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Mekanisk analyse av kamaksling på en skipsmotor Steffen Sunde Produktutvikling og produksjon Innlevert: juni 2015 Hovedveileder: Bjørn Haugen, IPM Norges teknisk-naturvitenskapelige universitet Institutt

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Mekanisk analyse av kamaksling på en skipsmotor Steffen Sunde Produktutvikling og produksjon Innlevert: juni 2015 Hovedveileder: Bjørn Haugen, IPM Norges teknisk-naturvitenskapelige universitet Institutt for produktutvikling og materialer Faculty of Engineering Science and Technology Department of Engineering Design and Materials Mechanical analysis of camshaft system on a marine engine Author: Steffen L. Sunde Supervisor: Bjørn Haugen June 10, 2015 Preface This master thesis was written at the department of Engineering Design and Materials at the Norwegian University of Science and Technology during the spring semester The report consists of a mechanical study of a mediumspeed diesel engine camshaft. Special attention is paid to the dynamical behaviour during engine operation. The new B33:45 engine series from Bergen Engines AS is under development, and a study of the camshaft dynamics for different cylinder configurations is desirable. Investigating suitable analysis software and methods is an integrated part of the project. The project is in cooperation with Bergen Engines - Rolls-Royce. I would like to express my gratitude to the main supervisor for this project work, Bjørn Haugen; His knowledge and insight was crucial for the project to finish on time. I would also like to thank Harald Berland and Jos van der Plas at Bergen Engines for providing technical information and help when needed. Steffen Loen Sunde Trondheim, June 10, 2015 iii Abstract This project is concerned with studying the dynamical behaviour of a mediumspeed diesel engine camshaft. As an important part of the reciprocal engine, the camshaft is subjected to increased loading due to the growing demand of higher engine performance. The time-varying forces which acts on the camshaft in operation give rise to oscillatory motion and these vibrations are important to account for in the overall mechanical analysis. The angular deflection of the camshaft is a key parameter in camshaft design as it affects the valve timings and hence the engine performance. The torsional vibration of camshaft is studied with special attention paid to the angular deflections. The camshaft is driven by the crankshaft through a two-stage gear drive. This gear drive has significant dynamic contributions to the camshaft performance and is included in the total model of the camshaft. The total model is spatially discretized as a multibody system with lumped masses and interconnecting idealised springs and dampers. An integration routine was written in Matlab to simultaneously solve the equations of motion based on the Newmark β-algorithms. The nonlinearities introduced to the systems by the time-varying stiffness and backlash in the gears required an iterative solving scheme to ensure equilibrium. One of the main goal of this thesis is to develop a dynamic simulation model and investigate to which extent the different mechanical systems on the engine needs to be included in order to get a sufficiently accurate simulation model. The dynamic response in the camshaft is found to be strongly dependent on the first natural frequency and the impulse loads introduced by the injection pumps. The results from the simulations must nonetheless be validated against physical testing before the accuracy of the model may be determined. The results were generally in concurrence with expectations and serves well as a starting point for more accurate simulation models. iv Sammendrag Dette prosjektet er et studium av den dynamiske adferden til en medium-speed diesel motor kamaksel. Kamakselen er en viktig komponent i stempelmotoren og blir utsatt for stadig høyere belastninger på grunn av økende krav til motorytelse. Den tidsvarierende belastningen under drift fører til svingende bevegelser og disse vibrasjonene er viktig å ta hensyn til i den mekaniske analysen av kamakselen. Vinkelutslag i kamaksel er en av de viktigste parameterene i designprosessen siden den påvirker ventilstyringen og dermed også motorytelsen. Torsjonsvibrasjoner i kamaksel er studert med spesielt fokus på vinkelutslag av fri ende. På motoren er kamakselen drevet av veivakselen via en to-stegs tennhjulsdrift som viser seg å bidra med dynamiske effekter på kamakselen. Kamakselen ble diskretisert til punktmasser med idealiserte fjærer og dempere. En integrasjonsrutine ble skrevet i Matlab for å løse systemet av differensialligninger basert på Newmarks β-metoder. Dødgang og varierende stivhet i tannhjulsdriften var blant ikke-lineære effekter som gjorde det nødvendig å implementere iterasjoner for likevekt. Et av hovedmålene i oppgaven er å utvikle en dynamisk simuleringsmodell og utforske til hvilken grad de ulike mekaniske systemene på motoren bør inkluderes for å oppnå realistiske og