Dynamic Modeling of High-Speed Impulse Turbine with

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

Master of Science in Mechanical Engineering

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

JUNE 2003

©Massachusetts Institute of Technology 2003

Signature of Author............................

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

1.2.1 Bearings ........................................................................................................ 111.2.2 Rotors............................................................................................................... 111.2.3 Vibration Isolation M ethods ........................................................................ 12

1.4 Thesis Structure ................................................................................................... 15Chapter 2: Theory ............................................................................................................. 15

2.1 Finite Elem ent Analysis...................................................................................... 172.2 Axial Vibration................................................................................................... 19

Chapter 3: Experim ental M ethods ................................................................................. 213.1 Sensors .................................................................................................................... 213.2 Flexible bearing supports.................................................................................... 233.3 High-speed rotor test-bed.................................................................................... 263.2 Spectral analysis with parametric bearing support variation ............................... 29

Chapter 4: Results and Discussion................................................................................ 314.1 Experim ental Results ........................................................................................... 31

4.1.1 V iton -70 ...................................................................................................... 324.1.2 Buna-N ............................................................................................................. 374.1.3 Silicone ........................................................................................................ 42

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.

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).

x 2xl

H-*M ring

0.5*Kwasher 0. 5 *Kearing, axial

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 ) 1 -kbearing X2 = 0

Mrotori2+ kbearing X2 -kbearingXi = f(t)

Written in matrix notation:

Mouter ring

0

0 1 1 kwasher bearing bearing X -

m, _ 2 _ -kbearing kbearing [X 2 _ f(t))

Mi+ Kx = F

Equation 2

Equation 3

or

Equation 4

Equation 5

19

Is

The homogenous solution is found by settingf(t) = 0. Assume that Equation 5

has the harmonic solutions:

x(t) = ek _j k Xj ckke )kt k =1,2 Equation 6X2k

where Xk is the mode shape and (Ok is the modal frequency. Substitution of Equation 6

into Equation 5 yields:

M( 2 cok ktce' + K(cke')' =0 Equation 7

MK- Mo4k = 0 Equation 8

Fkahe+ k 2i -kwashe, beainng otkeg bearing 2 Ilk =[G(O)XX)=o0 Equation 9kbeang kbearing k] Moto, W 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 J " e' + l 12 C2)e = 1 c sin(cwt + , )+ X 2 c, sin(co2t +$2) Equation 11(X21 )(X22

where c], c2, # , and #2 are constants determined by the initial conditions x(O) and i(o).

Thus, the mode shapes and modal frequencies can be calculated for the 2 degree-of-

freedom 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.

Figure 8: Solid model of canister-type high speed air driven turbine assembly.

23

7/00

. ............ .... ...... . ...... ... ... .... - -- ------ ___ ----_-- - - --

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 post-

manufacturing 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

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

0.005'" maxA

0.003" max

AI.D.0.238" ±0.005"

Cross Section0.030" ±0.003",

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 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

frequencies are:

RPM _ RPMFundamental train frequency: RPM 1 - cos(#) ~ (0.3987)

60 2 D 60

Defect on outer race: RPM 1 cos() ~(0.3987) RPM60 2 D, 60

RPM n D RPMDefect on inner race: - 1+ Db cos(#) (4.2090) 60

60 2 DP 60

RPM D rD1 RPM_Ball defect frequency: D- Il- cos2(0) = (4.6621) RPM

60 Db D 60

Equation 18

Equation 19

Equation 20

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 -15 dBV STATUSs PAUSED2' 63 A& MAG RMSs 10

d6

/DIV

STARTv 0 Hz SW* 190.97 Hz STOP2 20 000 Hz

RANGE -51 dBV STATUS PAUSEDA&MAG 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 withViton®-70 bearing supports. X-axis represents the frequency domain: 0 Hz - 20 kHz. Y-axisrepresents 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

1

0.1

0 1000 2000 3000 4000

Rotation rate [cycles/sec]

5000 6000

Viton-70: Phase

0

-90

-180

0 1000 2000 3000 4000


5000 6000

Figure 13: Vibration and phase of testbed, measured in the lateral direction. Phase measuredrelative to shaft tachometer.

35

0

00C)C)

* *.4 ***44**

*4

4.

