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Rotor-Bearing System Dynamics of a High-Speed Micro End Mill Spindle

HIDE
 Title Page
 Acknowledgement
 Table of Contents
 List of Tables
 List of Figures
 Abstract
 Introduction
 Development of equation of motion...
 Rotor dynamic analysis of micro...
 Experimental identification of...
 Conclusion and recommendations
 Appendices
 References
 Biographical sketch
 

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ROTOR-BEARING SYSTEM DYNAMICS OF A HIGH SPEED MICRO END MILL SPINDLE By VUGAR SAMADLI A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2006

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Copyright 2006 by Vugar Samadli

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iii ACKNOWLEDGMENTS I would like to first thank my advisor, Dr. N. Arakere, for guiding me through this work. I feel that I have learned much work ing on this project and that would not have been possible without his help. Similarly, the collaboration with Dr. J. Zeigert and Dr. L. William and their students Scott Payne, Eric Major and Andrew Riggs was fruitful. I want to thank Scott Payne for his help and it was always a friendly environment to work with him in his lab. My company, BP Azerbaijan unit, was a main factor in making this whole effort possible so I would like to th ank them all, and specifica lly Ralph Ladd, learning and development coordinator, and Kevi n Kennelley, engineering manager. Lastly, I would like to thank my parent s because they were supportive of this effort and always encouraged my education.

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iv TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iii LIST OF TABLES.............................................................................................................vi LIST OF FIGURES..........................................................................................................vii ABSTRACT....................................................................................................................... ix CHAPTER 1 INTRODUCTION........................................................................................................1 2 DEVELOPMENT OF EQUATION OF MOTION OF ROTOR-BEARING SYSTEM AND PARAMETER IDENTIFICATION...................................................4 2.1 Rigid Rotor Analysis..............................................................................................4 2.2 Finite Element Analysis..........................................................................................7 2.3 Identifying Bearing Paramete rs By Experimental Method....................................8 2.3.1 Methods Using Incremental Static Load...............................................9 2.3.2 Methods Using Dynamic Load.............................................................9 2.3.3 Methods Using an Excitation Force....................................................11 2.3.4 Method Using Unbalance Mass..........................................................14 2.3.5 Methods Using an Impact Hammer....................................................16 2.3.6 Methods Using Unknown Excitation..................................................17 3 ROTOR DYNAMIC ANALYSIS OF MICRO SPINDLE........................................19 3.1 Rigid Rotor Analysis............................................................................................20 3.2 Finite Element Analysis........................................................................................31 3.3 Summary...............................................................................................................42 4 EXPERIMENTAL IDENTIFICATI ON OF BEARING PARAMETERS................44 4.1 Method of Measurement.......................................................................................45 4.2 Design Process......................................................................................................48 5 CONCLUSION AND RECOMMENDATIONS.......................................................50

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v APPENDIX A CHRONOLOGICAL LIST OF PAPERS ON THE EXPERIMENTAL DYNAMIC PARAMETER IDENTIFICATION OF BEARINGS............................51 B GENERAL RIGID ROTOR SOLUTION 1...............................................................55 C GENERAL RIGID ROTOR SOLUTION 2...............................................................62 D EXAMPLE OF FINDING BEARING PARAMETERS............................................69 E KISTLER DYNANOMETER....................................................................................72 LIST OF REFERENCES...................................................................................................74 BIOGRAPHICAL SKETCH.............................................................................................77

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vi LIST OF TABLES Table page 3-1 Bearing parameters at 500,000 rpm ( =52360 rad/sec)..........................................23 3-2 Bearing Parameters for different rotor speeds..........................................................29

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vii LIST OF FIGURES Figure page 1-1 Commercial micro-tool..............................................................................................3 2-1 Rigid rotor schematic.................................................................................................5 2-2 A non-floating bearing housing and a rotating journal............................................11 2-3 A floating bearing housing and a fixed rotating shaft..............................................12 3-1 Micro-spindle..........................................................................................................20 3-2 Bearings location on spindle....................................................................................20 3-3 Rotor orbits at two be aring supports at 500,000 rpm...............................................24 3-4 Rotor orbits at the two b earing supports at 1,000,000 rpm......................................25 3-5 Rotor orbits at the tool end at 500,000 rpm..............................................................26 3-6 Orbit amplitude at 1st, 2nd bearing and tool tip.........................................................27 3-7 Stiffness, Kyy versus rotor eccentricity...................................................................28 3-8 Whirl map.................................................................................................................3 0 3-9 Finite element model................................................................................................31 3-10 Rigid body mode 1, 107758 rpm..............................................................................31 3-11 Rotor analysis at 115298 rpm. A) Rigid body mode 2, B) Potential energy distribution...............................................................................................................32 3-12 Rotor analysis at 2,042,349 rpm. A) Flexural mode 1, B) potential energy distribution...............................................................................................................32 3-13 Flexural mode 2, 4,955,776 rpm..............................................................................33 3-14 Critical speed map....................................................................................................33

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viii 3-15 1st bearing. A) and B) Unbalance respons e, C) Amplitude and phase lag, D) Nyquist plot for displacement..................................................................................34 3-16 2nd bearing. A) and B) Unbalance respons e, C) Amplitude and phase lag, D) Nyquist plot for displacement..................................................................................36 3-17 Tool tip. A) Unbalance response, B) Am plitude and phase lag, C) Nyquist plot for displacement.......................................................................................................38 3-18 Shaft orbits. A), B), C), D), E), F) ar e shaft orbits as a function of speed...............39 3-19 Stability Map (Note: Negative l og decrements indi cate instability)........................42 3-20 Orbit amplitude at 1st, 2nd bearing and tool tip, =0.1..............................................43 4-1 Flexure supported ball bearing: 1) Flexure, 2) Ball bearing....................................45 4-2 A chassis.................................................................................................................. .45 4-3 Schematic of the test rig...........................................................................................46 4-4 Test Setup: 1) Base, 2) Dyno, 3) Chassis, 4) Test bear ing, 5) Shaft, 6) Spindle.....48 4-5 Main Chassis: 1) Stator, 2) Di splacement Probe, 3) Test Bearing...........................49 4-6 Measurment system..................................................................................................49

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ix Abstract of Thesis Presen ted to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science ROTOR-BEARING SYSTEM DYNAMICS OF A HIGH SPEED MICRO END MILL SPINDLE By Vugar Samadli August 2006 Chair: Nagaraj Arakere Major Department: Mechanic al and Aerospace Engineering Current micro-scale manufacturing technolog ies find limited application in a wide range of high strength engineering material s because of the difficulties encountered in creating complex three dimensional structures and features. Alt hough milling is one of the most widely used processes for this type of manufacturing at th e macro scale, it has yet to become an economically viable te chnology for micro-s cale manufacturing. For optimal chip formation using very small di ameter cutters, and to achieve economical material removal rates combined with good surface finish, high spindle speeds are needed. In addition, a low runout is desired to prevent premat ure tool breakage. However, the lack of suitable spindles capable of achieving rota tional speeds in excess of 500,000 rpm coupled with sub-micrometer runout at the tool tip makes micro-scale milling commercially unviable.

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x This thesis demonstrates several means of analyzing the rotor dynamic behavior of a spindle in order to find cr itical speeds, unbalance response and linear stability margins. Experimental testing is performed to estim ate bearing dynamic behavior at high speed. The results of this study provide parame ters for bearing stiffness and damping, bearing span and balancing limits to achieve sub-micrometer r unout of tool tip for speeds upto 1 million rpm.

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1 CHAPTER 1 INTRODUCTION The technology development in the field of miniaturization has become a global phenomenon. Its impact is far and widespread across a broad application domain that encompasses many diverse fields and industr ies, such as telecommunications, portable consumer electronics, defense, and biomedical. The perfect example is the field of computers where modern computers which pos sess greater processing power and can fit under a desk or on a lap have replaced the bulky computers of the past such as the ENIAC (electronic numerical in tegrator and computer) which once filled large rooms. In recent times, more and more attention is being paid to the issues involved in the design, development, operation, and an alysis of the equipment and processes of manufacturing micro components since the global trend toward the increased integration of miniaturized technology into society has gained enormous momentum. Currently, common techniques utilized in the fabrication of micro-components are based on the techniques developed for the silicon wafer processing industry. Unfort unately these processes are limited to production of simple planar geometries in a narrow range of material and are cost effective only in large volume [1]. Even though non traditional fabrication methods, such as focused ion beam machining, laser mach ining, and electrodischarge machining, are capable of producing high-precision micro-co mponents, they have limited potential as mass production techniques due to the high initial cost, poor productivity, and limited material selection [2]. Micro milling has the potential to fabricate micro components and is capable of machining complex 3D shapes fr om wide variety of shapes and materials.

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2 The objective of this research is to devel op a micro-milling spindle which will rotate at over 500,000 rpm range with sub-micrometer runout, and thus become a commercially usable and cost effective manufacturing tec hnology. Most machine tools such as lathes, milling machines, and all types of grinding machines, use a spindle or an axis of rotation for positioning work pieces or tools or machining parts and thus a large part of their accuracy can be attributed to the spindle. C onsequently, the accuracy of the spindles used in their design directly influences the accur acy of the entire machine and thus can be considered as one of the most important components in the overall accuracy and operation of a machine tool. Most commercial micro-tools have a 1/8th inch diameter shank (see Figure 1-1). This must be of utmost importance when de signing the micro spindle. Another functional requirement is the ease of tool changing with minimal time and effort. The only viable way to meet the above design requirements wh ile still obtaining sati sfactory runout is to concentrate on designs incorporating the use of tool shank itself as the spindle shaft. To achieve desired performance, the follow ing three functions must be satisfied: 1. Bearing subsystem. The bearing system must be so designed that it meets the following requirements. Firstly and fore mo stly, it must be capable of supporting the tool shank without causing excessive runout. Also, it must support both radial and axial loads, and support rapid tool ch anges. Flexure Pivot Tilting Pad Bearings (FPTPB) are being studied as a potential bearing subsystem. 2. Drive subsystem. The tool drive system must be able to drive the tool at the required speed with enough torque and pow er to perform the desired machining operations. Also it should not introduce dist urbance forces that cause excessive tool point runout. These requirements make air turbine drive as a potential drive subsystem. The system would incorporate th e turbine blades directly into the tool shank of the micro-tool. 3. Monitoring subsystem. Theoretical and sc ientific understanding of micro-milling requires monitoring and recording of cutti ng forces. However, in the measurement bandwidth (that is 1,000,000 rpm with a 2flute cutter and tooth passing frequency

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3 of 33 KHz), the force measurement is extremely difficult because of the high frequencies encountered even thoug h the cutting forces are low. The goals of this proj ect are the following: To extend the capability to model and predic t rotordynamics and be aring behavior at small sizes and high speeds. Development of a procedure for identifi cation of dynamic stiffness and damping coefficients for the bearing. Figure 1-1. Commercial micro-tool

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4 CHAPTER 2 DEVELOPMENT OF EQUATION OF MO TION OF ROTOR-BEARING SYSTEM AND PARAMETER ID ENTIFICATION It is of utmost importance in many compan ies that not only the operation should be uninterrupted and reliable but also it should be carried out at hi gh power and high speed. Another vital requirement is the accurate pr ediction and control of the dynamic behavior (unbalance response, critical speeds and instability). These factors were the motivations for this research wherein rigid rotor analysis and finite element analysis was used to investigate bearing coefficient parameters a nd the rotordynamics of micro spindle. Both the rigid rotor analysis and finite element analysis have been performed simultaneously. The tungsten carbide spindle has a first be nding or flexure natu ral frequency of 2.2 million rpm for a bearing span of 1 inch. The spindle operating speeds are expected to be about 500,000 rpm. Hence rigid ro tor analysis can be justifie d. Finally, the experimental setup was designed to find bearing paramete rs which were compared with analytical results. 2.1 Rigid Rotor Analysis In order to get generalized rotor dynamic m odels, the Jeffcott roto r is extended to a four degree of freedom rigid ro tor system as shown in the schematic diagram of Figure 21. The four coordinates, whic h are the two geometric center translations (V, W) and the two rotation angles (B, ) describe the rotor c onfiguration relative to the fixed reference (X, Y, Z). Bearing 1 and bearing 2 are located at an axial distance a1 and a2 from the

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5 center of mass, respectively. Both these distan ces are defined as positive in the plus X direction. The rotor configuration is always defined so that a1 is positive. YZXV Wa1a2Brg.1 Brg.2B Z YV W m b c =t Figure 2-1. Rigid rotor schematic. Where (a, b, c) geometric center body reference ( ) eccentricity components ( ) spin angle = t ( ) constant spin frequency ()(cossin) ()(cossin) VtV m WtW m (2-1) The angular rate of the rigid body is sin cossincos coscossina b ctt tt (2-2)

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6 and the kinetic energy of the rigid body is 11 22222 ()() 22 TmVWII mmpac db (2-3) By considering the variational work of th e bearing forces, they are included in the equation of motion. The bearing force is a function of lateral sh aft translations and velocities at the bearing location. (,,,) (,,,)FFVWVW YY FFVWVW Z Z (2-4) Upon Taylor’s series expansion of eq. (2-4) about the origin, the force components in eq. (2-4) are approximated by th eir corresponding linear forms. At the ith typical bearing, forces are expr essed in the following equation: kkcc F V V iYYiYZiiYYiYZ Y i FkkWcc W i ZiZYiZZiZYiZZ i (2-5a) or Fkrcr iiiii (2-5b) where 100 010a i rqAq ii a i (2-6) T qVWB (2-7) The variational work done by bearing forces on the rotor is given by the following expression 24 1 1 WFrQq ii kkk i k (2-8) where Qk represents the genera lized bearing forces.

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7 Lagrange’s equations are of the following form: () dTT Q k dtqq kk k=1,2,3,4 (2-9) Using the above set of equations and insert ing in eq (2-9) the following set of rigid rotor equations of motion has been obtained: () M qCGqKqQ (2-10) where 000 000 000 000 m m M I d I d 0000 0000 000 000 G I p I p 2 1 T KAkA iii i 2 1 T CAcA iii i 22 cossin 00 00 Qmtmt (2-11) 2.2 Finite Element Analysis Typically, it is not possibl e to obtain analytical so lutions for problems involving complicated geometries, loadings and mate rial properties. Based on the study and inspection of various approaches available for modeling, one of the most appropriate methods for modeling of high-speed micro spindl e is the FEA, finite element method. It is also the only feasible type of computer simulation available for this purpose. The finite element method is generally a numerical method used for solving engineering and mathematical physics problems. The following steps are used in the FEA for dynamic response solution [3]:

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8 Form element stiffness matrix. Form element mass matrix. Assemble system stiffness matrix and incorporate constraints. Assemble system mass matrix and incorporate constraints. Solve eigenproblem and obtain a vector of frequencies and mode shapes. Form excitation vector in physical coordinates. 2.3 Identifying Bearing Paramet ers By Experimental Method The estimation of the dynamic bearing char acteristics using theoretical methods usually results in an error in the predic tion of the dynamic behavior of rotor-bearing systems. Reliable estimates of the bearing operating condition in actual test conditions are difficult to obtain and, therefore to re duce the discrepancy betw een the measurements and the prediction, physically meaningful and accurate parameter identification is required in actual test conditions. There are some similarities between various experimental methods for the dynamic characte rization of rolling element bearings, fluidfilm bearings and magnetic bearings. These me thods require forces as input signals and displacement/velocities/accelerat ions of the dynamic system to be measured are usually the output signals, and inputoutput relationships are us ed to determine the unknown parameters of the system models. There are a lot of identification techniques of bearing parameters, which are based on methods used to excite the system [4], such as the following: 1. Methods using Incremental Static Load 2. Methods using Dynamic Load 3. Methods using an Excited Load 4. Method using Unbalance Mass

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9 5. Methods using an Impact Hammer 6. Methods using Unknown Excitation Appendix A summarizes the source ma terial on the experimental dynamic parameter identification of bearings. 2.3.1 Methods Using Incremental Static Load Mitchell et al. (1965-66) [5] performed e xperiments to incrementally load the bearing and measuring the change in position, and obtained the four stiffness coefficients of fluid-film bearings. They obtained th e following simple relationships using the influence coefficient approach to / / k zz yy k zyzy / / k yzyz k zz yy (2-12) where zz yyyzzy / 1 / 2 yF yyy yF z yz / 1 / 2 zF zyy zF zzz (2-13) Here y1 and z1 are displacements of the journal center from its static equilibrium position in vertical and horizontal directions respectively, on the appl ication of a static incremental load F y in the vertical direction; and y2 and z2 are displacements corresponding to a static incremental load F z in the horizontal di rection. This method can be applied to any type of bearing sin ce the estimation of stiffness requires the establishment of a relationship between th e force and the corres ponding displacement. 2.3.2 Methods Using Dynamic Load Dynamic load methods have been the most researched and widely used in the identification of dynamic bearing parameters in the last 45 years [4]. Their major

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10 advantages are that they can be readily im plemented on a real mach ine and the excitation can be applied either to the journal or to the bearing housing depending on practical constraints. For the rigid rotor case, when the excitati on is applied to the journal (Figure 2-2), the fluid-film dynamic equation can be written as m() yy () mcckkfmyy yyy yzyyyzyyyzy B mmzcczkkzfmzz zzzzzzz zyzyzy B (2-14) where m is the mass of the journal, y and z represent the motion of the journal center from its equilibrium position relative to the bearing center, and yB and zB are the components of the absolute displacement of th e bearing center in ve rtical and horizontal directions, respectively. In this case, the origin of the coordinate system is assumed to be at the static equilibrium position, so that grav ity does not appear explicitly in the equation of motion. There will be one equation of this form for each of the bearings and the terms yB, zB represent the motion of the supporting struct ure. For the case of a rigid rotor with bearings on a rigid support, equation (2-14) can be expressed in the form M qCqKqfMq B BBR (2-15) The subscripts R and B refer to the rotor and bearings, respectively. On collecting the terms together, we get () M MqCqKqf B RBB (2-16) The overall system mass, damping and sti ffness matrices can be formed by adding the separate contributions of the bearings and rotor in equation (2 -16). This form was used by Arumugam et al. (1995) [6] to extract KB and CB in terms of the known and measurable quantities such as the rotor m odel, forcing and corre sponding response. The

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11 sinusoidal response of a rotor at speed is studied using the modified form of this equation (2-16), and the response is of the form j t qQe The governing equation of motion is given by 2 () M jCKQFZQ u (2-17) where [Z( )] is the dynamic stiffness matrix, Fu is the unbalance force, and is the rotational freque ncy of the rotor. Fluid Journal Non floating bearing housing fy(t) fz(t) Figure 2-2. A non-floati ng bearing housing and a rotating journal. 2.3.3 Methods Using an Excitation Force The application of a calibrate d force to the journal can only rarely be applied in practical situations. Glienicke (1966–67) [7 ] adopted the technique of exciting the floating bearing bush (housing) sinusoidally in two mutually perpendicular directions (Figure 2-3) and measuring the amplitude and phase of the resulting motions in each case. The stiffness and damping coefficients were then calculated from the frequencydomain equations.

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12 Morton (1971) [8] devised a measurement using the receptance coefficient method procedure for the estimation of the dynamic bearing characteristics. He excited the lightweight floati ng bearing bush by using very low forcing frequencies, (10 and 15 Hz). Assuming the inertia force due the fluid film and bearing housing masses to be negligible, and for sinusoidal motion, equation (2-14) may be written as zz F Y yyyz y zzZ F zz zy z with (2-18) zkjc where Y and Z are complex displacements and Fy and Fz are complex forces in the vertical and horizontal direct ions, respectively. In equati on (2-18) k represents the effective bearing stiffness coefficient, since while estimating the bearing dynamic stiffness, z, the fluid-film added-mass and j ournal mass effects contri bute to the real part of the dynamic stiffness and the e ffective stiffness is estimated. Fluid Journal Floating bearing bush fy(t) fz(t) Figure 2-3. A floating bearing housing and a fixed rotating shaft.

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13 Someya (1976) [9], Hisa et al. (1980) [10] and Saka kida et al. (1992) [11] identified the dynamic coefficients of largescale journal bearings by using simultaneous sinusoidal excitations on the bearing at tw o different frequencies and measuring the corresponding displacement responses. This is called the two-directional compound sinusoidal excitation method and all eight be aring dynamic coefficients can be obtained from a single test. When the journal is vi brating about the equilibrium position in a bearing, the dynamic component of the reaction force of the fluid film can be expressed by equation (2-18). If the excitation force a nd dynamic displacement are measured at two different excitation frequencies under the same static state of e quilibrium and ignoring the fluid-film added-mass effects equation (2-18) can be solved for the eight unknown coefficients as 2 111 111111 2 222222 222 2 111111111 2 222222 222 k yy FmY YZjYjZ k B yzyB YZjYjZc yy FmY B yB c yz k zy YZjYjZFmZ k B zB zz c YZjYjZ zy FmZ B zB c zz (2-19) where is the external excitation frequency a nd the subscripts 1 and 2 represent the measurements corresponding to two different excitation frequencies. Since equation (219) corresponds to eight real equations, th e bearing dynamic coefficients can be obtained on substituting the measured values of the complex quantities Fy, Fz, Y, Z, YB and ZB,

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14 2.3.4 Method Using Unbalance Mass From a practical point of view, the simp lest method of excitation is to use an unbalance force as this requires no sophisti cated equipment for the excitation, and it is relatively easy to identify the rotational speed dependency of the bearing dynamic characteristics. However, the disadvantage is that information is limited to the synchronous response. Nevertheless, since th is is the predominant requirement, the application of forces due to unbalance is extremely us eful. Hagg and Sankey (1956, 1958) [12-13] were among the first to use th e unbalance force only for experimentally measuring the oil-film elasticity and dampi ng for the case of a full journal bearing. They used the experimental measurement technique of Stone and Underwood (1947) [14] in which they used the vibration diagram to m easure the vibration amplitude and phase of the journal motion relative to the bearing housing. The direct stiffness and damping coefficients were only considered along the pr incipal directions in their study (i.e., major and minor axes of the journal elliptical orbit). The measured unbalance response whirl orbit gives the stiffness and damping coefficients. However, the results represen t some form of effective rotor-bearing coefficients and not the true film coeffici ents as the cross-coupled coefficients are ignored. Duffin and Johnson (1966–67) [15] employed a similar approach to that of Hagg and Sankey to identify bearing dynamic coef ficients of large journal bearings. They proposed an iterative procedure to calculate al l eight coefficients. Four equations can be written relating the measured values of displacement amplitude and phase Y, Z, y and z, together with the known value of the unbalance force, F, and four stiffness coefficients (obtained from st atic locus curve method; Mitchell et al., 1965–66) used to obtain the four unknown damping coefficients. This allows the solution of two sets of

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15 simultaneous equations having two equations in each set. The results had a greater accuracy than the method (Glienecke, 1966–67 ) in which two sets of four simultaneous equations were used to obtain the stiffness and damping coefficients. Murphy and Wagner (1991) [16] pres ented a method using a synchronously orbiting intentionally eccentric journal as the sole source of excitation for the extraction of stiffness and damping coefficients for hydrostatic bearings. The relative whirl orbits across the fluid film were made to be ellipt ic with asymmetric stiffness in the test bearing’s supporting structure. The study consid ered the bearing coefficients to be skewsymmetric and the elliptic nature was utilized in the data reduction process. Adams et al. (1992) [17] and Sawicki et al. (1997) [18] utilized experimentally measured responses corresponding to at least three discrete orbital frequencies, for a given operating condition to obtain twelve dynamic coefficien ts (stiffness, damping and added-mass) of hydrostatic and hybrid journa l bearings, respectively. They assumed that the bearing dynamic coefficients are independent of fr equency of excitation. The estimation equation was similar to equation (2-19) except the rotor mass was ignored and fluid film addedmass coefficients were considered. A confidence in the measurements was obtained by employing dual piezoelectric/strain gage load/displacement measuring systems. The difference between these two sets of dynamic force measurements was typically less than 2%. The test spindle (double-spool-shaft) ha d a provision for a circular orbit motion of adjustable magnitude with independent contro l over spin speed, orb it frequency and whirl direction. The least-squares linear regression fit to all frequency data points over the tested frequency range was used to obt ain the bearing dynamic coefficients.