nøyaktige resultater. Modellen kan så brukes til videre parameterstudie. Den dynamiske responsen i kamaksel er funnet å være dominert av den første egenfrekvensen, samt inpulslastene fra insprøytningspumpene. Resultatene som er funnet bør valideres mot fysiske målinger og tester på motor før nøyaktigheten av modellen kan bestemmes. Resultatene er forøvrig i henhold til det man på forhånd kunne forvente. v Contents Preface iii Abstract iv Sammendrag v 1 Introduction Scope Report outline Software used Theory The reciprocal engine and its camshaft Operating cycle Injection system Gears Profile shift Tip relief Transmission error Gear backlash vi CONTENTS vii 2.3 Mechanical vibrations Euler-Lagrange Nonlinear dynamics Free vibration Finite element method Element types Numerical methods Explicit methods Implicit methods Newton-Raphson iteration Discrete Fourier transform Camshaft modelling Introduction Material One dimensional model Kinetic and Potential Energy Equations of motion Damping Excitation forces Finite element assessment Adding complexity Gear transmission model Introduction CONTENTS viii 4.2 Two Degrees of freedom-model Backlash Mesh cycle stiffness FEM modelling of gears Numerical time integration Validation backlash model Combining the camshaft model and gear model Results Eigenvalue analysis Mode shapes Constrained camshaft Transient response of camshaft Undamped forced response The effect of damping Effect of engine firing order Transient response of camshaft with gear drive Effect of gear backlash Effect of time-varying mesh stiffness Discussion Natural frequencies Transient response of camshaft Gear backlash Gear meshing stiffness CONTENTS ix 6.5 Numerical procedures Validating the results Conclusion 72 A Appendix 78 A.1 Python code A.1.1 MacroReadDataLine.py A.2 Matlab routines A.2.1 TimeStepping.m A.2.2 CheckContact.m A.2.3 FFT.m A.2.4 GenerateStiffnessMatrix.m A.2.5 GenerateMassMatrix.m A.3 Results A.3.1 Eigenvalue analysis A.3.2 Transient response List of Figures 2.1 Gear nomenclature [26] The Newmark integration scheme for linear structural dynamics The Newmark integration scheme for nonlinear structural dynamics Schematic of the lumped mass model of the camshaft Variation of damping ratio with natural frequency for a typical system Torques on the camshaft section unit Frequency response function of simple harmonic oscillator cylinder camshaft modelled in Abaqus Gear drive assembly Gear drive model with two degrees of freedom Small elements in the areas with Hertzian contact Single and double tooth contact in the first gear stage Fourier approximation of time-varying stiffness for the two gear pairs using 10 harmonics and zero phase Discontinuous stepping function for describing the time-varying stiffness Simple two degree of freedom system with backlash x LIST OF FIGURES xi 4.8 Comparison of position of masses for Fedem and Matlab with zero backlash Comparison of position of masses for Fedem and Matlab with backlash Fundamental torsional mode for 9 cylinder camshaft using finite element (120.6 Hz) Second normal torsional mode for 9 cylinder camshaft using finite element (284.1 Hz) Third torsional normal mode for 9 cylinder camshaft using finite element (456.1 Hz) Fourth torsional normal mode for 9 cylinder camshaft using finite element (626.1 Hz) Fundamental mode for discrete model of camshaft (9 cyl.) Second normal mode for discrete model of camshaft (9 cyl.) Third normal mode for discrete model of camshaft (9 cyl.) Fourth normal mode for discrete model of camshaft (9 cyl.) Constrained 6 cylinder camshaft natural frequencies Constrained 8 cylinder camshaft natural frequencies Constrained 9 cylinder camshaft natural frequencies Time history response of free end of undamped 9 cylinder camshaft at 375 rpm Frequency domain response undamped 9 cylinder camshaft (free end) Frequency domain response undamped 8 cylinder camshaft (free end) Frequency domain response undamped 6 cylinder camshaft (free end) Angular deflection of free end during engine startup (6 cyl.). 60 LIST OF FIGURES xii 5.17 Angular deflection of free end during engine startup (8 cyl.) Angular deflection of free end during engine startup (9 cyl.) Effect of gear backlash on angular deflections in camshaft (damped 8 cyl.) Meshing stiffness at engine speed 750 rpm A.1 Fundamental mode for discrete model of camshaft (6 cyl.).. 84 A.2 Second normal mode for discrete model of camshaft (6 cyl.). 84 A.3 Third normal mode for discrete model of camshaft (6 cyl.).. 85 A.4 Fourth normal mode for discrete model of camshaft (6 cyl.). 85 A.5 Fundamental mode for discrete model of camshaft (6 cyl.).. 86 A.6 Second normal mode for discrete model of camshaft (8 cyl.). 86 A.7 Third normal mode for discrete model of camshaft (8 cyl.).. 86 A.8 Fourth normal mode for discrete model of camshaft (8 cyl.). 