444*44444

Viton-70: Testbed Axial Vibration

____________ 4

4 _______________ _______________ _______________

II _

4

4

2000 3000 4000


Viton-70: Phase

2000 3000 4000


Figure 14: Vibration and phase of testbed, measured in the axial direction. Phase measured relativeto shaft tachometer.

36

10

1

0.1

0

C)

0.010 1000 5000 6000

90

0

-e0

0

-90

*4 *4.. .4.

4 4 **4

* 444

444.44

4

-1800 1000 5000 6000

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 thedisintegration of flash during turbine operation.

38

5

dE

E TARTa Z zRO ~~ zSOS~ COO Hz

MAG R?4Sv 10

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 withBuna-N bearing supports. X-axis represents the frequency domain: 0 Hz - 20 kHz. Y-axisrepresents the changing rotor speed: speed range 4,800 rpm - 480,000 rpm. Acceleration ismeasured on the Z-axis: OG - 5G.

39

Buna: Testbed Lateral Vibration

4 4 *

*. *4 4 4_______ .'-4- ~- ~--

** 4*44 4

*44

.4 4

q4

4*4

2000 3000 4000 5000 6000 7000 8000


1000

Buna: Phase

0

-90

-180

-270

-360

-450

-540

-630

-7200 1000 2000 3000 4000 5000 6000


7000 8000


40

10

0-4

0.10

r--

to)U.

, - --......

.4.444*

Buna: Testbed Axial Vibration

I__ffi __-9-

_ 1.9 9. 9*99

99~~~~; ___ 9

-9-----

9

9 .9 *9

0 1000 2000 3000 4000 5000


Phase

6000

0 1000 2000 3000 4000 5000 6000

7000 8000

7000 8000



41

10

10

C.)C.)

0.1

90

0-

0

-90

9

9

;9999,***

9

99 9

--- 9..-

9*

9 9

99 *** * 9

1* 9-180

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 non-

integer 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 1020

dB/DIV

START# 0 Hz BW. 190.97 Hz STOP. 20 000 Hz

A MAG RMSt 102G

5dB

/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 withsilicone bearing supports. X-axis represents the frequency domain: 0 Hz - 20 kHz. Y-axis representsthe changing rotor speed: speed range 3,960 rpm - 492,000 rpm. Acceleration is measured on the Z-axis: OG - 2G.

43

Silicone: Testbed Lateral Vibration

10

1

0.1

0 1000 2000 3000 4000


Silicone: Phase

0

-180

-360

-540

-720

-900

-1080

5000 6000 7000 8000

0 1000 2000 3000 4000 5000 6000 7000 8000



44

0

049 9*

r-I

0

..

49* *'..*** -;..-.-. *. , *

Silicone: Testbed Axial Vibration

4000 6000


Phase

9 999

4140to +

80004000

I.- *9 94

9 *4

999

6T00



45

10

1

0.1

0

00

* -.9

99

Th _

0.01

0 2000 8000 10000

180 -

90 -

0-..r

-90

-180

-270

-360

10000

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 = N - s/m or Lbf - s/in Equation 242rf

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 materialproperties. (Smalley, Darlow, and Mehta 3-25)

Material: Viton"-70 Stiffhess Coefficient Loss CoefficientTemperature Al B1 A2 B2

25 C 4.520 x 10" 0.3747 0.255 0.1746

38 C 1.694 x 10 5 0.4195 0.1103 0.239266 *C 8.850 x 10' 0.1406 0.0226 0.3227

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

KO

0 ci

Figure 22: Voigt viscoelastic model with stiffness and damping coefficients. (Atkurk and Gohar 187)

Equivalent stiffness and damping factors are given by:

K e (K0 + K, )KOK, + K ccij 2

(K + K1 2 +C2

(K + K )2 + C2

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

Material Viton-70

KO 3.40 x 101

KI 6.05 x 106

ci 6050

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:

G"ri = 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 frequency-

dependent 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 Aff)A(fm A(f2) " = " Equation 30

solving for f allows determination of system damping:

( Br Equation 312Mc>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 f, [Hz] f, [Hz] f2 [Hz] Q B, [Lbf-s/in.]Viton-70 4050 3475 4350 4.6286 0.1080 0.0093

49

Viton-70 Stiffness Prediction

1.OOE+08

1.OOE+07

1.OOE+06100 1000 10000


Atkurk & Gohar extrapolated - Smalley, et al. o extrapo

(a)

Viton-70 Loss Coefficient Prediction

2 -

1.81.61.41.2

r 10.80.60.40.2

0 _

100 1000 10000


Atkurk & Gohar - extrapolated , Smalley, et al. e extrapolated

(b)

Figure 23: Comparison between two curve-fits for estimation of dynamic properties of Viton -70elastomer. (a) Stiffness (b) Loss Coefficient.

