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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 rotorbearing 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 Page 57 Page 58 Page 59 Page 60 Page 61 Page 62 Page 63 Page 64 Page 65 Page 66 Page 67 Page 68 Page 69 Page 70 Page 71 Page 72 Page 73 References Page 74 Page 75 Page 76 Biographical sketch Page 77 

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ROTORBEARING 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 ROTORBEARING 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 31 Bearing parameters at 500,000 rpm (Q=52360 rad/sec) .......................................23 32 Bearing Parameters for different rotor speeds............................... ............... 29 LIST OF FIGURES Figure page 11 C om m ercial m icrotool .............................................................. ....................... 3 21 Rigid rotor schematic. ................................... ... ....... ................. .5 22 A nonfloating bearing housing and a rotating journal ................. .... ........... 11 23 A floating bearing housing and a fixed rotating shaft....................... ...............12 31 M icrospindle. ..................................................... ................. 20 32 Bearings location on spindle. ............................................................................. 20 33 Rotor orbits at two bearing supports at 500,000 rpm .........................2...............24 34 Rotor orbits at the two bearing supports at 1,000,000 rpm .....................................25 35 R otor orbits at the tool end at 500,000 rpm ..................................... .....................26 36 Orbit amplitude at 1st, 2nd bearing and tool tip.................................................... 27 37 Stiffness, Kyy versus rotor eccentricity. ...................................... ............... 28 3 8 W h irl m ap ....................................................................... 3 0 39 Finite elem ent m odel ........................ ..... .......... ........... .... .... .. ........ .... 31 310 Rigid body mode 1, 107758 rpm ............................ ....... ...............31 311 Rotor analysis at 115298 rpm. A) Rigid body mode 2, B) Potential energy distribution. .......................................... ............................ 32 312 Rotor analysis at 2,042,349 rpm. A) Flexural mode 1, B) potential energy distribution. .......................................... ............................ 32 313 Flexural m ode 2, 4,955,776 rpm ........................................ ........................ 33 314 Critical speed m ap. .................................... .. .. ...... ...............33 315 1st bearing. A) and B) Unbalance response, C) Amplitude and phase lag, D) N yquist plot for displacem ent. ............................................................................ 34 316 2nd bearing. A) and B) Unbalance response, C) Amplitude and phase lag, D) N yquist plot for displacem ent. ............................................................................ 36 317 Tool tip. A) Unbalance response, B) Amplitude and phase lag, C) Nyquist plot for displaced ent. ..................................................................... 38 318 Shaft orbits. A), B), C), D), E), F) are shaft orbits as a function of speed ..............39 319 Stability Map (Note: Negative log decrements indicate instability) ......................42 320 Orbit amplitude at 1st, 2nd bearing and tool tip, e=0.1 ................................................43 41 Flexure supported ball bearing: 1) Flexure, 2) Ball bearing. ..................................45 42 A chassis .......................................................................................................... 45 43 Schem atic of the test rig. ............................................................................. ..... 