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Dynamic Modeling of High-Speed Impulse Turbinewith Elastomeric Bearing Supports
by
Abraham Schneider
B.S. Mechanical Engineering,Massachusetts Institute of Technology, 2002
Submitted to the Department of Mechanical Engineeringin partial fulfillment of the requirements for the degree of
C ertified by .......................................
Pap
Accepted by...................
Department of Mechanical Engineering
M*9, 2003
-- ----I .... ..................Woodie C. Flowers
palardo Professor of Mechanical Engineering
#',Oesis Supervisor
Ain A. SoninProfessor of Mechanical Engineering
Chairman, Department Committee on Graduate Students
MASSACHUSOF TEC
B3PKER JUL
LIBR
ETTS INSTITUTEHNOLOGY
0 8 2003
ARIES
DYNAMIC MODELING OF HIGH-SPEED IMPULSE TURBINE WITHELASTOMERIC BEARING SUPPORTS
by
ABRAHAM SCHNEIDER
Submitted to the Department of Mechanical Engineeringon May 9, 2003 in partial fulfillment of the
requirements for the degree of Master of Science inMechanical 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, elastomericO-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 studythe dynamic characteristics of this high speed rotating machinery. The rotor systemshave been found to traverse a number of critical frequencies during normal operatingconditions. The use of different 0-ring materials has been found to affect the rotorresponse and placement of critical frequencies. Rotordynamics have shown that selectionof bearing and support stiffness and damping can have a major effect on the dynamicbehavior 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 ............................................................................................................................... 2Acknowledgem ents........................................................................................................ 3Table of Contents .......................................................................................................... 4List of Figures ..................................................................................................................... 5List of Tables ...................................................................................................................... 8Chapter 1: Background .................................................................................................... 9
1.1 Introduction............................................................................................................... 91.2 System Com ponents................................................................................................ 10
4.2 Finite Elem ent M odel .......................................................................................... 464.2.1 Viton*-70 ...................................................................................................... 464.2.2 Buna-N ............................................................................................................. 614.2.3 Silicone ........................................................................................................ 61
4.3 Axial Dynam ic Behavior .................................................................................... 64Chapter 5: Results ............................................................................................................. 66W orks Cited ...................................................................................................................... 67Appendix A : Instrum entation ........................................................................................ 70Appendix B: Additional Figures.................................................................................... 73Appendix C: M atlab Script........................................................................................... 77
4
List of Figures
Figure 1: Schematic representation of a typical high-speed turbine and its housing.......... 9Figure 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 diameterof 0.25". The shaft is 0.0625" diameter stainless steel. Each bearing is supported byan O-ring with a cross-section of 0.030". .............................................................. 10
Figure 3: Common impulse turbine designs include (a) flat blade (b) double curved (c)sim ple curved (d) split cup.................................................................................... 12
Figure 4: Schematic of rotor model with reduced degrees of freedom.......................... 16Figure 5: Modeshapes of simply supported shaft with varying levels of support flexibility
relative to shaft stiffness. With flexible bearing supports, first and second modes arerigid-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 13substations. Two imbalances 1800 apart on shaft. 4 lumped inertial stations......... 17
Figure 7: Schematic representation of rotor/bearing assembly for axial vibrationm o d ellin g . ................................................................................................................. 19
Figure 8: Solid model of canister-type high speed air driven turbine assembly........... 23Figure 9: Schematic of O-ring, showing flash dimensions........................................... 25Figure 10: Exploded view of testbed setup: 1. Turbine canister 2. Inlet air path 3. Exhaust
air path 4. T est block............................................................................................. 26Figure 11: Schematic of experimental setup: Front view showing lateral and axial
accelerometers, magnetic pickup, and the low-stiffness open cell foam base..... 28Figure 12: Waterfall plot showing (a) lateral acceleration and (b) axial acceleration for
rotor with Viton -70 bearing supports. X-axis represents the frequency domain: 0Hz - 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. Phasem easured relative to shaft tachometer.................................................................... 35
Figure 14: Vibration and phase of testbed, measured in the axial direction. Phasem easured relative to shaft tachometer.................................................................... 36
Figure 15: Buna-N 0-ring after testing and disassembly. Black debris resulted from thedisintegration of flash during turbine operation.................................................... 38
Figure 16: Waterfall plot showing (a) lateral acceleration and (b) axial acceleration forrotor 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. Phasem easured relative to shaft tachometer.................................................................... 40
Figure 18: Vibration and phase of testbed, measured in the axial direction. Phasem easured relative to shaft tachometer.................................................................... 41
