Department of Industrial and Materials Science CHALMERS UNIVERSITY OF TECHNOLOGY Gothenburg, Sweden 2018 Spline optimization tool using Finite Element Analysis Master’s thesis in Applied Mechanics Anton Stenstrand MASTER’S THESIS IN APPLIED MECHANICS Spline optimization tool using Finite Element Analysis ANTON STENSTRAND Department of Industrial and Materials Science Division of Materials & Computational Mechanics CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg, Sweden 2018 Spline optimization tool using Finite Element Analysis ANTON STENSTRAND © ANTON STENSTRAND, 2018 Master’s thesis 2018 Department of Industrial and Materials Science Division of Materials & Computational Mechanics Chalmers University of Technology SE-412 96 Göteborg Sweden Telephone: +46 (0)31-772 1000 Cover: 3-part involute spline coupling showing the effective Von Mises stress distribution Chalmers Reproservice Göteborg, Sweden 2018 Spline optimization tool using Finite Element Analysis Master’s thesis in Applied Mechanics ANTON STENSTRAND Department of Industrial and Materials Science Division of Materials & Computational Mechanics Chalmers Univeristy of Technology Abstract The current method for spline design at the Driveline department consists of several steps. First the initial design is established using analytical calculations, then a 3-dimensional CAD- model is created based on the analytically derived geometrical data. If deemed necessary this model is then manually discretisized with necessary boundary conditions introduced and solved using the Finite Element Method. If needed additional changes are made to the CAD- model and the final design is obtained by through iteration. The main goal of this thesis is to create a program that can automate the pre-processing process and act as an intermediate tool between the analytical calculations and the full Finite Element models during spline design. This report covers the steps taken in order to create a standalone application with a graphical user interface that can generate meshed models of two and three component spline couplings, with all necessary boundary conditions and setup parameters, ready to be solved in the Finite Element program Abaqus. This report also includes a parameter study that investigates the effects different geometrical parameters have on the strength of spline couplings. Lastly the effects of manufacturing tolerances and assembling deviations are studied. Keywords: Spline, Gear, Involute, Trochoid, FEM, MATLAB i ii PREFACE This master thesis has been carried out as the last part of the masters program in Applied Mechanics at Chalmers University of Technology for the degree of Master of Science. The work has been performed during the spring of 2018 at the Concept and Simulation group under the Driveline department at Volvo Group Trucks Technology. M.Sc Sven Andersson and M.Sc Jon Elfridsson have been the supervisors at Volvo and Associate Professor Ralf Jänicke has been the examiner at Chalmers. ACKNOWLEDGEMENTS I would first like to thank all employees at the Concept and simulation group for their helpfulness and for creating a great and welcoming work environment for me to perform my thesis work. A special thanks goes to my supervisors Jon Elfridsson and Sven Aronsson for their continuous support and helpful insights throughout this project. I would also like to give a special thanks to Lennart Johansson at the Rotating parts and housings group for enduring all my questions regarding spline design and manufacturing techniques. Finally I would like to thank my examiner Ralf Jänicke for his support and the manager at the Concept and Simulation group Erik Nordlander for selecting me and giving me the opportunity to work on this master thesis project. Anton Stenstrand Göteborg 2018 iii iv NOMENCLATURE Spline specific z Number of teeth m Module α0 Pressure angle x Profile shift coefficient Dii Minor diameter internal spline Die Minor diameter external spline Dei Major diameter internal spline Dee Major diameter external spline Dfi Form diameter internal spline Dfe Form diameter external spline Dbase Base diameter Dpitch Pitch diameter E Space width (internal spline) S Tooth thickness (external spline) Ev Space width effective (internal spline) Sv Tooth thickness effective (external spline) wi width internal spline we width external spline inv () Involute function General PTO Power Take-Off CAD Computer Aided Design FEM Finite Element Method Abaqus Finite element software MATLAB Programming language Python Programming language ANSA Pre-processing software GUI Graphical User Interface Pa Pascal SS Swedish Standard SMS Sveriges Maskinindustriförenings Standardcentral (Swedish mechanical standard) .inc file Include file .inp file Analysis input file .txt file Text file .mat file MATLAB MAT-file v vi Contents 1 Introduction 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Spline coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Uses in gearboxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Scope and limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Theory 7 2.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Module system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.1 Primary units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.2 Profile shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.3 Secondary units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Basic geometric description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3.1 External involute spline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3.2 Internal involute spline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.3 Involute profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3.4 Trochoid curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4 Mathematical geometric description . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4.1 Trochoid curve for the external involute spline . . . . . . . . . . . . . . . . 16 2.4.2 Involute profile for the external involute spline . . . . . . . . . . . . . . . . 21 2.4.3 Root radius for the internal involute spline . . . . . . . . . . . . . . . . . . 21 2.4.4 Involute profile for the internal involute spline . . . . . . . . . . . . . . . . . 24 2.5 Contact mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.5.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.5.2 Contact mechanics in Abaqus . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3 Method 27 3.1 Meshing strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2.1 External geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2.2 Internal geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3 Finite Element discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3.1 Element definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3.2 2 dimensional mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3.3 3 dimensional mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.3.4 Node and element sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.4 Mesh quality improvement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.4.1 Element angle adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.4.2 Progressive element sizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.4.3 Element adjustment internal spline . . . . . . . . . . . . . . . . . . . . . . . 42 3.5 Program output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.5.1 Output files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.5.2 Simulation setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4 Program structure 48 4.1 Program overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.2 Graphical user interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.2.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.2.2 Mesh settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.2.3 Simulation setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 vii 4.2.4 3-part configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.3 Supporting scripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.3.1 Preview scripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.3.2 Help buttons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.4 Subprogram overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.4.1 Geometry creation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.4.2 Mesh algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.4.3 Creating Abaqus files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5 Geometrical verification 76 6 Testing and results 77 6.1 Element analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 6.2 Tool tip radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 6.3 Coupling effect on misalignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 6.4 Misalignment study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.4.1 Angular misalignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.4.2 Eccentric misalignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.5 Manufacturing tolerances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.5.1 Sinusoidal pitch deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.5.2 Random pitch deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.6 Convergence study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 7 Discussion 104 7.1 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 A Mesh quality check I A.1 Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I A.2 Aspect ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III B Subscripts IV viii 1 INTRODUCTION 1 Introduction 1.1 Background This chapter gives an introduction to the machine element known as a spline coupling and gives a brief overview of its history, main function and its applications in powertrain design. 1.1.1 Spline coupling A spline coupling is a machine element that is used in order to transmit torque between two components that are not permanently joined together. This feature makes them useful in vast number of applications and they are frequently used in the automotive industry in everything from driveshafts to clutches. Spline couplings ability to be translationally moved while rotating also makes them useful for connecting and disconnecting parts in a rotating assembly. This is utilized in powertrain engineering in for instance slip yokes and engaging sleeves. The most simple form of a torque transmitting coupling that works on the same principle as a spline coupling is a simple keyway joint, where torque is transfered between a hub and an axle trough a key. The design features a slot cut out of both the axle and the hub with a key fitted in order to limit rotational movement between the two components, as shown in Figure 1[1]. This is a joint that has been used frequently in the past and is still used today for low torque applications, usually featuring a flat-key or a Woodruff-key. Figure 1: Keyway coupling [1] The geometrically most simple spline is the rectangular spline where the axle have teeth of rectan- gular shape, much like the flat-key joint, but with the keys incorporated in the axle. The hub in its turn has the inverse geometry with broached slots on the inside. This results in higher possible torque transfer compared to a key joint design since the stress is distributed around the entire joint. These types of joints where used frequently in the past, but have since been surpassed by involute splines. The reason for this is that the rectangular shape with sharp edges results in high stress concentrations. Since the superior involute splines can be manufactured with the same machines and tools as involute gears at a lower cost there are few reasons to implement rectangular splines in industrial components. The same goes for splines with a triangular cog shape. Figure 2 shows a cross section of a rectangular spline coupling and in Figure 3 a cross section of a triangular spline coupling is shown [2][3]. 