PERPUSTAKAAN KUiTTHO ilill IIiiillili lilli ilill lliil liiii llii IIIi U\'12')b4~

Transcription

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2 PERPUSTAKAAN KUiTTHO ilill IIiiillili lilli ilill lliil liiii llii IIIi U\'12')b4~

3 KOLEJ UNIVERSITI TEKNOLOGI TUN HUSSEIN ONN BORANG PENGESAHAN STATUS TESIS JUDUL: SUCCESSIVE OVER RELAXATION TECHNIQUE FOR STEADY STATE AND DYNAMIC CHARACTERISTICS OF A CYLINDRICAL BORE BEARING SESI PENGAJIAN: MEl 2005/2006 Saya ZAIHARBIN YAACOB mengaku membenarkan tesis Sarjana ini disimpan di Perpustakaan dengan syarat syarat kegunaan seperti berikut: I. Tesis adalah hakmilik Kolej Universiti Teknologi Tun Hussein Onn. 2. Perpustakaan dibenarkan membuat salinan untuk tl(juan pengajian sahaja. 3. Perpustakaan dibenarkan membuat salinan tesis ini sebagai bahan pertukaran antara institusi pengajian tinggi. D SULiT D TERHAD IIv II TlDAK TERHAD (TA~ULIS) Alamat Tetap: 419 Jln Springhill 10/24 Bandar Springhill POl1 Dickson, N.S. Prof. Madya Dr. Ing. Ir. Darwin Sebayang Nama Penyelia Tarikh: Tarikh:

4 Pcngcsahan Pcnyclia "Saya aklli bahawa saya telah membaca karya ini dan pad a pandangan saya karya ini adalah memadai dari segi skop dan kllaliti llntllk tlljllan penganugerahan ijazah Smjana Kejllruteraan Mekanikal" Tandatangan Nama Penyelia Tarikh

5 SUCCESSIVE OVER RELAXATION TECHNIQUE FOR STEADY STATE AND DYNAMIC CHARACTERISTICS OF A CYLINDRICAL BORE BEARING ZAIHAR BIN Y AACOB A thesis submitted in fulfillment of the requirement for the award of the Degree of Master of Mechanical Engineering Faculty of Mechanical and Manufacturing Engineering Kolej Universiti Teknologi Tun Hussein Onn DECEMBER, 2005

6 11 "I declare that the work presented in this thesis, except where otherwise stated, is based on my own research, and has not been submitted previously for a degree in this or any other university. Signature Name Date. ~... ZAIHAR BIN Y AACOB 13/;i/ /.!.. /... 'S....

7 III DEDICA nons TO MY FAMILY, THANK YOU FOR BEING THERE FOR ME.

8 1 \ AKNOWLEDGEMENTS First of all I would like to express my gratitude to the most Gracious and i\!os! Merciful ALLAH S.W.T. Praiseworthy to Almighty Allah for giving me the will and strength to complete this thesis. Thank you to my family and friends for their love and support. To my supervisors Prof. Madya Dr. Ing. Ir. Darwin Sebayang and Prof. Ir. Dr. ShahNor I3asri thank you for their ideas, support and guidance. Special thanks to my colleagues at Kolej Uniti. staff of the faculty and graduate school in Kuittho for their help and support. May ALLAH bless you all.

9 v ABSTRACT The knowledge of static and dynamic characteristics of journal bearings is crucial for the accurate detennination of the critical speed of a shaft and also for studying the stability of the rotating shaft against self-excited vibrations. These characteristics are detennined from the solutions of Reynolds equation numerically using finite difference methods with successive over relaxation technique (SOR). In order to implement SOR effectively, the optimum value for over relaxation factor Q had to be found first. In this thesis, the exact value of Q was calculated by using a fornmla proposed by G.D. Smith. Khonsari and Booser (K&B) found the value of Q, by trial and error which is not exact and time consuming. While Orcutt and Arwas (O&A) used Gauss-Seidel technique which has a much slower convergence rate compared to SOR, also they used two convergence limits which had to be satisfied before tenninations of the iteration procedure. This thesis is intended to improve both works by calculating the exact value for Q and employed the SOR technique using only one convergence limit. The dynamic coefficients were then used as an input data for studying the stability characteristics of the rotor-bearing system and the threshold of instability were also plot. The computer program was written using FORTRAN 95 programming language and run in the Microsoft Developer Studio environment. Method in this thesis shows that the time taken for a complete solution for the steady state and dynamic characteristics of a cylindrical bore bearing were greatly shortened in tenns of number of iterations (about 90%) and the automatic calculation of Q. The accuracy of the results were good with less than 10% in difference when compared to results from both K&B and O&A. It is then concluded that the finite difference method and successive over relaxation technique used in this thesis can predict accurately and effectively the static and dynamic characteristics of a cylindrical bore bearing.

