AN EFFICIENT ALGORITHM FOR BLADE LOSS SIMULATIONS APPLIED TO A HIGH-ORDER ROTOR DYNAMICS PROBLEM. A Thesis NIKHIL KAUSHIK PARTHASARATHY

Transcription

1 AN EFFICIENT ALGORITHM FOR BLADE LOSS SIMULATIONS APPLIED TO A HIGH-ORDER ROTOR DYNAMICS PROBLEM A Thess by NIKHIL KAUSHIK PARTHASARATHY Submtted to the Offce of Graduate Studes of Texas A&M Unversty n partal fulfllment of the requrements for the degree of MASTER OF SCIENCE December 3 Major Subject: Mechancal Engneerng

2 AN EFFICIENT ALGORITHM FOR BLADE LOSS SIMULATIONS APPLIED TO A HIGH-ORDER ROTOR DYNAMICS PROBLEM A Thess by NIKHIL KAUSHIK PARTHASARATHY Submtted to the Offce of Graduate Studes of Texas A&M Unversty n partal fulfllment of the requrements for the degree of MASTER OF SCIENCE Approved as to style and content by: Alan Palazzolo (Char of Commttee) Ch-Der Suh (Member) Moo-Hyun Km (Member) Denns O Neal (Interm Department Head) December 3 Major Subject: Mechancal Engneerng

3 ABSTRACT An Effcent Algorthm for Blade Loss Smulatons Appled to a Hgh-Order Rotor Dynamcs Problem. (December 3) Nkhl Kaushk Parthasarathy, B.S., Bangalore Unversty, Inda Char of Advsory Commttee: Dr. Alan B. Palazzolo In ths thess, a novel approach s presented for blade loss smulaton of an arcraft gas turbne rotor mounted on rollng element bearngs wth squeeze flm dampers, seal rub and enclosed n a flexble housng. The modal truncaton augmentaton (MTA) method provdes an effcent tool for modelng ths large order system wth localzed nonlneartes n the ball bearngs. The gas turbne engne, whch s composed of the power turbne and gas generator rotors, s modeled wth 38 lumped masses. A nonlnear angular contact bearng model s employed, whch has ball and race degrees of freedom and uses a modfed Hertzan contact force between the races and balls and for the seal rub. Ths combnes a dry contact force and vscous dampng force. A flexble housng wth seal rub s also ncluded whose modal descrpton s mported from ANSYS. Predcton of the maxmum contact load and the correspondng stress on an ellptcal contact area between the races and balls s made durng the blade loss smulatons. A fnte-element based squeeze flm damper (SFD), whch determnes the pressure profle of the ol flm and calculates damper forces for any type of whrl orbt s utlzed n the smulaton. The new approach s shown to provde effcent and accurate predctons of whrl ampltudes, maxmum contact load and stress n the bearngs, transmssblty, thermal growths, maxmum and mnmum damper pressures and the amount of unbalanced force for ncpent ol flm cavtaton. It requres about 4 tmes less computatonal tme than the tradtonal approaches and has an error of less than 5 %.

4 v ACKNOWLEDGMENTS I would lke to thank Dr. Alan Palazzolo for hs countless hours of nstructon, patence and dedcaton n my research. My apprecaton goes to Dr. Ch-Der Suh and Dr. Moo-Hyun Km for ther gudance and for servng on my advsory commttee. I gratefully acknowledge Andy Provenza, Charles Lawrence and Kelly Carney of NASA Glenn, Oho for provdng support to ths research. I extend my grattude to the personnel of the Vbraton Control and Electromechancs Laboratory, specfcally, Dr. Guangyoung Sun, for provdng much needed assstance durng the course of ths nvestgaton. Last but defntely not least, I would lke to especally thank my parents for all ther support and encouragement throughout my master s program.

5 v TABLE OF CONTENTS Page ABSTRACT ACKNOWLEDGMENTS v TABLE OF CONTENTS v LIST OF FIGURES v LIST OF TABLES... x CHAPTER I INTRODUCTION Overvew Lterature Revew Objectves Outlne... 6 II FINITE ELEMENT MODEL OF THE SYSTEM Evaluaton of System Matrces Tmoshenko Beam Element Stffness Matrx Mass Matrx Dampng Matrx Descrpton of the Dual Rotor System Descrpton of the Gas Turbne Rotor System Support System FE Model of Complete System... 1 III PRELIMINARY STRUCTURAL ANALYSIS Statc Analyss Undamped Modal Analyss Gas Generator Rotor Undamped Modes Power Turbne Rotor Undamped Modes... 7

6 v TABLE OF CONTENTS (CONTINUED) CHAPTER Page 3.3 Gyroscopc Damped Mode Shapes Crtcal Speed Analyss Steady State Harmonc Response Analyss IV BLADE-OUT RESPONSE ANALYSIS USING MODAL BASED SOLUTION ALGORITHMS Modal Dsplacement Method Mode Acceleraton Method Modal Truncaton Augmentaton Method Theortcal Comparson of the Methods Smulaton Results Parametrc Study... 5 V STAGGERED ANALYSIS SCHEME Descrpton of Staggered Analyss Scheme Verfcaton of Staggered Analyss Scheme Smulaton Results wth Thermo-Mechancal Integraton Smulaton Results Usng Staggered Analyss Scheme Summary VI HOUSING, CONTACT AND THERMAL MODEL Descrpton of Flexble Housng Casng Undamped Modes from ANSYS Seal Rub Contact Model Thermal Model VII OVERALL SYSTEM SIMULATION FE Model of Complete System Smulaton Results VIII CONCLUSIONS AND FUTURE WORK REFERENCES APPENDIX A

7 v TABLE OF CONTENTS (CONTINUED) Page APPENDIX B VITA... 18

8 v LIST OF FIGURES FIGURE Page.1 Tmoshenko beam element Gyroscopc moments Two spool arcraft turbne engne wth 8 bearngs....4 Two spool arcraft turbne engne lumped parameter FE model Gas generator undamped mode at N = 1,41 RPM Gas generator undamped mode at N = 5,9 RPM Gas generator undamped mode at N = 34,685 RPM Gas generator undamped mode at N = 5857 RPM Gas generator undamped mode at N = 9,49 RPM Power turbne undamped mode at N = 38 RPM Power turbne undamped mode at N = 6943 RPM Power turbne undamped mode at N = 14,37 RPM Power turbne undamped mode at N = 7,563 RPM Power turbne undamped mode at N = 43,676 RPM Power turbne undamped mode at N =54,341 RPM Power turbne undamped mode at N = 68,4 RPM Power turbne undamped mode at N = 9,8 RPM Power turbne undamped mode at N = 93,481 RPM D and 3D mode shapes (wth whrl drecton) at N = RPM D and 3D mode shapes (wth whrl drecton) at N = 1,397 RPM D and 3D mode shapes (wth whrl drecton) at N =,54 RPM Campbell dagram natural frequences vs speed plot Steady state harmonc response of second stage power turbne (a) Transent response and (b) orbt plot of the second stage of the power turbne for 35 rev.:, y-axs; - -, z-axs (a) Transent response and (b) orbt plot of the SFD journal at brg # for 35 rev.:, y-axs; - -, z-axs... 44

9 x LIST OF FIGURES (CONTINUED) FIGURE Page 4.3 (a) Transent response and (b) orbt plot of the SFD journal at brg #4 for 35 rev.:, y-axs; - -, z-axs Transmssblty (a) at brg # and (b) at brg # Maxmum contact loads (a) at brg # IR, (b) at brg # OR, (c) at brg #4 IR, and (d) at brg #4 OR Maxmum contact stress (a) at brg # IR, (b) at brg # OR, (c) at brg #4 IR, and (d) at brg #4 OR Maxmum and mnmum pressures n brg # SFD Maxmum and mnmum pressures n brg #4 SFD Whrl ampltude of the power turbne Maxmum whrl ampltude of the SFD journal Maxmum ball contact loads at (a) brg # and (b) brg # Maxmum contact stress vs unbalanced load Pressure n SFD n (a) brg # and (b) brg # Transmssblty plots Computatonal flow dagram Staggered analyss scheme tme lne: --, Thermal only regon;, Thermo-mechancal regon (a) Transent response and (b) orbt plot of the second stage of the power turbne:, y-axs; - -, z-axs (a) Transent response and (b) orbt plot at brg # :, y-axs; - -, z-axs (a) Transent response and (b) orbt plot at brg #4 :, y-axs; - -, z-axs Power loss n:, brg #; - -, brg #

10 x LIST OF FIGURES (CONTINUED) FIGURE Page 5.7 Temperature of bearng nner race and ball at:, brg #; - -, brg # Temperature of bearng outer race and ol flm at:, brg #; - -, brg # Transmssblty at:, brg #; - -, brg # (a) Transent response and (b) orbt plot of the second stage of the power turbne:, y-axs; - -, z-axs (a) Transent response and (b) orbt plot at brg # :, y-axs; - -, z-axs (a) Transent response and (b) orbt plot at brg #4 :, y-axs; - -, z-axs Power loss n:, brg #; - -, brg # Temperature of bearng nner race and ball at:, brg #; - -, brg # Temperature of bearng outer race and ol flm at:, brg #; - -, brg # Transmssblty at:, brg #; - -, brg # Flexble housng from ANSYS Casng undamped mode at N =,13 RPM Casng undamped mode at N = 9,5 RPM Casng undamped mode at N = 4,33 RPM Casng undamped mode at N = 48,98 RPM Casng undamped mode at N = 15,84 RPM Rub rng contact model Cross-sectoned bearng wth thermal nodes Heat transfer network... 76

11 x LIST OF FIGURES (CONTINUED) FIGURE Page 6.1 1D radal heat flow through cylnder and electrcal analogy Full thermal model for power turbne rotor Dsplacements of ball center, nner and outer races ncludng thermal expanson Schematc dagram of the full system Tmelne for overall system smulaton (a) Transent response and (b) orbt plot of brg # :, y-axs; - -, z-axs (a) Transent response and (b) orbt plot of brg #4 :, y-axs; - -, z-axs (a) Transent response and (b) orbt plot of brg #ND :, y-axs; - -, z-axs (a) Transent response and (b) orbt plot of brg #1 :, y-axs; - -, z-axs (a) Transent response and (b) orbt plot of brg # :, y-axs; - -, z-axs (a) Transent response and (b) orbt plot of brg #3 :, y-axs; - -, z-axs Transmssblty : brg # -- brg # Maxmum contact stress at nner race : brg # -- brg # Maxmum contact stress at outer race : brg # -- brg # Shakedown dagram for bearng stress lmts Maxmum pressure at: brg # -- brg # Mnmum pressure at: brg # -- brg # Heat loss at: brg # -- brg # Bearng power loss at: brg # -- brg #