1 1.1 Background 1 INTRODUCTION Figure 2: Spline coupling with straight sided teeth[2] Figure 3: Spline coupling with triangular teeth[3] The by far most used type of spline couplings today is the involute spline, which as previously mentioned has the same geometry as involute gears. This geometry, which is shown in Figure 4[4] beneath results in much lower stress concentrations and thus also results in the ability to transfer higher torque in relation to the physical size of the coupling. Furthermore the involute profile makes the coupling self align under load even if it is designed as a loose fit coupling which also benefits the torque transmitting capabilities. Figure 4: Involute spline coupling [4] 2 1.1 Background 1 INTRODUCTION 1.1.2 Uses in gearboxes In most gearboxes there exists mainly two types of spline couplings. Permanent spline couplings that are constantly engaged such as the clutch disc to input shaft interface or the interface between the counter shaft and the power take off outlet (PTO). The other type are spline couplings that are disconnectable, such as the engaging sleeves that are used to change gear. Permanently engaged spline couplings are couplings between one external spline and one internal spline that is used to transfer torque between two shafts or rotating parts. The main reason for incorporating these types of couplings is to simplify production and assembly as well as allow for modularity. For instance in the case of the PTO, the manufacturing and assembly of the gearbox would be more complicated if it was permanently attached. Additionally it is a feature that is only used by some customers and therefore the spline coupling allows for gearboxes both with and without PTO outlets. Figure 5 beneath shows the principal working of a two component permanent spline coupling and and how it can be used in practice in a clutch assembly[5]. Figure 5: 2-component spline configuration[5] 3 1.1 Background 1 INTRODUCTION The engaging sleeve spline coupling is usually a coupling between four splined parts, although a maximum of three parts are meshing simultaneously. Engaging sleeves utilizes the spline couplings ability to perform translational movement while rotating. In a gear changing mechanism there are usually three external splines and one internal spline where the internal spline can be slided in either direction in order to engage a gear, Figure 6 shows one side of such a coupling consisting of three parts. In the figure the yellow part represents a shaft to which the input torque is supplied, the red part represents the splines on the gear wheel and the green part represents the engaging sleeve. In the left Figure the engaging sleeve is disconnected and no torque is transferred to the gear wheel. By sliding the engaging sleeve over the output spline (red spline) as shown in the right Figure torque is transmitted to the output spline through the engaging sleeve. This way of transferring torque between shafts and gear wheels through spline couplings is the basic functioning of all manual constant meshing gearboxes. Figure 6: Principal working of an engaging sleeve 4 1.1 Background 1 INTRODUCTION Figure 7 and 8 shows a principal sketch of a longitudinal constant meshing manual gearbox [6], where the parts in Figure 6 have been highlighted. The gearwheels on the counter shaft are solid mounted on the shaft and follows the rotation of the shaft, whilst the gearwheels on the main shaft are fitted with bearings and are free to rotate relative to the main shafts rotation. In Figure 7 the gearbox is in neutral and no torque is transmitted through the gearbox. In Figure 8 the engaging sleeve for the first and reverse gear have been slided to engage first gear and the red arrow shows the torque path through the gearbox. Figure 7: Gearbox in neutral Figure 8: Gearbox in 1st gear 5 1.2 Problem description 1 INTRODUCTION 1.2 Problem description Spline couplings are used for many different types of torque transmitting couplings in the pow- ertrain of a vehicle. Currently there exists a in house developed script that can generate basic discretisized geometries of straight involute splines. The generated geometry only includes the flange surfaces i.e. the involute contact profile. This script can therefore only be used to study the stress levels at the contact surfaces of the spline, which means that areas of interest such as the root radius is not included. In order to be able to analyze the spline joint in its entirety a CAD-model has to be created and then discretisized manually. This is a very time consuming way of analyzing a spline coupling, especially if it is in an early stage of design when the scope of possible spline combinations is large. 1.3 Objective The objective of this master thesis is to develop a program that can speed up the design process of spline couplings as well as work as a tool for further analysis of current spline configurations, in terms of quickly investigating sensitive design parameters. The program shall be well tested on a wide range of spline configurations in order to secure a stable algorithm. 1.4 Scope and limitations The created program should generate the correct geometry of the entire spline including the root radius/trochoid based on standardized input parameters, such as module and number of teeth, for both external and internal splines. The output from the program should be a complete three dimensional meshed model with all necessary boundary conditions and contact definitions in Abaqus syntax ready for solving. The program should be able to handle both two and three part configurations with user defined assembly options such as overlap, misalignment and desired outputs. The program shall be able to handle this for all typical straight involute spline geometries. The project will also include a parameter study in order to identify sensitive design parameters in spline design. This also applies to manufacturing tolerances where different tolerance outcomes can be studied in order to gain further knowledge of what tolerances are the most important. The script will not be able to perform complete fatigue analysis due to the time frame of the project, but instead comparison of stress levels will be used in order to find the sensitive parameters. 6 2 THEORY 2 Theory This chapter describes the theory behind spline couplings, their uses and the mathematical descrip- tion of their geometry. The chapter first gives an introduction to the basic parameters of involute spline and gear design together with the nomenclature used and lastly an explicit derivation of the mathematical functions needed is given. 2.1 General An involute spline, both external and internal shares all of its fundamental geometry with an invo- lute gear. The mathematical functions describing them are the same and they are manufactured with the same techniques, the only thing that usually distinguishes them from one another are the dimensions in terms of limiting circles, such as the major diameter for the external spline. The involute spline profile has a lot of advantages compared to rectangular and triangular spline couplings. The width of the profile increases from the top diameter down to the end of the involute profile and the transition from the involute profile to the bottom radius is smooth which helps lowering the stress concentrations in the root of the teeth. As mentioned previously the profile makes the coupling self align under load even if the two parts are fitted loosely together. Figure 9 shows an example of an external and an internal spline where the dimensions on the axles are in millimeters as they are for all following figures in this report unless otherwise stated. −15 −10 −5 0 5 10 15 15 20 25 30 35 Internal External Figure 9: Internal and external spline section 2.2 Module system The far most used system in terms of involute gear and spline design, and the only one covered in this report is the standardized module system. The module is a normalized dimension that is used to define the profile of a gear rack. Since the gear rack is a tool that can be used to manufacture gears the profile of the gear rack in itself defines the geometry of the gear. The dimensions of the gear rack profile varies depending on which standard that is used for the gear or spline design. Volvo has their own standardized rack profiles and tools for both gears and splines, but since they are classified and can not be shown in this report the gear rack profile according to Swedish standard (SMS 296 ) will be referred to instead. Figure 10 shows the reference profile according to SMS 296 together with its dimensions [[7] p.422 ]. All derivations as well as the method used 7 2.2 Module system 2 THEORY is the same regardless of the gear rack profile and only the actual dimension convention of the parts differ. For gears the pressure angle, α0 is specified at 20 degrees according to the standard, however for splines used within Volvo this angle can vary. In gear terminology the units in the module system are often divided into primary and secondary units, where the primary units are the units needed to describe the function of a gear transmission, for instance the speed ratio. The secondary units are the units needed in order to fully describe the geometry of the gear tooth including the root radius. These units are shown in Table 1 in Section 2.2.1 and Table 2 in Section 2.2.3. With these standard units known all other dimensions for a gear profile and thus also for a spline profile can be derived. Figure 10: Reference profile according to SMS 296 [[7] p.422 ] 2.2.1 Primary units The primary units of the module system are shown in Table 1, as mentioned in the previous section these four units are adequate in order to fully describe the function of a gear transmission. The basis of the module system is the gear module m, as can be seen in Figure 10 it specifies the overall shape of the gear rack and in extension the gear itself. The unit of the module is in millimeters and is chosen from a set of predefined standard modules, in the Swedish standard (SS 52 ) these range from 0,5mm to 32mm. There is of course the option to use modules outside of the predefined modules, however for small volumes that can increase the cost vastly and is avoided unless necessary. As mentioned earlier the pressure angle is fixed at 20◦ for SMS 296, this effectively means that all gearwheels manufactured according to the standard will work together given the same module. The condition for this to work is that the two gear wheels have the same base pitch, which is defined according to Equation 3. As can be seen this criterion will be met if the two wheels have the same pressure angle. For splines this is of less importance since all teeth are engaged at once and the entire coupling therefore has to be designed as a unit. The standard used at Volvo specifies the pressure angle for splines at 30◦, 37,5◦ and 45◦ with a corresponding set of modules ranging from 0,25mm to 10mm. This standard is usually followed during spline design, although deviations from this standard is quite common due to the many special applications of spline couplings used. In this case this is of limited importance from an economic perspective due to the large production volumes, moreover the majority of the tools used for both splines and gears are in Volvos case custom made for their own purposes. Regardless of to which extent the standard is used the mathematics and the way of describing the geometry stays the same. 8 2.2 Module system 2 THEORY Table 1: Primary units z Number of teeth m Gear module α0 Pressure angle x Profile shift coefficient From the primary units the two key diameters known as the base diameter and the pitch diameter can be derived. The base diameter is the diameter from which the involute profile is unrolled, this is explained further in Section 2.3.3. The pitch diameter is the diameter of the correct path of contact for a gearwheel, effectively the rolling diameter which for a gear transmission defines the speed ratio. The base diameter is defined according to Equation 1 and the pitch diameter is defined according to Equation 2. Figure 11 shows an example of the