10 \'} ABSTRAK Pengetahuan mengenai ciri-ciri statik dan dinamik galasjumal adalah pcnting untuk penentuan secara tepat halaju kritikal bagi suatu galas dan juga untuk mcngkaj i kestabilan galas yang berpusing terhadap getaran. Ciri-ciri ini ditentukan melalui penyelesaian berangka persamaan Reynolds menggunakan kaedah beza terhingga serta teknik santaian secara berturutan (SOR). Untuk menggunakan SOR secara cfektif. nilai optimum faktor santaian Q harus dicari terlebih dahulu. Dalam tesis ini nilai Q dicari menggunakan formula yang diperkenalkan oleh G.D. Smith. Khonsari dan Booser (K&B) mencari nilai Q dengan kaedah cuba jaya yang memakan masa dan tidak tepa!. Orcutt dan Arwas pula menggunakan teknik Gauss-Seidel yang mempunyai kadar penumpuan yang jauh lebih perlahan berbanding SOR, juga dua had penumpuan terpaksa dipenuhi sebelum prosedur lelaran ditamatkan. Tesis ini bertujuan untuk memperbaiki kedua-dua kerja tersebut dengan mengira nilai Q yang tcpat dan menggunakan teknik SOR dengan hanya satu had penumpuan diperlukan. Pemalarpemalar dinamik yang diperolehi digunakan sebagai data input untuk mengkaji kestabilan sistem rotor-galas dan juga memplotkan kemasukan ketidakstabilan. Program komputer ditulis menggunakan bahasa pengaturcaraan FORTRAN 95 dan dilarikan di dalam persekitaran Microsoft Developer Studio. Kaedah di dalam tesis ini mcnunjukkan masa yang diambil untuk penyelesaian penuh ciri-ciri statik dan dinamik suatu galas bergerek silinder dapat dikurangkan dengan ketara dari segi bilangan lelaran (kira-kira 90%) dan pengiraan Q secara automatik. Ketepatan keputusan adalah baik dcngan kurang dari 10% perbezaan apabila dibandingkan dengan kcdua-dua keputusan K&B dan O&A. Maka dapat disimpulkan bahawa kaedah unsur tcrhingga dan tcknik santaian secara berturutan yang digunakan di dalam tesis ini dapat mengagak dcngan tcpat dan berkesan ciri-ciri statik dan dinamik suatu galas bergerek silinder.

11 vii TABLE OF CONTENTS ITEM PAGE THESIS STATUS APPROVAL SUPERVISOR APPROVAL TITLE DECLARATION DEDICATION AKNOWLEDGEMENT ABSTRACT ABSTRAK TABLE OF CONTENTS LIST OF FIGURES NOMENCLATURE LIST OF APPENDIX 11 III IV v VI VII XI XV111 xxi CHAPTER I INTRODUCTION l.1 Introduction 1.2 Literature Survey Experimental Studies Theoretical Studies 1.3 Objective of Study 1.4 Scope 1.5 Overview

12 VIII ITEM PAGE CHAPTER II THEORETICAL BACKGROUND 2.1 Introduction II 2.2 Hydrodynamic Lubrication II 2.3 Reynolds Equation Journal Bearing Oil film Thickness in Journal Bearing Steady State Characteristics Load Carrying Capacity Lubricant Flow Rate Friction Coefficients Dynamic Characteristics Chapter Summary 31 CHAPTER III NUMERICAL SCHEMES 3.1 Introduction Finite Difference Method Nondimensionalization Finite Difference Schemes Pressure Boundary Conditions Evaluation of the Main Parameters Pressure Distribution Load Carrying Capacity Lubricant Flow Rate Friction Coefficients Stiffness and Damping Chapter Summary 47

13 IX ITEM PAGE CHAPTER IV COMPARATIVE STUDY 4.1 Introduction Convergence Characteristics Steady State Characteristics Pressure Distribution Load Carrying Capacity Maximum Pressure Attitude Angle Sommerfeld Number Side Leakage Inlet Flow Rate Friction Coefficients Dynamic Characteristics Chapter Summary 71 CHAPTER V RESULTS AND DISCUSSIONS 5.1 Introduction Pressure Load Capacity and Sommerfeld Number Attitude Angle and Friction Coefficients Inlet Flow Rate and Side Leakage Dynamic Characteristics Chapter Summary 90 CHAPTER VI CONCLUSIONS AND RECOMMENDATIONS 6.1 Conclusions Recommendations 93 REFERENCES 94