87 A.9 Constrained 7 cylinder camshaft natural frequencies A.10 Constrained 10 cylinder camshaft natural frequencies A.11 Maximum dynamic torque in camshaft during startup (6 cyl.) 88 A.12 Maximum dynamic torque in camshaft during startup (8 cyl.) 89 A.13 Maximum dynamic torque in camshaft during startup (9 cyl.) 89 List of Tables 3.1 Camshaft material Rayleigh damping coefficients assuming ξ 1 = Rayleigh damping coefficients assuming ξ 1 = 0.01 and ξ 2 = Camshaft gear drive data Measurements of stiffness on gear pairs Spring characteristics for a spring with k = 2000N/m and backlash of 0.05 m System values for comparison of results in Matlab and Fedem System values for comparison of results in Matlab and Fedem with backlash First five torsional eigenfrequencies of camshaft with 6 cylinder units First five torsional eigenfrequencies of camshaft with 7 cylinder units First five torsional eigenfrequencies of camshaft with 8 cylinder units First five torsional eigenfrequencies of camshaft with 9 cylinder units Simulation summary for 9 cylinder undamped camshaft xiii LIST OF TABLES xiv 5.6 Simulation summary for 8 cylinder undamped camshaft Simulation summary for 6 cylinder undamped camshaft Dynamic amplification factor (DAF) and angular deflection for different damping cases Different firing orders for 6 cylinder engine Different firing orders for 8 cylinder engine Simulation summary for 9 cylinder undamped camshaft including gear drive, backlash: 0.15 mm Simulation summary for 9 cylinder camshaft-gear, mass proportional damped (ξ 1 = 0.01), backlash: 0.15 mm Gear backlash effect on camshaft angular amplitude Comparing results for constant average stiffness in meshing gears and time varying stiffness (damping case 1) Chapter 1 Introduction Bergen Engines AS develops medium-speed reciprocal engines for marine and power generation running on either liquid fuel or pure gas. It started out as Bergen Mekaniske Verksted (BMV) which was founded in 1855 and has since 1943 developed, manufactured and installed diesel engines for the marine industry including ferries, offshore support vessels, passenger ships and more. Increasingly stringent environmental requirements forces engine producers to continuously improve engine design to reduce emissions but also increasing efficiency. The new B33:45 engine series from Bergen Engines is designed for efficiency and satisfies the International Maritime Organisation Tier II and tier III regulations for emissions. The gas exchange process is important for the internal combustion engine to achieve high efficiency and low emissions and this is largely affected by the cylinder valve timings. In modern medium-speed diesel engines, including the B33:45 engine series, the cylinder exhaust and air valves are controlled mechanically by a rotating camshaft. This camshaft flexes during operation, and to have a sufficient degree of control of its motion it is necessary to study its dynamic behaviour. 1.1 Scope The main objective of this thesis is to study the mechanical property of the camshaft design in the new B33:45 engine series. The work will investigate to which extent the different mechanical systems on the engine needs to be included in order to get a sufficiently accurate simulation model. The 1 CHAPTER 1. INTRODUCTION 2 simulated results should be validated against physical measurements made on the engine while on testbed. By validating and calibrating the dynamic model against measurements allows the model to be used as a parameter study on the camshaft performance. 1.2 Report outline The underlying theory for the project is first presented in chapter 2. This includes a short introduction to the reciprocal engine and the role of the camshaft. The camshaft gear drive is regarded an important factor in the camshaft dynamic behaviour and is therefore included in the theory chapter. Mechanical vibrations and numerical methods for solving dynamic problems is outlined. A short introduction to finite element methods is also mentioned due to its importance in spatial discretization. In chapter 3, a discrete model of the camshaft is presented. In chapter 4, the camshaft gear drive is analysed to be added to the camshaft model. A short literature study on gear dynamics is included to summarise the common methods and assumptions. Simulation results are presented in chapter 5, followed by a discussion of the results and finally conclusions. The appendix section includes further simulation results and extracts of the code developed in this thesis. 