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16 2.3.5 Methods Using an Impact Hammer Until the early 1970s, the common method to obtain the dynamic characteristics of systems involved using sinusoidal excitatio n [4]. Downham and Woods (1971) [19] proposed a technique using a pendulum hammer to apply an impulsive force to a machine structure. Although they were interested in vibration monitoring rather than the determination of bearing coefficients, their work led to the idea that impulse testing could be capable of exciting all the modes of a linear system. Nordmann (1975) [20] and Nordmann and Schllhorn (1980) [21] identified the stiffness and damping coefficients of journa l bearings by modal testing by means of the impact method wherein, a rigid rotor, runni ng in journal bearings was excited by an impact hammer. Two independent impacts first in the vertical direct ion and then in the horizontal direction were applied to the rotor and the corresponding responses were measured. A transformation of input signals (f orces) and output signals (displacements of the rotor) into the frequency domain was th en carried out and the four complex FRFs were calculated. The bearing dynamic parameters were assumed to be independent of the frequency of excitation. The analytical FRFs, which depend on the bearing dynamic coefficients, were fitted to the measured FRFs An iterative fitting process results in the stiffness and damping coefficients. Zhang et al. (1992a) [22] fitted the measured FRFs to those calculated theoretically so as to obtain the eight bearing dynamic co efficients. They also quantitatively analyzed the influence of noise and meas urement errors on the estima tion in order to improve the accuracy of estimated bearing dynamic coeffici ents. They used a half-sinusoid impulse excitation and with a different level of noise added to the resulting response to test their algorithm and averaged the frequency responses to reduce the uncertainty due to noise in

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17 the response. To reduce the effect of phase -measurement errors, they defined an error function using just the amplitude component s of the FRFs. This non-linear objective function was then used to estimate the beari ng parameters by an itera tive procedure. It was also demonstrated by then that it was necessary to remove the unbalance response from the signal when an impact test was us ed, especially at higher speeds of operation, and they concluded this to be the reason for the scatter in the resu lts by impact excitation, as compared to the discrete frequency harmonic excitation. This method is time-consuming though since impact tests have to be conducted for each rotor speed at which bearing dynamic para meters are desired. In general, the amount of information that can be extracted from a single impulse test is limited as the governing equations for a bearing include coupling betwee n the two perpendicular directions. Errors in the estimation will be greater for the case when bearing dynamic coefficients are functions of external excitation frequency as compared to the estimation from functions of rotor rotational frequency. Also impulse testing may lead to unde restimation of input forces when applied to a rotating shaft as a result of the generation of friction-related tangential force components and, further, is pr one to poor signal-to-noise ratios because of the high crest factor. 2.3.6 Methods Using Unknown Excitation In industrial machinery, the application of a calibrated force is difficult to apply. Due to residual unbalance, misalignment, rubbing between the rotor and stator, aerodynamic forces, oil whirl, oil whip and in stability, inherent forces are present in the system and these render the assessment of the forcing impossible. Adams and Rashidi (1985) [23] used the static loading method to measure bearing stiffness coefficients and determined orbital motion at an adjustable threshold speed to extract bearing damping

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18 coefficients by inverting the associated ei genproblem. The approach stems from the physical requirement for an exact internal energy balance between positive and negative damping influences at an instability thresh old. The approach was illustrated by simulation and does not require the measurement of dynamic forces. Lee and Shih (1996) [24] found roto r parameters including bearing dynamic coefficients, shaft unbalance distribution and disk eccentricity in flexible rotors by presenting an estimation procedure based on the transfer matrix method. The relations between measured response data and the known system parameters were used to formulate the normal equations. The parameter estimation was then performed using the least squares method by assuming that the bearing dynamic coefficients were constant at close spin speeds.

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19 CHAPTER 3 ROTOR DYNAMIC ANALYSI S OF MICRO SPINDLE The aim of this project is to rotate a sp indle supported by air bearings at up to 500,000 rpm, with sub-micrometer runout. The 1/ 8th inch diameter tool shank is used as a spindle shaft. As mentione d before, the only viable wa y to obtain satisfactory runout was to use the tool shank itself as the spindle shaft. An air turbine is used as a driving system for the spindle. Thus, the only vi able way to assemble air turbine is to manufacture the turbine integral with the spin dle, which is shown in Figure 3-1. Also from the practicality point of view the micr o-spindle must accept a variety of tools with minimal time and effort required for tool ch ange. Rotordynamics of high-speed flexible shafts is influenced by the complex interaction between the unbalance forces, bearing stiffness and damping, inertial properties of the rotor, gyroscopi c stiffening effects, aerodynamic coupling, and speed-dependent syst em critical speeds. For stable high-speed operations, bearings must be designed w ith the appropriate stiffness and damping properties, selected on the basis of a detailed rotordynamic analysis of the rotor system. The two types of rotor dynamic analyses that are used for high-speed thin spindle are rigid rotor analysis and finite element anal ysis. The dynamic behavior of a spindle is analyzed in order to find critical speeds unbalance response and lin ear stability margins by these methods.

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20 Figure 3-1. Micro-spindle. 3.1 Rigid Rotor Analysis Rigid rotor analysis was initially used to get the rotor unbalance response. The air bearings were located on either side of th e center of mass, as shown in Figure 3-2. In addition, center of mass is found by solid model ProE software. a1 a2 18.1 mm 26.18 mm 32.28 mm 34.81 mm 38.1 mm 3 1 7 5 mmbrg#1 brg#2 Figure 3-2. Bearings location on spindle.

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21 The rigid rotor has 4 degrees of freedom (DOF) represented by two displacements (V, W) and two rotations ( ) of the center of mass. The following equation, (derivation can be found in chapter 2), was us ed for a rigid rotor subjected to unbalance. () M qCGqKqQ (2-1) where M C, G, K are mass, damping, gyroscopic and stiffness matrices, respectively, Q is the force vector. Expressions of thes e matrices can be found in chapter 2. is constant spin frequency. In order to use equation (2-1) in the roto r orbit analysis, following procedure is applied. From chapter 2, it is known that disp lacement vector for 4 DOF is the following: T qVWB The shaft unbalance leads to harmonic synchronous excitation. Hence the displacement or response vector can be expressed as the following: cos()sin() VV cs WW cs qtt BB cs cs (3-1) As a result, first and second derivatives will have the following forms, respectively. sin()cos() VV cs WW cs qtt BB cs cs (3-2) 22 cos()sin() VV cs WW cs qtt BB cs cs (3-3)

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22 After substituting for M C G K Q into equation (2-1) using (3-1), (3-2), (3-3) and rearranging sine and cosine terms, and using harmonic balance, the following expression can been obtained: 122121212 () 1212 121221212 () 1212 1212212222122 () 12121212 1212 ()()( 1212 kkmkkakakakak yyyyyzyzyzyzyyyy kkkkmakakakak zyzyzzzzzzzzzyzy akakakakakakakak zyzyzzzzyyzzzyzy d akakakak yyyyyzyz 212221222 ) 1212 12121212 ()()()() 1212 12121212 ()()()() 1212 121212 ()()( 121212 akakakak yzyzzzzz d ccccacacacac yyyyyzyzyzyzyyyy ccccacacacac zyzyzzzzzzzzzyzy acacacacacac zyzyzzzzyyz 21222 )() 12 12122122212 ()()()() 12121212 acac zzyzyp acacacacacacacac yyyyyzyzyzyzpzzzz 12121212 ()()()() 1212 12121212 ()()()() 1212 121221222122 ()()()() 12121212 12 () 12 ccccacacacac yyyyyzyzyzyzyyyy ccccacacacac zyzyzzzzzzzzzyzy acacacacacacacac zyzyzzzzyyzzzyzy acac yyyy 1221222122 ()()() 121212 122121212 () 1212 121221212 () 1212 121221222 121212 acacacacacac yzyzyzyzzzzz kkmkkakakakak yyyyyzyzyzyzyyyy kkkkmakakakak zyzyzzzzzzzzzyzy akakakakakak zyyyzzzzyyzz 2122 () 12 1212212221222 ()()() 12121212 akak zyzy d akakakakakakakak yyyyyzyzyzyzzzzz d 0 0 2 0 0 V c W c B c c m V s W s B s s (3-4)

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23 where a1 and a2 are the distance of the bearings from center of mass. kyy, kyz, kzy, kzz, and cyy, cyz, czy and czz are stiffness and damping coefficients of each bearing, respectively. Id and Ip are the polar and diametric inertia, respectively. The charts from ‘Rotor-Bearing Dynamics Design Technology’ [25], design handbook for fluid film type bearings were in itially used to evaluate the damping and the stiffness coefficients. In order to use these charts, bearing length and diameter ratio was assumed to be two. Mass and gyroscopic matr ices were found by hand calculation, which were later entered in the MathCAD program. The two types of forces on the system are the unbalance force and the cutting force. As the cutting force is much smaller than the unbalance force, it was neglecte d. An unbalance eccentricity of eu=0.000002 inch was used initially to evaluate the unbalance for ce. The solution procedure was implemented in MathCAD to find the unbalance respond at the bearing locations. The rotor orbits at the two bearing supp orts at 500,000 rpm for a rotor with a mass=2.525x10-5 lbf-sec^2/in, Ip=4.187x10-6 lbf-sec^2-in, Id=4.894x10-8 lbf-sec^2-in and an unbalance eccentricity of 0.000002 inch are shown in Figure 3-3. The air bearing for this configuration has the following parameters, Table3-1. Table 3-1. Bearing parameters at 500,000 rpm ( =52360 rad/sec). Kyy1,2=950.4 lbf/in Cyy1,2=0.021 lbf-sec/in Kyz1,2=99.1 lbf/in Cyz1,2=-0.004 lbf-sec/in Kzy1,2=93.46 lbf/in Czy1,2=0.005 lbf-sec/in Kzz1,2=1050.4 lbf/in Czz1,2=0.106 lbf-sec/in

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24 Figure 3-3. Rotor orbits at tw o bearing supports at 500,000 rpm.

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25 Even for the 1,000,000 rpm and same tool with same rigid rotor parameters, displacement of rotor in beari ngs is small (see figure 3-4). Figure 3-4. Rotor orbits at the two bearing supports at 1,000,000 rpm.

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26 One of the most critical places is also tool end, next to the air turbine. Thus, rotor orbits should be found at tool end. As it can be seen from Figure 3-5, runout in both directions is small at the tool end (6 210 inch). Figure 3-5. Rotor orbits at the tool end at 500,000 rpm. The radial clearance was chosen 0.0002 inch. The eccentricity ratio will be 0.01 ( =e/c) for the chosen unbalance. Figure 3-6 shows that why figure 3-3 and fi gure 3-4 is same. It can be seen that after 500,000 rpm runout is nearly same and it is 6 210 inch. In addition this figure shows that highest amplitude is between 100,000 rpm and 120,000 rpm. Later, it will be shown that the rigid body critical speeds are 105,042 rpm and 115,231 rpm.

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27 Orbit Amplitude-1.00E-06 0.00E+00 1.00E-06 2.00E-06 3.00E-06 4.00E-06 5.00E-06 6.00E-06 7.00E-06 8.00E-06 9.00E-06 0100000200000300000400000500000600000700000 rotor speed (rpm)orbit amplitude (inch) brg#1 & 2 tool tip Figure 3-6. Orbit amplitude at 1st, 2nd bearing and tool tip. Linear stability analysis has been perform ed based on energy dissipated at bearings. System is stable, if total energy dissipated is negative. Based on this criteria the system is stable over the entire speed range. For 500, 000 rpm, total work (energy dissipated) per cycle at each bear ing is found –4.24253x10-8 (see Appendix B). Sample MathCAD program of rotor orbits can be found in Appendix B for a given spindle speed, 500,000rpm. The bearing analysis code was coupled w ith the rotordynamics code in MATLAB, to efficiently explore the bearing/rotor de sign space. A direct in terface to the bearing code used in XLTiltPadHGB was achieve d by writing a MATLAB function. Input structures, viz: geometry, fl uid, dynamic, operating, numerical were used to submit bearing parameters. Thus the data plotting and vi sual inspection of bearing performance was facilitated by the MATLAB interface. This helped in the determination of factors that determine stiffness and damping. For example, stiffness as a function of the rotor eccentricity is

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28 plotted in Figure 3-7. In this case, a consta nt gravity force was applied to the rotor and the bearing contained four pads. The bearing stiffness and da mping coefficients were passed from the bearing design code to the rotordynamic code. These co efficients are influenced by the bearing design parameters, and influence the rotordynamics. The rotor orbits within the bearings are calculated using the rotordynamic model and these orbits and their characteristics determine the rotor performance. Figure 3-7. Stiffness, Kyy versus rotor eccentricity. One of the most important aspects of res earch was to analyze rotor performance at critical speeds. Thus, rotor performance has been investigated at critical speeds. In order to find critical speeds, the following proce dure has been adopted. It is known that eigenvalues can be found from equation with first order form. However equation of motion for rotor-bearing system is in second order. So equation (2-1) was modified in order to find eigenvalues. () M qCGqKqQ (2-1) 0 M M M C 0 0M K K

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29 q x q 0 Q X (3-5) After entering equations (3-5 ) into (2-1), results in: ** M xKxX (3-6) In order to find eigenvalues, right hand side of equation (3 -6) is set to zero, and a harmonic solution j t xe and j t xje is assumed. After rearranging equation (36), the standard E.V.P (eigenvalue problem) of the term Axx can be set up, as shown below: 00*1* () j KMxx where 1 (3-7) In order to find all eight ei genvalues (four of which are complex conjugates of the other four) for a given spindle speed, a MathCAD program was generated to solve expression obtained above. The rotor whir l map was finally generated using these eigenvalues. The whirl map for a rotor with a mass=2.525x10-4 lbf-sec^2/in, Ip=4.187x10-6 lbfsec^2-in, Id=4.894x10-8 lbf-sec^2-in and an unbalance eccentricity of 0.000002 inch is shown in Figure 3-12; wherein the air beari ng has the following parameters, Table 3-2. Table 3-2. Bearing Parameters for different rotor speeds. Kyy1,2 Kyz1,2 Kzy1,2 Kzz1,2 [lbf/in] [rad/sec] 1069.2 99.1 -67.52 1069.2 62831 950.4 99.1 -93.46 1050.4 52000 831.6 105.04 -109.4 831.6 42000 751.9 115.92 -151.28 751.9 32000 700.8 118.8 -118.8 700.8 22000 623.4 178.2 -115.8 623.4 12000 594 190.08 -91.28 594 9500 159.2 137.22 -85.34 159.2 2000

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30 Table 3-2. Continued Kyy1,2 Kyz1,2 Kzy1,2 Kzz1,2 [lbf/in] [rad/sec] 36.7 83.76 -53.76 36.7 500 Cyy1,2 Cyz1,2 Czy1,2 Czz1,2 [lbf-sec/in] [rad/sec] 0.015 -0.003 0.004 0.088 62831 0.021 -0.004 0.005 0.106 52000 0.033 -0.005 0.007 0.131 42000 0.053 -0.009 0.01 0.158 32000 0.086 -0.014 0.025 0.23 22000 0.18 -0.042 0.077 0.383 12000 0.287 -0.08 0.129 0.402 9500 0.965 -0.011 0.018 0.322 2000 1.838 -0.011 0.017 0.147 500 whirl map0 5000 10000 15000 20000 25000 010000200003000040000500006000070000 rotor speed (rad/sec)eigenvalues (rad/sec) spin/whirl ratio=1 w1 w2 w3 Figure 3-8. Whirl map. It can be seen from graph (Figure3-8), critical speeds ar e 11000 rad/sec (105,042 rpm) and 12067 rad/sec (115,231 rpm). MathCAD program for different air beari ng stiffness and damping values, found by using XLTiltPadHGB, can be found in Appendix C.

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31 3.2 Finite Element Analysis The rotordynamics analysis of the roto r-bearing system was performed using a special-purpose code. The model shown in Figure 3-9 has 10 finite elements, 11 nodes, with each node having 4 degrees of free dom. The rigid body and flexural natural frequencies and model shapes can be computed from this model. To use finite element method it is assumed stiffness coefficient fixed and no cross coupling stiffness. The model parameters are bearing stiffness = 1,000 lb/in (fixed), bearing span = 0.5 in, shaft dia = 0.125 in, shaft length = 0.925 in and th e shaft material is tungsten carbide. In addition, 610ume oz-in is used in FE model. As a result four critical speeds are found for the following parameters. Figure 3-9. Finite element model. The rigid body critical speeds are 107,758, 115,298, while the flexural critical speeds are 2,042,349 and 4,955,776 rpm. Figure 3-10. Rigid body mode 1, 107758 rpm.

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32 A. B. Figure 3-11. Rotor analysis at 115298 rpm. A) Rigid body mode 2, B) Potential energy distribution. A. Figure 3-12. Rotor analysis at 2,042,349 rpm. A) Flexural mode 1, B) potential energy distribution.

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33 B. Figure 3-15. Continued Figure 3-13. Flexural mode 2, 4,955,776 rpm. Figure 3-14. Critical speed map.

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34 The critical speed map in Figure 3-14 show s that the flexural modes are unaffected by bearing stiffness, while the two rigid body modes increase with bearing stiffness. There are three critical points on the tool such as; two bearing positions and tool tip. As a result the following analys is are found for these positions. A. B. Figure 3-15. 1st bearing. A) and B) Unbalance respons e, C) Amplitude and phase lag, D) Nyquist plot for displacement.

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35 C. D. Figure 3-15. Continued

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36 A. B. Figure 3-16. 2nd bearing. A) and B) Unbalance respons e, C) Amplitude and phase lag, D) Nyquist plot for displacement.

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37 C. D. Figure 3-16. Continued

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38 A. B. Figure 3-17. Tool tip. A) Unbalance response, B) Amplitude and phase lag, C) Nyquist plot for displacement.

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39 C. Figure 3-17. Continued A. Figure 3-18. Shaft orbits. A), B), C), D), E), F) are shaft orbits as a function of speed.

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40 B. C. Figure 3-18. Continued

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41 D. E. Figure 3-18. Continued

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42 F. Figure 3-18. Continued. Figure 3-19. Stability Map (Note: Negativ e log decrements indicate instability). 3.3 Summary Rigid body critical speeds (107,758 and 115,298 rpm) found by FEA are very close to the rotor analysis criti cal speeds (105,042 and 115,231 rpm) calculated by rigid rotor

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43 analysis. The difference occurs, because in FEA analysis cross coupling stiffness terms and damping terms are eliminated and stiffness doesn’t change with rotor speed. For rigid rotor analysis eccen tricity ratio is calculated =0.01, which can be questionable to get this value in practice, and it leads 8 1.2210me u oz-in. So unbalance response analysis is performed again with =0.1. Figure 3-20 shows that even with this eccentricity ratio sub micrometer runout has been obtained for a given stiffness and damping parameters. Orbit Amplitude-1.00E-05 0.00E+00 1.00E-05 2.00E-05 3.00E-05 4.00E-05 5.00E-05 6.00E-05 7.00E-05 8.00E-05 9.00E-05 0100000200000300000400000500000600000700000800000 Rotor speed (rpm)Amplitude (inch) brg#1&2 tool tip Figure 3-20. Orbit amplitude at 1st, 2nd bearing and tool tip, =0.1. Both rigid rotor analysis (see appendix B) and FEA (see figure 3-19) shows system is stable.

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44 CHAPTER 4 EXPERIMENTAL IDENTIFICATI ON OF BEARING PARAMETERS In order to ensure proper operation, th e vibration phenomena, which the rotors supported by fluid film bearings are especially subject to, has to be properly predicted. Knowledge of dynamic coefficients, stiffne ss and damping of a bearing prior to its installation and operatio n can be highly influential in the operation costs of the final machine. Both an analytical and experime ntal approach can be employed to study the dynamic behavior of bearings. Numerical t echniques and computer-based simulation are usually used to perform analytical studies The current research has used XlTiltPad program to identify bearing parameters. The test set up aims at analyzing dynami c behavior of a tilted pad air bearing, which will be used in this research, and comparing the experimental results with analytical results. For the time being, experi ments used a commercial air bearing as the tilted pad air bearing is in the process of being designed and for the initial studies; a flexure-supported ball bearing (see figure 4-1) is being used. Due to the difficulties found in exciting the rotor-bearing system and in the measurement of force and displacement data, experimental testing on fluid film bear ings is known to be complex. The bearing parameters can be identified experimental ly by six different methods as mentioned before. Since the research deals with a micr o spindle supported by a small air bearing, the test setup will be also small making it extremely hard to use loading in order to excite the system. As a result, the viable wa y is using unbalance mass method.

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45 Figure 4-1. Flexure supported ball bearin g: 1) Flexure, 2) Ball bearing. 4.1 Method of Measurement The principle of operation of the test rig is rather simple: The bearing to be tested is placed on a chassis (see figure 4-2), which se rves to support displacement probes and is mounted on dynamometer (load cell), used for m easuring forces. A shaft is placed inside the bearing and one side of shaft is mounted to the high-speed spindle. Figure 4-3 shows a schematic of the test rig. Z Y displacement probe load sensor Figure 4-2. A chassis.