relation between these circles for a spline with a pressure angle, α0, of 30◦. Dbase = mz · cos (α0) (1) Dpitch = mz (2) −10 −5 0 5 10 12 14 16 18 20 22 24 26 28 30 Pitch diameter Base diameter (a) External involute spline −15 −10 −5 0 5 10 15 15 20 25 30 35 Pitch diameter Base diameter (b) Internal involute spline Figure 11: Base and pitch diameter With the primary units known it is possible to derive both the pitch and the base thickness of the teeth, which are needed in order to describe the spline geometry. The pitch is defined according to Equation 3 for both external and internal splines. The base thickness is the thickness of the teeth on the base diameter and is defined according to Equation 4 for an external spline and according to Equation 5 for an internal spline [7]. These dimensions are shown in Figure 12 for both types of splines. Pb = πm cos (α0) (3) Sb,ext = ( π 2 + 2x tan (α0) + z · inv (α0) ) m cos (α0) (4) Sb,int = ( π 2 − 2x tan (α0)− z · inv (α0) ) m cos (α0) (5) 9 2.2 Module system 2 THEORY −10 −5 0 5 10 12 14 16 18 20 22 24 26 28 30 Base diameter P b S b (a) External involute spline −15 −10 −5 0 5 10 15 15 20 25 30 35 Base diameter P b S b (b) Internal involute spline Figure 12: Pitch and base thickness for involute splines 2.2.2 Profile shift The last of the primary units is the profile shift coefficient, x, which is a dimensionless coefficient defined according to equation 6. A profile shift of a gear or spline means as the name would suggest that the profile is shifted either outwards or inwards in the radial direction. This is in practice done by shifting the gear cutting tool towards or away from the stock that is being cut. In the case of a gear rack being used for manufacturing, the profile shift is created by offsetting the distance between the gear racks pitch line and the pitch diameter of the gear as can be seen in Figure 13[8]. A positive profile shift is defined as moving the gear rack outwards radially from the spline or gear wheel and analogously moving it inwards for a negative profile shift, this is the case for both external and internal splines. Obviously a gear rack can not be used to cut internal gears and instead a circular gear shaper is usually used, but for the sake of the theory an imaginary gear rack works fine. A gear without profile shift is often called an uncorrected gear. For gears profile shifting is used in order to get the desired backlash between two gearwheels or to get the correct centre distance for multiple gear pairs of different size that are mounted on the same axle as they would in a gearbox. For splines it is commonly used in order to get the desired fit between the mating splines in a coupling. For an external spline a positive profile shift results in thicker teeth and a negative profile shift will result in thinner teeth. This can sometimes be utilized for spline wheels with a low number of teeth where a positive profile shift can be used in order to lower the stress levels in the root of the cog. X = x ·m [mm] (6) 10 2.2 Module system 2 THEORY Figure 13: Profile shift using gear rack cutting[8] 2.2.3 Secondary units The secondary units are the dimensions needed in order to describe the geometric representation of the involute spline and are listed in Table 2 as radiuses, it is also common to refer to these dimensions diametrically. For an external spline as shown in Figure 14a the major radius, ree, is the outermost radius of the spline and is machined prior to the spline cutting. The minor radius, rie is the innermost radius and is effectively the maximum cutting depth of the gear rack. For the internal spline, shown in Figure 14b, it is the other way around. The minor diameter, rii, is done prior to the gear cutting and the major diameter, rei, is the result of the maximum cutting depth. The last dimension that is needed in order to fully describe the spline profile is the form radius or form diameter that describes where the transition between the trochoid and involute function is located, which is explained further in Section 2.3. This means that the radial length of the active involute profile becomes rmajor − rform. This dimension can be used in order to define the geometry of the spline and thereafter derive the effective tool tip radius for the gear rack. According to the standard used at Volvo this is done the other way around where the form radius is a dimension derived from the effective tool tip radius. The pitch radius, rpitch, is also a derived dimension and is defined diametrically according to Equation 2. Table 2: Secondary units External Internal rmajor ree rei rminor rie rii rform rfe rfi rpitch r r Width we wi 11 2.3 Basic geometric description 2 THEORY −15 −10 −5 0 5 10 15 0 5 10 15 20 25 r ee r ie r r fe (a) External −15 −10 −5 0 5 10 15 0 5 10 15 20 25 30 r ei r ii r r fi (b) Internal Figure 14: Secondary units 2.3 Basic geometric description 2.3.1 External involute spline The external involute spline, as shown in Figure 15 can be described by two different functions. From the start of the involute, also known as the form radius (rfe), to the major radius (ree) it is described by the involute function shown in Equation 10. From the minor radius (rie) to the form radius (rfe) it is described by a trochoid function, which represents the radius created by the tool tip between two teeth during gear cutting. These two functions are explained explicitly in section 2.4. −5 0 5 19 20 21 22 23 24 25 26 D ie Trochoid Involute Figure 15: Buildup of an external involute spline tooth 12 2.3 Basic geometric description 2 THEORY 2.3.2 Internal involute spline An internal spline can for obvious reasons not be manufactured in the same way as an external spline. The most common way of manufacturing internal splines and gears is with a gear shaper. This means that theoretically there is a lot more design freedom when it comes to the geometry since it is effectively the inverse geometry of the tool. Additionally this manufacturing method does not result in the complex trochoid shape of the root radius. The common practice in internal involute spline design and the one practiced according to the Volvo standard is to have a root radius that is tangent to the the minor radius, rii, and intersects the involute profile, as can be seen in Figure 16. In technical specifications this is specified with a minimum tool tip radius and a maximum value for the form radius, rfi. This effectively specifies the minimum root radius of the spline as well as the minimum length of the involute profile. −5 −4 −3 −2 −1 0 1 2 3 4 5 21 22 23 24 25 26 27 28 29 D ei Root radius Involute Figure 16: Buildup of an internal involute spline tooth 13 2.3 Basic geometric description 2 THEORY 2.3.3 Involute profile The involute profile can be described as a string rolled of a stationary cylinder with the string kept tensioned, the resulting path of the end of the string will result in an involute profile with the diameter of the cylinder as the base diameter. This is illustrated in Figure 17 below where the vector rk represents the tensioned string, B the start point and P the end of the string. From Figure 17 the definition of the involute function can be derived which is done in Equation 7 through 10 according to MÄGI [7]. Figure 17: Involute profile rolled of the base circle From Figure 17 the lengths of the vectors r and rk can be derived according to Equation 7 r = rb cos (α) rk = rb tan (α) (7) The length of the vector rk will be equal to the length of the involute according to the definition. rk can therefore be expressed according to Equation 8 rk = (θ + α) rb (8) Using Equation 7 and 8 rk can be eliminated according to Equation 9 rk = rb tan (α) = (θ + α) rb −→ θ = tan (α)− α (9) θ in Equation 9 is called the involute of the angle α which is a function frequently used in gear and spline terminology and has its own function defined according to Equation 10 inv(α) = tan (α)− α (10) 14 2.3 Basic geometric description 2 THEORY 2.3.4 Trochoid curve A trochoid curve is defined as the path created from a point on a circle or connected to a circle that rolls along a straight line. In gear terminology the path that the tool tip of the cutter takes while cutting the profile of a gear tooth results in a trochoid curve between two teeth of a gear wheel. Since involute splines are manufactured in the same way as involute gears the root radius of a spline is also represented by a trochoid curve. Figure 18 shows the basics of how a trochoid curve is generated. 2 3 4 5 6 7 8 9 10 11 −3 −2 −1 0 1 2 3 4 5 v v v Figure 18: Trochoid curve 15 2.4 Mathematical geometric description 2 THEORY 2.4 Mathematical geometric description This chapter covers the derivation of the angles and distances needed in order fully describe the geometry of both the internal and external involute spline. 2.4.1 Trochoid curve for the external involute spline In order to determine the coordinates needed for creating the trochoid curve, a set of angles and distances needs to be introduced. For the derivation of the trochoid curve of the external involute spline a modified version of Von Ingo Maier will be used[9]. This method is originally developed for involute gears with a helix angle, but can with some alterations be applied to straight involute splines. The sketches shown in Figure 20 and 21 are made in the transverse plane and the tool is normal to the normal plane. Because of this the tool tip in the sketches have an elliptical form as it would assuming a non zero helix angle. For a gear with a helix angle the transverse plane is orthogonal to the symmetry axis of the gearwheel, whereas the normal plane is rotated with the helix angle and thus orthogonal to the direction of the gear teeth. Figure 19 shows the definition of the two planes for a spur gear with a helix angle of 10◦. For straight involute splines the helix angle is 0◦ and the transverse plane becomes coincident with the normal plane. In Figure 20 the trochoid profile with the angles and distances needed in order to calculate it is shown for an involute gear, Figure 21 shows the same geometry as Figure 20 but is zoomed in on the tool tip of the cutting rack. Since this derivation is done for a straight spline the geometry of the tool tip will in reality be a circle and not an ellipse as shown in Figure 20 and 21. Figure 19: Normal (2) and transverse (1) plane for a spur gear with a helix angle 16 2.4 Mathematical geometric description 2 THEORY Figure 20: Trochoid curve 17 2.4 Mathematical geometric description 2 THEORY Figure 21: Tool tip Equations 11 through 21 shows the necessary steps in order to calculate the curve for a straight involute spline. Equations 22 and 23 describes the geometry between the two trochoids in case of a specified radius smaller than the distance between two teeth at the form radius rfe. First the tool height hk is calculated according to equation 11 hk = r + x ·m− rie (11) Where: r = Pitch radius x = Profile shift coefficient m = Gear module rie = Minor radius (12) Now the distance from the center of the trochoid to the center of the tool can be calculated E = π ·m 4 − hk · tan (α0)− R cos(α0) ( 1− sin (α0) ) (13) Where: α0 = Preassure angle R = Tool tip radius (14) 18 2.4 Mathematical geometric description 2 THEORY The coordinate u in the local coordinate system showed in Figure 21 is defined according to equation 15. This is used as the governing coordinate in order to create points on the trochoid curve in the program. −R cos (α0) ≤ u < 0 (15) In the normal plane and without a helix angle the tool tip is a circle, which means that the coordinate v can be solved according to Equation 16. v becomes negative since the point of engagement will always be in the third or fourth quadrant. v2 + u2 = R2 → v = − √ R2 − u2 (16) Now the angle ν, which is the angle between the symmetry line of the tool and the vector stretching from the roll point to the engagement point as can be seen in Figure 21, can be calculated. tan (ν) = ∣∣∣∣dvdu ∣∣∣∣ = |u|√ R2 − u2 ν = arc tan ( |u|√ R2 − u2 ) (17) The distance from the bottom of the tool tip to the point where the vector −−→BF intersects the v-axis is calculated according to equation 18 W = (R+ v)− u · cot (ν) (18) With W known the roll angle φ can now be determined. φ = 1 r ( π ·m 2 − E + (hk − x ·m−W ) tan (ν) ) (19) The distance from the roll point to the point of engagement described by the vector −−→BF is given by equation 20 −−→ BF = hk − x ·m−W cos (ν) − u sin (ν) (20) The coordinates of the trochoid curve in the global Cartesian coordinate system can now be calculated according to Equation 21 x = r · sin (φ)−−−→BF · sin (φ+ ν) y = r · cos (φ)−−−→BF · cos (φ+ ν) (21) If the spline is not designed with a full radius an additional curve needs to be calculated for the part between the two trochoids. Figure 22 shows the difference between a spline with a full radius and a spline designed with smaller tool tip radius resulting in a flat part between the trochoids of two teeth. 