14 ITEM PAGE APPENDIX A. REYNODS EQUATION B. SAMPLE OUTPUT C. FLOW CHART & SOURCE CODE

15 xi LIST OF FIGURES FIGURE TITLE PAGE (a) 208(b) Journal bearing geometry and nomenclature 15 Schematic view of journal bearing 15 Schematic view of a journal bearing with radial (r) and tangential (t) 16 coordinate systems Triangle connecting bearing and journal center with some point 18 on the bearing surfaces Schematic view of a journal bearing with radial x-y coordinate 21 system and components of eccentricity Pressure distribution around a journal bearing Shear stresses acting on the journal and bearing 26 Idealized rotor of weight 2W supported on two journal bearings 28 Stiffness and damping coefficients of a journal bearing 29 Small amplitude journal motions about an equilibrium position 30 Unwrapped journal bearing 35 Finite difference grid for the oil film mesh 36 Typical boundary conditions on the Reynolds equation 40 Convergence characteristics for LID = 2 49 Convergence characteristics for LID = los 50 Convergence characteristics for LID = 1 50 Variations of dimensionless pressure with angular position. 51 c = 006. LID =1 Variations of load capacity with eccentricity ratio, LID = 1 52

16 xii FIGURE TITLE PAGE Variations of maximum pressure with eccentricity ratio, LID = 2 53 Variations of maximum pressure with eccentricity ratio, LID = 1 54 Variations of maximum pressure with eccentricity ratio, LID = Variations of attitude angle with eccentricity ratio, LID = 2 55 Variations of attitude angle with eccentricity ratio, LID = 1 56 Variations of attitude angle with eccentricity ratio, LID = 1/2 56 Variations of Sommerfeld number with eccentricity ratio. LID = 2 57 Variations of Sommerfeld number with eccentricity ratio, LID = 1 58 Variations of Sommerfeld number with eccentricity ratio, LID = Variations of side leakage with eccentricity ratio, LID = 2 59 Variations of side leakage with eccentricity ratio, LID = 1 60 Variations of side leakage with eccentricity ratio, LID = 1/2 60 Variations of inlet flow rate with eccentricity ratio, LID = 2 61 Variations of inlet flow rate with eccentricity ratio, LID = 1 62 Variations of inlet flow rate with eccentricity ratio, LID = 1/2 62 Variations offriction coefficients with eccentricity ratio. LID = 2 63 Variations of friction coefficients with eccentricity ratio, LID = 1 64 Variations of friction coefficients with eccentricity ratio, LID = Variations of feu with eccentricity ratio, LID = 1 67 Variations of k,!' with eccentricity ratio, LID = 1 67 Variations of KXl' with eccentricity ratio, LID = 1 68 Variations of - k n with eccentricity ratio, LID = 1 68 Variations of B with eccentricity ratio, LID = 1 69.a: Variations of B with eccentricity ratio, LID = 1 69 xy Variations of BJ)' with eccentricity ratio, LID = 1 70 Variations of Byx with eccentricity ratio, LID = 1 70

17 XIII FIGURE TITLE PAGE Variations of midplane pressure profile with angular position. & = 0.05 Variations of midplane pressure profile with angular position. &=0.5 Variations of midplane pressure profile with angular position. & = 0.8 Variations of maximum pressure with eccentricity ratio Variations of maximum pressure location with eccentricity ratio Variations of cavitation location with eccentricity ratio Variations ofload carrying capacity with eccentricity ratio Variations of Sommerfeld number with eccentricity ratio. LID = 2, 3/2, 1 Variations of Sommerfeld number with eccentricity ratio, LID = 3/4, 1/2, 1/3 Variations of Sommerfeld number with eccentricity ratio, LID = 1/4, 1/6, 1/8 Variations of attitude angle with eccentricity ratio Variations of friction coefficient with eccentricity ratio, LID = 2, 3/2, 1, 3/4 Variations offriction coefficient with eccentricity ratio, LID = 112, 1/3, 1/4, 1/6, 1/8 Variations of inlet flow rate with eccentricity ratio Variations of side leakage with eccentricity ratio Variations of K with eccentricity ratio,or Variations of K yt with eccentricity ratio Variations of K ty with eccentricity ratio Variations of k,y with eccentricity ratio Variations of ij.u with eccentricity ratio

18 \1\' FIGURE TITLE PAGE Variations of fl." with eccentricity ratio 88 Variations of EX) with eccentricity ratio 88 Variations of fl,) with eccentricity ratio 89 Non-dimensional critical speed (w) versus eccentricity ratio k) 90