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46 Load cell spindle displacement probe test bearing rotor Chasis disk Figure 4-3. Schematic of the test rig. The rotating part of the test bearing has been given an intentional eccentricity at the test bearing location. When th e shaft rotates, the eccentricity generates an orbital pattern synchronous with shaft speed. Since the shaft is driven with a synchronous harmonic load, the resulting shaft motion will, in general, be elliptic. Therefor e, relative displacements describing ellipse as a function of time is in following form: ()(cos)(sin)ytatbt (4-1) ()(cos)(sin)ztgtht (4-2) Following equations will be us ed in equation of motion: ()(sin)(cos)ytatbt (4-1a) ()(sin)(cos)ztgtht (4-2a)

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47 The four coefficients a, b, g and h are termed Fourier coefficients, and is the tester speed in radians per second. They are obtained from the synchronous components of complex frequency spectrums computed for the y and z displacements. The air bearing will respond with reaction fo rces which will be read by dyno to the off centered rotor movement. The same proce dure applied to the load data which load cells read and following equations are derived: ()(cos)(sin)yFtmtnt (4-3) ()(cos)(sin)zFtptqt (4-4) Equation of motion is the following: yyyzyyyzyyyz y zyzzzyzzzyzz zKKCCMM F yyy KKCCMM F zzz (4-5) Inertia term can be neglected in equation, because shaft relative displacement is so small that inertia force will not contribute a lot for air film force. So new equation is the following: yyyzyyyz y zyzzzyzz zKKCC F yy KKCC F zz (4-6) After substituting (4-1), (4-2), (4-1a), (4-2a), (4-3) and (4-4) into (4-6), the following equation is obtained: 0000 0000 0000 0000yy yz yy yz zy zz zy zzC C K bhagm K agbhn C bhagp C agbhq K K (4-7)

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48 Since all these bearing parameters are func tions of speed, it is necessary that the speeds span as wide a range as possible to give the best definition of the coefficients [16]. There is an example how to us e equation (4-7) in Appendix D. 4.2 Design Process The test rig was designed and developed relying on an existin g dynamometer, as seen in Figure 4-4. Detailed information a bout dynamometer is presented in Appendix E. In order to facilitate a better understanding of the setup, it was divi ded in to four parts that is the dyno; a main chassis (see Figur e 4-5), which serves to mount displacement probes in order to measure shaft displacement and to mount the test bearing; a shaft; and a spindle, which is an electric driven moto r and rotates at 50000 rpm (max.). Figure 4-4. Test Setup: 1) Base, 2) Dyno, 3) Chassis, 4) Te st bearing, 5) Shaft, 6) Spindle

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49 Figure 4-5. Main Chassis: 1) Stator, 2) Displacement Probe, 3) Test Bearing. The shaft motion is determined by displacem ent probes, and results can be read on PC by using fiber optic sensors (RC20) connected to the PC. Dyno is used to measure the forces in x and y directions re spectively, and the data is se nt to the PC using a charge amplifier. A NSK spindle drive controller adjusts the spindle speed. The complete measurement system is illustrated in Figure 4-6. Figure 4-6. Measurment system

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50 CHAPTER 5 CONCLUSION AND RECOMMENDATIONS The current research provides initial results for the development of a comprehensive rotordynamics analysis to de scribe high-speed micr o spindle vibrations. Results demonstrate that the air bearing ch aracteristics and spi ndle residual unbalance levels dictate the critical speed placement, unbalance response, and th e ability to limit the tool tip runout to sub-micrometer levels. Results demonstrate that the 1/8th inch tungsten carbide micro spindle with the integrally machined air turbine at one end and the end mill cutter at the other end can indeed operate with sub-micrometer runout at the tool tip for spindle speeds up to 1 million rpm. Air bearing stiffness used is about 2000 lb/in. Spindle residual unbalance level assumed is 10-6 oz-in (me=. 000001 oz-in). Residual unbalance two to three times this level will also be adequate with highe r bearing stiffness. Turbine engine shafts weighing 100 lbs or greater are routinely balanc ed to 0.1 oz-in. Considering that the end mill cutter weighs about 0.07 lb, the recomm ended unbalance limits can be achieved in practice. Next step in this resear ch is experimental analys is, which was undertaken to understand behavior of the air b earing. Since the air bearing is yet to be fabricated, a simple test rig has been built to test the sp indle mounted on ball bear ings with a flexure support. The x-y displacements at the bearings as a function of speed, are being used to evaluate the support stiffness ch aracteristics. Our goals are to demonstrate the feasibility of the proposed parameter estimation sc heme to evaluate support stiffness.

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51 APPENDIX A CHRONOLOGICAL LIST OF PAPERS ON THE EXPERIMENTAL DYNAMIC PARAMETER IDENTIFICATION OF BEARINGS

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52

PAGE 63

53

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54 The following abbreviations are used in th e table: AGS, annular gas seal; ANS, annular seal; ALS, annular liquid seal; BS, brush seal s; DS, damper seals; ER, electrorheological fluid; FAB, foil air; FTB, foil thrust; GDS, gas damper seal; GJ, gas journal; GS, gas seal; HCS, honeycombed seal; HDJ, hydrodyna mic journal; HSJ, hydrostatic journal; HYJ, hybrid journal; LGS, long gas seal; LS long seal; MB, magnetic; MD, metal mess bearing damper; PLS, plain liquid seal; RB, recirculating ball; RE, rolling element; SPR, springs; SQF, squeeze film; TPJ, tilting pad journal; TR, tapered roller.

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APPENDIX B GENERAL RIGID ROTOR SOLUTION 1

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56 Input Rigid Rotor Parameters lbf-sec^2/in a10. 6 inch m 0.0974 8 386 a20. 6 lbf-sec^2-in Ip 0.00161 6 386 eu0.00000 2 unbalance eccentricity, in lbf-sec^2-in Id 1.889105 386 eu 4 eucoseu eusineu 52359. 9 rad sec 1.41421106 1.41421106 Kyy12055. 3 Kyy22055. 3 lbf/in Cyy10.01 4 Cyy20.01 4 lbf-sec/in Kyz1147. 6 Kyz2147. 6 Cyz10.007 1 Cyz20.007 1 Kzy191.3 6 Kzy291.3 6 Czy10.00 5 Czy20.00 5 Kzz11713.1 3 Kzz21713.1 3 Czz10.03 5 Czz20.03 5

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57 A Kyy1Kyy2 2m Kzy1Kzy2 a1Kzy1 a2Kzy2 a1Kyy1 a2Kyy2 () Cyy1Cyy2 () Czy1Czy2 () a1Czy1 a2Czy2 () a1Cyy1 a2Cyy2 () Kyz1Kyz2 Kzz1Kzz2 2m a1Kzz1 a2Kzz2 a1Kyz1 a2Kyz2 () Cyz1Cyz2 () Czz1Czz2 () a1Czz1 a2Czz2 () a1Cyz1 a2Cyz2 () a1Kyz1 a2Kyz2 a1Kzz1 a2Kzz2 a12Kyy1 a22Kzz2 2Id a12Kyz1 a22Kyz2 a1Cyz1 a2Cyz2 () a1Czz1 a2Czz2 () a12Cyy1 a22Czz2 a12Cyz1 a22Cyz2 2Ip a1Kyy1 a2Kyy2 () a1Kzy1 a2Kzy2 () a12Kzy1 a22Kzy2 a12Kzz1 a22Kzz2 2Id a1Cyy1 a2Cyy2 () a1Czy1 a2Czy2 () a12Czy1 a22Czy2 2Ip a12Czz1 a22Czz2 Cyy1Cyy2 () Czy1Czy2 () a1Czy1 a2Czy2 () a1Cyy1 a2Cyy2 () Kyy1Kyy2 2m Kzy1Kzy2 a1Kzy1 a2Kzy2 a1Kyy1 a2Kyy2 () Cyz1Cyz2 () Czz1Czz2 () a1Czz1 a2Czz2 () a1Cyz1 a2Cyz2 () Kyz1Kyz2 Kzz1Kzz2 2m a1Kzz1 a2Kzz2 a1Kyz1 a2Kyz2 () a1Cyz1 a2Cyz2 () a1Czz1 a2Czz2 () a12Cyy1 a22Czz2 a12Cyz1 a22Cyz2 2Ip a1Kyz1 a2Kyz2 a1Kzz1 a2Kzz2 a12Kzz1 a22Kzz2 2Id a12Kyz1 a22Kyz2 a1Cyy1 a2Cyy2 () a1Czy1 a2Czy2 () a12Czy1 a22Czy2 2Ip a12Czz1 a22Czz2 a1Kyy1 a2Kyy2 () a1Kzy1 a2Kzy2 () a12Kzy1 a22Kzy2 a12Kyy1 a22Kyy2 2Id (RHS unbalance forcing terms) RHSm 2 0 0 0 0 RHS 0.09791 0.09791 0 0 0.09791 0.09791 0 0

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58 A 67306.42171 18.944 0 0 3141.594 52.3599 0 0 25.2 66987.02171 0 0 429.35118 3508.1133 0 0 0 0 617.62206 9.072 0 0 1196.94731 11323.04899 0 0 6.81984 675.11406 0 0 11458.76585 1262.92079 3141.594 52.3599 0 0 67306.42171 18.944 0 0 429.35118 3508.1133 0 0 25.2 66987.02171 0 0 0 0 1196.94731 11323.04899 0 0 675.11406 9.072 0 0 11458.76585 1262.92079 0 0 6.81984 560.13006 qA1RH S Bearing Orbits Bearing #1 V1cq0a1q 3 W1cq1a1q 2 V1sq4a1q 7 W1sq5a1q 6 q 1.37609 106 1.53315 106 0 0 1.50867106 1.37988 106 0 0 Bearing #2 V2cq0a2q 3 W2cq1a2q 2 V2sq4a2q 7 W2sq5a2q 6

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59 2 10 6 1 10 601 10 62 10 6 2 10 6 1 10 61 10 62 10 6 shaft displacement in y directionshaft displacement in z direction2.06256106 2.06256 106 W2t () 2.04177106 2.04177 106 V2t () PLOT BEARING ORBITS T 2 T0.00012 t0 T 200 T Bearing #1 Bearing #2 V1t ()V1ccostV1ssint V2t ()V2ccost V2ssint W1t ()W1ccostW1ssint W2t ()W2ccost W2ssint 2 10 6 1 10 601 10 62 10 6 2 10 6 1 10 61 10 62 10 6 shaft displacement in y directionshaft displacement in z directionW1t () V1t ()

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60 Work done per cycle at the bearings Bearing #1 Kd1 Kyz1Kzy12 Cm1 Cyy1Czz 1 2 WK1Cir2 Kd1V1cW1sV1sW1c () Work done by Circulation Force at Bearing #1 WK1Diss Cyy1V1c2W1s2 2Cm1V1cW1cV1sW1s () Czz1W1c2W1s2 Work done by dissipation forces WK1TotalWK1CirWK1Diss Total work done at the bearing #1 Bearing #2 Kd2 Kyz2Kzy22 Cm2 Cyy2Czz 2 2 WK2Cir2 Kd2V2cW2sV2sW2c () Work done by Circulation Force at Bearing #2 WK2Diss Cyy2V2c2W2s2 2Cm2V2cW2cV2sW2s () Czz2W2c2W2s2 Work done by dissipation forces WK2TotalWK2CirWK2Diss Total work done at the bearing #2

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61 Bearing #1 Bearing #2 WK1Cir4.139061011 WK2Cir4.139061011 WK1Diss4.24666 108 WK2Diss4.24666 108 WK1Total4.24253 108 WK2Total4.24253 108 N OTE: Total work done at each bearing/cycle must be nega tive(subtracting energy from rotor), for stable operation

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APPENDIX C GENERAL RIGID ROTOR SOLUTION 2

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63 Bearing parameters f ound by using XLTiltPadHGB: Input Rigid Rotor Parameters lbf-sec^2/in a10. 6 inch m 0.0974 8 386 a20. 6 lbf-sec^2-in Ip 0.00161 6 386 eu0.00000 2 unbalance eccentricity, in lbf-sec^2-in Id 1.889105 386 eu 4 eucoseu eusineu 52359. 9 rad sec 1.41421106 1.41421106 Kyy12055. 3 Kyy22055. 3 lbf/in Cyy10.01 4 Cyy20.01 4 lbf-sec/in Kyz1147. 6 Kyz2147. 6 Cyz10.007 1 Cyz20.007 1 Kzy191.3 6 Kzy291.3 6 Czy10.00 5 Czy20.00 5 Kzz11713.1 3 Kzz21713.1 3 Czz10.03 5 Czz20.03 5

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64 A Kyy1Kyy2 2m Kzy1Kzy2 a1Kzy1 a2Kzy2 a1Kyy1 a2Kyy2 () Cyy1Cyy2 () Czy1Czy2 () a1Czy1 a2Czy2 () a1Cyy1 a2Cyy2 () Kyz1Kyz2 Kzz1Kzz2 2m a1Kzz1 a2Kzz2 a1Kyz1 a2Kyz2 () Cyz1Cyz2 () Czz1Czz2 () a1Czz1 a2Czz2 () a1Cyz1 a2Cyz2 () a1Kyz1 a2Kyz2 a1Kzz1 a2Kzz2 a12Kyy1 a22Kzz2 2Id a12Kyz1 a22Kyz2 a1Cyz1 a2Cyz2 () a1Czz1 a2Czz2 () a12Cyy1 a22Czz2 a12Cyz1 a22Cyz2 2Ip a1Kyy1 a2Kyy2 () a1Kzy1 a2Kzy2 () a12Kzy1 a22Kzy2 a12Kzz1 a22Kzz2 2Id a1Cyy1 a2Cyy2 () a1Czy1 a2Czy2 () a12Czy1 a22Czy2 2Ip a12Czz1 a22Czz2 Cyy1Cyy2 () Czy1Czy2 () a1Czy1 a2Czy2 () a1Cyy1 a2Cyy2 () Kyy1Kyy2 2m Kzy1Kzy2 a1Kzy1 a2Kzy2 a1Kyy1 a2Kyy2 () Cyz1Cyz2 () Czz1Czz2 () a1Czz1 a2Czz2 () a1Cyz1 a2Cyz2 () Kyz1Kyz2 Kzz1Kzz2 2m a1Kzz1 a2Kzz2 a1Kyz1 a2Kyz2 () a1Cyz1 a2Cyz2 () a1Czz1 a2Czz2 () a12Cyy1 a22Czz2 a12Cyz1 a22Cyz2 2Ip a1Kyz1 a2Kyz2 a1Kzz1 a2Kzz2 a12Kzz1 a22Kzz2 2Id a12Kyz1 a22Kyz2 a1Cyy1 a2Cyy2 () a1Czy1 a2Czy2 () a12Czy1 a22Czy2 2Ip a12Czz1 a22Czz2 a1Kyy1 a2Kyy2 () a1Kzy1 a2Kzy2 () a12Kzy1 a22Kzy2 a12Kyy1 a22Kyy2 2Id (RHS unbalance forcing terms) RHSm 2 0 0 0 0 RHS 0.09791 0.09791 0 0 0.09791 0.09791 0 0

PAGE 75

65 A 65124.42171 182.72 0 0 1466.0772 523.599 0 0 295.2 65808.76171 0 0 743.51058 3665.193 0 0 0 0 1222.46886 106.272 0 0 923.62864 11209.95161 0 0 65.7792 1099.28766 0 0 11289.11978 1319.46948 1466.0772 523.599 0 0 65124.42171 182.72 0 0 743.51058 3665.193 0 0 295.2 65808.76171 0 0 0 0 923.62864 11209.95161 0 0 1099.28766 106.272 0 0 11289.11978 1319.46948 0 0 65.7792 1345.65006 qA1 RH S Bearing Orbits Bearing #1 V1cq0a1q 3 W1cq1a1q 2 V1sq4a1q 7 W1sq5a1q 6 q 1.46054 106 1.5494 106 0 0 1.51235106 1.39413 106 0 0 Bearing #2 V2cq0a2q 3 W2cq1a2q 2 V2sq4a2q 7 W2sq5a2q 6

PAGE 76

66 2 10 6 1 10 601 10 62 10 6 2 10 6 1 10 61 10 62 10 6 shaft displacement in y directionshaft displacement in z directionW1t () V1t () PLOT BEARING ORBITS T 2 T0.00012 t0 T 200 T Bearing #1 Bearing #2 V1t ()V1ccos t V1ssin t V2t ()V2ccos t V2ssin t W1t ()W1ccos t W1ssin t W2t ()W2ccos t W2ssin t 2 10 6 1 10 601 10 62 10 6 2 10 6 1 10 61 10 62 10 6 shaft displacement in y directionshaft displacement in z directionW2t () V2t ()

PAGE 77

67 Work done per cycle at the bearings Bearing #1 Kd1 Kyz1Kzy1 2 Cm1 Cyy1Czz 1 2 WK1Cir2 Kd1 V1cW1s V1sW1c () Work done by Circulation Force at Bearing #1 WK1Diss Cyy1V1c2W1s2 2Cm1 V1cW1c V1sW1s () Czz1W1c2W1s2 Work done by dissipation forces WK1TotalWK1CirWK1Diss Total work done at the bearing #1 Bearing #2 Kd2 Kyz2Kzy2 2 Cm2 Cyy2Czz 2 2 WK2Cir2 Kd2 V2cW2s V2sW2c () Work done by Circulation Force at Bearing #2 WK2Diss Cyy2V2c2W2s2 2Cm2 V2cW2c V2sW2s () Czz2W2c2W2s2 Work done by dissipation forces WK2TotalWK2CirWK2Diss Total work done at the bearing #2

PAGE 78

68 Bearing #1 Bearing #2 WK1Cir7.737681010 WK2Cir7.737681010 WK1Diss3.56451 108 WK2Diss3.56451 108 WK1Total3.48713 108 WK2Total3.48713 108 N OTE: Total work done at each bearing/cycle must be nega tive (subtracting energy from rotor), for stable operation

PAGE 79

69 APPENDIX D EXAMPLE OF FINDING BEARING PARAMETERS Let's choose two different speed, and different displacement components for each speed. So by assuming all variables, forces can be find by using equation of motion (4-6), where Fy()=m*cos()+n*sin() and Fz()=p*cos()+q*sin(). 210 4 rad/sec a0.0 3 b0.0 1 g0.0 2 h0.0 3 12.110 4 a10.03 6 b10.01 6 g10.02 6 h10.03 6 02 100 2 where = t z1 g1cos h1sin 0.0500.05 0.05 0 0.05 z () y ()z gcos hsin y acos bsin y1 a1cos b1sin 0.0500.05 0.05 0 0.05 z1() y1()Let Kyy100 0 Kzy50 0 lbf/in Cyy1 5 Czy 4 lbf-sec/in Kyz50 0 Kzz90 0 Cyz 4 Czz1 5

PAGE 80

70 Now by using force and displacement compone nts, lets find stiffness and damping coefficients. For each speed, there are set of four e quations, expressed above. Thus for two different speeds, following expression is obtained: As it can be seen, stiffness and damping coe fficients are the same values as assumed. maKyy gKyz b Cyy h Cy z m1a1Kyy g1Kyz b1 1 Cyy h1 1 Cy z nbKyy hKyz a Cyy g Cy z n1b1Kyy h1Kyz a1 1 Cyy g1 1 Cy z paKzy gKzz b Czy h Cz z p1a1Kzy g1Kzz b1 1 Czy h1 1 Cz z qbKzy hKzz a Czy g Cz z q1b1Kzy h1Kzz a1 1 Czy g1 1 Cz z P b a 0 0 h g 0 0 a b 0 0 g h 0 0 0 0 b a 0 0 h g 0 0 a b 0 0 g h T m n p q T1 m n p q m1 n1 p1 q1 P1 b a 0 0 b1 1 a1 1 0 0 h g 0 0 h1 1 g1 1 0 0 a b 0 0 a1 b1 0 0 g h 0 0 g1 h1 0 0 0 0 b a 0 0 b1 1 a1 1 0 0 h g 0 0 h1 1 g1 1 0 0 a b 0 0 a1 b1 0 0 g h0 0 g1 h1 AnsP11T1 Ans 15 4 1000 500 4 15 500 900 Cyy Cyz Kyy Kyz Czy Czz Kzy Kzz

PAGE 81

71 Let's assume force data were read in error, because of noise. Forces in y direction and forces in z direction are read in ten and four percent error, respectively. The result shows that the stiffness and damping coefficients are changed by the same percentage as forces. So it means noise in force affects bearing parameters at the same percentage as in force itself. mm1. 1 nn1. 1 pp1.0 4 qq1.0 4 m1m11. 1 n1n11. 1 p1p11.0 4 q1q11.0 4 P1 b a 0 0 b1 1 a1 1 0 0 h g 0 0 h1 1 g1 1 0 0 a b 0 0 a1 b1 0 0 g h 0 0 g1 h1 0 0 0 0 b a 0 0 b1 1 a1 1 0 0 h g 0 0 h1 1 g1 1 0 0 a b 0 0 a1 b1 0 0 gh 0 0 g1 h1 T1 m n p q m1 n1 p1 q1 AnsP11 T1 Ans 16.5 4.4 1.1103 550 4.16 15.6 520 936

PAGE 82

72 APPENDIX E KISTLER DYNANOMETER

PAGE 83

73

PAGE 84

74 LIST OF REFERENCES [1] Kussul, E. M., Rachkovskij, D. A., Baidyk, T. N., and Talayev, S. A., 1996, “Micromechanical engineering: A basis for the l0w-cost manufacturing for mechanical microdevices using microequi pment,” Journal of Micromechanical Microengineering, vol. 6, pp. 410–425. [2] Masuzawa, T., and Tonshoff, H. K., 1997, “Three-dimensional micromachining by machine tools,” Annals of the CIRP, vol. 46, pp. 621–628. [3] Roy R. Craig, Jr., 1981, “Structural Dynamics,” 1st edition, pp. 430-434. [4] Tiwari, R., Lees, A. W., and Friswell M. I., 2004, “Identif ication of Dynamic Bearing Parameters: A review,” The Shoc k and Vibration Digest, vol. 36, pp. 99124. [5] Mitchell, J.R., Holmes, R., and Ballegooyen, H.V., 1965–66, “Experimental Determination of a Bearing Oil Film S tiffness,” in Proceedings of the 4th Lubrication and Wear Convention, IMechE, Vol. 180, No. 3K, 90–96. [6] Arumugam, P., Swarnamani, S., a nd Prabhu, B. S., 1995, “Experimental Identification of Linearized Oil Film Coe fficients of Cylindrical and Tilting Pad Bearings,” ASME Journal of Engineering for Gas Turbines and Power, Vol. 117, No. 3, 593–599. [7] Glienicke, J., 1966–67, “Experimental Inve stigation of the Stiffness and Damping Coefficients of Turbine Bearings and Thei r Application to Instability Prediction,” Proceedings of IMechE, Vol. 181, No. 3B, 116–129. [8] Morton, P.G., 1971, “Measurement of the Dynamic Characteristics of Large Sleeve Bearing,” ASME Journal of Lubrication Technology, Vol. 93, No.1, 143–150. [9] Someya, T., 1976, “An Investigation into the Spring and Damping Coefficients of the Oil Film in Journal Bearing,” Transact ions of the Japan Society of Mechanical Engineers, Vol. 42, No. 360, 2599–2606. [10] Hisa, S., Matsuura, T., and Someya, T., 1980, “Experiments on the Dynamic Characteristics of Large Scale Journal Bearings,” in Proceedings of the 2nd International Conference on Vibration in Rotating Machinery, IMechE, Cambridge, UK, Paper C284, 223–230.

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75 [11] Sakakida, H., Asatsu, S., and Tasa ki, S., 1992, “The Static and Dynamic Characteristics of 23 Inch (584.2 mm) Diam eter Journal Bearing,” in Proceedings of the 5th Interna tional Conference on Vibration in Rotating Machinery, IMechE, Bath, UK, Paper C432/057, 351–358. [12] Hagg, A.C., and Sankey, G.O., 1956, “Some Dynamic Properties of Oil-Film Journal Bearings with Reference to the Unbalance Vibration of Rotors,” ASME Journal of Applied Mechanics, Vol. 78, No. 2, 302–305. [13] Hagg, A.C., and Sankey, G.O., 1958, “Ela stic and Damping Properties of Oil-Film Journal Bearings for Application to U nbalance Vibration Calculations,” ASME Journal of Applied Mechanics, Vol. 80, No. 1, 141–143. [14] Stone, J.M., and Underwood, A.F., 1947, “Load-Carrying Capacity of Journal Bearing,” SAE Quarterly Transactions, Vol. 1, No. 1, 56–70. [15] Duffin, S., and Johnson, B.T., 1966–67, “Some Experimental and Theoretical Studies of Journal Bearings for Large Tu rbine-Generator Sets,” Proceedings of IMechE, Vol. 181, Part 3B, 89–97. [16] Murphy, B.T., and Wagner, M.N., 1991, “Measurement of Rotordynamic Coefficients for a Hydrostatic Radial Bear ing,” ASME Journal of Tribology, Vol. 113, No. 3, 518–525. [17] Adams, M.L., Sawicki, J.T., and Capald i, R.J., 1992, “Experimental Determination of Hydrostatic Journal Bearing Rotordynami c Coefficients,” in Proceedings of the 5th International Conference on Vibration in Rotating Machinery, IMechE, Bath, UK, Paper C432/145, 365–374. [18] Sawicki, J.T., Capaldi, R.J., and Ad ams M.L., 1997, “Experimental and Theoretical Rotordynamic Characteristics of a Hybrid Journal Bearing,” ASME Journal of Tribology, Vol. 119, No. 1, 132–142. [19] Downham, E., and Woods, R., 1971, “T he Rationale of Monitoring Vibration on Rotating Machinery in Continuously Opera ting Process Plant,” in ASME Vibration Conference, Paper No. 71-Vibr-96. [20] Nordmann, R., 1975, “Identification of Stiffness and Damping Coefficients of Journal Bearings by Means of the Impact Method,” in Dynamics of Rotors: Stability and System Identification, O. Mahrenholtz, ed., Springer-Verlag, New York, 395–409. [21] Nordmann, R., and Schllhorn, K., 1980, “Identification of Stiffness and Damping Coefficients of Journal Bearings by Mean s of the Impact Method,” in Proceedings of the 2nd International C onference on Vibration in Rotating Machinery, IMechE, Cambridge, UK, Paper C285, 231–238.