46 44 Test Setup: 1) Base, 2) Dyno, 3) Chassis, 4) Test bearing, 5) Shaft, 6) Spindle .....48 45 Main Chassis: 1) Stator, 2) Displacement Probe, 3) Test Bearing...........................49 46 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 ROTORBEARING 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 microscale 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 microscale 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 submicrometer runout at the tool tip makes microscale 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 submicrometer 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 microcomponents 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 highprecision microcomponents, 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 micromilling spindle which will rotate at over 500,000 rpm range with submicrometer 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 microtools have a 1/8th inch diameter shank (see Figure 11). 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 microtool. 3. Monitoring subsystem. Theoretical and scientific understanding of micromilling requires monitoring and recording of cutting forces. However, in the measurement bandwidth (that is 1,000,000 rpm with a 2flute 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 11. Commercial microtool Irmsllslllllll~ CHAPTER 2 DEVELOPMENT OF EQUATION OF MOTION OF ROTORBEARING 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 21. 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 (21) (22) and the kinetic energy of the rigid body is 2 +W2) 2 2+)2] T=m( 2+Wm2)+ Ipa2+I 2+c2) (23) 2 2 L(23) 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) (24) Fz = F (V,W,V,W) Upon Taylor's series expansion of eq. (24) about the origin, the force components in eq. (24) 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 (25a) or F. = _k. c7 = ki t i (25b) where S0 0 a =O 1 a. 0 q A i (26) 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= (28) 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 (29) Using the above set of equations and inserting in eq (29) the following set of rigid rotor equations of motion has been obtained: MY + (C QG)K + = 0 (210) 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 (211) 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 highspeed 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 rotorbearing 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 inputoutput 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. (196566) [5] performed experiments to incrementally load the bearing and measuring the change in position, and obtained the four stiffness coefficients of fluidfilm bearings. They obtained the following simple relationships using the influence coefficient approach to kyy = azz / kyz = ayz / 7 (212) kzy = azy /7 kzz= ayy / Y where S= ayyazz ayzzy ayy = l / AFy azy = zl / AFy (213) 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 22), the fluidfilm dynamic equation can be written as myy my, zY ~Cyy cyz~ +Ikyy kyz] l(Y=fym + YB (214) 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 (214) can be expressed in the form MBq +CBq+KBq = f MRq (215) 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 (216) The overall system mass, damping and stiffness matrices can be formed by adding the separate contributions of the bearings and rotor in equation (216). 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 (216), 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 (217) 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 22. A nonfloating 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 (196667) [7] adopted the technique of exciting the floating bearing bush (housing) sinusoidally in two mutually perpendicular directions (Figure 23) 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 (214) may be written as ~zyy zyz fY SFi Zzy Zzz Z Fz with (218) 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 (218) k represents the effective bearing stiffness coefficient, since while estimating the bearing dynamic stiffness, z, the fluidfilm addedmass 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 23. 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 largescale journal bearings by using simultaneous sinusoidal excitations on the bearing at two different frequencies and measuring the corresponding displacement responses. This is called the twodirectional 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 (218). If the excitation force and dynamic displacement are measured at two different excitation frequencies under the same static state of equilibrium and ignoring the fluidfilm addedmass effects equation (218) 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 (219) 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) [1213] were among the first to use the unbalance force only for experimentally