Figure 19 Waterfall plot showing (a) lateral acceleration and (b) axial acceleration forrotor 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. Phasem easured relative to shaft tachom eter.................................................................... 44
Figure 21: Vibration and phase of testbed, measured in the axial direction. Phasemeasured relative to shaft tachom eter.................................................................... 45
Figure 22: Voigt viscoelastic model with stiffness and damping coefficients. (Atkirk andG oh ar 187) ................................................................................................................ 4 7
Figure 23: Comparison between two curve-fits for estimation of dynamic properties ofViton*-70 elastomer. (a) Stiffness (b) Loss Coefficient...................................... 50
Figure 24: Modeshapes and modal frequencies for rotor with Viton*-70 bearing supports.................................................................................................................................... 5 1
Figure 25: Bode plot of undamped response to imbalance for rotor with Viton®-70bearing supports..................................................................................................... 5 1
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 bearingsupports, modeled at 25'C, and incorporating damping estimates developed byAtkurk 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 firstrigid-body critical speed. Rotor achieves limit-cycle (stable) motion at all speedsw ithin norm al operating range ............................................................................... 53
Figure 28: Whirl mode shapes of rotor with Viton*-70 bearing supports. Backward whirlat 190,000 rpm and 500,000 rpm indicate too high a level of predicted damping infinite elem ent m odel. ............................................................................................. 54
Figure 29: Rotor system with Viton*-70 bearing supports (a) Stability map (b) Whirlspeed map showing damped natural frequencies................................................. 55
Figure 30: Transmitted force at bearing 1 and 2 for rotor with Viton*-70 bearingsu p p orts..................................................................................................................... 56
Figure 31: Comparison chart of thermal conductivity of Viton*, Buna, and siliconeelastom ers versus other m aterials. ......................................................................... 58
Figure 32: Temperature dependence of elastic modulus of Viton*-70 according toSm 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 experimentalre su lts. ....................................................................................................................... 5 9
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 444,000rpm. (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 200,000rpm. (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, limit-cycle 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............. 65Figure 38: Accelerometer calibration certificate .......................................................... 70
6
Figure 39: Accelerometer calibration certificate .......................................................... 71Figure 40: Magnetic pickup (tachometer) specifications............................................. 72Figure 41: Frequency-dependent elastic and loss moduli of Buna elastomer, referenced to
elastic modulus of Viton-70. Predictions based on Smalley, Darrow, and Mehta. . 73Figure 42: Dynamic force/deformation properties of natural rubber, illustrating different
regions of material behavior. (Source: Freakley 68) ............................................ 74Figure 43: Dynamic Elastic and Loss Modulus for Viton B (durometer 75±5) (Source:
Jon es 4 4 ) ................................................................................................................... 7 5Figure 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 dynamicmaterial properties. (Smalley, Darlow, and Mehta 3-25) ................................... 47
Table 2: Damping values for Viton®-70, based on half-power method applied toexperim ental data................................................................................................... 49
Table 3: Model coefficients for frequency-dependent stiffness of silicone elastomer..... 62Table 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 pre-load spring(30-35 psi) O-ring bearing supports,,,_,.,,,
airflow
handpiece exhaust pipeimpulseturbine
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 high-
speed 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 retainer-
pass, 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 outsidediameter of 0.295". The rolling element bearings have an outside diameter of 0.25". The shaft is0.0625" diameter stainless steel. Each bearing is supported by an O-ring with a cross-section of0.030".
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 = munbalancer C2 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 smooth-
running 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 (189-
190). 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 121-
131). 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 0-
ring 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
0 X
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 toshaft 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.
1 6
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.
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 2 _4x 2)
J(x)= 48EIS=P(LxXL2 - 8xL+4x2)
48E1
L
2
2 < x < L2
where I is the area moment inertia of the beam:
bh3
12
The maximum deflection of the beam is:
max = = - = -2.09 x 10 7'in4-2 48EI
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)2Equation 16
27
Equation 12
Equation 13
Equation 14
Setting Ep=Ek yields:
Co, = g_: = -54,270.09Hzm - 3max
Equation 17
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 24-
inch 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
Tnasdn Response vs. Tknex di6p; mi' -0 00036, Mex- 000031977
O d-M0.0031072 Max- 0.000319
0 000 3 ---- -- ----
000 0 0 0 00100.00 0.00 R,010o = 0.001now 00
F#.*. C.AyRoG*9_Row&deHCVOsdwfto25.not nsS4
2. (b)
T0ran.00 Respone vs. Tm
X d-W Mm - 4222E0. M- : 5.5474E-000FY I-: &Mn - . 4338E.00& U-x S .984SE-00
4.000-00 -4~ -4~~
000 P 1.0 O 0.W
00. 0000t.00100 0 D15 a MW 0 D25
FiW CA~yRo.8 M+ChedonK
ah oo~
3. (a) 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) 190,000rpm: 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 allspeeds within normal operating range.