50

. .. . ........... .. ...... ....... . ...... ... ...... _--- ...... ......

(a) Mode No. 1 (b) Mode No. 2 (c) Mode No. 3

361,007 rpm 654,975 rpm 1,220,675 rpm

Figure 24: Modeshapes and modal frequencies for rotor with Viton®-70 bearing supports.

Bode PtoIStation: 4, Sub-Station: I - (0-pk)

probe 1 (x) 0 deg - max amp = 0.042536 at 354000 rpmprobe 2 (y) 0 deg - max amp = d04253 at 354000 rpm

...................... . . . .

I I -I -I

0.05

0.04

0.02

0.01

0.001

File: C:\DyRoSeS_

).XEE+00 I 20E+05

RotoFHHCahcsdvton25.rot

2A4E+06 3.80E+05

RoatIcnal Speed (rpm)

Figure 25: Bode plot of undamped response to imbalance for rotor with Viton®-70 bearing supports.

51

cm

5/

I WEFLc~u 7i~jI -

S

II

- -- - - - - - - - - - - - - - - - - I - - - - - - I- - - - - -

- - - - - - - - - -- - - I - - - - - - r - - - - -I -- - - - - -

--------- 4------4----------L.I------ -- -----I I I

----------------- ------------ L..--------------------

4O0E+05 &90DE+05

H--L 7tL---

Bode PlotStation: 4. Sub -Station: I - (0-pk}

pfobe 1 (x) 0 deg - max amp = 000047926 at 386000 rpm10 probe 2 (y) 0 deg - max amp = 0.00047928 at 386000 rpm

8I . . .

720 --------------------- - -

630 -- - - -- - - -

40 -- - - - - -- - - - - - -- - -- - -

m 5 0 -1 -- - - - - - T -- - - - - - -r - - - - - - -- - - - - -

2730 ------- T------r-------I------

1I - -- - - - I -- - - - - - -- - - - - -- - - - - -

20

180- - - - - - - - - - - - - - - - - - - - -

I I

901 --- ,- - - - - - -- ---- -- - - L -- - --- J- - - - -- -

0.00072I.OE0 1.0E0 2.0E0 3.0+0 -0E0 O

Fie I.~~oe RoMI~s~n5 o oainlSed(~n

Fiur 26 Bod poofrsostoiblneoroowihVo 0eainsupr.Moe

inopoaesdmpn0stmte6roie by Atur and L...

52

X-Y Plot

X"00 i. -. 380M.00 MA .21S3M-006t tp: Mi - -1 1679E - m6 = I.390E-006

22E-00

-2120E-Ce ------------ -------------------------- ------------

F. C3DyRo6.SR0o000HC00aio4020oo00.

1. (a)

X-Y Plot000600 3

x itp. mi" 400031635,0 .0000319770 dap 4i=0003072M000.00001010a0 C3 . . . . ... ... . .... . ...

0.000S 1-----

2. (a)

X-Y P101X diq. Min 42220. Mo - 0.5474E-00Y disp.Min . .54330E-0 -,06... 64E-005

4 - ---- -

400E45 -0- ---- - -

F0. C '0e*S Rotmo7CV00A8o0n250rat

F4* C

Transierd Response V& Tenn

x T op Min - -. 6op01E M100 2.8108E-306

V06010.p n-tf 4;2-000. Ma I 5154E-006

50E-06 ------------ -- - ---- - - --- -- -- 0 ---- - - -- ----- -----

- -.00--06 -------- --- ---- -

400E-0.' -------- --- --- - -- - ---

0 ,0000 0o0, -C io.0..20 06

1. (b)

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.