measuring the oilfilm 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 rotorbearing coefficients and not the true film coefficients as the crosscoupled coefficients are ignored. Duffin and Johnson (196667) [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., 196566) 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, 196667) 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 addedmass) 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 (219) 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 (doublespoolshaft) had a provision for a circular orbit motion of adjustable magnitude with independent control over spin speed, orbit frequency and whirl direction. The leastsquares 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 halfsinusoid 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 phasemeasurement errors, they defined an error function using just the amplitude components of the FRFs. This nonlinear 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 timeconsuming 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 frictionrelated tangential force components and, further, is prone to poor signaltonoise 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 submicrometer 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 31. Also from the practicality point of view the microspindle must accept a variety of tools with minimal time and effort required for tool change. Rotordynamics of highspeed 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 speeddependent system critical speeds. For stable highspeed 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 highspeed 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 31. Microspindle. 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 32. 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 32. 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 = (21) 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 (21) 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 (31) As a result, first and second derivatives will have the following forms, respectively. q= Q c sin(fQt)+ Ws cos(Qt) Bc (32) FcJ FSJ (32) =n2 (33) After substituting for MA, C, G, K, Q into equation (21) using (31), (32), (33) 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 zz2Id 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 (34) 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 'RotorBearing 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=2.525x105 lbfsecA2/in, Ip=4.187x106 lbfsec^2in, Id=4.894x108 lbfsec^2in and an unbalance eccentricity of 0.000002 inch are shown in Figure 33. The air bearing for this configuration has the following parameters, Table31. Table 31. Bearing parameters at 500,000 rpm (Q=52360 rad/sec). Kyyl,2=950.4 lbf/in Cyyl,2=0.021 lbfsec/in Kyzl,2=99.1 lbf/in Cyzl,2=0.004 lbfsec/in Kzy1,2=93.46 lbf/in Czy1,2=0.005 lbfsec/in Kzz1,2=1050.4 lbf/in Czz,2=0.106 lbfsec/in x 106 1st bearing 2.56, 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 33. 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 34). 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 34. 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 35, runout in both directions is small at the tool end (2 x 106 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 106 Figure 35. 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 36 shows that why figure 33 and figure 34 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 9.00E06 8.00E06 E 7.00E06 6.00E06 S5.00E06 S4.00E06 E 3.00E06 2.00E06 0 1.00E06 0.00E+00 1.00E06 < 1 0 000 P0 0000 2 Q900 rotor speed (rpm) brg#1 & 2 tool tip Figure 36. 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 4.24253x108 (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 37. 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 37. 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 rotorbearing system is in second order. So equation (21) was modified in order to find eigenvalues. 0 M [ M 0 M C 0 K Slo0 (35) After entering equations (35) into (21), results in: M x+K x=X X(36) In order to find eigenvalues, right hand side of equation (36) 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 = (37) 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=2.525x104 lbfsec^2/in, Ip=4.187x106 lbf secA2in, Id=4.894x108 lbfsec^2in and an unbalance eccentricity of 0.000002 inch is shown in Figure 312; wherein the air bearing has the following parameters, Table 32. Table 32. 