Design of a Weight-Optimized Gearbox for a Formula Student Car Bachelor’s Thesis in Mechanics and Maritime Sciences EMIL ALEXSSON ERIK HENRIKSON CARL LUND CHRISTIAN TSOBANOGLOU Department of Mechanics and Maritime Sciences (M2) CHALMERS UNIVERSITY OF TECHNOLOGY Gothenburg, Sweden 2021 www.chalmers.se www.chalmers.se Bachelor’s thesis 2021 REPORT NO 2021:05 Design of a Weight-Optimized Gearbox for a Formula Student Car EMIL ALEXSSON ERIK HENRIKSON CARL LUND CHRISTIAN TSOBANOGLOU Department of Mechanics and Maritime Sciences (M2) Chalmers University of Technology Gothenburg, Sweden 2021 Design of a Weight-Optimized Gearbox for a Formula Student Car EMIL ALEXSSON ERIK HENRIKSON CARL LUND CHRISTIAN TSOBANOGLOU © EMIL ALEXSSON, ERIK HENRIKSON, CARL LUND and CHRISTIAN TSOBANOGLOU, 2021. Supervisor: BJÖRN PÅLSSON, Senior Lecturer at Mechanics and Maritime Sci- ences, Division of Dynamics Examiner: HÅKAN JOHANSSON, Associate Professor at Mechanics and Maritime Sciences, Division of Dynamics Bachelor’s thesis 2021:05 Department of Mechanics and Maritime Sciences (M2) Division of Dynamics Chalmers University of Technology SE-412 96 Gothenburg Telephone +46 31 772 1000 Cover: Visualization of gearbox development, from sketch to structurally analyzed. Typeset in LATEX, template by Magnus Gustaver Printed by Chalmers Reproservice Gothenburg, Sweden 2021 iv Design of a Weight-Optimized Gearbox for a Formula Student Car EMIL ALEXSSON ERIK HENRIKSON CARL LUND CHRISTIAN TSOBANOGLOU Department of Mechanics and Maritime Sciences (M2) Chalmers University of Technology Abstract Chalmers Formula Student is a team consisting of student engineers that design and build race cars to compete against other universities’ teams. Since 2019, Chalmers Formula Student has been designing electric powered four-wheel drive cars. In 2020, Chalmers Formula Student began utilizing electric machines and gearboxes powering all wheels at the wheel hub. This design has many advantages in terms of mechanical packaging and lowering the center of gravity, but the design also means higher mass for the shock absorbers to handle, also called unsprung mass. Higher unsprung mass has a negative effect on the vehicle dynamics. The project purpose is therefore to design a lighter gearbox than the most recently manufactured Chalmers Formula Student gearbox to reduce the unsprung mass. This report covers the process of developing a gearbox for the Formula Student car with the purpose previously mentioned. The project starts with investigating different gearbox configurations via a concept generation, resulting in that a single stage planetary gearbox is the most suitable concept. Then a method for deriving the dimensioning load cases for the gears, bearings and gearbox housing is developed by translating sensor data from competition to resulting forces. The load cases are then used for dimensioning the gears with the gear dimen- sioning software KISSsoft and for stress and deformation analysis on the gearbox housing and bearings with the FEM software ANSYS and 3D-modelling software CATIA V5. The result is a single stage planetary gearbox with a gear ratio of 11.5:1 that is 2.4% lighter than the most recent manufactured Chalmers Formula Student gearbox. However, if only the shared components are compared the final design is 12.3% heavier. The high gear ratio resulted in problematic gear design resulting in heavy gears in order to achieve acceptable safety factors. The conclusion is that even though the developed gearbox became marginally lighter, a single stage planetary gearbox is not a suitable gearbox layout for the application. However, the methods for developing the dimensioning load cases are considered accurate and suitable for dimensioning future gearboxes. Keywords: Gearbox, Formula Student, Gears, Planetary gearbox, Unsprung mass, Design, Dynamics . v Acknowledgements The project has had great assistance of the main supervisor Björn Pålsson and the examiner Håkan Johansson. Per Forsberg at Atlas Copco has offered support and advise in the gear design. Peter Wittke and Tommie Hall at Volvo Cars have assessed the project in an early stage and given guidance concerning the safety factors. Finally, it is noteworthy to thank the Chalmers Formula Student alumnus Erik Lund assisting with converting and managing the logged data for the gearbox housing and bearing load case. Emil Alexsson Erik Henrikson Carl Lund Christian Tsobanoglou Gothenburg, May 2021 vii Acronyms Acronym Description ADC Analog to Digital Converter CAD Computer Aided Design CANBUS Controller Area Network CFS Chalmers Formula Student CFS19 The Chalmers Formula Student 2019 Team CFS20 The Chalmers Formula Student 2020 Team CFS21 The Chalmers Formula Student 2021 Team COG Center Of Gravity DIN Deutsches Institut für Normung FEM Finite Element Method FL Front Left FR Front Right FS Formula Student FSA Formula Student Austria FSN Formula Student Netherlands GPS Global Positioning System MPT Mechanical Power Train RPM Revolutions Per Minute RL Rear Left RR Rear Right SKF Svenska Kullagerfabriken OEM Original Equipment Manufacturer ix Symbols Symbol Description Unit a Acceleration m/s2 A The area between the ground and rear tire - AD Frontal Area m2 a1 Life adjustment factor for reliability - aSKF SKF modification factor - B Bearing width mm C Basic dynamic load rating kN C0 Basic static load rating kN CD Drag coefficient - d Bearing inner diameter mm D Bearing outer diameter mm dm Average diameter mm e Limit for the load ratio - f0 Calculation factor - Fa Axial force on bearings kN FD Drag force N FL Lateral force N FL,FL Lateral force on front left tier N FL,FR Lateral