1.3 Software used Choosing suitable software is an important part of the project. To solve a complex dynamic problem, the methods and assumptions for spatial and time discretization is crucial for sufficiently accurate results to be obtained in a reasonable time frame. MATLAB 2013b is the main tool in this project, used for numerical computations, time integration and post-processing of simulation results. SolidWorks 2014 was used to produce three dimensional models to be studied and used as validation of results. The finite element software suite Abaqus 6.14 was used to assess the spatial CHAPTER 1. INTRODUCTION 3 discretization with numerical values obtained from the models generated in SolidWorks. FEDEM was used as a validation tool. Chapter 2 Theory 2.1 The reciprocal engine and its camshaft The reciprocal engine converts pressure in cylinders into rotational motion through the reciprocating pistons to the crankshaft. In the diesel engine this pressure is generated by fuel which is injected into the cylinder and ignited by the temperatures generated by greatly compressing the mixture of air an fuel. Due to its high compression ratio, the diesel engine have has thermal efficiency Operating cycle During the four stroke operating cycle, gases are exchanged through the valves in the cylinder head. As the piston moves away from the cylinder head, the air intake valves in the cylinder head are opened and air is introduced into the cylinder. The exhaust valves are closed. This is known as the induction stroke of the operating cycle. During the compression stroke all valves are closed and the piston is moved towards the cylinder head, compressing the air enclosed within the cylinder. The valves are kept closed and diesel fuel is injected into the chamber with the compressed air. The heat generated from the compression ignites the vaporised fuel particles which provides a pressure on the piston, thrusting it away from the cylinder head. This is the power stroke, and through the connecting rods the high pressure generated in the combustion chamber is transmitted to the crankshaft and into rotational energy. The last stroke, exhaust stroke, discards the exhaust gases from the combustion. The exhaust valves are opened and the piston moves towards the cylinder head again, pushing the exhaust gases through the exhaust valve 4 CHAPTER 2. THEORY 5 openings. The camshaft is responsible for operating the valves and fuel injection. Rotational motion is transmitted from the crankshaft to the camshaft usually through a set of gear wheels, belt or chain. As the camshaft rotates, cam lobes distributed along its axis acts, normally on pushrod systems which in turn forces the valves to open and close. The timing of these valves and injection pumps are of critical importance for the engine to have optimal gas exchange which in turn affects the efficiency and emissions of the engine. The increasingly strict emission requirements from International Maritime Organization (IMO) forces the engines to reduce NO x and particulate emission. This makes further control of the valve timings desirable (Variable valve timing). This permits the possibility to change the timing between the exhaust valves and inlet valves and to increase their overlap Injection system In Pump-Line-Nozzle (PLN) systems, a high pressure fuel line is responsible for the fuel injection. With the help from the camshaft the pump element generates high pressure fuel which is led to the injection nozzle to be sprayed into the combustion chamber. In newer diesel engines, the Pump-Line-Nozzle system is often replaced with common rail. This master thesis is focused on the camshaft of a four stroke marine diesel engine with PLN injection system. One single camshaft is responsible for all timings, i.e. the exhaust valves, inlet valves and the fuel injection pumps. 2.2 Gears Mechanical power is transmitted from the crankshaft to the camshaft through the gear drive. The gear drive consists of circular gear wheels with straight cut teeth of involute shape (spur gears). The involute gear tooth profile which was proposed by Leonhard Euler, is a spiral following a path traced by the end of a piece of string unwrapping from a cylinder. The involute gear profile have teeth which are involutes of a base circle. The benefits of having an involute profile is constant pressure angle, constant velocity ratio and less sensitivity to change of center distance [13]. The angle between the tooth normal and the radial line at some arbitrary point along the tooth face is called the pressure angle. The pressure line or CHAPTER 2. THEORY 6 line of action is normal to the tooth surface and tangent to the pitch circles. See figure 2.1. The main rule of gear toothing is expressed as i = ω 1 ω 2 = n 1 n 2 = d w2 d w1 = z 2 z 1 = T 2 T 1 (2.1) where ω j is the angular speed, n j is rotational speed, d wj is the pitch circle diameter, z j is the number of teeth and T j is the torque of the two mating gears (j = 1, 2). Figure 2.1: Gear nomenclature [26] The contact ratio, CR is defined as the average number of teeth in contact during mating. Due to the risk of deformation, contact ratios should be greater than 1.2 in order not to loose contact [27]. It is important to be aware of that the theoretical values of the contact ratio are greater then the actual values. Contact ratio can be calculated as [26], m p = ( dap 2 ) 2 ( dbp 2 ) 2 + ( dag 2 ) 2 ( ) Dbg 2 2 C sin ω (2.2) p b CHAPTER 2. THEORY 7 where, d ap, d ag = addendum diameter of pinion and gear d bp, d bg = base circle radii of pinion and gear C = operating center distance p b = base circle pitch High contact ratio can be achieved by increasing the number of teeth, lowering the pressure angle or increasing the

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