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 32. 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 [lbfsec/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 38. Whirl map. It can be seen from graph (Figure38), 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 rotorbearing system was performed using a specialpurpose code. The model shown in Figure 39 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 ozin is used in FE model. As a result four critical speeds are found for the following parameters. ti t ) 4 Figure 39. 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 310. 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 (sAf1) Overall: Shaft(S)= 0.04%, Bearing(Brg)= 99.96% M) Percentage: o Component: S No. or Stn: 1 Figure 311. 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 312. 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 (sAw1) Overall: Shaft(S)= 99.94%, Bearing(Brg)= 0.06% Percentage: Component: No. or Stn: B. Figure 315. Continued Critical Speed = 4955776 rpm = 82596.27 Hz I* I ~tF I I I 3 at. *a Figure 313. Flexural mode 2, 4,955,776 rpm. Bearing Stiffness Figure 314. Critical speed map. 34 The critical speed map in Figure 314 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, SubStation: 1 Peak disp: max amp = 5.3133E006 at 950000 rpm Negative (b) indicates backward precession 6.00 E0 .... .... ........ 2 4.BO0E06    E 3.60E06         2.40E06      S1.20E06  L  ..J  O.OOE+00 ..... .... ........ ........... .... .... ...... 0.00E + 00 ' 6.00E06 8 4.80E06    3.60E06   E 2.40E06     co 1.20E06     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 (semimajor 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 315. 1st bearing. A) and B) Unbalance response, C) Amplitude and phase lag, D) Nyquist plot for displacement. Bode Plot Station: 2, SubStation: 1 probe 1 (x) 0 deg max amp = 5.3133E006 at 950000 rpm 2 probe 2 (y) 90 deg max amp = 5.3133E006 at 950000 rpm a 240 I a 200   CM 160   . J 120    : 0 o 80     4 0 . . . . . . . . . . . 6.00E06 ....... 6 4.80E06     a 3.60E06    2.40E06 .    1.20E06    < 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, SubStation: 1 Speed range= 50000 1.3E+006 rpm probe 1 (x) 0 deg max amp = 5.3133E006 at 950000 rpm probe 2 (y) 90 deg max amp = 5.3133E006 at 950000 rpm 270 Rotation 1.3E+006 ,' 90 Full Scale = 6E006 (opk) D. Figure 315. Continued Elliptical Orbital Axes Station: 7, SubStation: 1 Peak disp: max amp = 1.8734E005 at 1.3E+006 rpm Negative (b) indicates backward precession 2.50E05 ......... ......... ....... ......... ....... 2.00E05      1.50E05    1.00E05    5.00E06 .................... ...... 0.OOE+00 * 2.00E05    O .O5 E + 0 F ... . . . . . . . . .. . 2. [ EE05 E EE  1.50E05    5.00E05  1.00E05 ....   I  5.00E06 , ^ ^  .   0.00E+00 .... ........... ......... ......... i .... . 5.00E06 '~ 0.00E+(3.00E+05 6.00E+05 9.00E+05 1.20E+06 1.50E Rotational Speed (rpm) Transmitted Force (semimajor 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 316. 2nd bearing. A) and B) Unbalance response, C) Amplitude and phase lag, D) Nyquist plot for displacement. Bode Plot Station: 7, SubStation: 1 probe 1 (x) 0 deg max amp = 1.8734E005 at 1.3E+006 rpm 2 probe 2 (y) 90 deg max amp = 1.8734E005 at 1.3E+006 rpm a 280 I 61 240     200  160 S 120    S 8 40 03 2.00E05   r n,  _I4 40 .............. ......... ......... ....... 2.50E05 ...................... 1.00E05   S1.50E05   t .   1.00E05    ............. S5.OOE0   < 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, SubStation: 1 Speed range = 50000 1.3E+006 rpm probe 1 (x) 0 deg max amp= 1.8734E005 at 1.3E+006 rpm probe 2 (y) 90 deg max amp = 1.8734E005 at 1.3E+006 rpm 270 S,,,. Rotation 1.3E+006"5 180  oo00 o  degree U 00000 1.3E+00oo 90 Full Scale = 2.5E005 (opk) D. Figure 316. Continued Elliptical Orbital Axes Station: 11, SubStation: 1 Peak disp: max amp = 0.00013149 at 1.3E+006 rpm Negative (b) indicates backward precession 2.00E04 ,............................ ....... ..... 1.60E04    1.20E04  L  8.00E05    4.00E05     O.OOE+00 .... .... ...... .. ......... ..... 1.80E04  1.50E04  L '  1.20E04       1.50E04 I  9.00E05   .00E05 . .................... L 0 5'i 3.00E05 . 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, SubStation: 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 . .. E04 E04 E04 E05 E05 E05 :+nn 0.00E+(3.OOE+05 6.00E+05 9.00E+05 1.20E+06 Rotational Speed (rpm) 1.50E+06 Figure 317. 