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76 [22] Zhang, Y.Y., Xie, Y.B., and Qiu, D .M., 1992a, “Identification of Linearized OilFilm Coefficients in a Flexible Rotor-Bearing System, Part I: Model and Simulation,” Journal of Sound and Vibration, Vol. 152, No. 3, 531–547. [23] Adams, M.L., and Rashidi, M, 1985, “O n the Use of Rotor-Bearing Instability Thresholds to Accurately Measure B earing Rotordynamic Properties,” ASME Journal of Vibration, Acoustics, Stress, a nd Reliability in Design, Vol. 107, No. 4, 404–409. [24] Lee, A.C., and Shih, Y.P., 1996, “Iden tification of the Unba lance Distribution and Dynamic Characteristics of B earings in Flexible Rotors,” Proceedings of IMechE, Part C: Journal of Mechanical Engin eering Science, Vol. 210, No. 5, 409–432. [25] Lund, J.W., Cheng, H.L., and Pan, H.T., 1965, “Rotor-Bearing Dynamic Design Technology, Part 3,” Mechanical Technology.

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77 BIOGRAPHICAL SKETCH The author of the thesis was born on July 2, 1981, in Azerbaijan. He grew up in Azerbaijan. In 1998 he moved to Turkey to attend Middle east T echnical University, where he received his Bachelor of Scien ce in Mechanical Engineering. In 2004 he traveled to United States of America for th e pursuit of a master’s degree in mechanical engineering at University of Fl orida. He is planning to comple te the degree of Master of Science in Mechanical Engi neering in August 2006.


Permanent Link: http://ufdc.ufl.edu/UFE0015756/00001

Material Information

Title: Rotor-Bearing System Dynamics of a High-Speed Micro End Mill Spindle
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0015756:00001

Permanent Link: http://ufdc.ufl.edu/UFE0015756/00001

Material Information

Title: Rotor-Bearing System Dynamics of a High-Speed Micro End Mill Spindle
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0015756:00001


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Table of Contents
    Title Page
        Page i
        Page ii
    Acknowledgement
        Page iii
    Table of Contents
        Page iv
        Page v
    List of Tables
        Page vi
    List of Figures
        Page vii
        Page viii
    Abstract
        Page ix
        Page x
    Introduction
        Page 1
        Page 2
        Page 3
    Development of equation of motion of rotor-bearing system and parameter identification
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
    Rotor dynamic analysis of micro spindle
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
    Experimental identification of bearing parameters
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
    Conclusion and recommendations
        Page 50
    Appendices
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
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        Page 70
        Page 71
        Page 72
        Page 73
    References
        Page 74
        Page 75
        Page 76
    Biographical sketch
        Page 77
Full Text












ROTOR-BEARING SYSTEM DYNAMICS OF A HIGH
SPEED MICRO END MILL SPINDLE















By

VUGAR SAMADLI


A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA


2006





























Copyright 2006

by

Vugar Samadli















ACKNOWLEDGMENTS

I would like to first thank my advisor, Dr. N. Arakere, for guiding me through this

work. I feel that I have learned much working on this project and that would not have

been possible without his help. Similarly, the collaboration with Dr. J. Zeigert and Dr. L.

William and their students Scott Payne, Eric Major and Andrew Riggs was fruitful. I

want to thank Scott Payne for his help and it was always a friendly environment to work

with him in his lab.

My company, BP Azerbaijan unit, was a main factor in making this whole effort

possible so I would like to thank them all, and specifically Ralph Ladd, learning and

development coordinator, and Kevin Kennelley, engineering manager.

Lastly, I would like to thank my parents because they were supportive of this

effort and always encouraged my education.
















TABLE OF CONTENTS
page

A C K N O W L E D G M E N T S ................................................................................................. iii

LIST OF TABLES ...................... .................. ........................ ....... vi

L IST O F FIG U R E S .... ...... ...................... ........................ .. ....... .............. vii

ABSTRACT ........ .............. ............. ...... ...................... ix

CHAPTER

1 IN T R O D U C T IO N ............................................................................. .............. ...

2 DEVELOPMENT OF EQUATION OF MOTION OF ROTOR-BEARING
SYSTEM AND PARAMETER IDENTIFICATION....................................4

2.1 Rigid R otor A analysis ..................... .. ...... ................... ........ .......... ...... .
2 .2 F inite E lem ent A naly sis........................ ................................................... ......7
2.3 Identifying Bearing Parameters By Experimental Method ..................................8
2.3.1 M ethods Using Incremental Static Load ..............................................9
2.3.2 M ethods U sing D ynam ic Load ........................................ ..................9
2.3.3 M ethods Using an Excitation Force.................................................. 11
2.3.4 M ethod U sing Unbalance M ass .................................. ............... 14
2.3.5 Methods Using an Impact Hammer .......................................... 16
2.3.6 Methods Using Unknown Excitation ...............................................17

3 ROTOR DYNAMIC ANALYSIS OF MICRO SPINDLE.............. ..................19

3 .1 R igid R otor A n aly sis ................................................................. ....................2 0
3.2 F inite E lem ent A analysis ......................................... .......... ............................... 3 1
3.3 Sum m ary ..................................... .................. ............. ........... 42

4 EXPERIMENTAL IDENTIFICATION OF BEARING PARAMETERS ................44

4 .1 M ethod of M easurem ent............................................................ .....................45
4.2 D design Process .................. .................. ................. ........ .............. .. 48

5 CONCLUSION AND RECOMMENDATIONS ................................................. 50









APPENDIX

A CHRONOLOGICAL LIST OF PAPERS ON THE EXPERIMENTAL
DYNAMIC PARAMETER IDENTIFICATION OF BEARINGS............................51

B GENERAL RIGID ROTOR SOLUTION 1 .................................... ...............55

C GENERAL RIGID ROTOR SOLUTION 2...........................................................62

D EXAMPLE OF FINDING BEARING PARAMETERS................ .............. ....69

E K ISTLER D Y N A N O M E TER ...................................................................................72

L IST O F R E FE R E N C E S ....................................................................... ... ...................74

B IO G R A PH IC A L SK E TCH ..................................................................... ..................77
















LIST OF TABLES

Table page

3-1 Bearing parameters at 500,000 rpm (Q=52360 rad/sec) .......................................23

3-2 Bearing Parameters for different rotor speeds............................... ............... 29
















LIST OF FIGURES

Figure page

1-1 C om m ercial m icro-tool .............................................................. ....................... 3

2-1 Rigid rotor schematic. ................................... ... ....... ................. .5

2-2 A non-floating bearing housing and a rotating journal ................. .... ........... 11

2-3 A floating bearing housing and a fixed rotating shaft....................... ...............12

3-1 M icro-spindle. ..................................................... ................. 20

3-2 Bearings location on spindle. ............................................................................. 20

3-3 Rotor orbits at two bearing supports at 500,000 rpm .........................2...............24

3-4 Rotor orbits at the two bearing supports at 1,000,000 rpm .....................................25

3-5 R otor orbits at the tool end at 500,000 rpm ..................................... .....................26

3-6 Orbit amplitude at 1st, 2nd bearing and tool tip.................................................... 27

3-7 Stiffness, Kyy versus rotor eccentricity. ...................................... ............... 28

3 -8 W h irl m ap ....................................................................... 3 0

3-9 Finite elem ent m odel ........................ ..... .......... ........... .... .... .. ........ .... 31

3-10 Rigid body mode 1, 107758 rpm ............................ ....... ...............31

3-11 Rotor analysis at 115298 rpm. A) Rigid body mode 2, B) Potential energy
distribution. .......................................... ............................ 32

3-12 Rotor analysis at 2,042,349 rpm. A) Flexural mode 1, B) potential energy
distribution. .......................................... ............................ 32

3-13 Flexural m ode 2, 4,955,776 rpm ........................................ ........................ 33

3-14 Critical speed m ap. .................................... .. .. ...... ...............33









3-15 1st bearing. A) and B) Unbalance response, C) Amplitude and phase lag, D)
N yquist plot for displacem ent. ............................................................................ 34

3-16 2nd bearing. A) and B) Unbalance response, C) Amplitude and phase lag, D)
N yquist plot for displacem ent. ............................................................................ 36

3-17 Tool tip. A) Unbalance response, B) Amplitude and phase lag, C) Nyquist plot
for displaced ent. ..................................................................... 38

3-18 Shaft orbits. A), B), C), D), E), F) are shaft orbits as a function of speed ..............39

3-19 Stability Map (Note: Negative log decrements indicate instability) ......................42

3-20 Orbit amplitude at 1st, 2nd bearing and tool tip, e=0.1 ................................................43

4-1 Flexure supported ball bearing: 1) Flexure, 2) Ball bearing. ..................................45

4-2 A chassis .......................................................................................................... 45

4-3 Schem atic of the test rig. ............................................................................. ..... 46

4-4 Test Setup: 1) Base, 2) Dyno, 3) Chassis, 4) Test bearing, 5) Shaft, 6) Spindle .....48

4-5 Main Chassis: 1) Stator, 2) Displacement Probe, 3) Test Bearing...........................49

4-6 M easurm ent system ........................................................... .. ............... 49















Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

ROTOR-BEARING SYSTEM DYNAMICS OF A HIGH
SPEED MICRO END MILL SPINDLE

By

Vugar Samadli

August 2006

Chair: Nagaraj Arakere
Major Department: Mechanical and Aerospace Engineering

Current micro-scale manufacturing technologies find limited application in a wide

range of high strength engineering materials because of the difficulties encountered in

creating complex three dimensional structures and features. Although milling is one of

the most widely used processes for this type of manufacturing at the macro scale, it has

yet to become an economically viable technology for micro-scale manufacturing. For

optimal chip formation using very small diameter cutters, and to achieve economical

material removal rates combined with good surface finish, high spindle speeds are

needed. In addition, a low runout is desired to prevent premature tool breakage. However,

the lack of suitable spindles capable of achieving rotational speeds in excess of 500,000

rpm coupled with sub-micrometer runout at the tool tip makes micro-scale milling

commercially unviable.









This thesis demonstrates several means of analyzing the rotor dynamic behavior of

a spindle in order to find critical speeds, unbalance response and linear stability margins.

Experimental testing is performed to estimate bearing dynamic behavior at high speed.

The results of this study provide parameters for bearing stiffness and damping,

bearing span and balancing limits to achieve sub-micrometer runout of tool tip for speeds

upto 1 million rpm.














CHAPTER 1
INTRODUCTION

The technology development in the field of miniaturization has become a global

phenomenon. Its impact is far and widespread across a broad application domain that

encompasses many diverse fields and industries, such as telecommunications, portable

consumer electronics, defense, and biomedical. The perfect example is the field of

computers where modem computers which possess greater processing power and can fit

under a desk or on a lap have replaced the bulky computers of the past such as the

ENIAC (electronic numerical integrator and computer) which once filled large rooms. In

recent times, more and more attention is being paid to the issues involved in the design,

development, operation, and analysis of the equipment and processes of manufacturing

micro components since the global trend toward the increased integration of miniaturized

technology into society has gained enormous momentum. Currently, common techniques

utilized in the fabrication of micro-components are based on the techniques developed for

the silicon wafer processing industry. Unfortunately these processes are limited to

production of simple planar geometries in a narrow range of material and are cost

effective only in large volume [1]. Even though non traditional fabrication methods, such

as focused ion beam machining, laser machining, and electrodischarge machining, are

capable of producing high-precision micro-components, they have limited potential as

mass production techniques due to the high initial cost, poor productivity, and limited

material selection [2]. Micro milling has the potential to fabricate micro components and

is capable of machining complex 3D shapes from wide variety of shapes and materials.









The objective of this research is to develop a micro-milling spindle which will rotate at

over 500,000 rpm range with sub-micrometer runout, and thus become a commercially

usable and cost effective manufacturing technology. Most machine tools such as lathes,

milling machines, and all types of grinding machines, use a spindle or an axis of rotation

for positioning work pieces or tools or machining parts and thus a large part of their

accuracy can be attributed to the spindle. Consequently, the accuracy of the spindles used

in their design directly influences the accuracy of the entire machine and thus can be

considered as one of the most important components in the overall accuracy and

operation of a machine tool.

Most commercial micro-tools have a 1/8th inch diameter shank (see Figure 1-1).

This must be of utmost importance when designing the micro spindle. Another functional

requirement is the ease of tool changing with minimal time and effort. The only viable

way to meet the above design requirements while still obtaining satisfactory runout is to

concentrate on designs incorporating the use of tool shank itself as the spindle shaft. To

achieve desired performance, the following three functions must be satisfied:

1. Bearing subsystem. The bearing system must be so designed that it meets the
following requirements. Firstly and fore mostly, it must be capable of supporting
the tool shank without causing excessive runout. Also, it must support both radial
and axial loads, and support rapid tool changes. Flexure Pivot Tilting Pad Bearings
(FPTPB) are being studied as a potential bearing subsystem.

2. Drive subsystem. The tool drive system must be able to drive the tool at the
required speed with enough torque and power to perform the desired machining
operations. Also it should not introduce disturbance forces that cause excessive tool
point runout. These requirements make air turbine drive as a potential drive
subsystem. The system would incorporate the turbine blades directly into the tool
shank of the micro-tool.

3. Monitoring subsystem. Theoretical and scientific understanding of micro-milling
requires monitoring and recording of cutting forces. However, in the measurement
bandwidth (that is 1,000,000 rpm with a 2-flute cutter and tooth passing frequency








of 33 KHz), the force measurement is extremely difficult because of the high
frequencies encountered even though the cutting forces are low.

The goals of this project are the following:

* To extend the capability to model and predict rotordynamics and bearing behavior at
small sizes and high speeds.

* Development of a procedure for identification of dynamic stiffness and damping
coefficients for the bearing.


Figure 1-1. Commercial micro-tool


Irmsllslllllll~














CHAPTER 2
DEVELOPMENT OF EQUATION OF MOTION OF ROTOR-BEARING SYSTEM
AND PARAMETER IDENTIFICATION

It is of utmost importance in many companies that not only the operation should be

uninterrupted and reliable but also it should be carried out at high power and high speed.

Another vital requirement is the accurate prediction and control of the dynamic behavior

(unbalance response, critical speeds and instability). These factors were the motivations

for this research wherein rigid rotor analysis and finite element analysis was used to

investigate bearing coefficient parameters and the rotordynamics of micro spindle. Both

the rigid rotor analysis and finite element analysis have been performed simultaneously.

The tungsten carbide spindle has a first bending or flexure natural frequency of 2.2

million rpm for a bearing span of 1 inch. The spindle operating speeds are expected to be

about 500,000 rpm. Hence rigid rotor analysis can be justified. Finally, the experimental

setup was designed to find bearing parameters which were compared with analytical

results.

2.1 Rigid Rotor Analysis

In order to get generalized rotor dynamic models, the Jeffcott rotor is extended to a

four degree of freedom rigid rotor system as shown in the schematic diagram of Figure 2-

1. The four coordinates, which are the two geometric center translations (V, W) and the

two rotation angles (B, F) describe the rotor configuration relative to the fixed reference

(X, Y, Z). Bearing 1 and bearing 2 are located at an axial distance ai and a2 from the









center of mass, respectively. Both these distances are defined as positive in the plus X

direction. The rotor configuration is always defined so that ai is positive.


TB


z
C

S\m b
p=Ot
Y
V


Figure 2-1. Rigid rotor schematic.

Where


(a, b, c) geometric center body reference
(1r, ) eccentricity components
(ip) spin angle = Qt
(Q) constant spin frequency

Vm (t)= V + (q cos ( s sin ()
Wm(t) = W + (; cos p + 7 sin ()

The angular rate of the rigid body is

)a = FsinB
0)b = P cos B sin Qt + B cos Qt
oc) F cos B cos Qt B sin Qt


(2-1)


(2-2)









and the kinetic energy of the rigid body is
2 +W2) 2 2+)2]
T=-m( 2+Wm2)+ Ipa2+I 2+c2) (2-3)
2 2 L(2-3)

By considering the variational work of the bearing forces, they are included in the

equation of motion. The bearing force is a function of lateral shaft translations and

velocities at the bearing location.

Fy = Fy(V,W,V,W)
(2-4)
Fz = F (V,W,V,W)

Upon Taylor's series expansion of eq. (2-4) about the origin, the force

components in eq. (2-4) are approximated by their corresponding linear forms. At the ith

typical bearing, forces are expressed in the following equation:

FY kIYY kiYZ ciYY ci YZ
FZ ki ki _cZZ i ciZY ciZZ.I (2-5a)

or
F. = -_k. c7
= -ki -t i (2-5b)

where

S0 0 -a
=O 1 a. 0 q- A
i (2-6)

7T=[v W B F] (

The variational work done by bearing forces on the rotor is given by the following

expression

2_ 4
gWk = Fi S= Y Qk 9qk
k =1 k= (2-8)

where Qk represents the generalized bearing forces.









Lagrange's equations are of the following form:

d OT OT
( )- =
k k k=1,2,3,4 (2-9)

Using the above set of equations and inserting in eq (2-9) the following set of rigid

rotor equations of motion has been obtained:

MY + (C QG)-K + = 0 (2-10)

where

m 0 0 0 0 0 0
0 m 0 0 0 0 0 0
0 0 Id o0 = 0 0 0 -Ip
0 0 0 Id 0 0 I 0

2 2 T
iK 1 i=1
K I k



Q = m2 cos ftt + m sin t
0 0

(2-11)
2.2 Finite Element Analysis

Typically, it is not possible to obtain analytical solutions for problems involving

complicated geometries, loadings and material properties. Based on the study and

inspection of various approaches available for modeling, one of the most appropriate

methods for modeling of high-speed micro spindle is the FEA, finite element method. It

is also the only feasible type of computer simulation available for this purpose. The finite

element method is generally a numerical method used for solving engineering and

mathematical physics problems. The following steps are used in the FEA for dynamic

response solution [3]:









* Form element stiffness matrix.

* Form element mass matrix.

* Assemble system stiffness matrix and incorporate constraints.

* Assemble system mass matrix and incorporate constraints.

* Solve eigenproblem and obtain a vector of frequencies and mode shapes.

* Form excitation vector in physical coordinates.

2.3 Identifying Bearing Parameters By Experimental Method

The estimation of the dynamic bearing characteristics using theoretical methods

usually results in an error in the prediction of the dynamic behavior of rotor-bearing

systems. Reliable estimates of the bearing operating condition in actual test conditions

are difficult to obtain and, therefore to reduce the discrepancy between the measurements

and the prediction, physically meaningful and accurate parameter identification is

required in actual test conditions. There are some similarities between various

experimental methods for the dynamic characterization of rolling element bearings, fluid-

film bearings and magnetic bearings. These methods require forces as input signals and

displacement/velocities/accelerations of the dynamic system to be measured are usually

the output signals, and input-output relationships are used to determine the unknown

parameters of the system models. There are a lot of identification techniques of bearing

parameters, which are based on methods used to excite the system [4], such as the

following:

1. Methods using Incremental Static Load

2. Methods using Dynamic Load

3. Methods using an Excited Load

4. Method using Unbalance Mass









5. Methods using an Impact Hammer

6. Methods using Unknown Excitation

Appendix A summarizes the source material on the experimental dynamic

parameter identification of bearings.

2.3.1 Methods Using Incremental Static Load

Mitchell et al. (1965-66) [5] performed experiments to incrementally load the

bearing and measuring the change in position, and obtained the four stiffness coefficients

of fluid-film bearings. They obtained the following simple relationships using the

influence coefficient approach to

kyy = azz / kyz = -ayz / 7
(2-12)
kzy = -azy /7 kzz= ayy / Y

where

S= ayyazz ayzzy

ayy = l / AFy azy = zl / AFy
(2-13)
ayz = Y2 / AFz azz = z2 / AFz
Here yi and zi are displacements of the journal center from its static equilibrium

position in vertical and horizontal directions respectively, on the application of a static

incremental load AFy in the vertical direction; and y2 and z2 are displacements

corresponding to a static incremental load AFz in the horizontal direction. This method

can be applied to any type of bearing since the estimation of stiffness requires the

establishment of a relationship between the force and the corresponding displacement.

2.3.2 Methods Using Dynamic Load

Dynamic load methods have been the most researched and widely used in the

identification of dynamic bearing parameters in the last 45 years [4]. Their major









advantages are that they can be readily implemented on a real machine and the excitation

can be applied either to the journal or to the bearing housing depending on practical

constraints.

For the rigid rotor case, when the excitation is applied to the journal (Figure 2-2),

the fluid-film dynamic equation can be written as

myy my, zY ~Cyy cyz~ +Ikyy kyz] l(Y=fy-m + YB (2-14)
mzy mzz [ czy czz J kzy kzz fz -m( +YB)

where m is the mass of the journal, y and z represent the motion of the journal center

from its equilibrium position relative to the bearing center, and yB and ZB are the

components of the absolute displacement of the bearing center in vertical and horizontal

directions, respectively. In this case, the origin of the coordinate system is assumed to be

at the static equilibrium position, so that gravity does not appear explicitly in the equation

of motion. There will be one equation of this form for each of the bearings and the terms

yB, ZB represent the motion of the supporting structure. For the case of a rigid rotor with

bearings on a rigid support, equation (2-14) can be expressed in the form

MBq +CBq+KBq = f -MRq (2-15)

The subscripts R and B refer to the rotor and bearings, respectively. On collecting

the terms together, we get

(MB +MR)q+CBq+KBq= f (2-16)

The overall system mass, damping and stiffness matrices can be formed by adding

the separate contributions of the bearings and rotor in equation (2-16). This form was

used by Arumugam et al. (1995) [6] to extract KB and CB in terms of the known and

measurable quantities such as the rotor model, forcing and corresponding response. The









sinusoidal response of a rotor at speed Q is studied using the modified form of this

equation (2-16), and the response is of the form

q = QejQt
The governing equation of motion is given by

[-M_ 2 +j2C+K Q=Fu= [Z(Q)]Q (2-17)

where [Z(Q)] is the dynamic stiffness matrix, Fu is the unbalance force, and Q is

the rotational frequency of the rotor.


Non floating
bearing housing
Fluid



Journal fz(t)







ffy(t)
Figure 2-2. A non-floating bearing housing and a rotating journal.

2.3.3 Methods Using an Excitation Force

The application of a calibrated force to the journal can only rarely be applied in

practical situations. Glienicke (1966-67) [7] adopted the technique of exciting the

floating bearing bush (housing) sinusoidally in two mutually perpendicular directions

(Figure 2-3) and measuring the amplitude and phase of the resulting motions in each

case. The stiffness and damping coefficients were then calculated from the frequency-

domain equations.