12 x LIST OF FIGURES (CONTINUED) FIGURE Page 7.17 Power loss due to rub at: brg # -- brg # Temperature of bearng nner race and bearng ball at:, brg #; - -, brg # Temperature of bearng outer race and ol flm at:, brg #; - -, brg # A.1 Dscretzed tme and response varables for newmark beta method... 1 B.1 SOLID45 3-D structural sold B. BEAM4 3-D elastc beam B.3 COMBIN14 sprng-damper... 16

13 x LIST OF TABLES TABLE Page.1 Lumped Parameters and Cross-Sectonal Propertes of the Power Turbne Rotor.... Lumped Parameters and Cross-Sectonal Propertes of the Gas Generator Rotor Statc Analyss Dsplacements n y Drecton Statc Analyss Dsplacements n y Drecton Specfcatons of Support Bearngs Stffness and Dampng of Support System Computatonal Tme Comparson Comparson of Results between Staggered and Non Staggered Analyss Thermal Resstances of Heat Transfer Network Specfcatons of Support Bearngs Stffness and Dampng of Support System... 85

14 1 CHAPTER I INTRODUCTION 1.1 Overvew Today, engneers are becomng ncreasngly valuable to the manufacturng ndustry because of ther ablty to analyze practcal machnes wth the effcency and accuracy that were not possble just a couple of decades ago. Each year, mllons of dollars are saved by the applcaton of computer technques to ad n the desgn and n the analyss of turbo machnery for better performance and relablty. The goal of achevng lghter weght constructon and hgher output power has resulted n more and more flexble rotor and housng support desgns. Vbraton dynamcs n ths new breed of machnery s mportant and must be analyzed wth sophstcated analytcal tools that are capable of handlng computer smulaton models consstng of more than a thousand degrees of freedom. In hgh performance rotatng machnery such as gas turbne engne, the desgn trend has been toward hgh power output and hgh effcency. The requrements for the desgn trends result n the desgn of rotors whch are lghter, flexble and operatng above bendng crtcal speeds. The most famlar rotor dynamc problem nvolves the analyss of a sngle rotor revolvng n flud flms or rollng element bearngs whch are themselves mounted on a rgd foundaton. Theoretcally, ths stuaton lends tself readly to the soluton of the beam equaton for the rotatng shaft and s convenently analyzed by methods developed for beam structures. Sometmes, more complex problems are encountered. For example, n aerospace applcatons, the development of the mult-spool gas turbne engnes requres two or more concentrcally rotatng rotors to be analyzed smultaneously. In some cases, hgh flexblty n the rotor housng and support system can have a sgnfcant effect on the vbraton characterstcs. Provsons must be gven such that the Ths thess follows the style and format of the Journal of Sound and Vbraton.

15 rotor and the support housng system may be treated together to generate acceptable theoretcal results. Therefore, t s rather complex to analyze two rotors rotatng at dfferent speeds and nteractng through the support system and the ntermedate bearngs effcently. An ncreasng amount of attenton for blade loss smulatons of arcraft gas turbne engne has been pad because of the ncreased operatng speed and need for relablty demand. From ths motvaton, a hgh fdelty ball bearng and SFD model s appled to blade loss smulaton n an arcraft turbne engne. To predct accurate dynamc response of an arcraft turbne engne under hgh mbalanced load, a good model of a hgh order flexble dual rotor s necessary n addton to a hgh fdelty bearng and SFD model. The gas turbne engne, whch s composed of the power turbne and gas generator rotors, s modeled wth 38 lumped masses. A flexble housng s ncluded whose modal descrpton has been mported nto TAMU code from ANSYS. The support system for the rotors s modeled wth a hgh fdelty ball bearng and SFD model, whle the ntermedate bearngs and support bearngs for the gas generator rotor are modeled wth lnear stffness and damper. There s also a seal rub ncluded n the smulaton whch protects the bearngs from permanent damage. The blade loss smulaton results provde the whrl ampltude of the power turbne rotor, contact load and stress n the bearngs, transmssblty, SFD pressures and temperature growths of the bearngs. 1. Lterature Revew Several methods of analyss are avalable to tackle these large and complex problems. Perhaps the most wdely used technque for general purpose structural analyss s the drect matrx approach n whch the structure s dvded nto a fnte number of elements. The equaton of moton s developed n matrx form by an assembly of the element mass, stffness and dampng matrces usng ether a fnte element or a dscretzed mass representaton. Ths matrx equaton of moton s then solved drectly wth hgh speed dgtal computers. The use of fnte element method n rotor dynamcs

16 3 was ntroduced by Ruhl and Booker [1]. Internal dampng effects were ncorporated nto the fnte element rotor model by Zorz and Nelson []. However, the drect matrx approach requres large amounts of computer storage and executon tme. A reducton of the system sze that the computer has to handle at one tme can be accomplshed by the component mode approach. In ths method, a large structure s parttoned nto a number of substructures. The modal nformaton for each ndvdual substructure s derved ether analytcally or from vbraton tests. The structure s reassembled n the modal coordnates by usng only a truncated number of modes for each substructure. In a large structure, the potental of ths method les not only n the ablty to represent substructures contanng thousands of degrees of freedom by a handful of normal modes, but also on the capablty that enable analysts to buld up accurate analytcal model of a complex structure by treatng only one porton of the total structure at a tme. Dynamc analyss by component mode synthess s used extensvely n the aerospace ndustry for the calculaton of undamped natural frequences of large ar-frame structures. Hurty [3] and Crag et al. [4] are among the early nvestgators usng ths method. The prmary emphass n ths analyss s drected towards the matchng of the dsplacements or forces exertng at the connecton boundares that bond the substructures together. To satsfy system contnuty at the connectons, t requres the substructures to share the common degrees of freedom at these ponts. Accordngly, a set of constrant equatons equal n number to the pars of common degrees of freedom s derved that expresses the knematc dependences among the generalzed coordnates relatng to the varous substructures. When the total number of degrees of freedom n a complex structure s too large for even modern dgtal computers to handle economcally, recourse s sometmes taken n whch a structure s sometmes represented by ts vbraton modes. The advantage of a modal representaton s that only a few of the lower frequency modes are usually requred to gve a good approxmate descrpton of a structure. The assumpton s that a dynamc system s unlkely to vbrate n ts hgher frequency modes because of the

17 4 relatvely hgh energy requrement. Thus, wth only a few of the vbraton modes, the problem sze s sgnfcantly decreased. Unfortunately, the reducton process alters the modal representaton of the appled loadng and can adversely affect the qualty of the calculated responses. Two methods that attempt to mprove the truncated modal representaton of the loadng are the mode acceleraton (MA) and the modal truncaton augmentaton (MTA) method. The mode acceleraton method s used prmarly by aerospace engneers for coupled load analyses and has not been used extensvely outsde of ths feld of specalzaton. The modal truncaton augmentaton method s much newer and s only begnnng to be mplemented n all felds of structural vbraton analyss. Dckens, Nakagawa and Wttbrodt [5] have compared both the Modal Truncaton Augmentaton (MTA) method [6] and the Modal Acceleraton (MA) method [7] theoretcally and numercally, and have concluded that the MTA method s superor to the MA method due to the followng reasons: (a) the MA method s an approxmaton of the MTA method; (b) due to added dynamcs, the MTA method gves overall better results than the MA method. There are few papers, n whch blade loss smulatons usng an arcraft power turbne model wth dual rotors were conducted. Stallone, M. J., et al. [8] developed an analytcal method based on the modal synthess to predct the transent response of an arcraft engne when a fan blade s lost and valdated ther results aganst expermental data. The analytcal tool accounts for rotor-casng rubs, hgh dampng and rapd deceleraton rates assocated wth blade loss events. Alam, M., and Nelson, H. D. [9] presented a shock spectrum procedure to estmate the peak dsplacement response of lnear flexble-rotor systems due to blade loss and appled to three types of rotors ncludng dual-shaft system. However, the detal of tme transent responses were not shown, and smple lnear stffness and dampng were used for the support system. Ths thess attempts to use the MTA method and further modfy t to perform an effcent and accurate smulaton of a blade loss event on a dual rotor arcraft gas turbne engne wth an enhanced support system.

18 5 The model for the support system s utlzed from Sun [1] whch employs a nonlnear angular contact bearng model whch has ball and race degrees of freedom and uses a modfed Hertzan contact force between the races and balls. A heat transfer network s establshed and the correspondng thermal equatons are derved usng thermal resstances and nodes so that the thermal growth s analyzed. The thermal model consders thermal heat sources such as power loss from mechancal contact between hgh-speed rotor and nner races and from drag torque. From the lterature revews n the blade loss smulaton, t becomes evdent that even though rollng element bearngs are one of essental components n an arcraft gas turbne rotor-support system, most papers on the numercal analyss for the blade loss smulaton have not consdered a large order model wth a detaled ball bearng and damper model. 1.3 Objectves A blade out event s dramatc, complex and fast movng. In ths thess, physcsbased modelng s used to predct the complex nteractons between the fan rotor, the hgh fdelty bearngs, and the engne casng durng a blade loss. Key objectves of ths research nclude mnmzng danger to the arcraft by accurately predctng destructve loads due to the mbalanced load followng a blade loss, preventng bearng damage after a blade out event and smulatng the response of the overall system n the most computatonally effcent manner. The prmary problem beng addressed s the accurate smulaton of a blade-out event n the most effcent manner. Relable smulatons of the loss of a blade are requred to ensure structural ntegrty durng flght as well as to guarantee successful blade-out certfcaton testng. The results generated by these analyses are crtcal for the teams desgnng several arplane components, ncludng the engne, nacelle, strut, and wng. In ths research, a novel approach s presented for the accurate blade loss smulaton of an arcraft gas turbne rotor mounted on hgh fdelty bearngs wth fnte

