Time Constant
seconds
Output Bias
89
VDC
Transverse Sensitivity
1.5
%
Resonant Frequency
905
kHz
Sensitivity Plot
Relative Humidity:
Temperaturs: 70 F (21 'C)
00
*___________________________
I___________
i-1.
dB
22%
M2o
__________
I0
10.0
Frequency (HZ)
10.0
15.0
30.0
50.0
100.0
Dev, (%)
10M0
Data Points
Dev. (%)
Frequency (Hz)
-1.2
300.0
-0.4
500.0
0.1
0.1
-0.3
1000,0
0.3
-0.1
3000.0
5000.0
0.2
0.0
10000.0
Frequency (Hz)
Dev. (%)
1.6
7000.0
10000.0
0.7
Condition of Unit
As Found:
n/a
As Left;
New Unit. in Tolerance
Notes
1. Calibration is NIST Traceable thru Project 822/267400 and PT8 Traceable thru Project 1055,
2, This certificate shall not be reproduced, except in full, without written approval from PCB Piezotronics, Inc.
3. Calibration is performed in compliance with 180 9001, ISO 10012-1, ANSI/NCSL Z540-1-1994 and ISO 17025.
4. See Manufacturers Specification Sheet for a detailed listing of performance specifications.
5. Measurement uncertainty (95% confidence level with coverage factor of 2) for reference frequency is +1- 1.6%,
Technician:
Mike Nowak
W
Date:
10/30/02
4PCBPIEZOTRONMC'
Cod
N 1SS
.4i 1
#AM I d
SI
VIBRATION DVISION
Depew,NY 14043
3423 Walden Avenue
www .peb.cm
TEL 11-64-0013 - FAX; 7165-336
I
Figure 38: Accelerometer calibration certificate
70
~
CalibrationCertificateISO16C3-21
Poor
352C22
Model Number:
Serial Number.
Deseriptio:
39916
Method:
ICP! Accelerometer
Manufacturer
Back-to-Back Comparison Calibration
PCB
Calibration Data
Sensitivity (@ 1.00.0 Hz
9.26
(0.944
Time Constant
23
8.5
VDC
Transverse Sensitivity
04
%
Resonant Frequency
91.5
kHlz
mV/g
Output Bias
mV/m/r0)
seconds
Sensitivity Plot
Thinperaiure: 70F
0-
'C )2
Relative Hunidity:
14%
--
I
30-s
dB
tIUlOUU
flz
Data Points
Frequency (Hz)
Dev. (50
Frequency (Hz)
10.0
0.6
300.0
15 0
1.4
500,0
30.0
-0.1
50,0
0.2
0.0
1000.0
3000.0
100.0
Frequency (Hx)
Dev. (%)
0.4
0.5
Dev, (%)
2.1
2.9
7000,0
10000.0
0.6
0.6
0.8
5000.0
Condition of Unit
As Found:
As Left:
na
New Unit. In Tolerance
Notes
1. Calibration is NIST Traceable thru Project 822/267400 and PTB Traceable thnr Project 1055.
2 This certificate shall not be reproduced, except in full, without written approval from PCB Piczotronics, Inc.
3. Calibration is performed in compliance with ISO 9001, ISO 10012.1, ANSI/NCSL Z540-1-1994 and ISO 17025.
4, See Manufacturer's Specification Sheet for a detailed listing of performance specifications.
5.
Measurement uncertainty (95% confidence level with coverage factor of 2) for reference frequency is +/-16%.
Technician:
Mike Nowak
SAut
Date:
12/10/02
*PCPIEZOTROYCS~
VIBRATlON
DISION
C.,t We *4a2-1
3423 Walden Aveaue
Depew, NY 14043
TEL :iV-610013
FAX 716-65-3896 - wwwpebcom
+
Figure 39: Accelerometer calibration certificate
71
10 -32 FAMILY
Passive Speed Sensors
High
Sensitivity
Thread Length (A)
(12.70)
.500
(31.75)
1.250
Ordering Part #
70085-1010-037
70085-1010-299
Performance Curves
.0101
A$
(0.25)
(4.B3)
~~
*
~O~
S
W AO
4t
ao
-
Ssod on 20 DR Gear
9
IS=tW
Specifications:
* Output Voltage (Standard): 13 V (P-P)
* Output Voltage (Guarantee Point): .6 V (P-P)
DC Resistance: 190 ohms max.
Typical Inductance: 10 mH, ref.
Output Polarity: White lead positive
Operating Temperature: -55 to +107*C
Lead Length: 18 in (45.7 cm)
Not Weight: 1 oz. max.