19 2.4 Mathematical geometric description 2 THEORY (a) Full radius (b) Non full radius spline root Figure 22: Different types of spline roots Figure 23 shows the introduced angles needed in order to define the geometry for this part of the root. Figure 23: Flat part between the trochoids The angle θ1 becomes the same as the angle φ for u → 0 and the two angles angles defining the flat part of the root are shown in Equation 22 θ1 = φ u→ 0 θ2 = 2 · π z · 2 = π z (22) The Cartesian coordinates in the global system for the part between the two trochoids can then be calculated according to equation 23. Where θ1 ≤ θ ≤ θ2. xf = rf · sin (θ) yf = rf · cos (θ) (23) 20 2.4 Mathematical geometric description 2 THEORY 2.4.2 Involute profile for the external involute spline When the coordinates for the trochoid curve is known then so is the form radius, rfe, defining the start point of the involute curve. The outer radius,ree, is a secondary unit and thus also specified and therefore the radius span for the entire involute profile is known. What remains to be determined is the corresponding angle (φ) in the polar coordinate system. The thickness for any given radius (Sr) on the involute profile is defined according to Equation 24[7]. Sr = ( pi+ 4x tan (α0) 2z + inv (α0)− arccos ( rb rr )) 2rr (24) The angle can then be calculated according to equation 25 φn = Sr 2 · r (25) 2.4.3 Root radius for the internal involute spline The first step in creating the geometry for the internal spline is to create the root radius. This is necessary because the size of the bottom radius defines the form radius rfi, which in itself defines the starting point of the involute profile as well as the starting point of the bottom part of the spline (rf ). Figure 24 shows a sketch of the root radius (red) and involute profile with the lengths and angles needed in order to determine its coordinates. Figure 24: Root radius for the internal involute spline 21 2.4 Mathematical geometric description 2 THEORY The radial coordinate of the origin of the root radius is calculated according to equation 26 ro = rf − rbot (26) The roll distance to the origin of the bottom radius, lo, can now be calculated according to equation 27 lo = √ r2 o − r2 b (27) The roll distance to the start of the involute, le, then becomes le = lo + rbot (28) The radial coordinate of the form radius can now be calculated according to equation 29 re = √ l2e + r2 b (29) The angles αe and αo then becomes: αe = arctan(rb re ) αo = arctan( rb ro ) (30) The angle dφ can be calculated as: dφ = αe − αo (31) The thickness at the form radius is given by equation 32[7] Se = π + 4 ∗ x ∗ tan(αpo) 2 · z + inv(αpo)− inv ( arccos ( rb re )) 2re (32) Where rb is the base circle radius defined as[7]: rb = ( m · z 2 ) · cos(αpo) (33) The angle φe can now be calculated according to equation 34 φe = Se 2re (34) Which gives the angle φo as: φo = φe − dφ (35) With both the angle and radius for the three points (o,e and f) known they can know be converted into cartesian coordinates according to equation 36 to simplify the creation of the bottom radius. 22 2.4 Mathematical geometric description 2 THEORY xe = re · sin(φe) ye = re · cos(φe) xo = ro · sin(φo) yo = ro · cos(φo) xf = rf · sin(φf ) yf = rf · cos(φf ) (36) Figure 25 shows the root radius zoomed in where the angles γe and γf have been introduced as well as the sweep angle for the root radius γef . Figure 25: Root radius for the internal involute spline Since the coordinates of all three points (o,e and f) are known the angles γe and γf can be calculated according to Equation 37. γe = arctan ( (ye − yo) (xe − xo) ) γf = arctan ( (xf − xo) (yf − yo) ) (37) The sweep angle for the bottom radius can now be defined according to Equation 38. γe ≤ γef ≤ π 2 − γf (38) 23 2.4 Mathematical geometric description 2 THEORY In a numerical enviroment the root radius will consist of a number of coordinates along the radius. These coordinates can now easily be created since the root radius rbot and the sweep angle γef are known. The coordinates will however be defined in the local coordinate system with its origin coinciding with the origin of the root radius and needs to be transformed to the global polar coordinate system. Let rbot and γef be vectors containing the coordinates for the root radius, then the coordinates in the global Cartesian coordinate system can be expressed according to Equation 39. Xbot = xo + rbot · cos(γef ) Ybot = yo + rbot · sin(γef ) (39) The global carteesian coordinates can then be transformed into global polar coordinates according to equation 40. rbotr = √ X2 bot + Y2 bot φbotr = arctan ( Xbot Ybot ) (40) 2.4.4 Involute profile for the internal involute spline With the root radius calculated the start point of the involute profile at the form radius, rfi, is now known as well as the end point at the major radius, rei. The remaining part is to determine the angle in the polar coordinate system φ for any given radius on the profile. The thickness for a given radius on the involute profile is given by Equation 41[7]. Sr = π + 4 · x · tan(αpo) 2 · z + inv(αpo)− inv ( arccos ( rb r )) 2r (41) With the thickness known the corresponding angle φ in the global polar coordinate system can be calculated according to equation 42 φinv = Sr 2 · r (42) 24 2.5 Contact mechanics 2 THEORY 2.5 Contact mechanics Contact mechanics in the finite element method becomes relevant whenever two different bodies interact with each other through a contact constraint. The contact could be between a deformable and rigid body or more commonly as in this case between two deformable bodies. This type of simulations adds more requirements to the simulation setup and are also much more demanding from a computational point of view compared to a single body analysis. This is due to the fact that the system of equations become nonlinear if contact is included even if dealing with small strain linear elasticity and therefore an iterative solver needs to be used in order to solve it. 2.5.1 Methods When applying contact theory for the finite element method there are two main methods that are used, the penalty method and the Lagrange multiplier method. Common for both these methods is that the solver needs to keep track of which nodes that comes in contact with each other when the load is applied. In some less determined simulations where its hard to predict what surfaces will come in contact during the simulation this is done with a global or semi global node search where large parts of the geometry needs to be checked for clashes like for instance in crash simulations. The method of searching the entire domain for contact between every loadstep would theoretically work fine for any contact simulation, but since it can be very computationally demanding it is much more efficient to predefine the contact surfaces if possible. In the case of a spline coupling this becomes a non issue since the contact surfaces of the spline is defined by its geometry and can easily be predefined in the pre-processing step. Figure 26 shows a simple 1-D contact mechanics problem taken from Ekh 2015 [10]. In this case the solver needs to keep track of the gap between nodes 3 and 4 with the initial gap g0 for each loadstep and when the two nodes come into contact with each other the iterative solving of the system can begin. Figure 26: 1-D contact mechanics problem [10] Penalty method The essential workings of the penalty method is that a stiffness is added to to the stiffness matrix for the nodes that clash together with the corresponding force added to the internal force vector. This can be visualized as spring added between two nodes of the parts which results in a force pushing them apart. For the system shown in Figure 26 the program would enter the iterative contact solver of the program when the absolute distance between node 3 and 4 becomes less than 0, i.e penetration occurs. The resulting system of equations then becomes according to Equation 43[10]. Ka +Kb −Kb 0 −Kb Kb + εN −εN 0 −εN Kc + εN u2 u3 u4  =  F εNg0 −εNg0  (43) 25 2.5 Contact mechanics 2 THEORY Lagrange multiplier method The Lagrange multiplier method is based on adding an external force to the force vector for the nodes that comes into contact. This force known as a Lagrange multiplier becomes an unknown in the system of equations and therefore a constraint equation needs to be added in order to make the system solvable. For the 1-D problem shown in Figure 26 assuming contact between nodes 3 and 4 have occurred the system of equations for the problem becomes according to Equation 44[10].  Ka +Kb −Kb 0 −Kb Kb 0 −1 0 0 Kc 1 0 −1 1 0   u2 u3 u4 λ  =  F 0 0 −g0  (44) Differences For the penalty method the resulting contact force is a function of the penalty stiffness and the penetration and thus there will be a small resulting penetration when the tolerance for equilibrium is reached by the solver. The only way of obtaining zero penetration for the penalty method is if the penalty stiffness goes towards infinity which in itself will result in an ill conditioned stiffness matrix and renders the problem unsolvable by numeric methods. This is the main disadvantage of the penalty method. The main advantage of the penalty method is that the size of the problem remains constant regardless of the amount of nodes in contact as can be seen in Equation 43. For the Lagrange multiplier method the system of equations needs to be redefined every time the number of nodes in contact changes as can be seen in Equation 44. On the other hand the Lagrange multiplier method results by definition in a gap of 0 between the contact nodes. This method will therefore result in a more accurate solution compared to the penalty method, but the penalty method is less computationally demanding since the number of equations are kept constant. 2.5.2 Contact mechanics in Abaqus In Abaqus there are several options when it comes to contact formulations, both in terms of solving method and surface interactions. The models created by the Optimization tool have the Penalty method as the default solving technique. The main reason for this is that some of the features included in the program are only compatible with the Penalty method definition in the Abaqus version used at Volvo. The contact formulation used in the Optimization tool is a finite sliding surface-to-surface formulation. The surface-to-surface option means that the contact surfaces are defined as surface-sets instead of node-sets which makes it possible for Abaqus to interpolate between the nodes of both the master and the slave surface. For the example shown in Figure 26 the geometrical formulation of the gap function is simple since its 1-dimensional, but for two 2- dimensional surfaces with different mesh refinement grade the gap function becomes more complex. By using the surface-to-surface definition instead of the regular node-to-node definition Abaqus is able to establish a more exact gap function through interpolation which results in a more accurate solution. The trade-off for this increased accuracy is the computational effort which means that it increases the solving time. 26 3 METHOD 3 Method In this chapter the theory and underlying structure of the program will be explained from defining the basic measurements of a single spline to a complete discretisized model. 