19 6 element squeeze flm dampers, seal rub and enclosed n a flexble casng. The Staggerng Analyss scheme mplementng the modal truncaton augmentaton (MTA) method provdes an effcent tool for modelng ths large order system wth localzed nonlneartes n the ball bearngs. Thus, the objectves for ths thess can be summarzed as follows: Integrate Hgh Fdelty Bearngs wth stress, temperature predctons and seal rub nto a large order modal based structural model. Import the modal descrpton of a flexble housng from any other commercally program lke ANSYS, NASTRAN, etc nto TAMU code. Apply Modal Truncaton Augmentaton method to hgh order rotor dynamcs problem and compare wth other exstng methods. Valdate the Staggerng Analyss scheme and mplement t to conduct long duraton smulatons (~ mn actual tme followng blade loss) wth temperature predcton. Increase computatonal effcency for repeated smulaton of system dynamcs. 1.4 Outlne Chapter II ntroduces the model used n the analyss and develops the fundamentals of the fnte element (FE) approach used n ths thess. It descrbes the Tmoshenko beam element n detal and uses t to develop the stffness matrx, the mass matrx and the gyroscopc dampng matrx. The second part of Chapter II descrbes the gas turbne rotor system, the support system and the FE model of the system used n the ntal analyss. The two spool arcraft gas turbne engne s taken as an example to llustrate the accurate and effcent predcton of the blade out event. In Chapter III, the prelmnary structural analyses are performed on the model to help understand the dynamcs of the model better. These nclude the statc analyss, the undamped modal analyss, the gyroscopc damped mode shape predcton, the crtcal

20 7 speed analyss and the steady state harmonc analyss. A bref descrpton wth the general equatons used s provded n ths chapter along wth the results. The blade-out response analyss usng modal based soluton algorthms are descrbed n Chapter IV. The response of a structure can be evaluated usng the modal response method whch s a relatvely smple and well establshed method. Ths method reduces the model to a much smaller number of dynamc degrees of freedom. Unfortunately, the reducton process alters the modal representaton of the appled loadng and can adversely affect the qualty of the results. Ths has been tred to overcome by mplementng and comparng three methods vz. Modal Dsplacement method, Mode Acceleraton method and the Modal Truncaton Augmentaton method. The response s predcted usng a non lnear equaton of moton and two hgh fdelty bearngs at the man power turbne locatons. A parametrc study s also done to understand the effect of varyng unbalanced load on dfferent output parameters. Chapter V dscusses the methodology and mplementaton of the staggered analyss scheme whch would enable effcent predcton of system response for an extended blade loss smulaton. Ths method enhances the advantages of the modal truncaton augmentaton method by performng ntermttent thermal and thermomechancal analyss. Ths method s also valdated by comparng t to a standard modal based soluton algorthm wthout any staggerng and the results wth the tme savngs are tabulated. Chapter VI brefly dscusses the flexble housng wth the undamped modes as t s mported from ANSYS. It also descrbes the modfed Hertzan contact force and equvalent dampng used n smulatng the seal rub n the model. In Chapter VII, the FE model of the overall system s dscussed n detal and an analyss s performed to smulate the overall system response when a blade s lost on the power turbne. Ths s done by sudden applcaton of a large unbalanced load at the blade loss locaton. The smulaton ncludes seal rub, temperature predcton, stress predcton, etc. The results of the smulaton are presented wth detaled explanatons.

21 8 VIII. The conclusons drawn from the work n ths thess are summarzed n Chapter

22 9 CHAPTER II FINITE ELEMENT MODEL OF THE SYSTEM The requrement of hgh specfc power output for gas turbne engnes has resulted n hghly flexble rotor desgns wth rotors typcally operatng above several crtcal speeds. The use of rollng element bearngs, wth low nherent dampng, makes t dffcult to reduce vbratonal ampltudes and dynamc loads transmtted to the rotor supportng structure. Operaton over a wde range of speed and power levels aggravates the dynamc problems that are often encountered. The analyss of an arcraft engne s consderably more complex than that of a conventonal turbo rotor. It s complcated by havng two rotors whch rotate at dfferent speeds and nteract through the ntermedate dfferental bearngs. Because the moton of one rotor s affected by the other dynamc analyss must be performed on both rotors smultaneously. The two spool arcraft engne s an example of a class of machnery that may be modeled by two rotatng shaft structures connected laterally n parallel wth each other. In ths chapter, the gas turbne rotor system s descrbed n detal along wth the methodology used for modelng the system. The system here refers to the rotatng shaft structures wth ther ntermedate bearngs and the support system.1 Evaluaton of System Matrces.1.1 Tmoshenko Beam Element The beam element [11] wll be assumed to be a straght bar of unform cross secton capable of resstng axal forces, bendng moments about the two prncpal axes n the plane of ts cross secton, and twstng moments about ts centrodal axs. The followng forces are actng on the beam: axal forces F 1 and F 7 ; shearng forces F, F 3, F 8 and F 9 ; bendng moments F 5, F 6, F 11 and F 1 ; and twstng moments (torques) F 4 and F 1. The locaton and postve drectons of these forces are shown n Fg..1. The correspondng dsplacements u 1 u 1 wll be taken to be postve n the postve drectons of the forces. The poston and atttude of the beam element n space wll be

23 1 specfed the co ordnates of the pth end of beam and by the drecton cosnes of the x axs (pq drecton) and the y axs, both taken wth respect to some convenent datum system co ordnate system, the latter beng requred to locate the drectons of prncpal axes of the cross secton. y x Fg.1. Tmoshenko beam element z Axal Forces (F 1 and F 7 ) The dfferental equaton for the axal dsplacement u of the unform beam s: du F1 = EA.1 dx where E Modulus of Elastcty A Area of cross secton l length of the element

24 11 Integratng drectly, F x = uea +. 1 C 1 where C 1 s a constant of ntegraton. Assumng that the left end of the beam at x= has a dsplacement u 1 whle the dsplacement s zero at x=l, C1 = F1l.3 Usng Eq.. and.3 for x=, we get, EA F 1 = u 1.4 l Also from the equaton of equlbrum n the x drecton t follows that F =.5 1 F 7 The ndvdual stffness coeffcents k j represent the element force F due to unt dsplacement u j when all other dsplacements are equal to zero. Hence, k k 1,1 7,1 F1 = u = 1 F7 = = u 1 EA l EA l whle all other coeffcents n the frst column of the stffness matrx are equal to zero. Smlarly, f u 1 = and we allow u 7 to be non zero, t can be shown from symmetry that EA k 7,7 =.7 l Twstng Moments (F 4 and F 1 ) The dfferental equaton for the twst θ on the beam s dθ F4 = GJ.8 dx where G Shear Modulus J Torsonal Moment of Inerta = (y + z ) da Integratng Eq..8, we get 4 C 1 A F x = GJθ +.9 and then by usng the boundary condton θ = at x = l we fnd that the constant of ntegraton s gven by.6

25 1 C1 = F4l.1 Snce θ = u 1 at x =, t follows from Eq..9 and.1 that GJ F 4 = u 4.11 l Usng the equlbrum condtons for the twstng moments, we have F =.1 1 F 4 Hence, F4 GJ k 4,4 = =.13 u l 4 F1 GJ k1,4 = =.14 u l 4 whle all other co-effcents n the fourth column of k are equal to zero. Smlarly, f u 4 = t can be demonstrated that GJ k 1,1 =.15 l Shearng Forces (F and F 8 ) The lateral deflecton v on the beam subjected to shearng forces and assocated moments s gven by: v = v b + v s.16 where v b s the lateral deflecton due to bendng strans and v s s the addtonal deflecton due to shearng strans, such that: dvs F dx =.17 GA s wth A s representng the beam cross sectonal area effectve n shear. The bendng deflecton for the beam held rgd at the rght hand sde s governed by the dfferental equaton d vb EI = F x F6.18 dx where I Normal Shear Area It has been assumed that the cross secton has the same propertes about the z and the y axes. Thus I s gven by

26 13 A I =.19 K where the shear co-effcent K s defned as [1] K (7 + 6µ )(1 + ( r / ro ) ) + ( + 1µ )( r = 6(1 + µ )(1 + ( r / r ) ) o / r o ). where µ - Posson s Rato r r o - Rato of nner radus to outer radus From ntegraton of Eq..17 and.18, t follows that EI v = F x 6 3 F6 x + C + C 1 F EI GA s x.1 where C 1 and C are the constants of ntegraton. Usng the boundary condtons, dv dvs F = = at x=, x=l. dx dx GA s v= at x=l.3 Eq..1 becomes 3 3 F x F6 x FΦxl l F EIv = + (1 + Φ) Fl where F 6 =.5 1EI and Φ =.6 GA l s The remanng forces actng on the beam can be determned from the equatons of equlbrum. and F =.7 8 F = F F l.8 F1 6 + Now at x =, v = u and hence from Eq..4 u 3 l F = (1 + ).9 1EI Φ Usng Eqs..5 and.7 to.9, we have

27 14 k, F 1EI = =.3 3 u (1 + Φ) l k k F F l 6EI 6 6, = = =.31 u u (1 + Φ) l F 1EI 8 8, = =.3 3 u (1 + Φ) l F1 F6 + Fl 6EI k1, = = u = u.33 (1 + Φ) l whle the remanng co-effcents n the second column are equal to zero. Smlarly, f the left hand sde of the beam s fxed, t can be shown from symmetry that k k 8,8 1,8 = k, = k 6, Bendng Moments (F 6 and F 1 ) In order to determne the stffness co-effcents assocated wth the rotatons u 6 and u 1, the beam s subjected to bendng moments and assocated shears. The deflectons can be determned from Eq..1, but the constants C 1 and C n these equatons must be evaluated from a dfferent set of boundary condtons whch are: and.34 v = at x=, x=l.35 dv dvs F = = at x=l.36 dx dx GA Eq..1 becomes F 3 EIv = ( x l 6 6F6 F = (4 + Φ) l s F6 x) + ( lx x ) As before, the remanng forces actng on the beam can be determned from the equatons of equlbrum,.e., Eq..7 and.8. Now at x= dvb dv dvs = = u 6.38 dx dx dx so that.37