Morton (1971) [8] devised a measurement using the receptance coefficient method

procedure for the estimation of the dynamic bearing characteristics. He excited the

lightweight floating bearing bush by using very low forcing frequencies, co (10 and 15

Hz). Assuming the inertia force due the fluid film and bearing housing masses to be

negligible, and for sinusoidal motion, equation (2-14) may be written as

~zyy zyz fY SFi
Zzy Zzz Z Fz

with (2-18)
z = k + joc
where Y and Z are complex displacements and Fy and Fz are complex forces in the

vertical and horizontal directions, respectively. In equation (2-18) k represents the

effective bearing stiffness coefficient, since while estimating the bearing dynamic

stiffness, z, the fluid-film added-mass and journal mass effects contribute to the real part

of the dynamic stiffness and the effective stiffness is estimated.


Floating bearing
bush
Fluid








fz(fy(tt)
Journal










Figure 2-3. A floating bearing housing and a fixed rotating shaft.









Someya (1976) [9], Hisa et al. (1980) [10] and Sakakida et al. (1992) [11]

identified the dynamic coefficients of large-scale journal bearings by using simultaneous

sinusoidal excitations on the bearing at two different frequencies and measuring the

corresponding displacement responses. This is called the two-directional compound

sinusoidal excitation method and all eight bearing dynamic coefficients can be obtained

from a single test. When the journal is vibrating about the equilibrium position in a

bearing, the dynamic component of the reaction force of the fluid film can be expressed

by equation (2-18). If the excitation force and dynamic displacement are measured at two

different excitation frequencies under the same static state of equilibrium and ignoring

the fluid-film added-mass effects equation (2-18) can be solved for the eight unknown

coefficients as

kyy 2
Y1 Z1 Jc Y J Z kyz IFl mBl YB1

SZ j2Y2 jo2Z2 cyy F -mBc2 YB2
Cyz
(2-19)
kzy
F Zl jw1 j1 1Z1 k zl B 1 B1
Y2 Z2 o2Y2 j2Z2_ czy Fz2 mB 2B2Z
Czz
where co is the external excitation frequency and the subscripts 1 and 2 represent the

measurements corresponding to two different excitation frequencies. Since equation (2-

19) corresponds to eight real equations, the bearing dynamic coefficients can be obtained

on substituting the measured values of the complex quantities Fy, Fz, Y, Z, YB and ZB,









2.3.4 Method Using Unbalance Mass

From a practical point of view, the simplest method of excitation is to use an

unbalance force as this requires no sophisticated equipment for the excitation, and it is

relatively easy to identify the rotational speed dependency of the bearing dynamic

characteristics. However, the disadvantage is that information is limited to the

synchronous response. Nevertheless, since this is the predominant requirement, the

application of forces due to unbalance is extremely useful. Hagg and Sankey (1956,

1958) [12-13] were among the first to use the unbalance force only for experimentally

measuring the oil-film elasticity and damping for the case of a full journal bearing. They

used the experimental measurement technique of Stone and Underwood (1947) [14] in

which they used the vibration diagram to measure the vibration amplitude and phase of

the journal motion relative to the bearing housing. The direct stiffness and damping

coefficients were only considered along the principal directions in their study (i.e., major

and minor axes of the journal elliptical orbit).

The measured unbalance response whirl orbit gives the stiffness and damping

coefficients. However, the results represent some form of effective rotor-bearing

coefficients and not the true film coefficients as the cross-coupled coefficients are

ignored. Duffin and Johnson (1966-67) [15] employed a similar approach to that of Hagg

and Sankey to identify bearing dynamic coefficients of large journal bearings. They

proposed an iterative procedure to calculate all eight coefficients. Four equations can be

written relating the measured values of displacement amplitude and phase Y, Z, py and

pz, together with the known value of the unbalance force, F, and four stiffness

coefficients (obtained from static locus curve method; Mitchell et al., 1965-66) used to

obtain the four unknown damping coefficients. This allows the solution of two sets of









simultaneous equations having two equations in each set. The results had a greater

accuracy than the method (Glienecke, 1966-67) in which two sets of four simultaneous

equations were used to obtain the stiffness and damping coefficients.

Murphy and Wagner (1991) [16] presented a method using a synchronously

orbiting intentionally eccentric journal as the sole source of excitation for the extraction

of stiffness and damping coefficients for hydrostatic bearings. The relative whirl orbits

across the fluid film were made to be elliptic with asymmetric stiffness in the test

bearing's supporting structure. The study considered the bearing coefficients to be skew-

symmetric and the elliptic nature was utilized in the data reduction process. Adams et al.

(1992) [17] and Sawicki et al. (1997) [18] utilized experimentally measured responses

corresponding to at least three discrete orbital frequencies, for a given operating

condition to obtain twelve dynamic coefficients (stiffness, damping and added-mass) of

hydrostatic and hybrid journal bearings, respectively. They assumed that the bearing

dynamic coefficients are independent of frequency of excitation. The estimation equation

was similar to equation (2-19) except the rotor mass was ignored and fluid film added-

mass coefficients were considered. A confidence in the measurements was obtained by

employing dual piezoelectric/strain gage load/displacement measuring systems. The

difference between these two sets of dynamic force measurements was typically less than

2%. The test spindle (double-spool-shaft) had a provision for a circular orbit motion of

adjustable magnitude with independent control over spin speed, orbit frequency and whirl

direction. The least-squares linear regression fit to all frequency data points over the

tested frequency range was used to obtain the bearing dynamic coefficients.









2.3.5 Methods Using an Impact Hammer

Until the early 1970s, the common method to obtain the dynamic characteristics of

systems involved using sinusoidal excitation [4]. Downham and Woods (1971) [19]

proposed a technique using a pendulum hammer to apply an impulsive force to a machine

structure. Although they were interested in vibration monitoring rather than the

determination of bearing coefficients, their work led to the idea that impulse testing could

be capable of exciting all the modes of a linear system.

Nordmann (1975) [20] and Nordmann and Schollhom (1980) [21] identified the

stiffness and damping coefficients of journal bearings by modal testing by means of the

impact method wherein, a rigid rotor, running in journal bearings was excited by an

impact hammer. Two independent impacts first in the vertical direction and then in the

horizontal direction were applied to the rotor and the corresponding responses were

measured. A transformation of input signals (forces) and output signals (displacements of

the rotor) into the frequency domain was then carried out and the four complex FRFs

were calculated. The bearing dynamic parameters were assumed to be independent of the

frequency of excitation. The analytical FRFs, which depend on the bearing dynamic

coefficients, were fitted to the measured FRFs. An iterative fitting process results in the

stiffness and damping coefficients.

Zhang et al. (1992a) [22] fitted the measured FRFs to those calculated theoretically

so as to obtain the eight bearing dynamic coefficients. They also quantitatively analyzed

the influence of noise and measurement errors on the estimation in order to improve the

accuracy of estimated bearing dynamic coefficients. They used a half-sinusoid impulse

excitation and with a different level of noise added to the resulting response to test their

algorithm and averaged the frequency responses to reduce the uncertainty due to noise in









the response. To reduce the effect of phase-measurement errors, they defined an error

function using just the amplitude components of the FRFs. This non-linear objective

function was then used to estimate the bearing parameters by an iterative procedure. It

was also demonstrated by then that it was necessary to remove the unbalance response

from the signal when an impact test was used, especially at higher speeds of operation,

and they concluded this to be the reason for the scatter in the results by impact excitation,

as compared to the discrete frequency harmonic excitation.

This method is time-consuming though since impact tests have to be conducted for

each rotor speed at which bearing dynamic parameters are desired. In general, the amount

of information that can be extracted from a single impulse test is limited as the governing

equations for a bearing include coupling between the two perpendicular directions. Errors

in the estimation will be greater for the case when bearing dynamic coefficients are

functions of external excitation frequency as compared to the estimation from functions

of rotor rotational frequency. Also impulse testing may lead to underestimation of input

forces when applied to a rotating shaft as a result of the generation of friction-related

tangential force components and, further, is prone to poor signal-to-noise ratios because

of the high crest factor.

2.3.6 Methods Using Unknown Excitation

In industrial machinery, the application of a calibrated force is difficult to apply.

Due to residual unbalance, misalignment, rubbing between the rotor and stator,

aerodynamic forces, oil whirl, oil whip and instability, inherent forces are present in the

system and these render the assessment of the forcing impossible. Adams and Rashidi

(1985) [23] used the static loading method to measure bearing stiffness coefficients and

determined orbital motion at an adjustable threshold speed to extract bearing damping









coefficients by inverting the associated eigenproblem. The approach stems from the

physical requirement for an exact internal energy balance between positive and negative

damping influences at an instability threshold. The approach was illustrated by simulation

and does not require the measurement of dynamic forces.

Lee and Shih (1996) [24] found rotor parameters including bearing dynamic

coefficients, shaft unbalance distribution and disk eccentricity in flexible rotors by

presenting an estimation procedure based on the transfer matrix method. The relations

between measured response data and the known system parameters were used to

formulate the normal equations. The parameter estimation was then performed using the

least squares method by assuming that the bearing dynamic coefficients were constant at

close spin speeds.














CHAPTER 3
ROTOR DYNAMIC ANALYSIS OF MICRO SPINDLE

The aim of this project is to rotate a spindle supported by air bearings at up to

500,000 rpm, with sub-micrometer runout. The 1/8th inch diameter tool shank is used as

a spindle shaft. As mentioned before, the only viable way to obtain satisfactory runout

was to use the tool shank itself as the spindle shaft. An air turbine is used as a driving

system for the spindle. Thus, the only viable way to assemble air turbine is to

manufacture the turbine integral with the spindle, which is shown in Figure 3-1. Also

from the practicality point of view the micro-spindle must accept a variety of tools with

minimal time and effort required for tool change. Rotordynamics of high-speed flexible

shafts is influenced by the complex interaction between the unbalance forces, bearing

stiffness and damping, inertial properties of the rotor, gyroscopic stiffening effects,

aerodynamic coupling, and speed-dependent system critical speeds. For stable high-speed

operations, bearings must be designed with the appropriate stiffness and damping

properties, selected on the basis of a detailed rotordynamic analysis of the rotor system.

The two types of rotor dynamic analyses that are used for high-speed thin spindle are

rigid rotor analysis and finite element analysis. The dynamic behavior of a spindle is

analyzed in order to find critical speeds, unbalance response and linear stability margins

by these methods.



























Figure 3-1. Micro-spindle.

3.1 Rigid Rotor Analysis

Rigid rotor analysis was initially used to get the rotor unbalance response. The air

bearings were located on either side of the center of mass, as shown in Figure 3-2. In

addition, center of mass is found by solid model ProE software.


38.1 mm
34.81 mm
32.28 mm
26.18 mm
18.1 mm







brg#1 al a2 brg#2


Figure 3-2. Bearings location on spindle.









The rigid rotor has 4 degrees of freedom (DOF) represented by two displacements

(V, W) and two rotations (B, F) of the center of mass. The following equation,

(derivation can be found in chapter 2), was used for a rigid rotor subjected to unbalance.

Mw2+ (C- QG)q + Kq = (2-1)

where M C, G, K are mass, damping, gyroscopic and stiffness matrices, respectively,

Q is the force vector. Expressions of these matrices can be found in chapter 2. Q is

constant spin frequency.

In order to use equation (2-1) in the rotor orbit analysis, following procedure is

applied. From chapter 2, it is known that displacement vector for 4 DOF is the following:

ST=[V W B F]

The shaft unbalance leads to harmonic synchronous excitation. Hence the

displacement or response vector can be expressed as the following:

vc 's
We Ws
= B cos(Qt) + sin(Qt)

tC Is 1 (3-1)

As a result, first and second derivatives will have the following forms, respectively.



q= -Q c sin(fQt)+ Ws cos(Qt)
Bc (3-2)
FcJ FSJ (3-2)


=-n2


(3-3)










After substituting for MA, C, G, K, Q into equation (2-1) using (3-1), (3-2), (3-3)

and rearranging sine and cosine terms, and using harmonic balance, the following

expression can been obtained:


k1 y+k2 yy- 2
1 2
k +k 2))

1l + 2k2zy
-(alk yy+a2k2yy)
-(1 2+ C2 )



-Qalc I +a2c 2z)

(alc y +ca2c2 )



c1 +c2y)

-clzy + C2zy)

(alclZY + a2c2 )



k lyy + k 2yy 2

k lzy +k 2yy
alk yI + a2k yy

-(alk1 2 )
-(alk ^+2^y


k1 +k2 yz
k zz +k 2= n22




1 2
-(alk z +a2k 2z)





-cl +c2z)
-Cclzz +2zz)

-(alc zz +a2c 2)








(c ly + c 2)
(alcl + a2c2 )

-Q(alc I +a2c 2)



k Z +k Z -m2
k lzz + k2zz m12

alkzz +a2k2zz

-(alklz +a2k 2,)


alk zz +a2k2 zz
a1 k +ya22k2 zz-2Id

1 22
-alkc l +a2c 2z)

Q(alcl +a2c2 z)
1 2
alc jy, +a2c 2z)
1 2
(al 2cIz+a2 2c 2z) 2



.(alclIyz + a2c2z )

Q(alclz +a2c2 )
21 222
"a, 2 Cyy+a2 c zz)

21 22
-Q(a c z + a2 c 2z)
1 2




a2kly + 222zz 2

-a2k1z r22
a1k z +a2k )


-(ak y+a2k2y)
1 2
-(alk z +a2k z)

-(a12kl +a22 k 2 )
a2k1 2k2 ,_2I
a1 k zz +a2 zz -




Cal2 1c ,zy +a22C2ZY) +2Ip

-"Xalclzz +a2c2zz)
1 2







--acc yy +a2c 2)
1 2

-(alc zy +a2c 2y)
-_a2c1 + a22c2zy)
21 22
-a1 c z +a2 c Z,)
Q(al c + a2 c ^)

-(alk y +a2k 2)

-(alk z +a2k2zy)

-(al k y +a22k y)

a2klzz+ 22k2 zz 2d


77

0




0
S0
~Q
i1


(3-4)









where al and a2 are the distance of the bearings from center of mass. ky, kyz, kzy, kzz, and

cyy, cyz, czy and czz are stiffness and damping coefficients of each bearing, respectively. Id

and Ip are the polar and diametric inertia, respectively.

The charts from 'Rotor-Bearing Dynamics Design Technology' [25], design

handbook for fluid film type bearings were initially used to evaluate the damping and the

stiffness coefficients. In order to use these charts, bearing length and diameter ratio was

assumed to be two. Mass and gyroscopic matrices were found by hand calculation, which

were later entered in the MathCAD program. The two types of forces on the system are

the unbalance force and the cutting force. As the cutting force is much smaller than the

unbalance force, it was neglected. An unbalance eccentricity of eu=0.000002 inch was

used initially to evaluate the unbalance force. The solution procedure was implemented in

MathCAD to find the unbalance respond at the bearing locations.

The rotor orbits at the two bearing supports at 500,000 rpm for a rotor with a

mass=2.525x10-5 lbf-secA2/in, Ip=4.187x10-6 lbf-sec^2-in, Id=4.894x10-8 lbf-sec^2-in

and an unbalance eccentricity of 0.000002 inch are shown in Figure 3-3. The air bearing

for this configuration has the following parameters, Table3-1.

Table 3-1. Bearing parameters at 500,000 rpm (Q=52360 rad/sec).
Kyyl,2=950.4 lbf/in Cyyl,2=0.021 lbf-sec/in
Kyzl,2=99.1 lbf/in Cyzl,2=-0.004 lbf-sec/in
Kzy1,2=93.46 lbf/in Czy1,2=0.005 lbf-sec/in
Kzz1,2=1050.4 lbf/in Czz,2=0.106 lbf-sec/in












x 106 1st bearing
2.5-6,

2-

1.5




0.6




-0.6



-1.5

-2 -

-2.5
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
V (inch) x 10



x 106 2nd bearing
2.5-



1.5



1c
0.5 -

0 0-

-0.5

-1

-1.5


-2.

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
V (inch) x 106

Figure 3-3. Rotor orbits at two bearing supports at 500,000 rpm.











Even for the 1,000,000 rpm and same tool with same rigid rotor parameters,


displacement of rotor in bearings is small (see figure 3-4).


x 10 1st bearing

2.5-


1.5



0.5

0-

-0.5

-1

-1.5

-2 -

-2.5
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
V (inch) x 10-



x 10i 2nd bearing
2.5-



1.5 -





-0.5


0 //


-1

-1.5

-2 -------

-2.5
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
V (inch) x 10-

Figure 3-4. Rotor orbits at the two bearing supports at 1,000,000 rpm.










One of the most critical places is also tool end, next to the air turbine. Thus, rotor

orbits should be found at tool end. As it can be seen from Figure 3-5, runout in both


directions is small at the tool end (2 x 10-6 inch).


2.65-----------

2.5

1.5



0.6 -

0

-0.5 -

-1

-1.5

-2

-2.5 I I I I I I- I
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
V x 10-6
Figure 3-5. Rotor orbits at the tool end at 500,000 rpm.

The radial clearance was chosen 0.0002 inch. The eccentricity ratio e will be 0.01

(e=e/c) for the chosen unbalance.

Figure 3-6 shows that why figure 3-3 and figure 3-4 is same. It can be seen that


after 500,000 rpm runout is nearly same and it is 2x106 inch. In addition this figure

shows that highest amplitude is between 100,000 rpm and 120,000 rpm. Later, it will be

shown that the rigid body critical speeds are 105,042 rpm and 115,231 rpm.











Orbit Amplitude

9.00E-06
8.00E-06
E 7.00E-06
6.00E-06
S5.00E-06
S4.00E-06
E 3.00E-06
2.00E-06
0 1.00E-06
0.00E+00
-1.00E-06 < 1 0 000 P0 0000 2- Q900
rotor speed (rpm)

brg#1 & 2 -tool tip

Figure 3-6. Orbit amplitude at 1st, 2nd bearing and tool tip.

Linear stability analysis has been performed based on energy dissipated at bearings.

System is stable, if total energy dissipated is negative. Based on this criteria the system is

stable over the entire speed range. For 500,000 rpm, total work (energy dissipated) per

cycle at each bearing is found -4.24253x10-8 (see Appendix B).

Sample MathCAD program of rotor orbits can be found in Appendix B for a given

spindle speed, 500,000rpm.

The bearing analysis code was coupled with the rotordynamics code in MATLAB,

to efficiently explore the bearing/rotor design space. A direct interface to the bearing

code used in XLTiltPadHGB was achieved by writing a MATLAB function. Input

structures, viz: geometry, fluid, dynamic, operating, numerical were used to submit

bearing parameters.

Thus the data plotting and visual inspection of bearing performance was facilitated

by the MATLAB interface. This helped in the determination of factors that determine

stiffness and damping. For example, stiffness as a function of the rotor eccentricity is









plotted in Figure 3-7. In this case, a constant gravity force was applied to the rotor and

the bearing contained four pads.

The bearing stiffness and damping coefficients were passed from the bearing

design code to the rotordynamic code. These coefficients are influenced by the bearing

design parameters, and influence the rotordynamics. The rotor orbits within the bearings

are calculated using the rotordynamic model and these orbits and their characteristics

determine the rotor performance.


x 10 ..... ..... :
7. --

















critical speeds. Thus, rotor performance has been investigated at critical speeds. In order

to find critical speeds, the following procedure has been adopted. It is known that
05. "".. .



Figure 3-7. Stiffness, Kyy versus rotor eccentricity.

One of the most important aspects of research was to analyze rotor performance at

critical speeds. Thus, rotor performance has been investigated at critical speeds. In order

to find critical speeds, the following procedure has been adopted. It is known that

eigenvalues can be found from equation with first order form. However equation of

motion for rotor-bearing system is in second order. So equation (2-1) was modified in

order to find eigenvalues.




0 M [ -M 0
M C 0 K











Slo0 (3-5)


After entering equations (3-5) into (2-1), results in:

M x+K x=X
X(3-6)

In order to find eigenvalues, right hand side of equation (3-6) is set to zero, and a

harmonic solution x = ejt and x = oete is assumed. After rearranging equation (3-

6), the standard E.V.P (eigenvalue problem) of the term Ax = Ax can be set up, as shown

below:


(-jK M*), = A, where A =- (3-7)
0)

In order to find all eight eigenvalues (four of which are complex conjugates of the

other four) for a given spindle speed, a MathCAD program was generated to solve

expression obtained above. The rotor whirl map was finally generated using these

eigenvalues.

The whirl map for a rotor with a mass=2.525x10-4 lbf-sec^2/in, Ip=4.187x10-6 lbf-

secA2-in, Id=4.894x10-8 lbf-sec^2-in and an unbalance eccentricity of 0.000002 inch is

shown in Figure 3-12; wherein the air bearing has the following parameters, Table 3-2.

Table 3-2. Bearing Parameters for different rotor speeds.
Kyyl,2 Kyzl,2 Kzyl,2 Kzzl,2 [lbf/in] Q [rad/sec]
1069.2 99.1 -67.52 1069.2 62831
950.4 99.1 -93.46 1050.4 52000
831.6 105.04 -109.4 831.6 42000
751.9 115.92 -151.28 751.9 32000
700.8 118.8 -118.8 700.8 22000
623.4 178.2 -115.8 623.4 12000
594 190.08 -91.28 594 9500
159.2 137.22 -85.34 159.2 2000










Table 3-2. Continued
Kyyl,2 Kyzl,2 Kzyl,2 Kzzl,2 [lbf/in] Q [rad/sec]
36.7 83.76 -53.76 36.7 500


Cyyl,2 Cyzl,2 Czyl,2 Czzl,2 [lbf-sec/in] Q [rad/sec]
0.015 -0.003 0.004 0.088 62831
0.021 -0.004 0.005 0.106 52000
0.033 -0.005 0.007 0.131 42000
0.053 -0.009 0.01 0.158 32000
0.086 -0.014 0.025 0.23 22000
0.18 -0.042 0.077 0.383 12000
0.287 -0.08 0.129 0.402 9500
0.965 -0.011 0.018 0.322 2000
1.838 -0.011 0.017 0.147 500


whirl map


25000

5 20000
a,

2 15000

S10000
>
C)
* 5000

0


0 10000 20000 30000 40000
rotor speed (rad/sec)


50000 60000 70000


spin/whirl ratio=1 wl - w3

Figure 3-8. Whirl map.

It can be seen from graph (Figure3-8), critical speeds are 11000 rad/sec (105,042

rpm) and 12067 rad/sec (115,231 rpm).

MathCAD program for different air bearing stiffness and damping values, found by


using XLTiltPadHGB, can be found in Appendix C.








3.2 Finite Element Analysis
The rotordynamics analysis of the rotor-bearing system was performed using a

special-purpose code. The model shown in Figure 3-9 has 10 finite elements, 11 nodes,

with each node having 4 degrees of freedom. The rigid body and flexural natural

frequencies and model shapes can be computed from this model. To use finite element

method it is assumed stiffness coefficient fixed and no cross coupling stiffness. The

model parameters are bearing stiffness = 1,000 lb/in (fixed), bearing span = 0.5 in, shaft

dia = 0.125 in, shaft length = 0.925 in and the shaft material is tungsten carbide. In

addition, me = 10 6 oz-in is used in FE model. As a result four critical speeds are found

for the following parameters.


ti t


-) 4


Figure 3-9. Finite element model.
The rigid body critical speeds are 107,758, 115,298, while the flexural critical

speeds are 2,042,349 and 4,955,776 rpm.