53

0.0024 OAM

Poces&oW ModA Swipe- STABLE BACKWARD Precession00.8 RAotSopeed =l8000rpm. Mode No.= 2

Wr Speed iDwrped NAtral Froq.) 103 p,, Lo% Do-rol 21W4.64.0

Pd.e C1Dy~o8.8Roto4Caed1od2SalOt

(a) 190,000 rpm: Stable backward whirlP1.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 whirlPoooooosio Mode Shap. - STABLE BACKWARD Ptoosoo

Shoft RooiDnig. Speed - SOO 8qom Mod. No.- 21hir0 Speed (Doaped NkrIM Freq.) 258 rpm. Log Decar*= 1075.980

Fda 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 190,000rpm 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------------------- ------------ ------------ ------------

5MI - - - - - - I --- - - - - -- - - - - - -- - - - - -

I.OE0 04A

1 3161 EI +1356hAj +0

Roaioa Spe (rm

File., O -Ayoe - - - -It IvO~25r~

(b)A

Figure~~~~~~~~ 2:RtrssewihVtn70baigspot a tblt a b hr pe a

shwn dape nauareunis

55

tIJZu+tJb 2,41R4O 3JYt+U5

Rottonal Speed (rpm)X~IM-iUo

S3,OOE*O7

IOOE+07

SFile-,

elk tvl=+C*Ml T. V - -:4- P.- RP., ikE q F -WIN +05

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 \

240 -r- e = 077 V 340 p

720 ------------- ----------------- - -------

I I{

1200 -- - - -- - - -- - - - -- - - - - - - - - - ----\

0 0 1 I I -. ... . I I I I I I I I I I I I I I I I I I I I0.00E+00 1.20E+05 240E05 3.60E+05 4 80E+05 QODE+05

Rotational Speed (rpm)File. CADyRo8eSRototHHCtmtsdvton25.rot

(a) Bearing 1

Transmitted Force (sent-major axis)Beauing/Support Station 1: = 4, Station J: = 10

Max Forces = 10 717 at 384000 rpmn12,501 ~ I n ~ T, rr-r ~i-r

10,00 ------- - --------- ---- I------

8 7.50--- --- ---- --- -- L K-- ---- -1- -- - --

(a Bern I

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 over-

damped. 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

0.1

0.01

----1

.co

Figure 31: Comparison chart of thermal conductivity of Viton0 , Buna, and silicone elastomers versusother materials.

Temperature Dependence of K

Viton-70, Smalley, et al.

____~~A t_4 __1i____ _____ ____I £iI;;;k-

100001000


-- 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

1.OOE+08

2 1.OOE+07

1.OOE+06

100

- = - . .. ... .....

N0

Bod. Pet

5tt0n. 4, Sub-8*a01 - tO-f)lpob. I ix) O g - S xmp0.01270 @1 44DOOmOrO

009 ----- ------- ------- ---- - ----- -+ --------- +--

II

ON -- 1.200.0 2105i06 200 4N00 &0t&.06

Rawls Speed (rpmFie Ci0yRoeeS_RoWo.tCcawsntot t

(a)

Bode PIgStem 4. ub-aton I - (-oki

p0b&1 (C) 0 d.9 - ap00 0007396 at 240000 poo

SProe2 (yU O d g o-m ap. 1 .00037200 oO 240000 Tpm

20000 ------- --------- + ---------- ------- -

ISaO ----------- ------ +- -- - -- ----- -

-00002 ----------- ------------------------- -- ----

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.

Whirl Speed Map

200E06

400.E-05

Rio CVyRfooSJ

2400+0 380E.00

Ratabonal Speed (4MM

(a)

Stablity Map

It.

0. 00E.00 1 0000 2AE.00 20E.06 40E+05 0.0.0 0RPgl.oCi $peed (pt o

(b)

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.

.0.. .axdp i. - D.00001204.. .00012903

fa 1 Y diY0 .p 4.000 0M.. 00001207

. . . . . . . . . . . . . . . . .O.0DE-05 -------- 7 6---

000DE.

40OE-05

FO. CSDA0 rot40 k..20Figure (y6: Saftcenter otonp(a)

e.ns.us..X isp. -401E0. Mw 45128E-00

Y dip M. -8.5011E-005. Mu 9 32740-

t 0.1%.......___ N6 OOE-05 -- ------ - ----------- -

400E-05 ------- --- - --------

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 motionat 200,000 rpm (b) Unstable elliptical motion at 500,000 rpm.