3.1 Meshing strategy The element type used for the discretization of the model is hexhedral elements, also known as brick elements. The choice of this element type was a demand from Volvo. The program is written in such a way that both 8-node and 20-node hex elements can be used in order to allow for flexibility with respect to precision and computational effort, these element types are shown in Figure 27 beneath. Since the geometry of a spline includes radiuses and to some extent relatively steep derivatives in its shape this element type creates some challenges in the discretization. (a) 8-node hex element (C3D8) (b) 20-node hex element (C3D20) Figure 27: Element types used for discretization In order to create a meshing algorithm that provides as good quality mesh as possible while at the same time being possible to implement in code for all different types of splines and inputs a study of different mesh schemes was made in the pre-processing program ANSA. A symmetry model of two geometrically different splines were created where different layouts could be tested quickly and effectively. After experimenting with different layouts and consulting with the supervisors at Volvo a promising meshing structure was obtained which can be seen in Figure 28. Figure 28 only shows the external spline, but since the internal spline is the inverse of the external spline it is sufficient in order to determine the mesh structure for both parts. 27 3.1 Meshing strategy 3 METHOD Figure 28: Mesh structure The greatest advantage of this mesh structure is that it provides high quality elements along the contact surfaces of the splines, as well as in the root radius which is advantageous since this is the area of most interest and with the largest discontinuities in the geometry. This layout will however result in some skewed elements especially between the sections around the projected root radius. This skewness of course varies with the geometry input parameters of the spline and one of the big challenges is to minimize the amount of skewed elements for a wide range of different spline geometries. Some bad elements are however unavoidable due to the basic geometry of a spline coupling and the shape of the elements used. The major reasoning behind this mesh structure is to make sure that these elements are located where they have a small impact on the discretization in its entirety. 28 3.2 Geometry 3 METHOD 3.2 Geometry This section describes the creation of the complete geometry of both the internal and external involute spline. The input data in terms of both geometrical data and mesh settings are used in order create points in space representing the node coordinates. Since the spline is rotationally symmetric it is enough to create the geometry for half a tooth, which can then be used to create the entire spline later in the mesh algorithm. 3.2.1 External geometry As mentioned earlier the node coordinates only have to be created for half of a tooth in two dimensions. The outer geometry is created based on the mathematical description explained in Section 2.4, this geometry is then divided into three different sections in order to imitate the mesh strategy shown in Figure 28. Figure 29 shows the geometry divided into the three different mesh sections. The size and position of these mesh sections are defined automatically based on the input, but can also be altered manually in order to allow for flexibility. −1 0 1 2 3 4 5 6 19 20 21 22 23 24 25 1 2 3 Figure 29: Mesh sections external spline Section one consists of the active involute profile together with the root of the tooth, which in itself consists of one or two parts depending on if the spline has a full radius or not. If the spline has a full radius the root is described solely by the trochoid function and if it is not it also has a flat part between the trochoids of two teeth with the diameter Die. The different parts of the profile and root is shown in Figure 30. 29 3.2 Geometry 3 METHOD 0 1 2 3 4 5 19 20 21 22 23 24 25 D minor Trochoid Involute Figure 30: Profile and root sections for an external spline The number of node points that are created along this profile is decided by user input. The user can either specify a total number of elements along the entire profile and the program automati- cally distributes them in order to obtain an equal element side length. It is also possible to specify the number of elements along the involute profile and the root and let the program distribute the number of elements for the two root sections or specify the number of elements for each section manually. The size of the section radially, i.e the position of the projected profile (red line in Figure 29) is determined by user input and is in the program created by adjusting the distance between the roll point and the cutting point, the vector −−→BF in Figure 20, in the trochoid function and projecting the involute in the circumferential direction. The number of node points created in radial direction is specified by user input determining the number of elements radially for section 1. The rest of the geometry is divided into section 2 and 3 by the black line in Figure 29, this line is a radial projection of the inner diameter and completes the outer boundary for both section 2 and 3. The nodal points for section 3 and thus also the resolution of the mesh are in the radial direction decided by the number of elements chosen for section 1. In the circumferential direction a new variable is introduced which sets the number of elements in this direction. The nodal points are in this case linearly spaced between the projected profile (red line in Figure 30) and the left boundary. This parameter then also sets the number of elements in the circumferential direction for the left part of section 2 (green part in Figure 31), whereas the right part of section 2 is set by the mesh resolution in section 1. For the radial direction a new variable is introduced which sets the number of projected nodal lines between the upper and lower boundary. Figure 31 shows the different mesh sections, their common node boundaries and the input variables that define the number of node coordinates along each section. In Figure 32 a basic coarse mesh for the symmetry part of the external spline is shown with the input parameters, as shown in Figure 31, set to the values displayed in Table 3. 30 3.2 Geometry 3 METHOD −1 0 1 2 3 4 5 6 7 19 20 21 22 23 24 25 26 Nr. of profiles Nel lower Nel Top N e l p ro file Figure 31: Meshing sections with input parameters for the external spline 0 1 2 3 4 5 20 20.5 21 21.5 22 22.5 23 23.5 24 Figure 32: Basic mesh for symmetry part of external involute spline defined by Table 3 Table 3: Mesh input parameters for Figure 32 Nel profile 10 Nr. of profiles 2 Nel top 2 Nel lower 3 31 3.2 Geometry 3 METHOD 3.2.2 Internal geometry The creation of the discretisized geometry for the internal involute spline is very similar to the external spline. The internal spline is as the external spline rotationally symmetric and therefore it is in the same way as for the external spline only necessary to create the geometry for half of a tooth explicitly. The mathematical description on how to derive the outer geometry of the internal involute spline is explained in Section 2.4. In the same way as for the external spline the internal spline has been divided into three different mesh sections according to Figure 33 in order to mimic the mesh mesh strategy set out in Section 3.1. The dividing line that makes up section 1 (red line in Figure 33) is a projection of the active profile consisting of the involute, the root radius and the major diameter (blue line in Figure 33) inwards. This is done by numerically differentiating the profile in order to obtain the unit normal orthogonal to the profile that specifies the projection direction. The line that splits up the remaining geometry into section 2 and 3 (black line in Figure 33) is simply a line that is drawn from a point on the dividing profile (red line in Figure 33) to a point on the outer diameter, the positioning and angle of this line is then altered in order to minimize the skweness of the elements in the intersecting boundary. The theory and application of this is explained further in Section 3.4. 0 1 2 3 4 5 6 22 23 24 25 26 27 28 1 2 3 Figure 33: Mesh sections internal spline As is the case for the external spline there is a set of input variables that sets the number of node coordinates both radially and circumferentially for each section. Section 1 has two variables that define these points. Section 2 has one variable that defines the number of points circumferentially and the number of points in the radial direction becomes specified by the input for section 1 due to their shared boundary. Section 3 has a variable that specifies the number of points circumferentially whilst the number of points radially are determined by the inputs for section 1 and 2. The input parameters determining the number of node coordinates for each mesh section are shown in Figure 34 where the boundaries with shared node coordinates have been highlighted. Figure 35 shows an example mesh of the internal involute spline with the input parameters defined according to Table 4. 32 3.2 Geometry 3 METHOD −2 −1 0 1 2 3 4 5 6 23 24 25 26 27 28 29 Nr. of profiles Nel upper Nel lower N e l p ro file Figure 34: Meshing sections with input parameters for the internal spline −1 0 1 2 3 4 5 6 7 23 24 25 26 27 28 29 30 Figure 35: Basic mesh for symmetry part of internal involute spline defined by Table 4 Table 4: Mesh input parameters for Figure 35 Nel profile 10 Nr. of profiles 2 Nel right 2 Nel upper 3 33 3.3 Finite Element discretization 3 METHOD 3.3 Finite Element discretization This chapter covers the method used in order to create a meshed 3 dimensional model of the spline coupling, from discretisizing the 2 dimensional mesh sections of the symmetric part of the spline shown in Section 3.2 to assembling it into a complete 3 dimensional model. As can be seen in Section 3.2 the creation of the node coordinates are done very similiarly for the internal and external spline, therefore the meshing technique is very similar for the two different parts. Because of this the meshing technique will only be shown for the external spline in order to avoid repetition. This chapter also includes the creation of the node and element sets that are needed in order to solve the Finite Element problem, as well as the sets that are needed for the desired outputs. 3.3.1 Element definition Figure 36 beneath shows the numbering and face convention that Abaqus uses for 8 and 20 node hex elements. As can be seen from the figure both elements have the same face definition and numbering convention, with the exception that the 20 node element has mid nodes. The numbering and face convention is of great importance in order to make sure that the node numbers are collected in the element coupling matrix in an order that allows for an easy extraction of element sets and faces after the discretization is done. For instance making sure that all elements along the involute profile have the same face pointing outwards makes defining the contact areas later much less cumbersome. (a) 8-node hex element (C3D8) (b) 20-node hex element (C3D20) Figure 36: Face definition 3.3.2 2 dimensional mesh As described in Section 3.2 the node coordinates are only explicitly derived for half of a tooth of the spline coupling since it is rotationally symmetric. This part is then in its turn divided into 3 sections which are meshed separately and then joined together. The output from the geometry program is one matrix containing the node coordinates for each mesh section. Figure 37 beneath shows a scatter plot of the output matrix created by the geometry subprogram which is used as input for the mesh subprogram. 