General Purpose - High Temperature
Thread Length (A)
Ordering Part #
.500
1.250
70085-1010-182
70085-1010-289
(12.70)
(31.75)
j
0.062
(1.57)
Performance Curves
0-32 UW 2A
.010 -
(0.25)
0190
(4.63)
Specifications:
eawd on 20 DR
Gear
* Output Voltage (Standard): 6 V (P-P)
* Output Voltage (Guarantee Point): .3 V (P-P)
DC Resistance: 45 ohms max.
Typical Inductance: 2 mH, ref.
Output Polarity: White lead positive
Operating Temperature: -73 to +150*C
Lead Length: 18 in (45.7 cm)
Net Weight: I oz. max.
I
I
a
I
C
I.
DimensionsIn Inches and (mm).
Figure 40: Magnetic pickup (tachometer) specifications
72
....
............
..
. ...
Appendix B: Additional Figures
Stiffness and Damping: Buna
0.9
3.OOE+07
0.8
2.50E+07
0.7
0.6
2.OOE+07
c
0.5
0.4
0.3
0.2
0.1
1.50E+07
1.OOE+07
5.OOE+06
0.OOE+00
100
1000
10000
Rotation rate [cycles/sec]
--- K [N/m]
extrapolated -*- Viton-70 --*--
extrapolated
Figure 41: Frequency-dependent elastic and loss moduli of Buna elastomer, referenced to elastic
modulus of Viton-70. Predictions based on Smalley, Darrow, and Mehta.
73
DYNAMIC FORCE-DEFORMATION PROPERTIES
LOG G',
MN m~
LOGG,. MN mt
TRANSITION REGiON
tI
'H 9
GLASSY REGION
RUBBERY ELASTIC
23] FLOW REGIOM
PLATE.AU
Ion
9-
0
2,0-
8-
7-
6- 10Ii
6G
5
4
.
.
0
-2
0
2
4
6
a
10
12
14
LOG (REDUCED FREQUENCY), CPS
Figure 42: Dynamic force/deformation properties of natural rubber, illustrating different regions of
material behavior. (Source: Freakley 68)
74
TEMPERATURE *F
150
100
50
VITON B:
Mg O:
t
0
4
100 PHR
20
M.T BLACK 5
DIAK #I1
I
CURE:
I HR AT 320 OF
ADHESIVE' CHEMLOK 607
E
102
0 (c
z
%N
M-
10
-j
)3
-
01
01
-1
1
10
102
3
&o
I
104
105
*6
107
lo
REDUCED FREQUENCY f ( - Hz
Figure 43: Dynamic Elastic and Loss Modulus for Viton B (durometer 75±5) (Source: Jones 44)
75
150
105
I-100I
-
TEMPERATURE
*F
100
0
50
-50
-00
GE RTV-630
-
(1)
(A)
0
/
I
/
LL)
E
*1%
/
0
Lii
Ijr
I!
I.
1 .1L
/
1J
100
10
I
1 102
/
REDUCED FREQUENCY
I
II
103 L 104
I
I05
1
106
1
10~
f cct -Hz
Figure 44: Reduced frequency - temperature nomogram for silicone (Source: Jones 46)
76
Appendix C: Matlab Script
%Abraham Schneider
%twodofthesis.m
%This code calculates the natural frequencies (eigenvalues)
%and eigenvectors of the 2 DOF axial vibration model for
%the high speed impulse turbine. Also, a bode plot is given
%for the system frequency response.
clear all
%Set up masses
ml=.0013122; %Mass of rotor (kg)
m2=.0002806; %Mass of bearing outer ring
(kg)
%Set up stiffnesses
k1=2128141; %Kbearing (N/m)
k2=17512; %Kwasher (N/m)
%Calculate system eigenvalues and eigenvectors
%Set up mass matrix
M=[m2 0;0 ml]
%Set up stiffness matrix
K=[kl+k2 -kl;-kl
kl]
[v,d]=eig(K,M); %v=eigenvectors. d=square of eigenvalues
wnatural=sqrt(d) %Natural frequencies=eigenvalues
%Transfer function Xl(s)/F(s)
numi=[(m2),0, (kl+k2)];
denl=[(ml*m2),O, (ml*kl+ml*k2+m2*kl),O, (kl*k2)];
hl=tf(numl, denl)
figure(1)
bode (hl)
grid on
%Transfer function X2(s)/F(s)
num2=[kl];
den2=[(ml*m2),O, (ml*kl+mi*k2+m2*kl),O, (kl*k2)];
h2=tf(num2, den2)
figure (2)
bode(h2)
grid on
77