Figure 28: Whirl mode shapes of rotor with Viton*-70 bearing supports. Backward whirl at 190,000rpm and 500,000 rpm indicate too high a level of predicted damping in finite element model.
54
________ ________Stability Map __________
-- -- ------- ------ ----------- j---------
I I I
-. - =O-eO
C:MDyRoBeSRoo'iHC~hcdvitof25.rot
Whirl Speed Map
Z 3GEtO------------------- ------------ ------------ ------------
00 00E+00 E+200.- - 24005 2000+- 40E0 t 00 00RS.BtonMlSpood~rpm)
Fit- CtDytO0000R0040C00ted.00.SUo
(b)
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.
Figure 34: Finite element analysis of rotor model with Viton®-70 bearing supports. (a) Whirl speedmap indicating damped natural frequencies at 231,000 rpm and 444,000 rpm. (b) Stability mapindicating stable rotor behavior.
59
------ -----------
-- - - L- - --- -
0O.w 12E i rt0WE+CVCd1W 1. 95
4 E5 06E 05
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):
k = 2.237x10 6(2f)0 ' 5 19 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] Stiffness [Lb-f/in]1000 19002000 20004000 22506000 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] Damping [Lbf-s/in]]1000 0.012000 0.0094000 0.0086000 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 - - - --
000E+00 520E0 2.0E0a5 30E.0a 40&E- &0Dt00E
Fikr CAyR0~S1_R0Wftom4CW4MKAVdn2r fts
(a)
Pi. CiVyR0o&8R..r04C5thcfSk.Z5 rd
(b)
Figure 35: Finite element analysis of rotor model with silicone bearing supports. (a) Whirl speedmap indicating damped natural frequencies at 120,000 rpm and 200,000 rpm. (b) Stability mapindicating stable rotor behavior up to 480,000 rpm.
Frequency (HZ) Dev, (%) Frequency (Hz) Dev. (%)10.015.0
30.050.0100.0
-1.2
-0.4
-0.3
-0.1
0.0
300.0500.0
1000,0
3000.05000.0
0.10.10.3
0.20.7
Frequency (Hz) Dev. (%)
7000.0
10000.0
1.6
Condition of UnitAs Found: n/a
As Left; New Unit. in Tolerance
Notes1. 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%,
Notes1. 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%.
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 positiveOperating Temperature: -55 to +107*CLead Length: 18 in (45.7 cm)Not Weight: 1 oz. max.
General Purpose - High Temperature
Ordering Part #70085-1010-18270085-1010-289
Performance Curves
Thread Length (A).500 (12.70)
1.250 (31.75)
eawd on 20 DR Gear
0.062 j 0-32 UW 2A(1.57)
.010 -(0.25)
0190(4.63)
Specifications:* 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 positiveOperating Temperature: -73 to +150*CLead Length: 18 in (45.7 cm)Net Weight: I oz. max.
Dimensions In Inches and (mm).
Figure 40: Magnetic pickup (tachometer) specifications
72
~~
* ~O ~ S 4t AO W 9 IS=tW
ao -
Ssod on 20 DR Gear
IIa
CII.
High Sensitivity
Appendix B: Additional Figures
Stiffness and Damping: Buna
3.OOE+07
2.50E+07
2.OOE+07
c 1.50E+07
1.OOE+07
5.OOE+06
0.OOE+00100
0.90.8
0.70.6
0.50.4
0.30.2
0.1
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~LOG G,. MN mttI
'H 9 23]
9-
8-
7-
6-
5
0
6-
4
2,0-
10-
0-2
TRANSITION REGiONRUBBERY ELASTIC
FLOW REGIOM PLATE.AU
G . .
GLASSY REGION
Ion
Ii
0 2 4 6 aLOG (REDUCED FREQUENCY), CPS
10 12 14
Figure 42: Dynamic force/deformation properties of natural rubber, illustrating different regions ofmaterial behavior. (Source: Freakley 68)
74
TEMPERATURE *F150 100
t0
0 (c
z%N
M--j
102
10
01 01 1 10 102 &o3 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
VITON B: 100 PHRMg O: 20M.T BLACK 5DIAK #I1 ICURE: I HR AT 320 OFADHESIVE' CHEMLOK 607
4 E
)3 - -1
50
I
TEMPERATURE *F
150 100 50 0 -50 -00- I I-100
- GE RTV-630
105
(1)
(A)
0E
I!
//
I /
/
I
LL)*1%0LiiIjr
1 .1L J 0 I. 1 I I II L 1 11 10 10 102 103 104 I05 106 10~
REDUCED FREQUENCY f cc t -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 eigenvalueswnatural=sqrt(d) %Natural frequencies=eigenvalues