34 3.3 Finite Element discretization 3 METHOD −1 0 1 2 3 4 5 20 20.5 21 21.5 22 22.5 23 23.5 24 24.5 Section 1 Section 2 Section 3 Figure 37: Scatter plot of node coordinates In order to define the geometry of the mesh two matrices are needed, one node coordinate matrix hereby referred to as node_coord and defined according to Equation 45 and one element cou- pling matrix hereby referred to as Enod and defined according to Equation 46 in 2 dimensions. node_coord defines the spatial coordinates for each node and Enod defines which nodes that make up each element. In Equation 45 and 46 n1,n2,.. represents arbitrary node numbers and e1,e2,... represents element numbers. node_coord =  n1 x1 y1 n2 x2 y2 . . . . . . . . .  (45) Enod =  e1 n1 n2 n3 ... nn e2 n1 n2 n3 ... nn . . . . . . . . .  (46) Figure 38 shows how the mesh algorithm is programmed for an example mesh, each section is divided into node lines which are then numbered chronologically. By using a set of loops to go through the node coordinate data the Enod matrix can be assembled. Equation 47 shows the first row of of the 2 dimensional Enod matrix for this example mesh. 35 3.3 Finite Element discretization 3 METHOD 0 1 2 3 4 5 20.5 21 21.5 22 22.5 23 23.5 24 24.5 25 123 21 22 33 3253 Nodes Line 1 Line 2 Line 3 Figure 38: Mesh technique Enod = e1 1 3 35 33 2 23 34 22 . .  (47) When the Enod and node_coord matrix have been created for each mesh section the matrices are merged together and the double nodes at the intersecting boundaries, as can be seen in Figure 31, are removed and the sections are connected in the Enod matrix. An example of the resulting 2 dimensional mesh for the symmetric part of the spline is shown in Figure 39. 0 1 2 3 4 5 20 20.5 21 21.5 22 22.5 23 23.5 24 Figure 39: 2 dimensional mesh of the symmetric part of the external spline 36 3.3 Finite Element discretization 3 METHOD The next step in the mesh program is to first mirror the symmetric part of the geometry in order to create a full tooth and then rotate it in a circular pattern in order to create the complete 2 dimensional geometry. The method of doing this is greatly simplified by defining all node coordinates throughout the geometry creation and mesh algorithm in polar coordinates. This means that mirroring can be done by coping the coordinates for the symmetry part of the tooth and changing the sign of the angle. The two parts can then be merged together at their common boundary. The same method can then be used for the circular pattern where only the angle coordinate needs to be altered for each tooth. Figure 40 beneath shows the mirrored geometry and Figure 41 shows the complete 2 dimensional mesh for the external spline. −4 −3 −2 −1 0 1 2 3 4 19 20 21 22 23 24 25 Figure 40: 2 dimensional mesh of a complete spline tooth −30 −20 −10 0 10 20 30 −20 −15 −10 −5 0 5 10 15 20 Figure 41: Complete 2 dimensional mesh 37 3.3 Finite Element discretization 3 METHOD 3.3.3 3 dimensional mesh The final step of creating the mesh is to extrude the 2 dimensional mesh consisting of quad elements and turning it into a 3 dimensional mesh built up by hex elements. This is in the program done by copying the 2 dimensional mesh and extruding it in the depth direction in two steps for each element row. The first extrusion is to create the mid nodes of the elements if 20-node hex elements are to be used and the second extrusion is to create the back side of the elements. This is then done for the number of elements specified in the depth direction where the intersecting boundaries in the layers between the element rows are merged together and the surplus nodes are deleted. The resulting 3 dimensional mesh can be seen in Figure 42. Figure 42: 3 dimensional mesh of external spline 3.3.4 Node and element sets In order to be able to set up the problem element sets needs to be extracted for the boundaries and the mating contact surfaces of the spline coupling. When creating the mesh for the spline care has been taken to make sure that all sets have an equal spacing between element and node numbers, this makes the sets easier to extract and makes post processing easier for the user. If the user for instance wants to investigate a set of elements or nodes outside of the predefined post processing sets it is very easy to find the corresponding sets for the other teeth. This step size is written out in the Abaqus input file for both node numbers and element numbers. For example if the user has performed a simulation where the stress distrubution is uneven, as it would be if there is a misalignment or eccentricity included and has defined a set containing the elements 1,2 and 3. Then the corresponding sets for the rest of the teeth can be found according to Equation 48, where s denotes the set and x the step size which is the number spacing between the symmetric parts that build up the mesh. s1 = [1 2 3] s2 = [1 2 3] + x · 1 s3 = [1 2 3] + x · 2 sn = [1 2 3] + x · (n− 1) (48) 38 3.3 Finite Element discretization 3 METHOD The default sets that are extracted for all simulations are shown in Table 5, some of these sets needs to be extracted in order to set up the simulation and some are extracted in order to aid in post processing. All sets are created for both parts except for the inner and outer boundary since they are part specific. From these sets the face definitions for the involute surfaces, the front and back surface and the inner and outer boundary surfaces are extracted. For the involute profile of the spline one set is created for both the left and the right side for each tooth individually. These sets are created primarily in order to be able to define the contact areas for the solver, but they can also of course be used to quickly extract stress or any other data from the contact surfaces of the spline. The reason for creating individual sets for each involute is to be able to emulate different contact ratios that can stem from manufacturing tolerances, which is explained further under Section 6.5 and to allow for simple extraction of the contact pressure. The inner boundary of the external spline and the outer boundary of the internal spline together with the front and back face of both parts are used for boundary conditions. The rest of the sets are created in order to aid in post processing. The use of the sets and the actual setup of the simulation is explained further under Section 3.5.2. Figure 43 shows an example of some of the default sets for the external involute spline. Table 5: Node and element sets Set Description Involute left side One element set for each tooth Involute right side One element set for each tooth All involutes One node set containing involute profiles (left and right) Root One node set and one element set for each root (tooth) Inner boundary An element set for the inner boundary of the external spline Outer boundary An element set for the outer boundary of the internal spline Front An element set for the front face of each part Back An element set for the back face of each part Figure 43: Root elements, root nodes, involute elements, involute surface, front surface, back elements and inner surface sets 39 3.4 Mesh quality improvement 3 METHOD 3.4 Mesh quality improvement The involute spline geometry is a circular geometry that contains radiuses and relatively steep derivatives in its shape, whilst the element type used in the program is rectangular. This means that simply building the spline geometry with elements defined in two directions, radial and circumferential, would result in a lot of elements with poor element geometry. The first step in addressing this is as mentioned earlier in the report to create three different mesh sections by projecting parts of the outer geometry of the spline in order to minimize the amount of bad elements and control where the worst elements end up. This also makes the mesh more flexible for the relatively large variations in geometry between different spline geometries. With the mesh sections defined there are still ways to further improve the mesh quality within the sections and to reduce the computational effort by allowing for finer mesh sizes in places of interest and coarser mesh in places where the need for resolution is low. 3.4.1 Element angle adjustment By design the elements of mesh sections 1 and 3 of the external spline will become of sufficient quality regardless of the spline geometry used given that the mesh settings have been set up properly. The elements of mesh section 2 however will become skewed along the upper boundary due to the geometry difference between the elements and the domain, this is especially prominent around the area where all sections intersect each other (the green dot in Figure 44a). A few elements with bad geometry in this area does not affect the solution as a whole to great extent and as mentioned earlier some bad elements are unavoidable due to the geometry. However by adjusting the radial element lines stretching from the upper boundary of mesh section 2 to the inner boundary the skewness of the elements can be minimized as well as the sharp corners of the elements close to the root radius. Figure 44a beneath shows an example mesh of the external spline where the radial lines of mesh section 2 are purely radial and Figure 44b shows the same mesh where the radial lines have been moved in the negative φ direction in the polar coordinate system in order straighten up the elements. 0 1 2 3 4 5 20 20.5 21 21.5 22 22.5 23 23.5 24 (a) Without angle adjustment 0 1 2 3 4 5 20 20.5 21 21.5 22 22.5 23 23.5 24 (b) With angle adjustment Figure 44: External mesh with and without angle adjustment The method used for adjusting the elements of mesh section 2 is to first calculate the element angle for the two elements with a common node in the point where all mesh sections intersect, the node highlighted in green in Figure 44a. The node points on the inner boundary of mesh section 2 are then moved in order to make these two angles equal up to a certain threshold depending on the size of mesh section 2 in comparison to the other mesh sections. If the inner diameter of the external spline, which can vary considerable between different spline configurations, becomes small mesh section 2 becomes large and it is not possible to have an equal angle between the two elements of 40 3.4 Mesh quality improvement 3 METHOD interest. If the corner angle of these two are held equal and the inner diameter becomes small the elements to the left in section 2 would end up outside of the domain and therefore a threshold on the angle of adjustment depending on the size of mesh section 2 has been implemented. Figure 45a shows an example of a symmetry section of an external spline with a large inner diameter and Figure 45b one with a small inner diameter. The mesh algorithm will not work for external splines on a solid axle i.e with an inner diameter of 0, but this is not necessary due to how the simulation is set up which is explained further in Section 3.5.2. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 21 21.5 22 22.5 23 23.5 24 (a) Large inner diameter −4 −2 0 2 4 6 8 14 16 18 20 22 24 (b) Small inner diameter Figure 45: Mesh behaviour for difference in inner diameter for the external spline The same problem that occurs for mesh section 2 for the external spline also occurs for mesh section 2 of the internal spline. It is in this case the elements along the lower boundary of mesh section 2 that obtains small corner angles if the element lines are purely radial. Therefore the same technique to improve the element geometry has been applied to the internal spline and the behaviour of the mesh corresponding to Figure 45 for the internal spline with a varying outer diameter is shown in Figure 46. 