Figure 3-10. Rigid body mode 1, 107758 rpm.


LU












SI I I I1


L I
A.

Mode No.= 2, Critical Speed= 115298 rpm = 1921.64 Hz
Potential Energy Distribution (sAf-1)
Overall: Shaft(S)= 0.04%, Bearing(Brg)= 99.96%

M)


Percentage: o
Component: S
No. or Stn: 1


Figure 3-11. Rotor analysis at 115298 rpm. A) Rigid body mode 2, B) Potential energy
distribution.
Critical Speed = 2042349 rpm = 34039.16 Hz


IZi I


I I I I


Figure 3-12. Rotor analysis at 2,042,349 rpm. A) Flexural mode 1, B) potential energy
distribution.









Mode No.= 3, Critical Speed = 2042349 rpm = 34039.16 Hz
Potential Energy Distribution (sA-w1)
Overall: Shaft(S)= 99.94%, Bearing(Brg)= 0.06%


Percentage:
Component:
No. or Stn:


B.

Figure 3-15. Continued

Critical Speed = 4955776 rpm = 82596.27 Hz


I* I


~t-F


I I I -3-


at. *a


Figure 3-13. Flexural mode 2, 4,955,776 rpm.


Bearing Stiffness


Figure 3-14. Critical speed map.







34


The critical speed map in Figure 3-14 shows that the flexural modes are unaffected

by bearing stiffness, while the two rigid body modes increase with bearing stiffness.

There are three critical points on the tool such as; two bearing positions and tool

tip. As a result the following analysis are found for these positions.


Elliptical Orbital Axes
Station: 2, Sub-Station: 1
Peak disp: max amp = 5.3133E-006 at 950000 rpm
Negative (b) indicates backward precession
6.00 E-0 .... .... ........
2 4.BO0E-06 ------------------------ --------- ---------
E 3.60E-06 --------- ----- -- ------- -- ---------- -- ----
2.40E-06 ----------- ------ ---------- ---------- ---------
S1.20E-06 -------------- -----------------L -- ..J-- ---
O.OOE+00 ..... .... ........ ........... .... .... ......
0.00E + 00 '
6.00E-06
8 4.80E-06 ------------ ------- -------
3.60E-06 ------------- ---------------------
E 2.40E-06 -------- ------------------- ---------- ------------
co 1.20E-06 ---- ---------- --------- ---------------
0.00E+00 .. .. ..
0.00E+(3.00E+05 6.00E+05 9.00E+05 1.20E+06 1.50E+06
Rotational Speed (rpm)

A.

Transmitted Force (semi-major axis)
Bearing/Support: 1 at Station: 2
Max Forces = 1.1752 at 1.1E+06 rpm

S 1.50 ----------- ----------- --------------------------





2 0.30 ---------- *---- ------*---------/--------------------------





0.00E+C3.00E+05 6.00E+05 9.00E+05 1.20E+06 1.50E+06
BLL
-o

'f>
I-

E 0 3 0 - - - - - - -

0.00
O.00E+C3.00E+05 6.00E+05 9.00E+05 1.20E+06 1.50E+06
Rotational Speed (rpm)

B.

Figure 3-15. 1st bearing. A) and B) Unbalance response, C) Amplitude and phase lag, D)
Nyquist plot for displacement.











Bode Plot
Station: 2, Sub-Station: 1
probe 1 (x) 0 deg max amp = 5.3133E-006 at 950000 rpm
2 probe 2 (y) 90 deg max amp = 5.3133E-006 at 950000 rpm
a 240 I
a 200 -------------------- -------
CM 160 ---------- ------------------------ ----------.
-J 120 --------------------- --------- -- ---------:

0
o 80 -------------- ---------- -- ---------
4 0 . . . . . . . . . . .
6.00E-06 .......
6 4.80E-06 ----------------- ----- ------ ------
a 3.60E-06 --------------- ------ ----------
2.40E-06 ---------.------ ----------- ------------ -----------
1.20E-06 ---- -------------------- ---------------------
< 0.00 1E 00 .... ..' *** ......... ** ,* ......... ......... .........
O.OOE+(3.00E+05 6.00E+05 9.00E+05 1.20E+06 1.50E+06
Rotational Speed (rpm)

C.

Polar Plot
Station: 2, Sub-Station: 1
Speed range= 50000- 1.3E+006 rpm
probe 1 (x) 0 deg max amp = 5.3133E-006 at 950000 rpm
probe 2 (y) 90 deg max amp = 5.3133E-006 at 950000 rpm
270



-Rotation






1.3E+006 ,'


90 Full Scale = 6E-006 (o-pk)

D.


Figure 3-15. Continued












Elliptical Orbital Axes
Station: 7, Sub-Station: 1
Peak disp: max amp = 1.8734E-005 at 1.3E+006 rpm
Negative (b) indicates backward precession
2.50E-05 ......... ......... ....... ......... .......
2.00E-05 ----------- ---------- ---------- --------- -----------
1.50E-05 --------------------- ---------- --------
1.00E-05 ------------ ---------- -------------------
5.00E-06 ----.................... ......
0.OOE+00 *
2.00E-05 ---- --------- ---------
O .O5 E + -0 F ... . . . . . . . . .. .


2. [ EE-05 E----------- EE---------------------- -----------------------
1.50E-05 -------- ------- -------
5.00E-05 -------
1.00E-05 ....---------- ----------- ---- -------I --------
5.00E-06 ,---------- --^ ---^ -- .----------- ------------- -
0.00E+00 .... ........... ......... ......... i .... .
5.00E-06 '-----~-----


0.00E+(3.00E+05 6.00E+05 9.00E+05 1.20E+06 1.50E

Rotational Speed (rpm)


Transmitted Force (semi-major axis)
Bearing/Support: 2 at Station: 7
Max Forces = 5.1010 at 1.3E+06 rpm
S......... ............. ..... ..... I ... ....


0) ----------- ------------I----------------------- -----------


o ---------- ------------------------ -------- ------------


S------------ --------------------- --- --------- -----------
0 -------------------

0------------- -------.--


0--------- --------------------- -


n
- -
i ~I
i ~I


0.00E+C3.00E+05


6.00E+05 9.00E+05

Rotational Speed (rpm)


1.20E+06


+06


1.50E+06


Figure 3-16. 2nd bearing. A) and B) Unbalance response, C) Amplitude and phase lag, D)
Nyquist plot for displacement.











Bode Plot
Station: 7, Sub-Station: 1
probe 1 (x) 0 deg max amp = 1.8734E-005 at 1.3E+006 rpm
2 probe 2 (y) 90 deg max amp = 1.8734E-005 at 1.3E+006 rpm
a 280 I
61 240 -------------------- ----------- ---------- -----------
200 ---------
160--
S 120 -------- ---------- --------------------
S 8 40



03 2.00E-05 ------------ --------- r---------- n, -----------
_I4- 40 .............. ......... ......... .......
2.50E-05 ......................
1.00E-05 --- --
S1.50E-05 ---------- ---------------------- t .----- ---- ---
1.00E-05 ----------- ------ ------------ ----.............----------
S5.OOE-0 -------- ---------------
< 0.OOE+.00 .. ...... .. .. ..
0.00E+(3.00E+05 6.00E+05 9.00E+05 1.20E+06 1.50E+06

Rotational Speed (rpm)

C.



Polar Plot
Station: 7, Sub-Station: 1
Speed range = 50000- 1.3E+006 rpm
probe 1 (x) 0 deg max amp= 1.8734E-005 at 1.3E+006 rpm
probe 2 (y) 90 deg max amp = 1.8734E-005 at 1.3E+006 rpm
270



S,,,.- Rotation
1.3E+006"5-

180 ------ oo00 o- ------- degree

U -00000

1.3E+00oo


90 Full Scale = 2.5E-005 (o-pk)

D.


Figure 3-16. Continued











Elliptical Orbital Axes
Station: 11, Sub-Station: 1
Peak disp: max amp = 0.00013149 at 1.3E+006 rpm
Negative (b) indicates backward precession
2.00E-04 ,............................ ....... .....
1.60E-04 --------------------- ----------- ---------------------
1.20E-04 ---------------- -----------L-------- -------
8.00E-05 ---------- ---------------------- --------
4.00E-05 ----------- ---------- ----- -------
O.OOE+00 .... .... ...... .. ......... .....
1.80E-04- --
1.50E-04 ----------- L ----------'----------- ----------------------
1.20E-04 ----------- ------------ -------- --- --- -------
1.50E-04 I -
9.00E-05 ----------- ------------
.00E-05 -----------. ....................------------
L 0 5'-i-----------
3.00E-05 .
O.OOE +O 0 :O ......
0.00E+(3.00E+05 6.00E+05 9.00E+05 1.20E+06 1.50E

Rotational Speed (rpm)


-a

a)
C=

013
a.
CL
-c
0-
-- 1.80E
- 1.50E
-- 1.20E
-g 9.00E

E 3.00E
< O.OOE


+06


Bode Plot
Station: 11, Sub-Station: 1
probe 1 (x) 0 deg max amp = 0.00013149 at 1.3E+006 rpm
200 probe 2 (y) 90 deg max amp = 0.00013149 at 1.3E+006 rpm

160 !
120 ---I------- ..--.--------

40 ---------- -- J------------------------
0. -.... ... -.I .. .. ,- -----
A . . I. . . I . ..


E-04
E-04
E-04
E-05
E-05
E-05
:+nn


0.00E+(3.OOE+05 6.00E+05 9.00E+05 1.20E+06

Rotational Speed (rpm)


1.50E+06


Figure 3-17. Tool tip. A) Unbalance response, B) Amplitude and phase lag, C) Nyquist
plot for displacement.


-------- --- --------- j ----------- L ---------- j -------------
---------------------------------- -------- -- ---------

----------- L ---------- ----------- -- ----- ------------
. -. -. -.-.-.-...-.-.-.----- ---- ---- ------ -
*t* * *t*


+00











Polar Plot
Station: 11, Sub-Station: 1
Speed range = 50000 1.3E+006 rpm
probe 1 (x) 0 deg max amp = 0.00013149 at 1.3E+006 rpm
probe 2 (y) 90 deg max amp = 0.00013149 at 1.3E+006 rpm
270



Rotation


180 -d------- degree

S1.3E+




90 Full Scale =0.00018 (o-pk)

C.



Figure 3-17. Continued



Shaft Response due to shaft 1 excitation
Rotor Speed = 50000 rpm, Response FORWARD Precession
Max Orbit at stn 11, substn 1, with a = 5.3751 E-007, b = 5.3751 E-007


Y


Figure 3-18. Shaft orbits. A), B), C), D), E), F) are shaft orbits as a function of speed.







40


Shaft Response due to shaft 1 excitation
Rotor Speed = 100000 rpm, Response FORWARD Precession
Max Orbit at stn 11, substn 1, with a = 1.1509E-006, b = 1.1509E-006


Shaft Response due to shaft 1 excitation
Rotor Speed = 200000 rpm, Response FORWARD Precession
Max Orbit at stn 11, substn 1, with a = 2.747E-006, b = 2.747E-006

Y


Figure 3-18. Continued










Shaft Response due to shaft 1 excitation
Rotor Speed = 300000 rpm, Response FORWARD Precession
Max Orbit at stn 11, substn 1, with a = 5.0653E-006, b = 5.0653E-006


Shaft Response due to shaft 1 excitation
Rotor Speed = 400000 rpm, Response FORWARD Precession
Max Orbit at stn 11, substn 1, with a = 8.2848E-006, b = 8.2848E-006

Y


Figure 3-18. Continued







42


Shaft Response due to shaft 1 excitation
Rotor Speed = 500000 rpm, Response FORWARD Precession
Max Orbit at stn 11, substn 1, with a = 1.2559E-005, b = 1.2559E-005

Y


















F.

Figure 3-18. Continued.



Stability Map


1.80 ------ ---- -----------------------------------------
S 000ooooo00 ooooo60oo0600000oo ooooo
1.50 '-- -2 E- 1-E 6
y s ( i
t 1.20 ...........rotor. analysi -----............................-
S1920------------
^ 0 ------------ I----------- --------------------------;
S0.90 ----------- --
t0 .i '
0.60 ,i

0.30 'J , ", ', ,n n,, n' a
0.OOE+C3.00E+05 6.00E+05 9.00E+05 1.20E+06 1.50E+06
Rotational Speed (rpm)

Figure 3-19. Stability Map (Note: Negative log decrements indicate instability).

3.3 Summary

Rigid body critical speeds (107,758 and 115,298 rpm) found by FEA are very close

to the rotor analysis critical speeds (105,042 and 115,231 rpm) calculated by rigid rotor










analysis. The difference occurs, because in FEA analysis cross coupling stiffness terms

and damping terms are eliminated and stiffness doesn't change with rotor speed.

For rigid rotor analysis eccentricity ratio is calculated e=0.01, which can be


questionable to get this value in practice, and it leads mx e. = 1.22 10-8 oz-in. So


unbalance response analysis is performed again with e=0.1. Figure 3-20 shows that even

with this eccentricity ratio sub micrometer runout has been obtained for a given stiffness

and damping parameters.


Orbit Amplitude


9.00E-05
8.00E-05
7.00E-05
S6.00E-05
5.00E-05
S4.00E-05
S3.00E-05
E 2.00E-05
< 1.00E-05
0.00E+00
-1 n00F-05f


Rotor speed (rpm)

I-- brg#1&2 -tool tip

Figure 3-20. Orbit amplitude at 1st, 2nd bearing and tool tip, e=0.1.

Both rigid rotor analysis (see appendix B) and FEA (see figure 3-19) shows system

is stable.


1














CHAPTER 4
EXPERIMENTAL IDENTIFICATION OF BEARING PARAMETERS

In order to ensure proper operation, the vibration phenomena, which the rotors

supported by fluid film bearings are especially subject to, has to be properly predicted.

Knowledge of dynamic coefficients, stiffness and damping of a bearing prior to its

installation and operation can be highly influential in the operation costs of the final

machine. Both an analytical and experimental approach can be employed to study the

dynamic behavior of bearings. Numerical techniques and computer-based simulation are

usually used to perform analytical studies. The current research has used XlTiltPad

program to identify bearing parameters.

The test set up aims at analyzing dynamic behavior of a tilted pad air bearing,

which will be used in this research, and comparing the experimental results with

analytical results. For the time being, experiments used a commercial air bearing as the

tilted pad air bearing is in the process of being designed and for the initial studies; a

flexure-supported ball bearing (see figure 4-1) is being used. Due to the difficulties found

in exciting the rotor-bearing system and in the measurement of force and displacement

data, experimental testing on fluid film bearings is known to be complex. The bearing

parameters can be identified experimentally by six different methods as mentioned

before. Since the research deals with a micro spindle supported by a small air bearing, the

test setup will be also small making it extremely hard to use loading in order to excite the

system. As a result, the viable way is using unbalance mass method.



























Figure 4-1. Flexure supported ball bearing: 1) Flexure, 2) Ball bearing.

4.1 Method of Measurement

The principle of operation of the test rig is rather simple: The bearing to be tested is

placed on a chassis (see figure 4-2), which serves to support displacement probes and is

mounted on dynamometer (load cell), used for measuring forces. A shaft is placed inside

the bearing and one side of shaft is mounted to the high-speed spindle. Figure 4-3 shows

a schematic of the test rig.



displacement load sensor
probe
Iz


Figure 4-2. A chassis.









spindle


rotor


test bearii

displacement
probe


disk


Chasis


Figure 4-3. Schematic of the test rig.

The rotating part of the test bearing has been given an intentional eccentricity at the

test bearing location. When the shaft rotates, the eccentricity generates an orbital pattern

synchronous with shaft speed.

Since the shaft is driven with a synchronous harmonic load, the resulting shaft

motion will, in general, be elliptic. Therefore, relative displacements describing ellipse as

a function of time is in following form:

y(t) = a(cos at) + b(sin at) (4-1)

z(t) = g(cos at) + h(sin at) (4-2)

Following equations will be used in equation of motion:

j(t) = -ao(sin mt) + b (cos ot) (4-1 a)


z(t) = -go(sin at) + ho(cos at)


(4-2a)









The four coefficients a, b, g and h are termed Fourier coefficients, and co is the

tester speed in radians per second. They are obtained from the synchronous components

of complex frequency spectrums computed for the y and z displacements.

The air bearing will respond with reaction forces which will be read by dyno to the

off centered rotor movement. The same procedure applied to the load data which load

cells read and following equations are derived:

Fy (t)= m(cos oit)+ n(sin oit) (4-3)

F (t) = p(cos oat) + q(sin oat) (4-4)

Equation of motion is the following:

K K+ + +CJ f, C (4-5)
F K Kz C C [" [ M j


Inertia term can be neglected in equation, because shaft relative displacement is so

small that inertia force will not contribute a lot for air film force. So new equation is the

following:

KY K, y+CY Cli, (4-6)


After substituting (4-1), (4-2), (4-la), (4-2a), (4-3) and (4-4) into (4-6), the

following equation is obtained:

yy
Cyz

bo ho a g 0 0 0 0 Ky m
-aa) -go b h 0 0 0 0 KYz n (4-7)
0 0 0 0 b) ha) a g Cz p
0 0 0 0 -aa) -go b h Cz q
Kz
Kzz









Since all these bearing parameters are functions of speed, it is necessary that the

speeds span as wide a range as possible to give the best definition of the coefficients [16].

There is an example how to use equation (4-7) in Appendix D.

4.2 Design Process

The test rig was designed and developed relying on an existing dynamometer, as

seen in Figure 4-4. Detailed information about dynamometer is presented in Appendix E.

In order to facilitate a better understanding of the setup, it was divided in to four parts

that is the dyno; a main chassis (see Figure 4-5), which serves to mount displacement

probes in order to measure shaft displacement and to mount the test bearing; a shaft; and

a spindle, which is an electric driven motor and rotates at 50000 rpm (max.).





























Figure 4-4. Test Setup: 1) Base, 2) Dyno, 3) Chassis, 4) Test bearing, 5) Shaft, 6) Spindle























Figure 4-5. Main Chassis: 1) Stator, 2) Displacement Probe, 3) Test Bearing.

The shaft motion is determined by displacement probes, and results can be read on

PC by using fiber optic sensors (RC20) connected to the PC. Dyno is used to measure the

forces in x and y directions respectively, and the data is sent to the PC using a charge

amplifier. A NSK spindle drive controller adjusts the spindle speed. The complete

measurement system is illustrated in Figure 4-6.


Figure 4-6. Measurment system














CHAPTER 5
CONCLUSION AND RECOMMENDATIONS

The current research provides initial results for the development of a

comprehensive rotordynamics analysis to describe high-speed micro spindle vibrations.

Results demonstrate that the air bearing characteristics and spindle residual unbalance

levels dictate the critical speed placement, unbalance response, and the ability to limit the

tool tip runout to sub-micrometer levels.

Results demonstrate that the 1/8th inch tungsten carbide micro spindle with the

integrally machined air turbine at one end and the end mill cutter at the other end can

indeed operate with sub-micrometer runout at the tool tip for spindle speeds up to 1

million rpm. Air bearing stiffness used is about 2000 lb/in. Spindle residual unbalance

level assumed is 10-6 oz-in (me=. 000001 oz-in). Residual unbalance two to three times

this level will also be adequate with higher bearing stiffness. Turbine engine shafts

weighing 100 lbs or greater are routinely balanced to 0.1 oz-in. Considering that the end

mill cutter weighs about 0.07 lb, the recommended unbalance limits can be achieved in

practice.

Next step in this research is experimental analysis, which was undertaken to

understand behavior of the air bearing. Since the air bearing is yet to be fabricated, a

simple test rig has been built to test the spindle mounted on ball bearings with a flexure

support. The x-y displacements at the bearings, as a function of speed, are being used to

evaluate the support stiffness characteristics. Our goals are to demonstrate the feasibility

of the proposed parameter estimation scheme to evaluate support stiffness.





















APPENDIX A
CHRONOLOGICAL LIST OF PAPERS ON THE EXPERIMENTAL DYNAMIC
PARAMETER IDENTIFICATION OF BEARINGS



References ring Type of excitation ration s Identified dynamic parameters
measured |dentlfted dynamic parameters


Hagg and Sankey (1956, 1958)
Mitchell at al. (1965-66)

Duftin and Johnson (1966-67)
Glienlcke (1966-67)
Woodcock and Holmes (19 P9-7i111
Black and Jenssen (1969-70)

Morton (1971)
Williams and Holmes (1971)
Thomsen and Andersen (1974)
Morton (1975a)
Bannister (1976)

Tonnesen (1976)
Fleming et al. (1977)
Wright (1978)
Parklns (1979)
Stanway et al. (1979a, 1979b)
Childs at al. (1980)
Diana at al. 1981:~1
Dogan et al. (1980)
HIsa et al. (1980)
lino and Kaneko (1980)
Nordmann and Schbllhorn (1980)
Walford and Stone (1980a, 1980b)
Burrows etaL (1981)
Parklns (1981)
Burrows and Sahlnkaya (1982a)
Diana at al. (1982)
Stone (1982)
Goodwin t al. (1983)
Wright (1983)
Burrows et al (1984)
Falco et al. (1984)
Goodwin et al. (1984)
Kankl and Kawakami (1984)
Nordmann and Massmann (1984)
Sahlnkaya et al. (1984)
Sahlnkaya and Burrows (1984a)
Burrows and Sahlnkaya (1985)

Chang and Zneng 1 1995i
Childs and Kim (1985)
Kaushal et al. (1985)
Childs and Kim (1986)
Childs and Scharrer (1986)
Nelson et al. (1986)
Roberts et al. (1986)
Childs and Garcia (1987)
Kraus et al. (1987)
Ramll et al. (1987)
Stanway et al. 11671
Burrows etal. (198 a)
Childs and Scharrer (1988)
Ellis et al. (1988)
Kang and JIn (1988)


HDJ Unbalance
HDJ Incremental static
load
HDJ Unbalance
HDJ Sinusoidal
HDJ Unbalance
PLS Incremental static
load
HDJ Sinusoidal
SOF Static load
SQF Unbalance
HDJ Step function
HDJ Unbalance

SQF Unbalance
GJ Unbalance
LS Impact
HSJ BI-dlrectlonal sine
SQF PRBS
Seals Eccentric shaft
HSJ Inertial exciter
HDJ PRBS
HDJ Sinusoidal
PLS Synchronous
HDJ Impact
Ball Sinusoidal
SQF PRBS/SPHS
HSJ BI-dlrectlonal sine I
SQF PRBS
PLS Synchronous
RE Impact and Sine
HSJ Unbalance
GS Synchronous
SQF SPHS
PLS Sinusoidal
HYJ Unbalance
PLS Sinusoldal
ANS Impact
SQF SPHS
SQF SPHS
SQF SPHS

HDJ Step function
ALS Eccentric rotor
SQF Unbalance
DS Eccentric rotor
GS Sinusoidal
ALS Sinusoidal
SQF Step function
DS Eccentric rotor
Ball Impact
SQF Transient
SQF PRBS
SQF Unbalance
LGS Uni-directional sine
SQF Step/sine
HDJ Impact


DispL I Iraeuenc'y
Displacement

Displ. (frequency)
Displ. Ire.uencyv
Displ. (frequency)
Displacement

Displ. Ira-.uencyi
Ecc., alt.. angle, vel.
Displ. Irreuency)
Displ. (frequency)
Displ. (frequency)