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, SubStation: 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 (opk) C. Figure 317. 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 E007, b = 5.3751 E007 Y Figure 318. 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 = 1.1509E006, b = 1.1509E006 Shaft Response due to shaft 1 excitation Rotor Speed = 200000 rpm, Response FORWARD Precession Max Orbit at stn 11, substn 1, with a = 2.747E006, b = 2.747E006 Y Figure 318. Continued Shaft Response due to shaft 1 excitation Rotor Speed = 300000 rpm, Response FORWARD Precession Max Orbit at stn 11, substn 1, with a = 5.0653E006, b = 5.0653E006 Shaft Response due to shaft 1 excitation Rotor Speed = 400000 rpm, Response FORWARD Precession Max Orbit at stn 11, substn 1, with a = 8.2848E006, b = 8.2848E006 Y Figure 318. 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 = 1.2559E005, b = 1.2559E005 Y F. Figure 318. Continued. Stability Map 1.80    S 000ooooo00 ooooo60oo0600000oo ooooo 1.50 ' 2 E 1E 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 319. 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. = 1.22 108 ozin. So unbalance response analysis is performed again with e=0.1. Figure 320 shows that even with this eccentricity ratio sub micrometer runout has been obtained for a given stiffness and damping parameters. Orbit Amplitude 9.00E05 8.00E05 7.00E05 S6.00E05 5.00E05 S4.00E05 S3.00E05 E 2.00E05 < 1.00E05 0.00E+00 1 n00F05f Rotor speed (rpm) I brg#1&2 tool tip Figure 320. Orbit amplitude at 1st, 2nd bearing and tool tip, e=0.1. Both rigid rotor analysis (see appendix B) and FEA (see figure 319) 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 computerbased 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 flexuresupported ball bearing (see figure 41) is being used. Due to the difficulties found in exciting the rotorbearing 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 41. 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 42), 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 highspeed spindle. Figure 43 shows a schematic of the test rig. displacement load sensor probe Iz Figure 42. A chassis. spindle rotor test bearii displacement probe disk Chasis Figure 43. 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) (41) z(t) = g(cos at) + h(sin at) (42) Following equations will be used in equation of motion: j(t) = ao(sin mt) + b (cos ot) (41 a) z(t) = go(sin at) + ho(cos at) (42a) 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) (43) F (t) = p(cos oat) + q(sin oat) (44) Equation of motion is the following: K K+ + +CJ f, C (45) 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, (46) After substituting (41), (42), (4la), (42a), (43) and (44) into (46), 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 (47) 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 (47) in Appendix D. 4.2 Design Process The test rig was designed and developed relying on an existing dynamometer, as seen in Figure 44. 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 45), 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 44. Test Setup: 1) Base, 2) Dyno, 3) Chassis, 4) Test bearing, 5) Shaft, 6) Spindle Figure 45. 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 46. Figure 46. Measurment system CHAPTER 5 CONCLUSION AND RECOMMENDATIONS The current research provides initial results for the development of a comprehensive rotordynamics analysis to describe highspeed 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 submicrometer 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 submicrometer 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 106 ozin (me=. 000001 ozin). 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 ozin. 