28 15 F (1 6 + Φ) l u 6 =.39 EI (4 + Φ) Hence, from Eqs..7,.8,.37 to.39, k k k 6,6 8,6 1,6 F = u 6 6 F = u 8 6 F = u 1 6 (4 + Φ) EI = (1 + Φ) l F = u 6 6EI = (1 + Φ) l F6 + Fl ( Φ) EI = = u 6 (1 + Φ) l If the deflecton of the left hand end of the beam s equal to zero, t s evdent from symmetry that 1,1 k6,6.4 k =.41 Shearng Forces (F 3 and F 9 ) The stffness co-effcents assocated wth the dsplacements u 3 and u 9 can be derved drectly from prevous results. It should be observed, however, that wth the sgn conventon adopted n Fg..1 the drectons of the postve bendng moments n the yx and zx planes are dfferent. Therefore k k k k k k 3,3 5,3 9,3 11,3 9,9 11,9 = k = k = k, = k 8, = k 8,8 = k 6, 1, 1,8.4 Bendng Moments (F 5 and F 11 ) Here the same remarks apply as n the precedng secton; thus we have k k k 5,5 9,5 11,5 = k = k 6,6 = k 8,6 1,6.43

29 16.1. Stffness Matrx The results obtaned n the prevous sub sectons can now be compled nto a matrx equaton relatng the element forces to ther correspondng dsplacements.e. {F}=[K]{u}. The stffness matrx from ths relatonshp s gven by where K AE AE L L a c b c a d b d GJ GJ L L d e c f = c e d f AE AE L L b d a d b c a c GJ GJ L L d f c e c f d e a = c = 1EI = b 3 L (1 + Φ) 6EI = d L (1 + Φ) (4 + Φ) EI e = L (1 + Φ) ( Φ) EI f = L (1 + Φ).1.3 Mass Matrx Consstent Mass Matrx

30 17 The mass matrx M derved from the weghted ntegral formulaton of the governng equaton s called the consstent mass matrx, and t s symmetrc, postve defnte and non dagonal. The components of the mass matrx are usually of the form Ω = d N N M j j ρ.46 where N s the shape functon and ρ the densty. Calculatng and assemblng these components, we get.47 Where.48 n the transverse drecton. = c b f e c b f e JL JL a e d a e d AL AL c b c b JL M Y S a a AL M 3 / 6 / 3 / 6 / 3 / 3 / ρ ρ ρ ρ ρ ρ 36 * * 156 θ m u m a Ψ + Ψ = L L b m u m 3 * * θ + Ψ = Ψ * 4 4 * L L c m u m θ + Ψ = Ψ *36 *54 θ m u m d Ψ = Ψ L L e m u m 3 * *13 θ Ψ = Ψ * 3 * L L f m u m θ Ψ = Ψ 4 AL / u m ρ = Ψ L I m 3 / ρ Ψ θ =

31 18 Lumped Mass Matrx Dagonal mass matrces are known as Lumped Mass Matrces. The use of a lumped mass matrx n a transent analyss saves computatonal tme n two ways. Frst, for forward dfference schemes, lumped mass matrces result n explct algebrac equatons, not requrng matrx nversons. Second, the crtcal tme step requred for condtonally stable schemes s larger, and hence less computatonal tme s requred when lumped mass matrces are used. The lumped mass matrx for the Tmoshenko Beam element s gven by: Mt Mt Mt Ip It 1 M = It Mt Mt Mt Ip It It where A cross sectonal area L Elemental length Mt = ρal.49 Ip Mass Moment of Inerta n the axal x drecton It Mass Moment of Inerta n the transverse y or z drectons.1.4 Dampng Matrx The Dampng matrx s a sum of the structural dampng matrx, [C b ] and the gyroscopc dampng matrx, [G]. [ C] = [ Cb ] + [ G].5

32 19 Gyroscopc Dampng Matrx I p ω θ y ω I p ω θ z Fg.. Gyroscopc moments The gyroscopc matrx for an ax symmetrc rgd body rotatng about ts axs of symmetry, (assumed to be the x-axs) as shown n fg.. s: = ω ω p p I I G.51 where, p I - Polar Moment of Inerta about the axs of spn ω Speed of rotaton about the x-axs Ths s a skew symmetrc matrx,.e. G G T =.. Descrpton of the Dual Rotor System..1 Descrpton of the Gas Turbne Rotor System

33 Fg..3 Two spool arcraft turbne engne wth 8 bearngs Fgure.3 represents a schematc drawng of the two-spool gas turbne engne [13] used n ths analyss. The basc engne conssts of an nner core rotor called the power turbne, whch s supported by two man bearngs located at shaft extremtes. There are two ntermedate dfferental bearngs connectng the core power turbne to the gas generator rotor. The gas generator rotor conssts of a two stage generator turbne whch drves an axal compressor. It s supported prncpally by rollng element bearngs at four locatons. The typcal operatng speed for the gas generator s 15, RPM and the power turbne s 16, RPM... Support System A nonlnear angular contact bearng model s employed, whch has ball and race degrees of freedom and uses a modfed Hertzan contact force between the races and balls. Ths combnes a dry contact force and an equvalent vscous dampng force. Predcton of the maxmum contact load and the correspondng stress on an ellptcal contact area between the races and balls s made durng the blade loss smulatons. A fnte-element based squeeze flm damper (SFD), whch determnes the pressure profle of ol flm and calculates damper forces for any type of whrl orbt s utlzed n the

34 1 smulatons. Thermal growths durng blade loss n the support bearngs and SFD ol flm of the gas turbne engne are also estmated...3 FE Model of Complete System The computer model shown n Fg..4 has a total of 38 lumped masses (for the rotors) wth power turbne dvded up nto nodes and the gas generator dvded up nto 16 nodes. Each node s allowed to have 3 translatonal and 3 rotatonal degrees of freedom. The system has a total of 8 degrees of freedom. Polar moments of nerta n the rotors are consdered only at the turbne and compressor stages. The bearngs # and #4 are modeled as hgh fdelty bearngs, whle four bearngs n the gas generator and two ntermedate dfferental bearngs are modeled wth smple lnear stffness and dampers. The arrows on Fg..4 ndcate the locatons of mbalanced load. Both the mbalanced loads are out of phase by 18. Tables.1 and. show the physcal and cross-sectonal propertes of the power rotor and the gas generator rotor respectvely. F F Hgh Fdelty Bearngs Fg..4 Two spool arcraft turbne engne lumped parameter FE model

35 Table.1 Lumped Parameters and Cross-Sectonal Propertes of the Power Turbne Rotor Staton No. Weght (lb) Length (n) Shaft da out Shaft da n I (n^4) Ip Lb-n^ It Lb-n^ E*1-6 (lb/n^) Table. Lumped Parameters and Cross-Sectonal Propertes of the Gas Generator Rotor Staton No. Weght (lb) Length (n) Shaft da outsde Shaft da nsde I (n^4) Ip (lb n^) It (lbn^) E*1-6 Lb/n^

36 3 CHAPTER III PRELIMINARY STRUCTURAL ANALYSES The prelmnary structural analyses are performed usng the full system structural matrces. The varous types of analyses performed and the respectve equatons used for them are gven below. 3.1 Statc Analyss The equaton of equlbrum for statc analyses can be wrtten as: [K]{Q}={F} 3.1 where, [K] System stffness matrx {F} Statc nput force vector {Q} Statc deflecton vector The followng load case was run for the statc analyss: Case 1: 1 lb force actng n the postve y drecton at bearng #4 Table 3.1: Statc Analyss Dsplacements n y Drecton Locaton Full Model wthout reducton (nches) # Bearng E-5 Rotor Md Span E-5 Second Stage Power Turbne.7 #4 Bearng.3 Case : 5 lb force actng downwards on the power turbne md span

37 4 Table 3.: Statc Analyss Dsplacements n y Drecton Locaton Full Model wthout reducton (nches) # Bearng E-5 Rotor Md Span E-4 Second Stage Power Turbne 6.59 E-4 #4 Bearng 6.36 E-4 3. Undamped Modal Analyss Undamped normal mode frequences and vectors are extracted by means of the followng equatons: ([K] - ω [M]) [Φ] = 3. where, [M] - System mass matrx [Φ] - Egen vector matrx ω - Normal mode frequences The gyroscopc and dampng matrces are gnored for ths analyss. The modes of each of the rotors and the casng were obtaned ndependently of each other by omttng the effects of dsk gyroscopcs and the ntermedate bearngs. The undamped normal mode shapes below 11, RPM are presented here. Ths covers the range up to about 6 tmes the top operatng speeds of the rotors Gas Generator Rotor Undamped Modes Fgures 3.1 to 3.5 show the frst fve gas generator modes.

38 5 Fg 3.1 Gas generator undamped mode at N = 1,41 RPM Fg 3. Gas generator undamped mode at N = 5,9 RPM

39 6 Fg 3.3 Gas generator undamped mode at N = 34,685 RPM Fg 3.4 Gas generator undamped mode at N = 5857 RPM

40 7 Fg 3.5 Gas generator undamped mode at N = 9,49 RPM 3.. Power Turbne Rotor Undamped Modes Fgures 3.6 to 3.14 show the frst nne power turbne modes. It s clear from the value of the frequency that the gas generator s stffer than the power turbne.

41 8 Fg 3.6 Power turbne undamped mode at N = 38 RPM Fg 3.7 Power turbne undamped mode at N = 6943 RPM

42 9 Fg 3.8 Power turbne undamped mode at N = 14,37 RPM Fg 3.9 Power turbne undamped mode at N = 7,563 RPM

43 3 Fg 3.1 Power turbne undamped mode at N = 43,676 RPM Fg 3.11 Power turbne undamped mode at N =54,341 RPM

44 31 Fg 3.1 Power turbne undamped mode at N = 68,4 RPM Fg 3.13 Power turbne undamped mode at N = 9,8 RPM

45 3 Fg 3.14 Power turbne undamped mode at N = 93,481 RPM 3.3 Gyroscopc Damped Mode Shapes The gyroscopc damped mode shapes are generated consderng the effect of the gyroscopc and dampng matrces. Ths s done usng the equaton: where [Λ] = [Λ][Φ] = [Φ] [λ] 3.3 M I 1 C 1 M K O [C] = System dampng matrx ncludng gyroscopc terms [I] = Identty matrx [O] = Zero Matrx [λ] = Egen values matrx (dagonal matrx) Fgures 3.15 to 3.17 show the D and 3D mode shapes as the dsplacement n the y drecton aganst the shaft length.