Displ. (frequency)
Displ. (frequency)
Displ. (time)
Dlspl., vol. (time)
DispL (time)
Displ. Ilrequencyi
Disp. IIrequency i
Displ. (frequency)
Vel. (frequency)
Displ. (frequency)
Displ. (frequency)
Displ., ace. (frequency)
Displ. (frequency)
Dlspl., vel. (time)
Displ. (frequency)
Displ. I frequency i
DispL *lime, frequency)
DispL (frequency)
Displ. (frequency)
Displ. (frequency)
Displ. (frequency)
Displ. (frequency)
Displ. (frequency)
Displ. (frequency)
Displ. (frequency)
Displ. (time)
Displ., vel. (time,
frequency)
*Dipl ace. irecluer.:y
Displ. (Irequencyi
Displ. Irequency I
DispL (frequency)
Displ. (frequency)
Displ. (frequency)
Displ. (time)
Displ. (frequency)
Displ. (frequency)
Displ. (time)
Displ. (frequency)
Displ. (Irequencyi
Displ. lrequencyi
Displ. (time)
Displ., acc. I limej


Direct damping and stiffness
Stiffness

Direct damping and stiffness
Damping and stiffness
Damping and uncertainty
Stiffness

Damping and stiffness
Damping
Damping
Damping and stiffness
Damping, stiffness and 28
non-linear coefficients
Damping
Damping and stiffness
Damping and stiffness
Damping and stiffness
Damping
Mass, damping and stiffness
Damping and stiffness
Damping and stiffness
Damping and stiffness
Mass, damping and stiffness
Damping and stiffness
Damping and stiffness
Damping
Damping and stiffness
Damping and uncertainty
Stiffness
Damping and stiffness
Damping and stiffness
Damping and stiffness
Damping
Damping and stiffness
Damping and stiffness
Mass, damping and stiffness
Mass, damping and stiffness
Damping and uncertainty
Damping and uncertainty
Damping and uncertainty

Damping and stiffness (0/4/4)
Mass, damping and stiffness
Damping 1v.'0 i and uncertainty
Mass, damping and stiffness
Damping and stiffness
Mass, damping and stiffness
Mass and damping
Mass, damping and stiffness
Mass, damping and stiffness
Direct damping and stiffness
Damping
Mass and damping
Damping, stiffness uncertainty
Direct damping and stiffness
Damping and stiffness (0/4/4)












References ar Type of excitaton Vibration response Identified dynamic parameters
type Typemeasured Identied dynamic parameters


Roberts et al. (1988)
Stanway et al. (1988)
Childs et al. (1989)
Ellis et al. 1989)
Elrod et al. (1989)
Hawkins et al (1989)
Kanemori and Iwatsubo (1989,
1992)
Someya (1989)
Brockwell et al. (1990)
Burrows et al. (1990)
Childs et al. (1990a)
Childs at al. (1990b)
Childs et al. (1990c)
Ellis et al. :19901
Frltzen and Selbold (1990)
Iwatsubo and Sheng (1990)
Kim et al. (1990)
Kostrzewsky et al. (1990)
Matsumoto et al. (1990)

Mohammad and Burdess 11990I
Muszynskaand Benty (1990)
Roberts et al. 11990.
Rouch (1990)
Yanabe et al. (1990)

Chan and White (1991)
Childs and Ramsey (1991)
Childs et a. (1991)
Imlach et al. (1991)

Jung et al. 11991a, 1991Di'
Murphy and Wagner (1991)
Wang and Llou (1991)
Adams et al. (1992)
Brown and Ismall (1992, 1994)
Childs and Kleynhans (1992)
Myllerup et al. (1992)

Rouvas et al. (1992)
Sakakida et al (1992)
Znang et al. 11 92a, 1992b'i
Conner and Chllds (1993)
Flack et al. (1993)
Jung and Vance (1993)
Muszynska et al (1993)
Parkins and Homer (1993)

Roberts et al. (1993)
Rouvas and Chllds (1993)
Childs and Hale (1994)
Franchek and Chllds (1994)
Heshmat and Ku (1994)


Kim and Lee (1994)
Kostrzewskv et al. (1994)


SOF
SQF
HCS
SQF
HCS
AGS
PLS

HDJ
HDJ
SQF
HCS
ANS
DS
SQF
ANS
DS
TR
HOJ
HDJ

HDJ
HDJ
SQF
SQF
HDJ

TPJ
AGS
AGS
MB

SQF
HSJ
Ball
HSJ
ANS
ANS
HDU

HSJ
HSJ
HDJ
BS
HDJ
SQF
Seals
HDJ

SQF
HSJ
HSJ
HYJ
HDJ
110 ii
Seals
HDJ


Sinusoidal
Synchronous
Sine-sweep
Bi-directional sinel
Uni-directional sine
Uni-directional sinel
Eccentric rotor

Sinusoldal
Bi-dlrectlonal sine
SPHS
Unl-directional sine
Eccentric rotor
Eccentric rotor
Random
Impact
Eccentric sleeves
Impulse
Bi-directional sine
Two-directional
sine
Random
Sine sweep
Step function
Sinusoidal
incremental static/
unbalancellmpact
Impact
Sine sweep
Sine sweep
Incremental static
load
Eccentric sleeve
Eccentric shaft
Impact
Unbalance
Multi-frequency
Sine sweep
incremental static
load
Impact
Sinusoidal
Impact
Sine-sweep
BI-directional sine I
Eccentric sleeve
Unbalance
Incremental static
load
Sinusoidal
PRBS and Sine
PRBS
PRBS
Sinusoidal


Displ. frequency)
Displ. (frequency)
Displ, acc. .requency)
Displ. nimel
Displ.; freq uenti y
Dispi. acc. !Irequency)
Displ. frequency)

Displ. Irequency)
Displ. frequency)
Displ. frequency)
Displ acc. (frequency)
Displ. itrequency)
Displ Ilrequency)
Displ. (time)
Dlspl. (tlme)
Displ. frequency)
Acc. (frequency)
Displ. frequency)
Displ. frequency )

Dispi itmei
Displ. i reauency)
Displ. lime)
Displ ilrequency)
Displacement
1rrequencyi
Displ. Frequency)
Displ. (frequency)
Displ. Ilrequencyi
Displacement

Displ. frequency)
Displ. (frequency)
Displ. i requency)
Displ. (Frequency)
DIspl. limej
Displ. ;irequenc yi
Displacement

Dispi. acc. (Irequencyi
Displ. acc. Ilrequencyl
Displ. (frequency)
Displ. (frequency)
Displ. IOrequency)
Displ. Ilrequency)
Displ. Frequency)
Displacement

Displacement ITire)
Displ., acc. Ilrequency)
Displ., acc. i requency)
Displ., acc. itrequencyl
Displ., acc. Itrequency)


Impact Displ i requency)
Bi-directional sine Displ. i reouencyv


Mass, damping and stiffness
nth-power velo, ity damping
Damping stiffness and uncertainty
Mass, damping and stiffness
Damping, stiffness and uncertainty
Damping, stiffness and uncertainty
Mass, damping and stiffness

Damping and stiffness
Damping and stiffness
Mass, damping and uncertainty
Damping, stiffness and uncertainty
Mass, damping and stiffness
Mass, damping and stiffness
Mass, damping and stiffness
Mass, damping and stiffness.
Mass, damping and stiffness
Direct stiffness and damping
Stiffness and uncertainty
Damping and stiffness

Damping and stiffness
Damping and effective stiffness
Mass, damping and uncertainty
Damping and stiffness
Damping and stiffness

Damping and stiffness
Damping, stiffness and uncertainty
Damping, stiffness and uncertainty
Direct stiffness

Mass and damping
Damping, stiffness and uncertainty
Damping and stiffness
Mass, damping and stiffness
Damping and stiffness
Damping, stiffness and uncertainty
Stiffness

Mass, damping and stiffness
Damping and stiffness
Damping, stiffness and uncertainty
Damping stiffness and uncertainty
Damping, stiffness, uncertainty
Mass and damping
Radial damping and stiffness
Stiffness

Non-linear model
Mass, damping and stiffness
Mass, damping and stiffness
Mass, damping and stiffness
Damping, stiffness and uncertainty

Mass, damping and stiffness
Mass, dampina and stiffness












References Beang Type of excitation Vibration response Identified dynamic parameters
type measured


Ku (1994)
Ku and Heshmat (1994)

Tieu and Qlu (1994)
Xu (1994)
Zhang and Roberts (1994)
Zhang et al (1994)
Alexander et al. (1995)
Arumugam et al. (1995)
Chen and Lee (1995)
Dmochowski and Brockwell 199E
Franchek et al (1995)
ParKins :1995,
San AndrBs et al. (1995)
Santos (1995)
Taylor et al. (1995)
Tiwarl and Vyas (1995, 1997a,
1996)
Childs and Gansle (1996)
Small and Brown (1996)
Kostrzewsky et al. (1996)
Lee et al. .199tC

Qiu and Tieu (1996)
Santos i1996)
Vance and Li (1996)
Znang and Roberts (1996)
Arumugam et al. (1997a, 1997b)
Chen and Lee (1997)
Goodwin at al. (1997)
Jiang etal. i19971
Marquette et al. (1997)

Marsh and Yantek (1997)
MQller-Karger et al. (1997)
Prabhu (1997)
Qlu and Tleu (1997)
Reddy et a. (1997)
San Andres and Childs (1997)

Sawlcki et al. (1997)

Tiwarl and Vyas 11997D1
Ismail and Brown (1998)

Kaneko et al. (1998)

Kostrzewsky et al. (1998)
Mitsuya at al. .19981
Mosher and Childs (1998)

Nikolakopoulos and
Papadopoulos 119981
Royston and Basdogan (1998)
Yu and Cnilas k19981


FTB
Foil
HDJ
HDJ
HYJ
SQF
SQF
AGS
HDJ
Ball
;5 HDJ
HYJ
HDJ
HSJ
HDJ
HOD
Ball

AGS
ANS
HDJ
MB

HDJ
HDJ
DS
SOF
HDJ
Ball
HDJ
HDJ
PLS

RB
HDJ
HDJ
HDJ
HDJ
HSJ


Slnusoldal Displ. (frequency) Damping and stiffness
Bi-aireclional sine Dispi. acc. (frequency) Damping srltrness and uncertainty


Unbalance
Slnusoidal
Slnusoidal
Slnusoldal
PRBS
Uni-directional sine
Unbalance
BI-directional sine
PRBS
BI-directional sine
PRBS
Slnusoidal
Slnusoidal
Random

Uni-directional sine
Multi-frequency
BI-directional sine
Random

Comp. sine
Sine sweep
Impact
Slnusoidal
Uni-directional sine
Unbalance
PRBS
Impulse
PRBS

Impact
Slnusoidal
Slnusoidal
Impulse
Slnusoidal
PRBS


HYJ Unbalance

Ball Ran/unbalance
PLS SPHS


Dlspl. (frequency)
Displ (frequency)
Displ. (frequency)
Displacement Ilme'
Dispi acc. (Irequency)
Dlspl. frequencyi
Olspl. (frequency)
Oispl., acc. (frequency)
Displ., acc. rrequencly
Velocity Ilimal
Displ acc. (frequency)
Displ. (frequency)
Displ. (frequency)
Dlspl., vel. (time)

Displ., acc. (trequericy)
Dlspl. vel., acc. (time)
DOspl. (frequency)
Current, volts, displ.
(frequency)
Dlspl. Irrequencyl
Displ. (frequency)
Olspl. (timel
Displ (frequency)
Displ. (frequency)
Displ. vel. (time)
Displ. (frequency)
Displ. frequencyi
Dlspl., acc. (frequency)

Acc. (frequency)
DOspl. (frequency)
Displ. (frequency i
Dlspl. (Frequencyi
Displ (frequency)
Displ., acc. irequen cy

Displ (frequency)

Dlspl., vel. (time)
Olspl., acc. (time)


ANS Eccentric sleeves Displ. i requency i


HDJ BI-dlrectlonal sine
Ball Impact
HYJ PRBS


HDJ
(ER)
Ball
GS
HCS


Displ. (frequency)
Displ. (frequency)
DOspl., acc. (frequency)


Incremental static Displacemrnir
load
Random Displ. frequentn
Uni-directional sine Displ., acc. (fre


Damping and stiffness
Damping and simness
Non-linear force coefficients
Mass, damping and stiffness
Damping and stiffness
Damping and stiffness
Damping and stiffness
Damping, stiffness and uncertainty
Mass, damping and stiffness
Damping and uncertainty
Mass, damping and stiffness
Damping and stiffness
Damping, stiffness and uncertainty
Non-linear stiffness

Damping, stiffness and uncertainty
Damping and stiffness
Damping silrtness and uncertainty
Damping and stiffness
Current stiffness
Damping, stiffness and uncertainty
Damping and stiffness
Damping
Mass and damping
Damping, stiffness and uncertainty
Damping and stiffness
Damping and stiffness
Damping and stiffness
Mass, damping, stiffness and
uncertainty
Direct stiffness
Damping, stiffness and uncertainty
Damping
Damping and stiffness
Damping and simness
Mass, damping, stiffness and
uncertainty
Mass, damping, srirfness and
uncertainty
Non-linear stiffness
Mass, damping, stiffness and
uncertainty
Mass, damping, stiffness and
uncertainty
Damping, stiffness and uncertainty
Damping and stiffness
Mass, damping, stiffness and
uncertainty
Stiffness


cy) Axial and radial stiffness
iquency) Damping, stiffness and uncertainty















References Be Type ofexcitation ma red Identified dynamic parameters

Chllds and Fayolle (1999) PLS PRBS Displ., acc. (frequency) Mass, damping, stiffness and


Fayolle and Chlds (1999)

Ha and Yang (1999)
Howard (1999)

Kim and Lee .1i 991
Laurant and Childs (1999)

LI et al. (1999)
Pettlnato and Choudhury (1999)

Ransom et al. (1999)
Soto and Chllds (1999)
Wygant at al. (1999)
Laos et al. (2000)
LI et al, (2000)
Lindsey and Childs (2000)

Shamlne et al. (2000)
Tlwarl (2000)
Vance and YIng (2000)
Zarzour and Vance (2000)
Howard et al. .2001 1
Nielsen etal (20:01,1
Pettinato and Flaci 20011
Pettlnato et al. (2001)
San Andr6s et al. (2001)
Tlwarl et al. (2002)
Dawson et al. (2002a, 2002b)
Holt and Childs (2002)

Laurant and Cnllds (200:21

Vazquez et al. (2002)
Chatterjee and Vyas 120:131
Kaneko et al. (2003)

Weatherwax and Chllds (2003)


PRBS


HYJ

HDJ
FAB

MB
HYJ

GDS
HDJ

GDS
HCS
HDJ
GDS
GDS
PLS

RE
Ball
ER
MD
FAB
AGS
HDJ
HDJ
SOF
SPR
AGS
AGS

HYJ

SPR
Ball
ALS


Dispi, acc. (frequency)

DispL, acc. (frequency)
Displacement

Control current, displ.
Displ., acc. (frequency)

Displ. (frequency)
Displ. (frequency)

Displ. (frequency)
Displ. ace. (frequency)
Displ. (frequency)
Displ., acc. (frequency)
Displ., acc. (frequency)
DispL, acc. (frequency)

Acc. trrequencyi
Displ., vel. (time)
Displacement (time)
Acc. IIimei
Displacement (time)
DispI, acc. (frequency)
Displ. frequencyi
Displ. (frequency)
Disp acc. (frequency)
Displ. lrequencyv
Dispi, acc. (frequency)
Displ., acc. (frequency)

Dispi acc. (frequency)

Acc. trrequency
Displ. !rr euenc y
Displ. frequencyy


AGS Eccentric sleeves DispL, acc. (frequency)


uncertainty
Mass, damping, stiffness and
uncertainty
Damping and stiffness
Damping and stiffness

Current and position stiffness
Mass, damping, stiffness and
uncertainty
Damping and stiffness
Damping, stiffness and uncertainty

Damping, stiffness and uncertainty
Damping, stiffness and uncertainty
Damping stiffness and uncertainty
Damping and uncertainty
Damping and stiffness
Mass, damping, stiffness and
uncertainty
Stiffness and damping (tilt)
Non-linear stiffness
Damping (0/2/0)
Damping and stiffness
Damping and stiffness (011/1)
Damping, stiffness and uncertainty
Damping, stiffness and uncertainty
Damping, stiffness and uncertainty
Damping, stiffness and uncertainty
Damping and stiffness
Damping, stiffness and uncertainty
Damping, stiffness and uncer-
tainty
Mass, damping, stiffness and
uncertainty
Damping and stiffness
Damping and non-linear stiffness
Mass, damping, stiffness and
uncertainty
Damping, stiffness and uncertainty


* The following abbreviations are used in the table: AGS, annular gas seal; ANS, annular


seal; ALS, annular liquid seal; BS, brush seals; DS, damper seals; ER, electrorheological


fluid; FAB, foil air; FTB, foil thrust; GDS, gas damper seal; GJ, gas journal; GS, gas


seal; HCS, honeycombed seal; HDJ, hydrodynamic journal; HSJ, hydrostatic journal;


HYJ, hybrid journal; LGS, long gas seal; LS, long seal; MB, magnetic; MD, metal mess


bearing damper; PLS, plain liquid seal; RB, recirculating ball; RE, rolling element; SPR,


springs; SQF, squeeze film; TPJ, tilting pad journal; TR, tapered roller.


Bi-directional sine
Incremental static
load
Magnetic
PRBS

Impact
Incremental static
load/unbalance
Impact
Swept sine
Bi-directional sine
Periodic chirp
Impact
PRBS

Impact
Random
Impact
Impact
Impact
Swept-sine
Bi-directional sine
Bi-directional sine
Impact and sine
Unbalance
PRBS
PRBS

PRBS

Swept-sine
Sinusoldal
Eccentric sleeves















APPENDIX B
GENERAL RIGID ROTOR SOLUTION 1














Input Rigid Rotor Parameters


0.0974
m :=
386
0.00161(
Ip :=
386
-5
1.88910




:= 52359. rad
sec


lbf-sec^2/in


lbf-sec^2-in


lbf-sec^2-in


al := 0.(

a2 := -0.(


eu := 0.00000:


eu9 := -
4
S:= eu-cos(euO)

= 1.41421x 10 6
] = 1.41421x 10


inch


unbalance eccentricity, in


:=eu-sin(euO)

= 1.41421x 10 6
S= 1.41421x 10


Kyy2 := 2055.: lbf/in

Kyz2:= 147.(

Kzy2:= -91.3(

Kzz2:= 1713.1'


Cyyl := 0.01/

Cyzl := -0.007

Czyl:= 0.00'

Czzl:= 0.03'


Kyyl :I

Kyzl :

Kzyl :=

Kzzl:=


S2055.:

147.(

-91.3(

1713.1.


lbf-sec/in


Cyy2 :-

Cyz2:=

Czy2:=

Czz2:=


0.01Z

S-0.007:

0.005

0.03,


















Kyyl+Kyy2 (0)2.m

Kzyl+ Kzy2

al-Kzyl+ a2-Kzy2

-(al-Kyyl+ a2Kyy

-Q-(Cyyl+ Cyy

-Q-(Czyl+ Czy)

-Q-(al-Czyl+ a2Czy)

O-(al-Cyyl+ a2Cyy


RHS:= m.-()2


Kyzl+ Kyz2 al-Kyzl+ a2Kyz2 (al-Kyyl+ a2Kyyl

Kzzl+ Kzz2 (0)2.m al-Kzzl+ a2.Kzz2 (al-Kzyl+ a2Kzy)

al-Kzzl+ a2Kzz2 al2Kyyl+ a22Kzz2 (0)2.Id -al2.Kzyl+ a22Kzy)

-(alKyzl+ a2Kyz -(al 2Kyzl+ a22Kyz) al2Kzzl+ a22Kzz2 (0)2.Id

-Q-(Cyzl+ Cyz -Q-(al-Cyzl+ a2Cyz4 .-(al-Cyyl+ a2Cyy)

-Q-(Czzl+ Czz4 -Q-(al-Czzl+ a2Czz4 .-(al-Czyl+ a2Czy)

-Q-(al-Czzl+ a2-Czz -2.(al2.Cyyl+ a22Czz al2Czy a2Cz .(a1Cy a2.Czy (2)2.Ip

0-(al-Cyzl+ a2-Cyz 2.(al2.Cyzl+ a22Cyzj + (n)2.Ip -.(al2.Czzl+ a22.Czz


0

0
0


0
o)
-i





[o}


(RHS unbalance forcing terms)


RHS


-.(Cyyl+ Cyy2
.-(Czyl+ Czy)

0-(al-Czyl+ a2Czy)

-.-(al-Cyyl+ a2Cyy

Kyyl+ Kyy2 (a)2.m

Kzyl+ Kzy2

al-Kzyl+ a2Kzy2

-(al-Kyyl+ a2Kyy2


0.09791

0.09791

0

0

0.09791

0.09791

0

0


0-(Cyzl+ Cyz3

0-(Czzl+ Czz)

.-(al-Czzl+ a2Czz4

-Q-(al-Cyzl+ a2Cyz4

Kyzl+ Kyz2

Kzzl+ Kzz2 (0)2.m

al-Kzzl+ a2.Kzz2

(al-Kyzl+ a2Kyz)


0-(al-Cyzl+ a2Cyz)

.-(al-Czzl+ a2Czz4

n-.(al2.Cyyl+ a22 Czz

--.(al2.Cyzl+ a22Cyz) (Q)2.Ip

al-Kyzl+ a2Kyz2

al-Kzzl+ a2Kzz2

al2.Kzzl+ a22.Kzz2 (Q)2 Id

(al2.Kyzl+ a22Kyz)


-Q-(al-Cyyl+ a2Cyy2

-Q-(al-Czyl+ a2Czy)

-.(al2.Czyl+ a22.Czy + (f)2.Ip

o (al2.Czzl+ a22Czzi

(al-Kyyl+ a2Kyy2

-(al-Kzyl+ a2Kzy)

(al2Kzyl+ a22Kzy)

al2.Kyyl+ a22.Kyy2 (Q)2 Id













-67306.42171
18.944
0
0
-3141.594
-52.3599
0
0


25.2
-66987.02171
0
0
429.35118
-3508.1133
0
0


0
0
617.62206
-9.072
0
0
-1196.94731
11323.04899


q := A 1RH


0
0
-6.81984
675.11406
0
0
-11458.76585
-1262.92079


Bearing Orbits


-1.37609x 10 6
-6
-1.53315x 10
0
0
-6
1.50867x 10 6
-6
-1.37988x 10
0
0 )


Bearing #1


Vlc:= q -al-q3

Vls:= q4 -al-q7


Wlc := q + al-q2

Wls := q5 + al-q6


Bearing #2


V2c := q -a2-q3

V2s := q4 a2-q7


W2c := q + a2q2

W2s := q5+ a2.q6


3141.594
52.3599
0
0
-67306.42171
18.944
0
0


-429.35118
3508.1133
0
0
25.2
-66987.02171
0
0


0
0
1196.94731
-11323.04899
0
0
675.11406
-9.072


0
0
11458.76585
1262.92079
0
0
-6.81984
560.13006)











PLOT BEARING ORBITS


2"7u
T:=
R


T = 0.00012


T
t := 0,-..T
200


Bearing #1


Vl(t) := Vlocos(O-t) + Vls.sin(Q.t)


Wl(t) := Wlc.cos(Q.t) + Wls.sin(O-t)


Wl(t)


V2(t) := V2ocos(O-t) + V2s.sin(Q.t)


W2(t) := W2c.cos(O-t) + W2s.sin(Q.t)


2.06256x10 6
...6 5x0


W2(t)