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 xy 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. (196566) Duftin and Johnson (196667) Glienlcke (196667) Woodcock and Holmes (19 P97i111 Black and Jenssen (196970) 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 BIdlrectlonal sine SQF PRBS Seals Eccentric shaft HSJ Inertial exciter HDJ PRBS HDJ Sinusoidal PLS Synchronous HDJ Impact Ball Sinusoidal SQF PRBS/SPHS HSJ BIdlrectlonal 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 Unidirectional 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 nonlinear 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 Sinesweep Bidirectional sinel Unidirectional sine Unidirectional sinel Eccentric rotor Sinusoldal Bidlrectlonal sine SPHS Unldirectional sine Eccentric rotor Eccentric rotor Random Impact Eccentric sleeves Impulse Bidirectional sine Twodirectional sine Random Sine sweep Step function Sinusoidal incremental static/ unbalancellmpact Impact Sine sweep Sine sweep Incremental static load Eccentric sleeve Eccentric shaft Impact Unbalance Multifrequency Sine sweep incremental static load Impact Sinusoidal Impact Sinesweep BIdirectional 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) Bidirectional sine Displ. i reouencyv Mass, damping and stiffness nthpower 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 Nonlinear 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) MQllerKarger 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 Biaireclional sine Dispi. acc. (frequency) Damping srltrness and uncertainty Unbalance Slnusoidal Slnusoidal Slnusoldal PRBS Unidirectional sine Unbalance BIdirectional sine PRBS BIdirectional sine PRBS Slnusoidal Slnusoidal Random Unidirectional sine Multifrequency BIdirectional sine Random Comp. sine Sine sweep Impact Slnusoidal Unidirectional 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 BIdlrectlonal sine Ball Impact HYJ PRBS HDJ (ER) Ball GS HCS Displ. (frequency) Displ. (frequency) DOspl., acc. (frequency) Incremental static Displacemrnir load Random Displ. frequentn Unidirectional sine Displ., acc. (fre Damping and stiffness Damping and simness Nonlinear 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 Nonlinear 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 Nonlinear 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) Nonlinear 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 nonlinear 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. Bidirectional sine Incremental static load Magnetic PRBS Impact Incremental static load/unbalance Impact Swept sine Bidirectional sine Periodic chirp Impact PRBS Impact Random Impact Impact Impact Sweptsine Bidirectional sine Bidirectional sine Impact and sine Unbalance PRBS PRBS PRBS Sweptsine 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 lbfsec^2/in lbfsec^2in lbfsec^2in al := 0.( a2 := 0.( eu := 0.00000: eu9 :=  4 S:= eucos(euO) = 1.41421x 10 6 ] = 1.41421x 10 inch unbalance eccentricity, in :=eusin(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. lbfsec/in Cyy2 : Cyz2:= Czy2:= Czz2:= 0.01Z S0.007: 0.005 0.03, Kyyl+Kyy2 (0)2.m Kzyl+ Kzy2 alKzyl+ a2Kzy2 (alKyyl+ a2Kyy Q(Cyyl+ Cyy Q(Czyl+ Czy) Q(alCzyl+ a2Czy) O(alCyyl+ a2Cyy RHS:= m.()2 Kyzl+ Kyz2 alKyzl+ a2Kyz2 (alKyyl+ a2Kyyl Kzzl+ Kzz2 (0)2.m alKzzl+ a2.Kzz2 (alKzyl+ a2Kzy) alKzzl+ a2Kzz2 al2Kyyl+ a22Kzz2 (0)2.Id al2.Kzyl+ a22Kzy) (alKyzl+ a2Kyz (al 2Kyzl+ a22Kyz) al2Kzzl+ a22Kzz2 (0)2.Id Q(Cyzl+ Cyz Q(alCyzl+ a2Cyz4 .(alCyyl+ a2Cyy) Q(Czzl+ Czz4 Q(alCzzl+ a2Czz4 .(alCzyl+ a2Czy) Q(alCzzl+ a2Czz 2.(al2.Cyyl+ a22Czz al2Czy a2Cz .(a1Cy a2.Czy (2)2.Ip 0(alCyzl+ a2Cyz 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(alCzyl+ a2Czy) .(alCyyl+ a2Cyy Kyyl+ Kyy2 (a)2.m Kzyl+ Kzy2 alKzyl+ a2Kzy2 (alKyyl+ a2Kyy2 0.09791 0.09791 0 0 0.09791 0.09791 0 0 0(Cyzl+ Cyz3 0(Czzl+ Czz) .(alCzzl+ a2Czz4 Q(alCyzl+ a2Cyz4 Kyzl+ Kyz2 Kzzl+ Kzz2 (0)2.m alKzzl+ a2.Kzz2 (alKyzl+ a2Kyz) 0(alCyzl+ a2Cyz) .(alCzzl+ a2Czz4 n.(al2.Cyyl+ a22 Czz .(al2.Cyzl+ a22Cyz) (Q)2.Ip alKyzl+ a2Kyz2 alKzzl+ a2Kzz2 al2.Kzzl+ a22.Kzz2 (Q)2 Id (al2.Kyzl+ a22Kyz) Q(alCyyl+ a2Cyy2 Q(alCzyl+ a2Czy) .(al2.Czyl+ a22.Czy + (f)2.Ip o (al2.Czzl+ a22Czzi (alKyyl+ a2Kyy2 (alKzyl+ 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 alq3 Vls:= q4 alq7 Wlc := q + alq2 Wls := q5 + alq6 Bearing #2 V2c := q a2q3 V2s := q4 a2q7 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(Ot) + Vls.sin(Q.t) Wl(t) := Wlc.cos(Q.t) + Wls.sin(Ot) Wl(t) V2(t) := V2ocos(Ot) + V2s.sin(Q.t) W2(t) := W2c.cos(Ot) + 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:= 27.Kdl(VloWls VlsWlc) Work done by Circulation Force at Bearing #1 WKlDiss := T. Cyy.Vc2 + W1s2) + 2Cml(VloWlc + VIsWls) + 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) + 2Cm2(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 lbfsec^2/in lbfsec^2in lbfsec^2in al := 0.( a2 := 0.