46 33 Fg 3.15 D and 3D mode shapes (wth whrl drecton) at N = RPM Fg 3.16 D and 3D mode shapes (wth whrl drecton) at N = 1,397 RPM

47 34 Fg 3.17 D and 3D mode shapes (wth whrl drecton) at N =,54 RPM 3.4 Crtcal Speed Analyss Ths s performed smlar to the Mode Shape generaton, where the reference spn speed s varyng. It loops through the Power Turbne spn speeds from to 3 rpm n steps of 5 rpm, calculatng the egen values at each step. The crtcal speed analyss s useful to show the followng system propertes as a functon of shaft speed: Sub/super-synchronous frequences exctable at a partcular operatng speed. Confrms sngle analyss computaton of crtcal speeds wth λ=ω (ω = Spn Speed) Increasng separaton of forward and backward whrl modes wth shaft speed. Fgure 3.18 show the natural frequences vs speed plot for the gven problem.

48 35 Fg Campbell dagram natural frequences vs speed plot 3.5 Steady State Harmonc Response Analyss The general form of the equlbrum equaton s wrtten as: (-ω [M] + ω [C] + [K] ) {Q(ω)} = {F(ω)} 3.4 where [C] = [C b ] + [G(ω)] [C b ] = System Dampng Matrx [G(ω)] = Gyroscopc matrx Q(ω) = Steady State Response vector F(ω) = Input lnear forcng functon

49 36 An mbalanced load equvalent to 85 N (64 lbs) force at 16 RPM was appled to the power turbne md-span and second stage turbne wth a phase dfference of Π rads. The result s shown n fg Fg 3.19 Steady state harmonc response of second stage power turbne

50 37 CHAPTER IV BLADE-OUT RESPONSE ANALYSIS USING MODAL BASED SOLUTION ALGORITHMS 4.1 Modal Dsplacement Method The equatons of moton of a complete structural system are gven by: [ M ]{ q& ( t)} + [ C]{ q& ( t)} + [ K]{ q( t)} = { F( t)} + { NF( q, q&, t)} 4.1 where [M], [C] and [K] are the NxN mass, dampng and stffness matrces respectvely; { q ( t)},{ q& ( t)},{ q& ( t)} are the physcal dsplacement, velocty and acceleraton vectors respectvely; {F(t)} s the appled lnear force vector and { NF ( q, q&, t)} s the appled non lnear force vector. The loadng may be comprsed of several spatal load vectors {F } o each wth a correspondng tme varyng porton, r (t). In ths analyss, there s one spatal load vector for the sne part and one for the cosne part. The non-lnear force vector s represented as a unt force at the selected DOF multpled by the approprate force component at each tme step. Here we have vectors (for the radal drectons) at each of the hgh fdelty bearngs. Let n { P( t)} = { Fo } g( t) + { NF( q, q&, t)} = { Ro} r ( t) = 1 4. where n s the total number of load vectors appled; {R o} s the nvarant spatal porton and r (t) s the tme varyng porton for each of the loads. For a modal dsplacement response analyss, the physcal coordnates of eqn (4.1) are transformed to modal coordnates by a retaned set of egenvectors of the system. { q( t)} = [ Φ]{ χ( t)} 4.3 where [Φ] are determned from the general egen value problem: [ K][ Φ] = [ M ][ Φ][ Ω ] 4.4

51 38 The number of retaned egenvectors, gven by n, s typcally much less than N. If eqn (4.) s used to transform eqn (4.1) to modal coordnates: [ M ˆ ]{&& χ ( t)} + [ Cˆ]{ & χ( t)} + [ Kˆ ]{ χ( t)} = { Fˆ ( t)} 4.5 where ˆ T [ M ] = [ Φ] [ M ][ Φ] ˆ T [ K] = [ Φ] [ K][ Φ] T [ Cˆ] = [ Φ] [ C][ Φ] { Fˆ T ( t)} = [ Φ] { P( t)} Mode Acceleraton Method Accordng to the Mode Acceleraton (MA) method, the number of modes retaned accurately spans the frequency range of nterest; any loadng represented by the nonretaned modes wll produce a quas-statc response. To develop the MA algorthm, eqn (4.1) s wrtten as [ K ]{ q( t)} = { P( t)} [ M ]{ q& ( t)} [ C]{ q& ( t)} 4.7 Rewrtng nto modal coordnates, the approxmate dsplacements usng the MA algorthm are gven by: { q u ( t)} = [ K] 1 { P( t)} [ K] 1 [ M ][ Φ]{ & χ ( t)} [ K] 1 [ C][ Φ]{ & χ( t)} 4.8 The MA physcal veloctes and acceleratons are determned usng: { q& u { q&& u ( t)} = [ Φ]{ & χ( t)} ( t)} = [ Φ]{ && χ( t)} Modal Truncaton Augmentaton Method Now The pseudo statc response s [ K ]{ q} = { } 4.1 R o

52 39 T T [ Φ] [ K][ Φ]{ χ} = [ Φ] { R } T ω { χ} = [ Φ] { R } { χ} = [ ω ] 1 o T [ Φ] { R } o o The pseudo statc force represented by the modes can be wrtten as: { 1 T R s} = [ K][ Φ]{ } = [ M ][ Φ][ ω ][ ω ] [ Φ] { Ro} T R s} [ M ][ Φ][ ] { Ro} 4.11 χ 4.1 { = Φ 4.13 The spatal load { R } does not completely represent the retaned modes n a modal o reducton technque. Let the porton of the load vector { R } that s not represented by the o modes be the force truncaton vector, { R } whch can be evaluated by subtractng the t modally represented load vector { R s} from the appled spatal load vector{ R o}. The force truncaton (FT) vector s gven by: { R } = { R } { R } 4.14 t o s The modal truncaton augmentaton (MTA) method attempts to correct for the nadequate representaton of the spatal loads n the modal doman by creatng addtonal pseudo egen or MT vectors to nclude n the modal set for the response analyss. Ths modal truncaton (MT) vector s derved from the statcally truncated dsplacement vector {X} obtaned from the force truncaton by solvng [K]{X}={R t } 4.15 Now form T [ K] = { X} [ K]{ X} T [ M ] = { X} [ M ]{ X} 4.16 and solve the reduced egenvalue problem as: [ K ][ Q] = [ M ][ Q][ ϖ ] 4.17 Form the MT vectors {P} whch have the assocated frequences [ϖ ] { P } = { X}[ Q] 4.18

53 4 The MT vector s appended to the modal set [Φ] for the modal response analyss. The augmented egenvector set that s used for the modal analyss s: ~ [ Φ ] = [ Φ P] 4.19 For several appled spatal load vectors, the vector {X} above s replaced by a matrx whose columns are equal to the number of ndependent spatal loads appled. Ths gves several MT vectors whch are appended to the reduced egen vector set. The physcal coordnates are now transformed nto the modal coordnates as: ~ { q( t)} = [ Φ]{ χ( t)} ~ { q& ( t)} = [ Φ]{ & χ( t)} 4. ~ { q&& ( t)} = [ Φ]{ && χ( t)} Usng Eq. (4.) to transform Eq. (4.1) yelds ~ ~ ~ ~ [ M ]{& χ ( t)} + [ C]{ & χ( t)} + [ K]{ χ( t)} = { F( t)} 4.1 where ~ ~ T ~ [ M ] = [ Φ] [ M ][ Φ] ~ ~ T ~ [ C] = [ Φ] [ C][ Φ] ~ ~ ~ 4. T [ K] = [ Φ] [ K][ Φ] ~ ~ T { F( t)} = [ Φ] { P( t)} The physcal acceleraton, velocty and dsplacements are found usng the pseudo modal set of vectors after completng the modal response analyss. 4.4 Theoretcal Comparson of the Methods To compare the MA and MT methods, the exact soluton of eqn. 4.1 s wrtten n two parts as: { q( t)} = { q ( t)} ( q ( t)} 4.3 s + t The frst part, { q s ( t)} s the porton of the dsplacement soluton obtaned from the retaned modes and the second part, { q t ( t)} s the porton of the dsplacement soluton lost due to the modal truncaton. Substtutng eqns and 4.3 nto equaton 4.1,

54 41 [ M ]{ q& ( t)} + [ C]{ q& ( t)} [ K]{ q ( t)} s s + s + [ M ]{ q& ( t)} + [ C]{ q& ( t)} + [ K]{ q ( t)} = ( R + R ) r ( t) 4.4 t t The modal response solved from equaton (4.5) yelds a soluton n the physcal doman that satsfes the equaton: n [ M ]{ q& ( t)} + [ C]{ q& ( t)} + [ K]{ q ( t)} = ( R ) r ( t) 4.5 s Subtractng equaton (4.5) from (4.4) we get s s = 1 n [ M ]{ q& ( t)} + [ C]{ q& ( t)} + [ K]{ q ( t)} = ( R ) r ( t) 4.6 t t t = 1 Equaton 4.6 s the porton of the soluton not represented by the modes retaned n the analyss. Both the MA and the MT methods attempt to fnd a soluton for eqn. 4.6 wthout calculatng the non retaned modes by approxmatng the non-modally represented soluton. The dfference between the two methods s n the approxmaton for {q t (t)}. The MA method approxmates {q t (t)} by standard modal truncaton plus a correcton for the statc response of the porton of P(t) that s omtted by the mode expanson. The MT method approxmates t by a dynamc soluton whch averages all the non retaned modes by a truncaton augmentaton vector. Thus, the MA method s an approxmaton to the MTA vector method. Also, the MA method corrects only the physcal dsplacement whereas the physcal velocty and acceleraton are smply represented usng only the retaned modes. The MTA method attempts to correct the physcal dsplacements, velocty and acceleraton. Due to the added dynamcs and the velocty and acceleraton correctons, t s expected that the MTA method would gve better results overall than the MA method. t t s n = 1 s t 4.5 Smulaton Results Blade loss smulatons are conducted to llustrate mplementaton of the hgh fdelty ball bearng FE SFD model and effcency of the MTA method [14]. Numercal ntegraton (Refer Appendx A) of the equatons of moton for arcraft turbne engne mounted on the support system s performed usng the Newmark Beta method [15]. The