63

Wirl Speed Map

510000

U1050505UaA

Stability Map

-- -- --- --- --- -- -- - -- -

' '- ' '- ' - ' - -' -'-'-'---

---------

--------------~~~~~~~ ------ ------------

-0301

0 +0 1W I E+-W 1ADE+-M 360E+0 440E+05booo~ ~~ ~MOW sn as(rPMo ako-4000

2500M ....... 1409a" LogDecorrAint ladcAn M51111044

-T ------------ r ------------ ------------ I ------------

T -------------.2

----------- I ------------ ----------- -------------------

------------ T ------------ r ------------ -------------------------

------------ ------------ ----------- I ----------- I ------------

...........6.00+05

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 10A 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 Diagram100

C - -- - . - ..--.--- -,.- --

-0 ------- ++- -t - t-------+-

S - I ------w-200 ~ ~ -- -----I I I I ------- - --- I ---

-3100

Fgr37 Bode po o ial virtino r s ingsyste eienles9 0 - - - - ----L .. - . . - - - .. - . J .. . - - - - -J - . J . . . - .. - A - .. . .. . .- ...-L - '- L - ' .. - 'L.. 6 -:-.. L. 4

a> 45 T T , - - -, -- - --- - -,r

> 90 ------+-i t te--- - + + + + - + + 9 M

10 3 10 4 10 5 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 low-

mass, 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 Turbo-

Alternator 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

Calibration Certificate -P4Wr IWW166-21

352C22

3948S

[CP* Accelerometer Method: Back-t-fack Comparison Cnlibration

PCB

Calibration Data

sensitivity ( 100.0 Hz

Time Constant

9.25 mV/g

(0.943 mV/m/s')

3.0 seconds

Output Bias

Transverse Sensitivity

Resonant Frequency

Sensitivity PlotTemperaturs: 70 F (21 'C) Relative Humidity: 22%

00

i-1. *___________________________M2o I___________ __________

10M0 10000.0

Data Points

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%,

Mike Nowak W Date: 10/30/02

4PCBPIEZOTRONMC'VIBRATION DVISION

3423 Walden Avenue - Depew, NY 14043TEL 11-64-0013 - FAX; 7165-336 www .peb.cm

Figure 38: Accelerometer calibration certificate

70

Model Number:

Serial Number:

Description:

Manufacturer:

8 9 VDC

1.5 %

905 kHz

dB

10.0

Technician:

Cod N 1SS SI

.4i 1#AM I d I

I0

Calibration Certificate ~Poor ISO16C3-21

352C22

39916

ICP! Accelerometer

PCB

Method: Back-to-Back Comparison Calibration

Calibration Data

Sensitivity (@ 1.00.0 Hz

Time Constant

9.26 mV/g

(0.944 mV/m/r0)

Output Bias

Transverse Sensitivity

23 seconds Resonant Frequency

Sensitivity PlotThinperaiure: 70F )2 'C Relative Hunidity: 14%

0 - -- I

30-s

Data Points

Frequency (Hz) Dev. (50 Frequency (Hz) Dev. (%)

0.6

1.4

-0.1

0.20.0

300.0

500,0

1000.03000.0

5000.0

0.4

0.5

0.60.60.8

tIUlOUU

Frequency (Hx) Dev, (%)7000,0

10000.0

2.1

2.9

Condition of UnitAs Found: na

As Left: New Unit. In Tolerance

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%.

Mike Nowak SAut Date: 12/10/02

*PCPIEZOTROYCS~VIBRATlON DISION

3423 Walden Aveaue - Depew, NY 14043TEL :iV-610013 + FAX 716-65-3896 - wwwpebcom

Figure 39: Accelerometer calibration certificate

71

Model Number:

Serial Number.

Deseriptio:

Manufacturer

8.5 VDC

04 %

91.5 kHlz

dB

flz

10.015 0

30.0

50,0

100.0

Technician:

C.,t We *4a2-1

Passive Speed Sensors 10 -32 FAMILY

Ordering Part #70085-1010-03770085-1010-299

Performance Curves

Thread Length (A).500 (12.70)

1.250 (31.75)

.0101 A$(0.25)

(4.B3)

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


--- 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

%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

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Documents

Dynamic Modeling of High-Speed Impulse Turbine with