19 xv NOMENCLATURE Finite difference parameters Damping coefficients Dimensionless damping coefficients C D e Radial clearance Journal diameter (= 2R ) Eccentricity Component of eccentricity ratio in x direction Component of eccentricity ratio in y direction Convergence factor Eccentricity ratio used in flowchart Component of eccentricity ratio in x and y direction used in flowchart F F Fr Oil film force Dimensionless oil film force Journal frictional force Dimensionless oil film forces in x and y direction f h Friction coefficients Oil Film thickness Dimensionless oil film thickness K xx ' KxyKyx, KJ)',...,...,..,... K xx ' KxyKyx, KJ~' L Stiffness coefficients Dimensionless stiffness coefficients Bearing axial length

20 XVI M,N Number of finite difference mesh in circumferential and axial direction Dimensionless critical mass of the rotor-bearing system Centre of the bearing Centre ofthe journal Rotational speed (rpm) Oil Film pressure Maximum oil film pressure Atmospheric pressure p Dimensionless oil film pressure Maximum of P Side leakage Dimensionless side leakage Journal radius Sommerfeld number Time T U v Variable in quadratic equation defined in text Surface speed of shaft Dimensionless squeeze film velocity Dimensionless velocity of rotor centre in x and y coordinates w W f).b, f1z Load carrying capacity Dimensionless load carrying capacity Fluid film mesh in circumferential and axial direction Difference of oil film forces in x and y direction Q f.1 /l. Optimum over relaxation factor Lubricant viscosity Length over diameter ratio

21 XVll ax. ~}' =-,y=al p al Journal center velocities Film density Eccentricity ratio z Component of eccentricity ratio in x and y direction Dimensionless length of bearing Deflections measured in x and y direction Attitude angle Angular velocity of journal/shaft = 2m7 e e Dimensionless angular velocity Angle measured from vertical Angle between position of maximum film thickness and some point around the bearing circumference SUBSCRIPTS r x.y ij a a 111 cav in 0111 L Radial coordinate Coordinate tangential to radial coordinate Cartesian coordinate Coordinate of mesh point in e and Z direction Equilibrium position Atmospheric Maximum Cavitation Inlet Outlet Leakage

22 xviii LIST OF APPENDICES TITLE A. REYNOLDS EQUATION B. SAMPLE OUTPUT C. FLOW CHART & SOURCE CODE

23 CHAPTER I INTRODUCTION In this chapter, the motivation of the research described in this thesis will be summarized and a brief survey of some literature on journal bearings and rotordynamics wiii be given. The objectives will be presented and an overview will explain the organization of the rest of the thesis. 1.1 Introduction If two bodies are in contact and in relative motion to each other, a tangential force, the force of sliding friction, results on the surfaces of contact. These surfaces will wear out rapidly which is the major cause of material wastage, loss of mechanical performance of machine elements and shortened the life of the machines used. Reduction in wear can be achieved by improving friction control and any reduction in wear can result in considerable savings. Lubrication is an effective means of controlling wear and reducing friction, and it has wide applications in the operation of machine element such as bearings.

24 2 Bearings are representative mechanical elements used in many classes of rotatin1! machinery. They are classified into rolling element bearings and plain bearings. depending on whether they are in rolling contact or in sliding contact. A sliding bearing typically uses a lubricant to reduced friction between the sliding surfaces. A shaft and bushing bearing are known as ajournal bearing. Cylindrical bore bearing is a journal bearing with plain cylindrical sleeve (bushing) wrapped around the journal (shaft). The journal is rotating inside the bore of the sleeve with a thin clearance. In journal bearings, the bearing surfaces are parallel to the axis of rotation. The journal and bearing surfaces are separated by a film of lubricant that is supplied to the clearance space between the surfaces through a hole or a groove. When there is a continuous fluid film separating the surfaces we speak of fluid film lubrication. When the journal bearing begin rotating there is very little lubricant between the bearing and shaft at some contact point and rubbing occurs. After the bearing has reached sufficient speed, the lubricants begins to wedge into the contact area and the relative motion of the surfaces causes the fluid pressure to support the load without metal to metal contact. This lubrication phenomenon is known as hydrodynamic lubrication. The understanding of hydrodynamic lubrication began with the classical experiments of Beauchamp Tower in 1883 in which the existence of a film was dctected from measurements of pressure within the lubricant, and ofnikilay Petroff in 1883 who reached the same conclusion from friction measurements (Hamrock, 1994). Tower's works was closely followed by Osborne Reynolds celebrated analytical papcr in 1886 in which he used a reduced form of the Navier-Stokes equations in association with the continuity equation to generate a second-order differential equation for the pressure in the narrow converging gap between bearing surfaces. He derived and published not only the descriptive differential equation that today bears his name but also certain solutions to this equation that agree well with the experimental measurements oftowcr (Gross, et ai., 1980). Since then Reynolds equation has become the foundation of hydrodynamic analysis of bearing performance.