55 4 bearng dmenson, materal characterstcs and SFD damper dmenson for the locatons # and #4 are lsted n Table 4.1. A smple support model wth lnear stffness and damper s utlzed for all the locatons except # and #4 snce they are the most crucal n the event of a blade loss. The stffness and dampng coeffcents are lsted n Table 4.. Table 4.1 Specfcatons of Support Bearngs Dmenson Brg # (1J) Brg #4 (17H) Geometry: Bore dameter, BD [cm] Outsde dameter, OD 9 11 [cm] Wdth [cm] 1.6 Number of balls 1 7 Dameter of a ball [mm] Intal contact angle [deg] Axal preload [N] 8 1,1 Number of rows 1 1 Materal: Densty of ball: steel [g/cm 3 ] Densty of races: steel [g/cm 3 ] Elastc modulus of ball 8 8 [GPa] Posson s rato of ball.3.3 Elastc modulus of races 8 8 [GPa] Posson s rato of races.3.3 Support system: Stffness of centerng sprng [N/mm] Radus of SFD journal 1 [cm] 8 [cm] Length of SFD journal 4 3. [cm] Vscosty of flud 7 7 lubrcant [cp] Radal clearance [mm]

56 43 Table 4. Stffness and Dampng of Support System Bearng No. Stffness (N/m) Dampng (N-sec/m) ND 1.751E E3 # E E3 # 8.756E E3 #3 7.5E E3 FDB 8.756E7 3.5E3 ADB 8.756E7 3.5E3 The pseudo modal set for the gven set of loads (lnear & non lnear) s determned as follows. There s one spatal load vector for the sne part and one for the cosne part of the appled mbalanced load. The non lnear force vector s represented as a unt force at selected DOF multpled by the approprate force component at each tme step. Here we have vectors (for the radal drectons) at each of the hgh fdelty bearngs. The power turbne s spnnng at 16, rpm. Before the blade loss occurs, several revolutons of tme transent solutons under low mbalanced load are performed to show proper status of the moton of the rotor-support system. Sudden ncrease of mbalanced load s then appled at the second stage (node #) of the power turbne and at the geometrc center of the power turbne (node #11). Numercal soluton s obtaned usng Newmark Beta method [15]. The total ntegraton tme s 35 rev. for each case and the tme step 1-7 sec s used. Fgure 4.1(a) shows the transent responses of the second stage power turbne rotor under an unbalanced force of,847 N. After several overshoots, steady-state peakto-peak ampltude of.9 mm s observed. The orbt plot n Fg. 4.1(b) shows the ntal several overshoots and steady-state whrls of an ellptcal shape. The transent response of the SFD journal at bearng # and steady-state peak-to-peak ampltude of.1 mm are shown n Fg. 4.(a), whle the transent response of the SFD journal at bearng #4 s shown n Fg It can be notced that the steady-state ampltude n x-axs at bearng #4

57 44 s about 4 tmes greater than that at bearng # and the ampltude n y-axs at bearng #4 s damped out. Fg. 4.1 (a) Transent response and (b) orbt plot of the second stage of the power turbne for 35 rev.:, y-axs; - -, z-axs Fg. 4. (a) Transent response and (b) orbt plot of the SFD journal at brg # for 35 rev.:, y-axs; - -, z-axs

58 45 Fg. 4.3 (a) Transent response and (b) orbt plot of the SFD journal at brg #4 for 35 rev.:, y-axs; - -, z-axs Transmssblty s defned as the rato of the force transmtted to the support system to the unbalanced force and s descrbed as: FX + Fs T = (4.7) F u where F X s the dampng force from the SFD, F s s the centerng sprng force and F u s the unbalanced force. The transmssblty at bearng # shown n Fg.4.4 does not have much dfference between the ntal and fnal values, whle that at bearng #4 has an ntal value of.3 and oscllates about.5 after 35 rev. Fgure 4.5 shows the maxmum contact loads between the races and balls at each tme step. The ntal load due to preload s appled before the blade loss occurs whch causes the ntal offset. It s observed that the contact loads at bearng # are hgher than those at bearng #4 even though mbalanced load locaton at the power turbne blade s closer to bearng #4. Ths s due to the fact that the number of balls of bearng # (1) are lesser than those of bearng #4 (7) and hence the contact load appled on each ball n bearng # s greater. The maxmum stress due to compressve contact load s shown n Fg.4.6. It has the same

59 46 trend as the contact load because the contact stress s a functon of the contact load. Contact stress σ,e s determned from 3Q σ, e = π a, e b, e, e (4.8) where a, b are the sem-major & sem-mnor axes of an ellptcal contact area and Q,e s nternal bearng load due to thermal expansons..5. Trans. at Brg # tme, sec.4 Trans. at Brg # tme, sec Fg. 4.4 Transmssblty (a) at brg # and (b) at brg #4

60 47 force, N IR contact, brg # force, lbs OR contact, brg # tme,sec tme,sec force, N force, lbs 5 IR contact, brg # OR contact, brg # force, N 3 5 force, lbs force, N 5 1 force, lbs tme, sec tme,sec Fg. 4.5 Maxmum contact loads (a) at brg # IR, (b) at brg # OR, (c) at brg #4 IR, and (d) at brg #4 OR

61 48 IR contact, brg # x OR contact, brg # x stress, MPa 15 stress, ps stress, MPa stress, ps tme,sec tme,sec 14 IR contact, brg #4 x OR contact, brg #4 x stress, MPa stress, ps stress, MPa stress, ps tme, sec tme,sec Fg. 4.6 Maxmum contact stress (a) at brg # IR, (b) at brg # OR, (c) at brg #4 IR, and (d) at brg #4 OR Fgures 4.7, 4.8 show the maxmum and mnmum pressure n the FE SFD. A supply pressure of.1 MPa s used. Due to the hgh eccentrcty, the maxmum pressure at bearng #4 s much hgher than that at bearng #, whle the mnmum pressure at bearng #4 s lower than that at bearng #. Zero pressure ndcates ol cavtaton.

62 max. pressure, Mpa tme, sec. mn. pressure, MPa tme, sec Fg. 4.7 Maxmum and mnmum pressures n brg # SFD.4 max. pressure, Mpa tme, sec.5 mn. pressure, MPa tme, sec Fg. 4.8 Maxmum and mnmum pressures n brg #4 SFD

63 5 Table 4.3 shows a comparson of the computatonal tme taken for the dfferent methods under dentcal loadng condtons on an Intel P3 933 MHz processor wth 51 MB RAM and GB hard dsk. Method Numercal Integraton of Physcal Co-ordnates Table 4.3 Computatonal Tme Comparson No. of Rotor Maxmum Tme taken (n Tme Savngs modes used Error (%) hrs) (%) - % 3.5 hrs % All Modes All (8).1 %.9 hrs.5 % Mode Dsplacement 15.3 % 14.9 hrs 36.6 % Mode Acceleraton 14. % 16.4 hrs 3. % Modal Truncaton Augmentaton 8.9 % 14.3 hrs 38.8 % 4.6 Parametrc Study: A study of the mportant parameters s performed wth the amount of unbalanced force varyng from 71 N to 11,387 N. The whrl ampltude change of the power turbne versus the unbalanced force s shown n Fg The very frst overshoot after blade loss occurs s called transent ampltude here.

64 Transent Steady State p.7.6 whrl ampltude, mm unbalance force, N Fg.4.9 Whrl ampltude of the power turbne The transent moton s greater than the steady-state moton up to 5694 N, whch means that the SFD damps out the transent vbraton but after 5694 N, the steady-state whrl moton s greater than the transent moton. The maxmum whrl moton of the SFD journal at bearng # versus the unbalanced force n Fg. 4.1 shows the lnear relatonshp between the ampltude and the force. A contact stffness of 1.75E+5 N/mm between the SFD journal and outer bearng s used. In the last two cases n Fg. 4.1, the SFD journal at bearng #4 bottoms out and rubs aganst the outer race. Ths results n no dampng n the system whch results n the non lnear relatonshp.

65 Bearng # Bearng #4 Nomnal Clearance 1 whrl ampltude, mm unbalance force, N Fg.4.1 Maxmum whrl ampltude of the SFD journal Fgure 4.11 shows the maxmum contact loads at bearng # and at bearng #4 respectvely. As observed n Fg. 4.11, the contact load at the outer race s slghtly hgher than that at the nner race because of centrfugal force on balls. However, the contact loads at both races are almost same n the last two hgh mbalanced cases. The contact loads at bearng #4 ncreases enormously n the two cases. Fgure 4.1 llustrates that the contact stress at bearng # ncreases lnearly and the contact stress at bearng #4 s lower than that at bearng # up to 5694 N due to the hgher number of balls but t ncreases enormously n the last two cases. The maxmum and mnmum SFD pressures at both bearngs are compared n Fg The maxmum pressure at bearng #4 ncreases to 1. MPa and that at bearng # ncreases to.6 MPa. The excessve nonlnear flm force produced by hgh eccentrcty can make the SFD lock up and cause an ncrease n the transmssblty as shown n Fg

66 53 Fg Maxmum ball contact loads at (a) brg # and (b) brg # Inner Race of brg # Outer Race of brg # Inner Race of brg #4 Outer Race of brg #4 Maxmum Contact Stress at Hgh Fdelty Bearngs 5 45 contact stress, MPa unbalance force, N Fg. 4.1 Maxmum contact stress vs unbalanced load

67 54 Fg Pressure n SFD n (a) brg # and (b) brg #4 1.4 Bearng # Bearng # transmssblty unbalance force, N Fg Transmssblty plots

68 55 The smulaton results show that the SFD damps out the transent whrl ampltude of the power turbne rotor up to an unbalanced force of 5694 N followed by an enormous ncrease n the steady state whrl ampltudes because the SFD wth hgh eccentrcty locks up the rotor support system. Ths phenomenon s also observed n the maxmum contact loads, whch ncrease over 5 tmes hgher than the contact loads at 5694 N, and n the transmssblty, whch ncreases from.6 to 1.1. The maxmum contact stress also ncreases proportonally wth the 1/3 power of the contact load as the unbalanced force ncreases. The ol flm cavtaton at bearng #4 close to the mbalanced load locaton occurs at an unbalanced force of 5694 N, whle the cavtaton at bearng # occurs at 8541 N. It s also observed n the transmssblty plot n Fg that the transmssblty at bearng #4 remans at.3 up to an unbalanced force of 5694 N and then ncreases to 1.3.