( eu := 0.00000: 71 eu0 :=  4 S:= eucos(euO) i = 1.41421x 10 o := 52359.' rad sec inch unbalance eccentricity, in :=eusin(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: lbfsec/in Cyy2 : Cyz2: Czy2 : Czz2:= =0.01 S0.007 S0.00' 0.03' Kyyl+ Kyy2 () m Kyzl Kyz2 alKyzl+ a2Kyz2 (alKyyl+ a2Kyyj n(Cyyl+ Cyy Kzyl Kzy2 Kzzk Kzz2 () 2m alKzzk a2Kzz2 (alKzyl a2Kzy7 .(Czyl+ Czy alKzyl+ a2Kzy2 alKzzKl a2Kzz2 al2Kyl+ ai2Kzz2 (n)2 Id al2Kzy ai2Kzy n.(alCzyl+ a2Czy (alKyyl+ a2Kyyj (alKyzl a2Kyz4 (al2Kyzh a22Kyz al2.Kzzl+a22.Kzz2()2.Id Q(alCyyl+ a2Cyy Q(Cyyl+ Cyy .(Cyzl+ Cyz .(alCyzl+ a2Cyz .(alCyyl+ a2Cyy Kyyl+ Kyy2 () m .(Czyl+ Czy) (Czzl+ Czz4 (alCzzl+ a2Czz7 .(alCzy+ a2Czy Kzyl+ Kzy2 .(alCzyl+a2Czy .(alCzzl+ 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 .(alCzzl+ a2Czz4 .(alCyzl+ a2Cyz4 Kyzl+ Kyz2 Kzzk Kzz2 () *m alKzzlk a2Kzz2 (alKyzl+ a2Kyz7 n(alCyzhl a2Cyz4 .(alCzzlk a2Czz7 n(alCyl1+ a2iCzz .(al2CyzI a9Cyz (n)2*Ip alKyzl+ a2Kyz2 aliKzzlk a2Kzz2 al2.Kzzk aiKzz2 (Q)2.Id a Kyzh a Kyz Q(alCyyl+ a2Cyyj .(alCzyl a2Czy7 n(alCzyvb a22Czy+ (n)2*Ip n(al.Czzl a2Czz alKyyl+ a2Kyyj alKzyl a2Kzy4 (a2Kzyv 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 0 106 10 1.51235x 10 6 6 1.39413x 10 0 0 Vlc:= q0 alq3 Vls:= q4 alq Wlc := q + alq2 Wls := q5 + alq6 Bearing #2 V2c:= q0 a2.q3 V2s := q4 a2.q7 W2c := q + a2q2 W2s := q5 + a2q6 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(Ot) + Vls.sin(Q.t) Wl(t) := Wlc.cos(Q.t) + Wls.sin(Ot) N5 W1(t) 2. V06 1_10 Bearing #2 V2(t) := V2ocos(Ot) + V2s.sin(Q.t) W2(t) := W2c.cos(Ot) + 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) + 2Cml(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) + bsinm() z(0) := gcos(0) + hsin(0) 0.05 1 0.05 I 0.05 0 0.05 Kyy := 100( Kzy := 50C Kyz:= 50C Kzz:= 90C yl(0) := alcos(0) + blsin(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 lbfsec/in Czz:= 15 So by assuming all variables, forces can be find by using equation of motion (46), 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:= aKyy + gKyz+ bQCyy + hQCy2 n := bKyy + hKyz aQCyy gQCy2 p := aKzy + g.Kzz+ bQCzy + h.QCz; q := bKzy + h.Kzz aQCzy gQCz ml:= alKyy + 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 + hl1Q.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:= mll.1 Sb. aQ 0 0 bl.1 alQ 0 0 n := nl.1 nl := nll.1 h. g.Q 0 0 hl.1 gl. 1 0 0 0 0 bQ aQ 0 0 bl.1 al 1 Ans := P 1.T1 p := p.l.OZ pl :=pl1.OZ 0 0 hQ 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 := ql1.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 3KomponentenDynamometer FF, F Fz Dynamombtre i 3 composantes Fx,,* F, 3Component Dynamometer Fx, Fy, Fz QuarzkristallDreikomponentenDynamofmeter 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 Lhreecomponent 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. c1 = .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 rsilsnee K .'C) Ground insulatlon Prolectin class Weight ) Pont d'applcanon de la force audedans et max 25 mm audessus 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, CH8408 Winerthur, Switzerland, Tel. (052) 224 11 11 Kisler Instument Cor., Amherst, NY 142282171, USA, Phone (716) 6915100 1 N (Newton) = 1 kg m s2 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 Porteoutil type 9403 Tool holder Type 9403 COble de connexion type 1687B5 / 168985 Connecting cable Type 168785 / 168985 Type 168785 Kistler InstrumenteAG Winterthu CH8408 Winterthur, Switzerland, Tel (052) 224 1111 Kistler Instrument Corp, Amherst, NY 142282171, USA, Phone (716) 6915100 LIST OF REFERENCES [1] Kussul, E. M., Rachkovskij, D. A., Baidyk, T. N., and Talayev, S. A., 1996, "Micromechanical engineering: A basis for the 10wcost manufacturing for mechanical microdevices using microequipment," Journal of Micromechanical Microengineering, vol. 6, pp. 410425. 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[23] Adams, M.L., and Rashidi, M, 1985, "On the Use of RotorBearing Instability Thresholds to Accurately Measure Bearing Rotordynamic Properties," ASME Journal of Vibration, Acoustics, Stress, and Reliability in Design, Vol. 107, No. 4, 404409. [24] Lee, A.C., and Shih, Y.P., 1996, "Identification of the Unbalance Distribution and Dynamic Characteristics of Bearings in Flexible Rotors," Proceedings of IMechE, Part C: Journal of Mechanical Engineering Science, Vol. 210, No. 5, 409432. [25] Lund, J.W., Cheng, H.L., and Pan, H.T., 1965, "RotorBearing Dynamic Design Technology, Part 3," Mechanical Technology. 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. 