2.06256x10 6
.-2.06256x10


Vl(t)
shaft displacement in y direction


-2.04177x10 6 V2(t) ,2.04177x10 6
shaft displacement in y direction


Bearing #2










Work done per cycle at the bearings


Bearing #1


Kdl : Kyzl + Kzyl


Cm Cyyl + Czzl
Cml:= ----


WK1Cir:= 2-7.Kdl-(VloWls Vls-Wlc)


Work done by Circulation Force at Bearing #1


WKlDiss := -T. Cyy-.Vc2 + W1s2) + 2-Cml-(VloWlc + VIs-Wls) + Czzl(Wlc2 + Ws )]


WKlTotal := WK1Cir+ WKlDiss


Kd2 Kyz2 + Kzy2
Kd2 := -


Work done by dissipation forces


Total work done at the bearing #1


C Cyy2 + Czz2
Cm2:=


WK2Cir:= 2.-.Kd2.(V2oW2s V2s.W2c)


Work done by Circulation Force at Bearing #2


WK2Diss := -7.Q.Cyy2.(V2c2 + W2s2) + 2-Cm2(V2oW2c + V2s.W2s) + Czz2(W2c2 + W2s2)]


Work done by dissipation forces


Total work done at the bearing #2


WK2Total := WK2Cir + WK2Diss











Bearing #1

WK1Cir = 4.13906x 10
-8
WKlDiss = -4.24666x 10 8

WKlTotal =-4.24253x 10 8
WK1Total = -4.24253x 10


Bearing #2

WK2Cir = 4.13906x 10
-8
WK2Diss = -4.24666x 10 8

WK2Total =4.24253x 10 8
WK2Total = -4.24253x 10


NOTE: Total work done at each bearing/cycle must be negative(subtracting energy from rotor), for stable operation















APPENDIX C
GENERAL RIGID ROTOR SOLUTION 2
















Input Rigid Rotor Parameters


0.0974S
m :=
386
0.00161(
Ip :=
386
-5
1.88910
Id :=
386


lbf-sec^2/in


lbf-sec^2-in


lbf-sec^2-in


al := 0.(

a2 := -0.(


eu := 0.00000:


71
eu0 := -
4
S:= eu-cos(euO)

i = 1.41421x 10


o := 52359.' rad
sec


inch


unbalance eccentricity, in


:=eu-sin(euO)

6 6
( = 1.41421x 10


Bearing parameters found by using XLTiltPadHGB:


Kyy2 := 2055.: lbf/in

Kyz2:= 147.(

Kzy2:= -91.3(

Kzz2:= 1713.1:


Cyyl := 0.01/

Cyzl:= -0.007

Czyl:= 0.00'

Czzl:= 0.03'


Kyyl :

Kyzl:=

Kzyl :=

Kzzl:=


S2055.:

147.(

-91.3(

1713.1:


lbf-sec/in


Cyy2 :

Cyz2:-

Czy2 :

Czz2:=


=0.01

S-0.007

S0.00'

0.03'











Kyyl+ Kyy2 () -m Kyzl Kyz2 al-Kyzl+ a2Kyz2 -(alKyyl+ a2Kyyj n-(Cyyl+ Cyy
Kzyl Kzy2 Kzzk Kzz2- () 2m al-Kzzk a2Kzz2 -(al-Kzyl a2Kzy7 -.(Czyl+ Czy
alKzyl+ a2Kzy2 al-KzzKl a2Kzz2 al2Kyl+ ai2Kzz2- (n)2 Id al2-Kzy ai2Kzy n.(al-Czyl+ a2Czy

-(alKyyl+ a2Kyyj -(alKyzl a2Kyz4 -(al2Kyzh a22Kyz al2.Kzzl+a22.Kzz2-()2.Id -Q-(al-Cyyl+ a2Cyy

-Q-(Cyyl+ Cyy --.(Cyzl+ Cyz --.(alCyzl+ a2Cyz -.(al-Cyyl+ a2Cyy Kyyl+ Kyy2- () m
--.(Czyl+ Czy) --(Czzl+ Czz4 --(al-Czzl+ a2Czz7 -.(al-Czy+ a2Czy Kzyl+ Kzy2
-.(al-Czyl+a2Czy --.(al-Czzl+ a2Czz7 -I.(a12Cyyl+ aiCzz n0.(al2.Czyl+ aiCzy- (n)2.Ip alKzyl+ a2Kzy2
n.(al.Cyyl+a2Cyyr n(al.Cyzl+a2Cyz n.(a12.Cyzl a2.Cyz+ (n)2.Ip -~.(a2.Czzl a22Czz -(alKyyl+ a2Kyy


n-(Cyzk- Cyz4
*.(Czzkl Czz7
-.(al-Czzl+ a2Czz4
--.(alCyzl+ a2Cyz4
Kyzl+ Kyz2
Kzzk Kzz2- () *m
al-Kzzlk a2Kzz2

-(alKyzl+ a2Kyz7


n-(alCyzhl a2Cyz4
-.(al-Czzlk a2Czz7
n-(al-Cyl1+ a2iCzz
-.(al2CyzI a9-Cyz -(n)2*Ip
al-Kyzl+ a2Kyz2
aliKzzlk a2Kzz2
al2.Kzzk aiKzz2- (Q)2.Id
-a -Kyzh a -Kyz


-Q-(al-Cyyl+ a2Cyyj
--.(al-Czyl a2Czy7
-n(alCzyvb a22Czy+ (n)2*Ip
n-(al.Czzl a2-Czz

-al-Kyyl+ a2Kyyj
-al-Kzyl a2Kzy4
(a2-Kzyv ai.Kzyl
al2Kyyl+ a22Kyy2- ()2*Id


RHS:= m.()2:


0
if






0

0

o)


(RHS unbalance forcing terms)


RHS


0.09791)
0.09791
0
0
-0.09791
0.09791
0

0











295.2
-65808.76171
0
0
743.51058
-3665.193
0
0


0 0
0 0
1222.46886 65.7792
-106.272 1099.28766
0 0
0 0
-923.62864 -11289.11978
11209.95161 -1319.46948


A *.RHS Bearing Orbits


Bearing #1


-1.5494x
0
0


10-6
10


1.51235x 10 6
-6
-1.39413x 10
0
0


Vlc:= q0 -al-q3

Vls:= q4 -al-q


Wlc := q + al-q2

Wls := q5 + alq6


Bearing #2


V2c:= q0 a2.q3

V2s := q4 a2.q7


W2c := q + a2q2

W2s := q5 + a2-q6


-65124.42171
-182.72
0
0
-1466.0772
-523.599
0
0


1466.0772
523.599
0
0
-65124.42171
-182.72
0
0


-743.51058
3665.193
0
0
295.2
-65808.76171
0
0


0
0
923.62864
-11209.95161
0
0
1099.28766
-106.272


0
0
11289.11978
1319.46948
0
0
65.7792
1345.65006)


1.46054x 10 6)











PLOT BEARING ORBITS


2"7u
T:=
R


T = 0.00012


T
t := 0,-..T
200


Bearing #1


Vl(t):= Vlocos(O-t) + Vls.sin(Q.t)


Wl(t) := Wlc.cos(Q.t) + Wls.sin(O-t)









N5


W1(t)
-2. V06 -1_10


Bearing #2

V2(t) := V2ocos(O-t) + V2s.sin(Q.t)


W2(t) := W2c.cos(O-t) + W2s.sin(Q.t)


W2(t)
-2


Vl(t)
shaft displacement in y direction


V2(t)
shaft displacement in y direction










Work done per cycle at the bearings


Bearing #1


Kdl : Kyzl + Kzyl


CmlCyyl + Czzl
Cml:= ----


WK1Cir:= 2.7.Kdl.(VloWls Vls.Wlc)


Work done by Circulation Force at Bearing #1


WKlDiss := -t..[Cyyl(VIc2 + W1s2) + 2-Cml(VloWlc + V1s.Wls) + Czzl(Wlc2 + WIs2)]


WKlTotal := WK1Cir + WKlDiss


Kd2 Kyz2 + Kzy2
Kd2 := -


Work done by dissipation forces


Total work done at the bearing #1


Cm2:Cyy2 + Czz2
Cm2 :=


WK2Cir:= 2- t.Kd2.(V2oW2s V2s.W2c)


Work done by Circulation Force at Bearing #2


WK2Diss := -7.Q.Cyy2.(V2c2 + W2s2) + 2.Cm2(V2oW2c + V2s.W2s) + Czz2(W2c2 + W2s2)]


Work done by dissipation forces


Total work done at the bearing #2


WK2Total := WK2Cir + WK2Diss














Bearing #1
10
WK1Cir = 7.73768x 10 10
-8
WKlDiss =-3.56451x 10

WKTotal =-3.48713x 10- 8
WK1Total = -3.48713x 10


Bearing #2

10
WK2Cir = 7.73768x 10
-8
WK2Diss = -3.56451x 10

WK2Total =3.48713x 10 8
WK2Total = -3.48713x 10


NOTE: Total work done at each bearing/cycle must be negative (subtracting energy from rotor), for stable operation
















APPENDIX D
EXAMPLE OF FINDING BEARING PARAMETERS

Let's choose two different speed, and different displacement components for each speed.


Q := 2.10 rad/sec

a := 0.0:


b := 0.01


g := 0.0;


h := 0.02


1 := 2.1.10


al := 0.03(


bl := 0.01(


gl := 0.02(


hi := 0.03(


where 0=ft


y(0) := acos(0) + b-sinm()
z(0) := gcos(0) + h-sin(0)


0.05 1


-0.05 I
-0.05 0 0.05


Kyy := 100( Kzy := -50C
Kyz:= 50C Kzz:= 90C


yl(0) := alcos(0) + bl-sin(0)
zl(0) := gl.cos(0) + hlsin(0)


0.05 1 I


zl(6)


-0.05
-0.05 0 0.05
yl(6)


lbf/in


Cyy := 15
Cyz:= 4


Czy := -4 lbf-sec/in
Czz:= 15


So by assuming all variables, forces can be find by using equation of motion (4-6),

where

Fy(0)=m*cos(0)+n*sin(0) and Fz(0)=p*cos(0)+q*sin(0).


S:= 0,2-- .. 2.7
100











m:= a-Kyy + g-Kyz+ b-Q-Cyy + h-Q-Cy2

n := bKyy + h-Kyz a-Q-Cyy g-Q-Cy2

p := aKzy + g.Kzz+ b-Q-Czy + h.-QCz;

q := bKzy + h.Kzz a-QCzy g-Q-Cz


ml:= al-Kyy + gl.Kyz + bl.Q1Cyy + hl.Q1.Cy2

nl := bl Kyy + hl.Kyz al.Q.1Cyy gl.-1.Cy2

pl :=al.Kzy + gl.Kzz+ bl .-1.Czy + hl-1Q.Cz7

ql := blKzy + hlKzz al.Q.1Czy gl.-1QCz


Now by using force and displacement components, lets find stiffness and damping

coefficients.


b.-
-a.Q
0
0


h.-

-g.Q
0
0


a g
b h
0 0
0 0


0
0
b.Q
-a.-


0
0
h.Q

-g.Q


m\
n
T :=
P
lq


For each speed, there are set of four equations, expressed above. Thus for two


different speeds, following expression is obtained:


b.Q
-a. Q

0
0
bl.Q1
-al.-1
0
0


Ans :=P- 1. T


As it can be seen, stiffness and damping coefficients are the same values as assumed.


h.Q

-g.Q
0
0
hl.-l
-gl. 1
0
0


0
0
b.Q
-a. Q
0
0
bl.-21
-al.Q1


0
0
h.Q
-g.Q
0
0
hl.21
-gl. 1


m^
n

P
q
ml
nl

pl
ql)


15
4
1000
500
-4
15
-500
900


Cyy
Cyz
Kyy
Kyz
Czy
Czz
Kzy
Kzz)







71


Let's assume force data were read in error, because of noise. Forces in y direction and


forces in z direction are read in ten and four percent error, respectively.


m:= m 1.1

ml:= ml-l.1


Sb.-
-a-Q

0
0
bl-.1
-al-Q

0
0


n := nl.1

nl := nll.1


h.-
-g.Q
0
0
hl.-1

-gl. 1
0
0


0
0
b-Q
-a-Q

0
0
bl-.1
-al- 1


Ans := P- 1.T1


p := p.l.OZ
pl :=pl-1.OZ


0
0
h-Q
-g.Q
0
0
hl.-1
-gl. 1


16.5
4.4

1.1x 103
550
-4.16
15.6
-520
936 )


q := q.1.0Z

ql := ql-1.OZ


m)
n

P
p
q
ml
nl

pl
qlI


The result shows that the stiffness and damping coefficients are changed by the


same percentage as forces. So it means noise in force affects bearing parameters at the


same percentage as in force itself.



























APPENDIX E

KISTLER DYNANOMETER


Force FMD




3-Komponenten-Dynamometer FF, F Fz
Dynamombtre i 3 composantes Fx,,* F,
3-Component Dynamometer Fx, Fy, Fz


Quarzkristall-Dreikomponenten-Dynamofmeter
zum Messen der drel orthogonalen Komponen-
ten einer Kraft. Das Dynamometer besitzt eine
grosse Steithei und demzutolge eine hohe Ei-
genfr"uen? D S '1mr" Aufltbsungsvermdgen
*...- .: .jl.c.' 1. r.e.::. .-. kleirnslen dynami-
r -, -, ....rg ,- 3 :-.Krafte.


KISTLER


9257B, 9403


Dynamomrtre a cristal de quariz A trois com-
posantes pour mesurer des trois composantes
orthogonales cdurne'force Le dynamomntre
: a; '. i.-r* d.-.._.: .:; : d:l 1.= i -or : '.'. .1.jl r i
,,.r.- ir ^^ : f r ;i..- .=1I. ?- .0 r.*:*. .. au = jlu
solution permet de mesurer les moindres varia-
lios de large forces.


Quartz Lhree-component dynamometer for
measuring the three orthogonal components of
a force. The dynamometer has a great rigidity
and consequently a high natural frequency. Its
high resolution enables the smallest dynamic
changes In large forces to be measured.


Type 9257B


Technische Daten
Baerslch

Kallbderter Trelberelch 1

Kalbbrdieer Teilbereich 2

Oberlast
F, bei F, und Fy O.5 F,
An prehach lwlle
Empflnualir.ken

LlnseriUI, jli. Bel.i:re r
Hysterese, ai- P--.-n--
Ubersprechen
Slefihmi

Elgentrequenz
Eigenlrequenz
Egenlnquen
iT....i..-rl an i'..OscP,-r.
SBetslrDstaeperaturbeil ch
STemperaurkoetrllient
os.. E.~lleail. hll.h Rl
Kapazltic:.r. .e.-
ol.dlattonse8dmratd I ; I
! Mlasselolaton
Schutahrl
Geinchil


*) *'na lre.-i.- t fra '/ 5 wa* ?
:os.rr at >* DL lr uIe
") BEreich beim Dreen, Krelatangrff
bei Prkt A.
"*) Mit AnsrmausskabelType 1687BS. 16a885


Donn6e* techniques
Gamma
FZ pour F, et Fy 50,5 F,
Gate partlelle etaloenne 1

Gaamme partile etalonn6e 2

Surcharge
Fz pour F et Fy 50, F,
Scull de r6ponse
Sensabilt6

Ln6aritt, toutes les garmmes
HySer6Esls. c-1- = -.a :. 5 "rrr-*
Cross lalk
Rlgidlt6

Fr6quence propre
Fr6quence propre
(Installf sur brides)
Game de ltemprature dulillsation
Coefaclent de temperature
de la sensibifitl
Capacity (de canal)
Resistance dlsalement (20 'C)
tlol6 & l masse
Case de protection
Polds


Technical
Range
F".;,' a F, 0:1.5 P
ClUmsraled partial range 1

Calibraled partial rnge 2

Overload
F, for F, and Fy i0,5 Fz
Threshold
SensivMty

Llneerity, all ranges
Hystereels, all ranges
Cross talk
Migiloty

Natural frequency
NhtueS frequency
(mounted on flanges)
Operating lernporaetir range
Temperature coe"ickent
of sensitivity
Capacitance ,i :r.".r.,n. '
Insaulmlln rsil-snee K .'C)
Ground insulatlon
Prolectin class
Weight


) Pont d'applcanon de la force au-dedans et
max 25 mm au-dessus de la plaque superieure.
") Gamme lrs du toumage, point drapplicaion
au point A.
") Ave tod dte connexion types 1687B5. 1609B5


IData
. F F, ir
F IN
F,. F, N
F N
F Fy N
F, N
F,. Fy. F, kN
F, IkN
N
F, F, pC/N
F, pCN
% FSO
% FSO
%
c,. c, krNlMn
c, kNfpm
It (x, y. z) kHz
If (x, y) kHz
If (z) kHz
*C
%/1C

pF

0g


*) AppScalton of force inside and mra
above top plate area.
") Range for turning, application of o
at point A.
*) With connect cable Types 16B7


0 ..500
S... 1000
0.. 50
0 .. 100
-7.5/7.5
-7.5/15
<0.01
--7.5
--3.7
Stl
S0.5
5st2
>1
>2
-3.5
-2.3
-3.5
0... 70
-0.02

-220
>1013
>108
IP 67 "")
7.3
x. 25 mm

mce

B5, 16BBS


Kistle InstrumenteAG Wintmrthur, CH-8408 Winerthur, Switzerland, Tel. (052) 224 11 11 Kisler Instument Cor., Amherst, NY 14228-2171, USA, Phone (716) 691-5100


1 N (Newton) = 1 kg m s-2 0,1019... kp = 0.2248... lf; I Ich = 25,4 mm; 1 kg 2,2046... Ib 1 Nm 0.73756... bll















Dynamometer Typ 9257B
Abmesumngn


FrAsen, Schlafen
Craisoey? frciainge


Dynamometre type 9257B


Dynam


Dwmensions Omn
If


ometer Type 9257B
kons


Dynamometer Typ 9257B
Abm"mung f mit menti oetm StlhhaHe
Stalhalter Typ 9403
Anschlusskabel Typ 1687B5 / 1689B5


Dehen
Tournae
Turning


Dynamometre type 9257B Dynamomete Type 257B
imenkons amvc port~-oul# month Dmnslonms with mounted tool hokler
Porte-outil type 9403 Tool holder Type 9403
COble de connexion type 1687B5 / 168985 Connecting cable Type 168785 / 168985


Type 168785


Kistler InstrumenteAG Winterthu CH-8408 Winterthur, Switzerland, Tel (052) 224 1111 Kistler Instrument Corp, Amherst, NY 14228-2171, USA, Phone (716) 691-5100
















LIST OF REFERENCES


[1] Kussul, E. M., Rachkovskij, D. A., Baidyk, T. N., and Talayev, S. A., 1996,
"Micromechanical engineering: A basis for the 10w-cost manufacturing for
mechanical microdevices using microequipment," Journal of Micromechanical
Microengineering, vol. 6, pp. 410-425.

[2] Masuzawa, T., and Tonshoff, H. K., 1997, "Three-dimensional micromachining by
machine tools," Annals of the CIRP, vol. 46, pp. 621-628.

[3] Roy R. Craig, Jr., 1981, "Structural Dynamics," 1st edition, pp. 430-434.

[4] Tiwari, R., Lees, A. W., and Friswell, M. I., 2004, "Identification of Dynamic
Bearing Parameters: A review," The Shock and Vibration Digest, vol. 36, pp. 99-
124.

[5] Mitchell, J.R., Holmes, R., and Ballegooyen, H.V., 1965-66, "Experimental
Determination of a Bearing Oil Film Stiffness," in Proceedings of the 4th
Lubrication and Wear Convention, IMechE, Vol. 180, No. 3K, 90-96.

[6] Arumugam, P., Swarnamani, S., and Prabhu, B. S., 1995, "Experimental
Identification of Linearized Oil Film Coefficients of Cylindrical and Tilting Pad
Bearings," ASME Journal of Engineering for Gas Turbines and Power, Vol. 117,
No. 3, 593-599.

[7] Glienicke, J., 1966-67, "Experimental Investigation of the Stiffness and Damping
Coefficients of Turbine Bearings and Their Application to Instability Prediction,"
Proceedings of IMechE, Vol. 181, No. 3B, 116-129.

[8] Morton, P.G., 1971, "Measurement of the Dynamic Characteristics of Large Sleeve
Bearing," ASME Journal of Lubrication Technology, Vol. 93, No.1, 143-150.

[9] Someya, T., 1976, "An Investigation into the Spring and Damping Coefficients of
the Oil Film in Journal Bearing," Transactions of the Japan Society of Mechanical
Engineers, Vol. 42, No. 360, 2599-2606.

[10] Hisa, S., Matsuura, T., and Someya, T., 1980, "Experiments on the Dynamic
Characteristics of Large Scale Journal Bearings," in Proceedings of the 2nd
International Conference on Vibration in Rotating Machinery, IMechE, Cambridge,
UK, Paper C284, 223-230.









[11] Sakakida, H., Asatsu, S., and Tasaki, S., 1992, "The Static and Dynamic
Characteristics of 23 Inch (584.2 mm) Diameter Journal Bearing," in Proceedings
of the 5th International Conference on Vibration in Rotating Machinery, IMechE,
Bath, UK, Paper C432/057, 351-358.

[12] Hagg, A.C., and Sankey, G.O., 1956, "Some Dynamic Properties of Oil-Film
Journal Bearings with Reference to the Unbalance Vibration of Rotors," ASME
Journal of Applied Mechanics, Vol. 78, No. 2, 302-305.

[13] Hagg, A.C., and Sankey, G.O., 1958, "Elastic and Damping Properties of Oil-Film
Journal Bearings for Application to Unbalance Vibration Calculations," ASME
Journal of Applied Mechanics, Vol. 80, No. 1, 141-143.

[14] Stone, J.M., and Underwood, A.F., 1947, "Load-Carrying Capacity of Journal
Bearing," SAE Quarterly Transactions, Vol. 1, No. 1, 56-70.

[15] Duffin, S., and Johnson, B.T., 1966-67, "Some Experimental and Theoretical
Studies of Journal Bearings for Large Turbine-Generator Sets," Proceedings of
IMechE, Vol. 181, Part 3B, 89-97.

[16] Murphy, B.T., and Wagner, M.N., 1991, "Measurement of Rotordynamic
Coefficients for a Hydrostatic Radial Bearing," ASME Journal of Tribology, Vol.
113, No. 3, 518-525.

[17] Adams, M.L., Sawicki, J.T., and Capaldi, R.J., 1992, "Experimental Determination
of Hydrostatic Journal Bearing Rotordynamic Coefficients," in Proceedings of the
5th International Conference on Vibration in Rotating Machinery, IMechE, Bath,
UK, Paper C432/145, 365-374.

[18] Sawicki, J.T., Capaldi, R.J., and Adams M.L., 1997, "Experimental and Theoretical
Rotordynamic Characteristics of a Hybrid Journal Bearing," ASME Journal of
Tribology, Vol. 119, No. 1, 132-142.

[19] Downham, E., and Woods, R., 1971, "The Rationale of Monitoring Vibration on
Rotating Machinery in Continuously Operating Process Plant," in ASME Vibration
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BIOGRAPHICAL SKETCH

The author of the thesis was born on July 2, 1981, in Azerbaijan. He grew up in

Azerbaijan. In 1998 he moved to Turkey to attend Middle east Technical University,

where he received his Bachelor of Science in Mechanical Engineering. In 2004 he

traveled to United States of America for the pursuit of a master's degree in mechanical

engineering at University of Florida. He is planning to complete the degree of Master of

Science in Mechanical Engineering in August 2006.