69 56 CHAPTER V STAGGERED ANALYSIS SCHEME The computaton tme that s requred to solve for the response of large order systems becomes prohbtve when an extended blade loss smulaton result s desred. Ths gans even more mportance when the analyss must be performed for many repettons n an optmal parameter search. Wth ths motvaton, a new scheme was developed wheren the computatonal effcency s greatly ncreased wth a neglgble loss n the accuracy of the soluton. 5.1 Descrpton of Staggered Analyss Scheme In ths scheme, the blade loss analyss s carred out usng the Modal Truncaton Augmentaton method ntermttently wth thermal only regons between the thermomechancal regons. A thermal only regon s defned as one where a constant power source s assumed and only the bearng thermal equatons are ntegrated wth a relatvely large tme step. A thermo-mechancal regon s defned as on where the complete system equatons and the bearng thermal equatons are ntegrated smultaneously at each tme step. The assumpton n formulatng ths scheme s that once the mechancal steady state s reached, there s no sgnfcant change n the response. Thus, a constant power loss (rms value for the last 5 cycles) s calculated and appled to the thermal only regon. After the completon of the thermal only regon, the fnal temperatures are appled back to the system whch then calculates the thermal growth, change n vscosty, etc and performs tme ntegraton wth the new values n thermo-mechancal regon. Ths process can be used alternately several tmes to obtan an extended blade loss smulaton for several mnutes. The length of each of the thermal only and the thermo-mechancal regons s decded on a tral and error bass. Fgure 5.1 shows the computatonal flow dagram used to mplement ths method.

70 57 Start Input Rotor data, casng data, bearng data Calculate system matrces and ntalze force vector Determne retaned modes and reduce the system EOM nto modal coordnates Integrate modal EOM ) ) ) ) [ M ]{&& χ( t)} + [ C]{ & χ( t)} + [ K]{ χ( t)} = { P( t)} Integrate thermal eqn dt m C p = Q + H dt where Q s determned from conducton and convecton eqns for each temperature node, and H s the power loss at node No Determne thermal expanson usng the thermo elastcty eqn ξ, e ε, e = ( T s + TL. e ) ( 1+ ν, e ) r, e 3 for nner / outer race and shaft εb = ξb rb Tb for balls Calculate contact force, stresses, change n ol vscosty, etc due to temperature change Tme = Thermal only tme? Yes Calc. avg. p loss No Tme = Fnsh tme? Perform ntegraton of thermal eqn only wth rms value of power loss Yes Output work-space data Plot results Stop Fgure 5.1. Computatonal flow dagram

71 58 5. Verfcaton of Staggered Analyss Scheme Fg.5. Staggered analyss scheme tme lne: --, Thermal only regon;, Thermomechancal regon. Fgure 5. shows the tme lne used for the verfcaton of the staggered analyss scheme. Snce the results from ths scheme had to be compared wth the results from full ntegraton, a short ntegraton was performed for a total of sec. Both the methods use a tme step of 1E-6 sec n the thermo-mechancal regon whle t s.1 s n the thermal only regon for the staggered analyss scheme. The power turbne s spnnng at 18, rpm, whle the gas turbne at 15, rpm wth an mbalanced load of 5 lbs each appled at the second stage and md of the power turbne. An ntal temperature of 3 C s used at all the temperature nodes Smulaton Results wth Thermo-Mechancal Integraton The followng plots show some of the mportant results obtaned from the Thermo-Mechancal only ntegraton scheme.

72 59 Fg. 5.3 (a) Transent response and (b) orbt plot of the second stage of the power turbne, y-axs; - -, z-axs Fg. 5.4 (a) Transent response and (b) orbt plot at brg # :, y-axs; - -, z-axs

73 6 Fg. 5.5 (a) Transent response and (b) orbt plot at brg #4:, y-axs; - -, z-axs Fgures 5.3, 5.4 and 5.5 show the transent response and the orbt plots of the second stage power turbne, bearng # and bearng #4 respectvely. Snce the mbalanced load s small, there s an ntal transent and then t reaches the steady state very quckly. The power loss at bearng #4 s hgher than that at bearng # as notced n fg Fgures 5.7 and 5.8 show the temperature plot of the bearng nner race, outer race, ball and ol. Snce the mbalanced loads appled are closer to bearng #4 the temperatures are hgher than that of bearng #. Ths s also notced n the transmssblty plot n fg 5.9.

74 Power Loss n Bearng, W tme, sec Fg 5.6 Power loss n:, brg #; - -, brg # Temperature, o F Temperature at bearng ball Temperature, o F tme, sec Fg 5.7 Temperature of bearng nner race and ball at:, brg #; - -, brg #4

75 Temperature, o F Temperature at ol flm Temperature, o F tme, sec Fg 5.8 Temperature of bearng outer race and ol flm at:, brg #; - -, brg # Transmssblty tme, sec Fg 5.9 Transmssblty at:, brg #; - -, brg #4

76 Smulaton Results Usng Staggered Analyss Scheme The followng plots show some of the mportant results obtaned from the Staggered Analyss scheme. Fg. 5.1 (a) Transent response and (b) orbt plot of the second stage of the power turbne, y-axs; - -, z-axs Fg (a) Transent response and (b) orbt plot of at brg # :, y-axs; - -, z-axs

77 64 From fg. 5.1, 5.11 and 5.1 we see that the transent response plots and the orbt plots at all the man locatons show very close concordance wth the plots obtaned n fg. 5.3, 5.4 and 5.5. Fgure 5.13 shows the power loss n the bearng usng the staggerng effect. Ths also follows the non staggerng plot n fg. 5.6 wth an ntal value of 13 W and 55 W and droppng down to 1 W and 5 W for bearng #4 and bearng # respectvely. Fg. 5.1 (a) Transent response and (b) orbt plot at brg #4:, y-axs; - -, z-axs

78 Power Loss n Bearng, W Power Loss n Bearng, W tme, sec tme, sec 1 Power Loss n Bearng, W tme, sec Fg 5.13 Power loss n:, brg #; - -, brg #4 95 Temperature, o F Temperature at bearng ball Temperature, o F tme, sec Fg 5.14 Temperature of bearng nner race and ball at:, brg #; - -, brg #4

79 Temperature, o F Temperature at ol flm Temperature, o F tme, sec Fg 5.15 Temperature of bearng outer race and ol flm at:, brg #; - -, brg #4.4.4 Transmssblty.3..1 Transmssblty tme, sec tme, sec.4 Transmssblty tme, sec Fg 5.16 Transmssblty at:, brg #; - -, brg #4

80 67 Fgures 5.14 and 5.15 show the temperature plot obtaned by usng the rms value of power loss n the thermal only regon. Comparng the fnal values of the temperature wth that obtaned from the non staggerng plot, we can say that the approxmaton of the power loss s not far from the true value. Ths s further proved by comparng the transmssblty plots of fg and Summary The followng table summarzes the results obtaned from both the staggerng and non staggerng analyss at the power turbne bearng locatons. From ths, we can conclude that the staggerng scheme can be appled to the present blade loss problem wthout any major loss n accuracy (less than 5 %) whle mprovng the computatonal speed by almost 5 tmes.

81 68 Table 5.1 Comparson of Results between Staggered and Non Staggered Analyss Smulaton Results Method Used Full Integraton Staggered Analyss Bearng # Bearng #4 Bearng # Bearng #4 Peak dsplacement of power turbne, mm Steady-state dsplacement of power turbne, mm Maxmum power loss n ol flm, watt Maxmum power loss n bearng, watt Peak Transmssblty Steady State Transmssblty Fnal temperatures, F (Intal temperature s 8 F) Inner race Ball Outer race Ol flm Tme Taken *, hrs 4.8 hrs 4.65 hrs * - On a P3 Intel 933 MHz processor wth 51 MB RAM.

82 69 CHAPTER VI HOUSING, CONTACT AND THERMAL MODEL In the prevous chapters, all the bearngs except the nter shaft bearngs were connected to the ground. But n realty, ths s not true snce the rotors and bearngs are enclosed n a flexble housng whch n turn s connected to the arcraft wng. Also, seals are bult nto the model whch rubs aganst the rotor durng hgh mbalanced load to protect the bearngs. Hence, n ths chapter both the flexble housng and seal rub s ncluded nto the model. 6.1 Descrpton of Flexble Housng Fg 6.1. Flexble housng from ANSYS

83 7 The housng s bult usng 14 SOLID45 (8 node sold element wth 3 DOF at each node), 8 BEAM4 ( node 3D elastc beam wth 6 DOF at each node) and 6 COMBIN14 (longtudnal sprng damper element) elements (Refer Appendx B for ANSYS element descrpton). It conssts of a total of 118 nodes wth a total of 34 degrees of freedom. The housng has a dameter of 1 n, length of 57 n wth a wall thckness of.5 n. The top face of the housng s rgdly fxed.e. all DOF at all nodes on top face s zero. The materal constants used for the housng are: Young s Modulus, E = 3E7 ps =.6E11N / m 3 Densty, ρ = 783kg / m =.83lb / n Posson s Rato, υ = Casng Undamped Modes from ANSYS Fgures 6. to 6.6 show the frst few undamped modes less than 11, rpm that are mported from ANSYS snce they are the contrbutng modes. They are shown below. Fg 6. Casng undamped mode at N =,13 RPM

84 71 Fg 6.3 Casng undamped mode at N = 9,5 RPM Fg 6.4 Casng undamped mode at N = 4,33 RPM

85 7 Fg 6.5 Casng undamped mode at N = 48,98 RPM Fg 6.6 Casng undamped mode at N = 15,84 RPM

86 73 6. Seal Rub Contact Model Rotor rub aganst a non rotatng part generates a very complex rotor vbraton, whch may lead to total destructon of the machne n merely a few rotatons. The most common type of rub n rotatng machnery s blade tp and seal rub both of whch can be caused by thermal expanson. z F t Rotor F r F r β ω F t y Rub Rng Fg 6.7 Rub rng contact model The fgure shows the rub rng contact model based on a modfed Hertzan contact force and equvalent dampng used n ths problem [16]. The radal dsplacement from the rotor center and ts dervatve s gven by: δ = Y + Z C 6.1 r where Z r,y r,- Radal rotor dsplacement C Rub rng clearance r