0 1 2 3 4 5 23 23.5 24 24.5 25 25.5 26 26.5 (a) Small outer diameter −2 0 2 4 6 8 23 24 25 26 27 28 29 30 31 32 (b) Large outer diameter Figure 46: Mesh behaviour for difference in outer diameter for the internal spline 41 3.4 Mesh quality improvement 3 METHOD 3.4.2 Progressive element sizing The inner diameter of the part with external splines and the outer diameter of the part with internal splines can vary substantially between different couplings. If the inner diameter of the external spline for instance is relatively small the total size of the part becomes large with a big portion of the part where resolution in the solution is of limited interest. This creates some challenges with respect to precision and computational effort when it comes to the discretization. If the elements stretching from the inner diameter to the upper boundary of mesh section 2 are of equal size the amount of elements will rapidly increase with a decreasing inner diameter if the elements are to be kept roughly equal in size to the rest of the part, this can be seen in Figure 47a beneath. On the other hand if the elements of this section are made large the precision and resolution at the upper boundary of mesh section 2 is compromised. In order to address this problem three different element sizing options have been implemented for this section. The user can choose from using linear element sizing, progressive element sizing or automatic element sizing, which is a progressive element sizing where the initial size is determined by the the mesh size used for mesh section 1. The automatic and progressive element sizing is based on an exponential function where the user can alter the progressiveness of the function by changing the exponent. An example mesh with linearly sized elements for section 2 is shown in Figure 47a, in Figure 47b the same geometry is meshed using the automatic element sizing with an exponent set to 2. It is clear from Figure 47 that the part using automatic element sizing will require less computational effort with small sacrifice to resolution close to the root. As mentioned the same problem occurs when the outer diameter of the part with internal splines becomes large and thus the same type of element sizing algorithms has been implemented for the internal spline. −2 −1 0 1 2 3 4 5 6 18 19 20 21 22 23 24 (a) Linear −2 −1 0 1 2 3 4 5 6 18 19 20 21 22 23 24 (b) Automatic Figure 47: Difference between linear and automatic element sizing 3.4.3 Element adjustment internal spline For mesh section 1 of the internal spline the element sides in the radial direction are based on the involute profile and the projected involute profile and the element sides in the circumferential direction are orthogonal to these two profiles. The elements in the lower part of mesh section 3 are in the radial direction based on projections of the minor diameter (Dii) and are linearly spaced in the circumferential direction between the projected involute profile and the left boundary. This method results in some skewness in the elements closest to the minor diameter as shown in Figure 49a. This skewness is small compared to many other elements in the discretization, but by comparing them to other elements in mesh section 1 the skewness is noticeable when calculating the determinant of the Jacobian in the element mapping. Since these elements are close to the 42 3.4 Mesh quality improvement 3 METHOD contact surface a function that adjusts these elements have been implemented and the result, although difficult to see, is shown in Figure 49b. Figure 48 shows the part of the spline covered in Figure 49. −2 0 2 4 6 8 23 24 25 26 27 28 29 30 Figure 48: Part of interest 2.5 3 3.5 4 4.5 5 21.8 22 22.2 22.4 22.6 22.8 23 23.2 23.4 23.6 23.8 (a) Before adjustment 2.5 3 3.5 4 4.5 21.8 22 22.2 22.4 22.6 22.8 23 23.2 23.4 23.6 23.8 (b) After adjustment Figure 49: Adjustment of lower elements for internal spline 43 3.5 Program output 3 METHOD 3.5 Program output The program generates output files in Abaqus syntax in two levels, it generates two or three part files (.inc) depending on the configuration and one assembly file (.inp). The program also generates a .mat file containing all the data specified in the GUI, this file is used in order to be able to reset the program to settings specified in a previous analysis. 3.5.1 Output files The setup of the simulation and the definition of parts are done in separate files in order to allow for flexibility, separating the part definition from the actual setup makes it possible to alter between a two part configuration and a three part configuration by only changing the assembly file. The part files are built up by three parts, the first is purely informational and contains the geometric description of the part, the mesh settings used in its creation and general information such as name, creation date and software version. The second part of the file is the actual geometric description of the mesh defined by the node coordinate matrix and the element coupling matrix. The last part is where all the node sets, element sets and surfaces are defined. The assembly file consists of all the information needed in order to setup the simulation as well as the output requests. 3.5.2 Simulation setup The program can in total generate four different types of analyses, first of all it can generate a spline coupling in either a 2 part configuration with an internal spline and an external spline or in a 3 part configuration with 2 external splines and one internal spline used to emulate an engaging sleeve. For both these two configurations it is possible to run them as either a standalone simulation or as part of a shaft assembly. The standalone simulation type introduces all necessary boundary conditions in order to solve the finite element problem, where as the shaft assembly option creates all necessary element and surface definitions as well as creates tie constraints in order to allow the user to quickly couple the spline coupling to an existing model. The assembly input files starts with reading the geometry and sets from the part files and places out the parts in space according to the users input, an example of this is shown for a 2 part configuration in Figure 50a and for a 3-part configuration in Figure 51b. (a) 2 part configuration (b) 3 part configuration Figure 50: Spline coupling assembly 44 3.5 Program output 3 METHOD If the simulation is of a standalone 2-part configuration the next step in the setup is creating couplings in the centre of each part on to which the boundary conditions are applied, these couplings are connected to the surface pertaining to the inner boundary of the external spline and the outer boundary of the internal spline respectively. Effectively this means that the boundary conditions applied to these couplings points to their respective surface. The program can create two different types of couplings, either kinematic couplings or distributing couplings which differ in the transfer of the boundary conditions and are explained more thoroughly under Section 6.3. The program then goes on to defining the contact surfaces for the simulation, i.e the involute profile of the teeth. If the simulation is of the shaft assembly type both the left and the right involute of the teeth are defined as contact surfaces since the loading is unknown. If on the other hand the simulation is of the standalone type the side that is defined as contact surface is dependent on the direction of the torque applied and to which part the torque is applied, both these parameters are user defined in the program. In both the 2 part configuration as well as the 3 part configuration the involute surfaces of the external spline are defined as slave surfaces and the corresponding surfaces of the internal spline are defined as master surfaces. Several different setups of master and slave surfaces have been tested with little to no difference in result, so in order to maintain consistency the internal spline involute surfaces are always defined as master surfaces. There are several guidelines to follow when it comes to defining these surfaces, for instance it is often common practice to set the surface with the finest mesh as the slave surface which in this special application is hard to determine. It is a fair assumption to assume that mesh size of the involute surfaces for the internal and the external involute profile to be of roughly equal size. Furthermore the contact type has been set to surface to surface which means that the solver will interpolate between contact nodes if necessary in order to get the contact normal and gap function as close to reality as possible. As mentioned previously all involute profiles both left and right for all parts are defined individually. This is done partly for ease of post processing since it makes it easy to extract contact force and stress level variation. The other reason for defining these surfaces individually is order to be able to alter the gap function for each contact pair independently. This feature allows the user to emulate the effect of manufacturing tolerances by manually changing the gap between each contact surface with the Abaqus command Clearance and effectively changing the contact ratio. Figure 51 shows the contact surfaces for both types of parts. (a) External (b) Internal Figure 51: Extracted contact surfaces 45 3.5 Program output 3 METHOD The boundary conditions used in this analysis are quite straight forward, for the two part config- uration the coupling corresponding to the spline without any torque applied to it is clamped in all six degrees of freedom. The other part is clamped in all degrees of freedom except for the one to which the torque is applied. The driven spline is only bound by its contacts with the clamped part and no inertia forces are taken into account due to it being a static analysis. This means that theoretically the part would rotate an infinite amount even by a infinitely small load step. Of course Abaqus has built in algorithms that establishes contacts and counteracts this problem, but to build in robustness in the program and speed up the solving process the program has a built in subroutine that automatically takes up any rotational gap in the coupling and establishes the contacts. An example of this is shown in Figure 52 where Figure 52a shows a spline coupling with a loose fit and Figure 52b how the program automatically rotates the parts in order to establish the contact. (a) Loose fit spline coupling (b) Contact established Figure 52: Automatic contact establishment The last part of the simulation setup covers the output requests, this part of the setup mostly consists of reading the output requested by the user and translating them into Abaqus syntax. The user can choose all outputs recognized by Abaqus for all predefined node sets, element sets and contact surfaces as well as the print frequency for these outputs. There is also an additional output parameter created specifically for the node and element set pertaining to the roots of the parts, the user can select the starting point for these sets in terms of number of elements from the involute profile. This has been created in order for the user to quickly be able to define the boundary of the root set, which is one of the parts of most interest in the post processing. By for instance setting the start point of these sets one or a few elements from the start of the involute profile stress concentrations due to the contact can be avoided when analyzing the stress levels in the root. Figure 53a shows an example of an element set corresponding to the root with the boundary set all the way to the start of the involute profile and Figure 53b shows an example where the root element set boundary is moved three elements down from the start of the involute profile. 