87 74 Y Y& r r + Z r Z& & r δ = 6. δ + C The radal contact force and the tangental force s gven by: where k 1 9 F r = k l δ (1.5α & δ + 1) l = ( lc ) ; l c Contact length α =. ~.8 for steel on steel rub F t = µ F c 6.4 where µ Co effcent of frcton =.1 The forces are resolved nto the y and z drectons as: Z β tan 1 r = Y r F = F cos β + F F y z c = F sn β F c t t sn β cos β These forces are added to the rght hand sde of the equatons of moton at the approprate degrees of freedom. The Power Loss due to the rub s defned as: H = r Ft rω 6.6 where r radus at the rub locaton ω Speed of the rotor, rad/s Thermal Model Actual heat transfer n a bearng needs 3 dmensonal analyss but assumng that the heat flux s unform n the radal drecton and symmetrc to the axal drecton one dmensonal radal heat transfer equatons are developed usng bulk heat masses. The heat transfer n the equatons s descrbed by thermal resstances as the connectors between nterested temperature nodes. Crucal temperature nodes n the cross-secton of an angular contact ball bearng supported on SFD are shown n Fg.6.8. The length L s s the dstance from the axal center of nner race to the end of rotor and the lengths L j, L h

88 75 denote the axal length of the SFD journal and housng, respectvely. The wdths w, w e of nner and outer rngs are assumed to be same. Fgure 6.9 shows the heat transfer network of the ball bearng wth grease packed, whch conssts of the thermal resstances and heat sources. The heat sources H,e and the heat source H sf due to vscous dsspaton energy n flud flm s obtaned from [1]. H r s the power loss due to rub whch s obtaned from Eqn T T T sf d m T Fg. 6.8 Cross-sectoned bearng wth thermal nodes

89 76 In addton, the followng assumptons are utlzed n modelng: (a) Heat generaton from the contacts acts on shaft, balls, nner race and outer race. (b) Ball bearng s modeled wth lumped heat mass elements. (c) Each heat mass has unform temperature dstrbuton. (d) Conducton heat transfer through flud flm. T Rh Th Rsf Hsf Tsf Rsf1 Tj Rj To Re He TLe RL RL Tb Rb / Rb / H TL R T Rsr Hr Ts Rsa T Fg. 6.9 Heat transfer network

90 77 The thermal resstances are a functon of materal propertes, geometry and heat transfer mode. For nstance, the thermal resstance R between the nner race contact wth balls and nner race bore radus s obtaned as follows. The radal heat flow of a cylnder of nsde radus r s, outsde radus r and length w can be descrbed as dt q = ka dr 6.7 or dt q = kπrw dr 6.8 where k s the thermal conductvty of the cylnder and A s the area normal to temperature gradent. Eqn. 6.8 has the boundary condtons as T = T at r = r s 6.9 T = T L at r = r 6.1 Snce the heat flow at any radal locaton s same, ntegratng Eqn. 6.9 becomes or r= r r= r s 1 q dr = πw k r T = T L T = T dt ( r r ) = πkw ( T T ) s q ln 6.11 Rearrangng Eqn q = TL T ln πkw L ( r rs ) R T = L T The electrcal analogy can be used to express the relatonshp between the temperature node and thermal resstance as shown n Fg Table 6.1 shows the thermal resstances n the thermal heat network. 6.1 ( r r ) ln R = πkw Fg D radal heat flow through cylnder and electrcal analogy s

91 78 Table 6.1 Thermal Resstances of Heat Transfer Network Ball/lubrcant Inner race/ shaft Outer race/journal R L rb k (πrw πnr l b ) R r ln r s = πk W R e r ln o r e = πk W e e R Le k (πr W πnr l e r b e b ) R sr 1 = πk W s R j rj ln r o = πk W j e R b 1 nk πr b b R sa = k L s sπrs 1 + h πr s s SFD Housng R R sf 1 sf ( rj + c ln = πk r ln = h ( r j sf πk sf sf / ) L L j + c sf j r j / ) R R hr ha r ln h r o 1 = + πk L h πr L = πk h h L ( r h h h o h h + r ) πh h h ( r 1 h r o ) R h = R R hr hr R ha + R ha Free convecton coeffcent h [1] s approxmated as a functon of the relatve temperatures, whch s ( T ). 5 h = 3 T [W/m - C] 6.13 From the heat transfer network and heat sources a thermal equaton of moton s developed. For the nner and outer races, half the mass s used to represent temperature node. To consder convecton heat transfer at the axal end of rotor and the ends of housng, small heat mass s assgned to these locatons and the convecton coeffcent h s changed accordng to the temperature dfference. The equaton for the axal end of rotor s

92 79 m dt T = T T T s1 s1 s s1 s1 C ps 6.14 dt Rsac Rsacv where m s1 s the heat mass, C p the specfc heat and T the ambent temperature. R sac, R sacv are the conducton and convecton thermal resstances of R sa, respectvely. The thermal equatons for the nodes T s to T sf are m C s ps dt dt T T R s s s s1 = 6.15 sr T T R sac m C p dt dt T T = R sr s T T R L 6.16 m C dt TL T TL Tb H = R R L p dt 1 dtb Tb TL Tb TLe H H e m bc pb = dt R R m C dt e Le pe dt 1 TLe Tb TLe Te H e = R R e where the thermal resstances R 1 = R R L L R b + R / b / and R = R R Le Le R b + R / b /. m e C pe dt dt e Te T = R e Le Te T j R j 6. Assumng the heat generaton due to vscous dsspaton n the squeeze flm s provded only to ol flm, m C j pj dt dt j T j Te T j Tsf = 6.1 R R j sf 1 dtsf Tsf T j Tsf Th m sf C psf = + H dt R R sf 1 sf sf 6. The thermal equaton for the housng s m h C ph dt dt h Th T = R sf sf Th T R hac ha Th T R hrc hr 6.3

93 8 where T ha, T hr are the temperatures at the axal and radal ends of the housng, respectvely, and R hac, R hrc are the conducton thermal resstances n the axal and radal drectons, respectvely. The thermal equatons for the ends of the housng are m m ha hr C C ph ph dt dt dt dt T ha ha h ha = 6.4 R T hac T T R hrc T R T hacv T T R hr hr h hr = 6.5 hrcv where R hacv, R hrcv are the convecton thermal resstances of R ha, R hr, each. Snce the temperature node T s at one rotor end s connected to the temperature node T s at the other end n the heat transfer network, Eqn should be updated to m C s ps dt dt s Ts T = R sr Ts T R sac s1 Ts T R where the thermal resstance R sc s descrbed usng the average rotor radus r and dstance L between two nodes as R sc sc s 6.6 L = 6.7 k πr s Fgure 6.11 shows the full thermal model used for the power turbne rotor. T T Rh Rh Th Th Rsf Rsf Brg # Hsf Tsf Brg #4 Hsf Tsf Rsf1 Rsf1 Tj Tj Rj Rj To To R e Re He TLe He TLe h = 3W / m o C RL RL H Rb / Tb Rb / TL RL RL H Rb / Tb Rb / TL h = 3W / m o C R R T T T Hr Rsr Ts Rsa T M = 84.5 lbs Hr Rsr Ts Rsa T T m=.79 lbs m=5.1 lbs Fg 6.11 Full thermal model for power turbne rotor

94 81 The thermal expanson of a hollow cylnder wth temperature dstrbuton T(r) n the radal drecton s estmated usng [1]. Assumng the temperature dstrbuton s lnear the thermal expanson ε e of the outer race and SFD journal s gven by ξ, e ε, e = ( T s + TL. e ) ( 1+ ν, e ) r, e where ξ e [m/m- C] s the thermal expanson coeffcent of the outer race, ν e s the Posson s rato and T ndcates the temperature ncrease from an ntal value. The thermal expanson of a ball wth unform temperature s ε = ξ r T 6.9 b b b b where r b s the ball radus. The thermal expansons of the bearng components are substtuted nto the calculaton of contact forces. From the geometrc relatonshp between the ball, nner race and outer race, l snα + u v o o z z tan α = 6.3 lo cosα o + ur + ε vr l snα + v w oe o z z tan α e = 6.31 loe cosα o + vr wr ε e l l e b ( l cosα + u + ε v ) + ( l sn + u v ) = ε + α 6.3 b o o r r ( l cosα + v w ε ) + ( l sn + v w ) = ε + α 6.33 oe o r r e o oe o o z z z z δ = l l 6.34 e e o oe δ = l l 6.35 e Fgure 6.1 descrbes the geometrc parameters used n the above equatons.

95 8 w z wr + ε e α o l oe v z α e le ε b v r l o u z l ε b α ur + ε Fg.6.1. Dsplacements of ball center, nner and outer races ncludng thermal expanson From the modfed Hertzan contact force [16], Q Q 3 / 3 = δ α & δ k 3/ 3 = δ e α & δ e e k e For an ellptcal contact area, the ball contact stress [17] at the geometrc center s 3Q, e σ, e = 6.38 π a b, e, e a, b are the sem-major and sem-mnor axes of the projected ellptcal area, whch are calculated as descrbed n [17]. The ncreased contact force causes more frcton heat generaton, the heat source s transferred to the bearng components, and the temperature ncrease brngs more thermal expanson. Ths feedback loop becomes unstable when the heat generaton ncreases so rapdly that the heat transfer mechansm can not redstrbute heat quckly enough and the thermally nduced load ncreases untl falure occurs. It s called thermally nduced bearng sezure or lockup.

96 83 CHAPTER VII OVERALL SYSTEM SIMULATION Ths chapter presents the system smulaton after a blade loss event of the arcraft gas turbne engne mounted on sx rollng element bearngs & squeeze flm dampers (SFD) wth seal rub and enclosed n a flexble housng. The approach beng used s to frst develop and nclude a detaled hgh-fdelty model to capture the structural loads resultng from blade loss, and then use these loads n an overall system model that ncludes complete structural models of both the engnes and arcraft structure. A hgh fdelty nonlnear ball bearng model and fnte element (FE) SFD model s employed n the smulaton. The contact stress on the ball s calculated usng the ball bearng model and permanent deformaton s predcted. The FE SFD determnes pressure profle of ol flm around SFD journal and calculates damper forces dependng on the journal moton and velocty. A bearng thermal model predcts temperature growths n support bearngs and ol n SFD. Smulaton results ncludng whrl ampltudes of power turbne, bearng contact load and stress, ol pressure n SFD, power loss n bearng and transmssblty. 7.1 FE Model of Complete System Fg 7.1 Schematc dagram of the full system