46 3.5 Program output 3 METHOD (a) 0 elements from start of involute (b) 3 elements from start of involute Figure 53: Root element set definition 47 4 PROGRAM STRUCTURE 4 Program structure This section deals with the overall structure of the program from reading the input parameters to creating the Abaqus input file. The chapter starts from an holistic perspective and briefly explains the function of the main subprograms and then explains how the subprograms are built up in terms of their subscripts. Included is also an overview and explanation of the graphical user interface (GUI) and the subprograms and sub GUI:s embedded into it. 4.1 Program overview Figure 54 beneath shows a flow chart of the main subprograms that works under the GUI. The user enters input values to the GUI and when the GUI gets the signal to create an input file it goes through the steps showed in Figure 54. It first saves all input data such as geometry and mesh settings as well as all logical data needed in order to remember the state of the interface. This data is then written to a .mat-file called raw_data which is saved in the output folder. The next step is saving the data necessary in order to set up the simulation so that the subprograms can reach it. Once this is done the geometry function is called which creates all node coordinates that are then sent to the mesh algorithm. The mesh algorithm creates the FE discretisized geometry and extracts all element and node sets. This data is then read by the program that generates the part input files (.inc) which translates the data to Abaqus syntax, defines all parameters needed on the part level and creates the part file. Lastly the program creating the assembly input file (.inp) is called which reads the user input data as well as the part files and sets up all parameters for the simulation which are then written into the assembly input file. As can be seen in Figure 54 the path and methods from the input data to the input files on the part level are very similar for the internal and external spline. Due to the modular build of the program the 3-part configuration spline coupling generation becomes very similar to the 2-part configuration. The major difference lays in what order the different subprograms and subscripts are called and in what order data is saved. In general the only difference is that the external section of Figure 54 is called twice and two external input files are created and a different program creating the assembly file for the 3-part configuration is called. 48 4.1 Program overview 4 PROGRAM STRUCTURE GUIInput data Geometry external Geometry internal Mesh external Mesh internal Create external input file Create internal input file External.inc Create assembly file Internal.incAssembly.inp raw_data.mat User input Figure 54: Program overview 49 4.2 Graphical user interface 4 PROGRAM STRUCTURE 4.2 Graphical user interface The GUI part of the program is designed to help the user to set up the simulation and help to make sure that the spline coupling, mesh settings and simulation settings are according to the desired specification before the creation of the input file is initiated. The GUI is shown in Figure 55 and as can be seen it is rather large due to the many input parameters and options. It is divided into three main sections: geometry creation, meshing and simulation setup with supporting control functions for each section in order to make sure that the output is according to what the user anticipates before any computational time is put into setting up the analysis. This section will cover the typical path from input parameters to a complete ready to be solved model in the order shown in Figure 55 for a 2-part spline coupling. The creation of a 3-part model requires a few more inputs compared to the 2-part configuration and the difference is explained further in section 4.2.4. Geometry Mesh settings Simulation setup Figure 55: Graphical user interface with highlighted main sections 50 4.2 Graphical user interface 4 PROGRAM STRUCTURE 4.2.1 Geometry This section deals with the geometry creation part of the GUI, this part in itself has been divided into three different parts as shown in Figure 56. A B C Figure 56: Geometry section of the GUI 51 4.2 Graphical user interface 4 PROGRAM STRUCTURE Part A This part consists of the Configuration pop up menu and the general data part of the geometric description. The Configuration menu lets the user select between a 2-part or 3-part configuration, triggering this menu sets the layout of the GUI between its 2-part layout and 3-part layout. The general data section is where the user sets the common geometric data that is the same for both the internal and external spline. This part also includes two check buttons where the user can choose to define the spline geometry either with a profile shift coefficient or by specifying the tooth thickness and space width. The profile shift coefficient is explained in more detail in Section 2.2.2 and the tooth thickness and space width is defined in Section 2.2.1. Part B In this section the geometric parameters specific to the internal spline are defined. These dimen- sions are quite self explanatory and have already been covered under Section 2 in this report. Part C This part is the external splines counterpart to part B where the geometric data for the external spline is defined. As with part B these dimensions have already been covered in Section 2. How- ever at the bottom of part C there is a push button and a check box that are coupled to the entire geometric part of the GUI. Pushing the Preview geometry button calls on a script that plots a two dimensional symmetry section of the spline coupling based on the inputs entered by the user in the geometric part of the GUI, as the one shown in Figure 57a. By enabling the check box, Show D_form, the plot also includes the form diameter for both the internal and external spline in order to simplify for the user to see if the path of contact is correct. Figure 57b shows an example plot where the form diameter is included. −2 −1 0 1 2 3 4 5 6 7 20 21 22 23 24 25 26 27 (a) Standard preview −2 −1 0 1 2 3 4 5 6 7 20 21 22 23 24 25 26 27 D form (b) Preview with Dform Figure 57: Preview of the spline coupling triggered by the Preview geometry button 4.2.2 Mesh settings After the geometry has been established the next step is to set the mesh settings for the discretiza- tion. The mesh control parameters are the ones within the green box in Figure 55, these have been expanded and are shown beneath in Figure 58. The mesh control parameters themselves have been divided into three parts consisting of element settings (part A), mesh section controls (part B) and visualization (Part C). The left column contains the settings for the internal spline and the right column contains the settings corresponding to the external spline. 52 4.2 Graphical user interface 4 PROGRAM STRUCTURE A B C Figure 58: Mesh section of the GUI Part A This part of the mesh settings controls the number of elements for the three different mesh sections specified in Section 3.2.1. In its standard state the GUI is set as displayed in Figure 58 where both the Optional mesh settings and Specified root buttons are turned off, additionally the Lower line type option in part B is set to auto. This means that the number of elements for the three different mesh sections are only governed by the active fields shown in Figure 58. As can be seen the field Nel lower is inactive, this is coupled to the fact that the field Lower line type is set to auto and is explained further under part B. Figure 59 shows the three different states of part A for the mesh settings of the external spline. 53 4.2 Graphical user interface 4 PROGRAM STRUCTURE Figure 59: Part A of the mesh settings The two check boxes in part A controls the element settings for the active profile, where the user have three different options as shown in Figure 59. The leftmost image in Figure 59 shows the standard state of the element settings where the number of elements are specified by the four active input parameters. In this case the number of elements gets distributed as shown in Figure 32 in Section 3.2.1, where a subscript calculates the length of the three different parts that make up the active profile and distributes the elements in order to get an equal element side length. If the check boxes are ticked according to the middle image in Figure 59 the user has the option to specify the involute profile and the root of the tooth separately where the distribution of elements among the two different parts of the root are done automatically based on element side length. The last option is shown in the right image of Figure 59 with both check boxes ticked. With the settings set in this manner the user can specify the number of elements for each of the three parts of the active profile as shown in Figure 30 in Section 3.2.1 Part B The second part of the mesh parameters are shown in Figure 60 for the external spline, this part includes the settings that determines the size and position of the three mesh regions of the spline as well as some mesh enhancing parameters. 54 4.2 Graphical user interface 4 PROGRAM STRUCTURE Figure 60: Part B of the mesh settings The first parameter labeled Divide base is used to determine the size of mesh section 1 of the external spline. This is done by setting the distance of which the active profile is projected onto the geometry which is defined by the two parameters Divide base and Divide fraction. The divide base sets which charecteristic length of the spline to base the distance on and the divide fraction is the fraction of this distance. Figure 61a shows the two lengths of which the divide base is based on and Figure 61b shows an example where the divide base is set to lower and the divide fraction is set to 0.3 as in Figure 60. For the divide base option there is a third alternative which is the auto setting. By setting the divide base to auto the program calculates the length of both characteristic lengths and selects the shortest of the two, this setting yields the most stable meshing and is therefore set as default. 0 1 2 3 4 5 19 20 21 22 23 24 25 Upper Lower (a) Divide base external 0 1 2 3 4 5 19 20 21 22 23 24 25 Dividing profile Divide fraction (b) Divide fraction example Figure 61: Mesh section 1 settings 55 4.2 Graphical user interface 4 PROGRAM STRUCTURE The settings within the box named Lower line type in Figure 60 are used to control the mesh scheme used in mesh section 2 (see Figure 29) of the external spline. The reasoning behind including this parameter is explained more thoroughly under Section 3.4.2, but in short it allows to set the element side length in the radial direction in a linear or progressive manner. The popup menu itself has three different settings: linear, progressive and auto, where the linear setting spaces the elements linearly in the radial direction and the progressive and auto setting spaces the elements progressively based on an exponential function. The parameter Exponent sets the exponent of the exponential spacing function and is therefore only active if the progressive or auto setting is enabled. The difference between the progressive and auto setting is that with the progressive setting enabled the user can freely choose the number of elements, but with the automatic setting enabled the number of elements are based on the element size used in mesh section 1. The program calculates the element side length of the elements in mesh section 1 and calculates the number of elements needed for mesh section 2 in order to get a smooth transition between the two mesh sections. Figure 62 beneath shows an example of the difference between the two mesh spacing functions, Figure 62a shows a linear spacing and Figure 62b shows a progressive spacing using the auto setting. 0 1 2 3 4 5 20 20.5 21 21.5 22 22.5 23 23.5 24 (a) Linear spacing 0 1 2 3 4 5 20 20.5 21 21.5 22 22.5 23 23.5 24 (b) Progressive spacing using automatic number of elements Figure 62: Mesh schemes for mesh section 2 The last setting included in Part B of the mesh settings is the Radius refinement option which is as the name would suggest a parameter that is used to alter the mesh size at the root radius (trochoid) of the spline tooth. This parameter is implemented in such a way that it refines the mesh along the profile based on the curvature, i.e the larger the spatial derivative of the profile is for a certain point the more the mesh will be refined in that point. This is done in practice by setting the projected distance on which the elements o