force on front right tier N FL,RL Lateral force on rear left tier N FL,RR Lateral force on rear right tier N FN Normal force N FNF Normal force front wheel axis N FNR Normal force rear wheel axis N Fr Radial force on bearings kN FT Tangential force N g Gravitational constant m/s2 i Gear ratio - igearbox Gear ratio for gearbox - itot Total gear ratio from motor to ground - itotnewlow New lower limit for gear ratio - iwheel Gear ratio that the wheel contributes with - Ka Application factor for gears - κ Viscosity ratio - L1 Distance m L10 Basic rating life (90% reliability) Million rev L10m Basic rating life Million rev L2 Distance m L3 Distance m m Mass of the car kg mwheel Mass of the wheel kg M Bearing mass kg n Rotational speed r/min ηc Contamination factor - x Symbol Description Unit Nlim Limiting speed r/min Nref Reference speed r/min Nspeed bins Number of bins of speed bins # Ntorque bins Number of bins of torque bins # ν Actual viscosity of the lubricant mm2/s ν1 Required viscosity of the lubricant mm2/s Oxy Point, Outer bearing in the xy-plane - Oyz Point, Outer bearing in the yz-plane - OW Offset wheel center m P Equivalent dynamic bearing load kN p Exponent of life equation - PU Fatigue load limit kN R Wheel radius m Rao Reaction force axially outer bearing N Rrix Reaction force radially inner bearing x N Rriz Reaction force radially inner bearing z N Rrox Reaction force radial outer bearing x N Rroz Reaction force radial inner bearing x N Rroz Reaction force radial outer bearing z N ρ Density kg/m3 T Applied torque NM T19 Torque from CFS19 NM Tnew New Torque NM Ttot Total Torque NM U Life cycle fraction % v Velocity m/s ω19 Rotational speed from CFS19 rad/s ωnew New rotational speed rad/s X Calculation factor for the radial load - Y Calculation factors for the axial load - z1 Number of teeth on the input gear # z2 Number of teeth on the output gear # zr Number of teeth of the ring gear # zs Number of teeth of the sun gear # xi xii Contents List of Figures xv List of Tables xvii 1 Introduction 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Unsprung Mass . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Formula Student . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.3 Transmissions for Electric Machines . . . . . . . . . . . . . . . 3 1.1.4 Previous CFS-Transmissions . . . . . . . . . . . . . . . . . . . 3 1.1.5 Gear Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Theory 7 2.1 Gear Geometry and Gearbox Kinematics . . . . . . . . . . . . . . . . 7 2.2 Load Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Concept Generation 11 3.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.1.1 Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.1.2 Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.1.3 Sub-Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.1.4 Concept Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2.1 Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2.2 Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2.3 Sub-Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2.4 Concept Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2.5 Final Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4 Transmission Load Cases 19 4.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.2.1 Data Sourcing . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 xiii Contents 4.2.2 Load Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.2.3 Expected Service Life . . . . . . . . . . . . . . . . . . . . . . . 22 4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5 Gear Design 25 5.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.2.1 Configurating Fundamental Geometry . . . . . . . . . . . . . 26 5.2.2 Defining parameters . . . . . . . . . . . . . . . . . . . . . . . 27 5.2.3 Contact, Stress and Load Analysis . . . . . . . . . . . . . . . 29 5.2.4 Planetary Bearings . . . . . . . . . . . . . . . . . . . . . . . . 31 5.2.5 Gear Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 6 Bearings & Housing Load Cases 37 6.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 6.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6.2.1 Data Sourcing . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6.2.2 Deriving Load Cases from Data . . . . . . . . . . . . . . . . . 40 6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 6.3.1 Data Interpretation . . . . . . . . . . . . . . . . . . . . . . . . 44 6.3.2 Deriving Loads From Data . . . . . . . . . . . . . . . . . . . . 46 6.3.3 Final Load Cases . . . . . . . . . . . . . . . . . . . . . . . . . 49 6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 7 Housing Design & Bearing Selection 51 7.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 7.1.1 Housing & Carrier Design . . . . . . . . . . . . . . . . . . . . 51 7.1.2 Bearing Selection . . . . . . . . . . . . . . . . . . . . . . . . . 51 7.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 7.2.1 Housing & Carrier Design . . . . . . . . . . . . . . . . . . . . 52 7.2.2 Bearing Selection . . . . . . . . . . . . . . . . . . . . . . . . . 55 7.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 7.3.1 Housing Design . . . . . . . . . . . . . . . . . . . . . . . . . . 59 7.3.2 Bearing Selection . . . . . . . . . . . . . . . . . . . . . . . . . 60 8 Final Design 61 9 Discussion 65 10 Conclusion 67 Bibliography 68 11 Appendices 71 xiv List of Figures 1.1 Madeleine, CFS20’s Car - Photographer: Eric Gustafsson . . . . . . . 1 1.2 Exploded view of the current powertrain assembly for the CFS car . . 3 1.3 Lap time simulation - Created by: Daniel Persson Ilonen, CFS19 [12] 4 2.1 Three examples of geared transmissions . . . . . . . . . . . . . . . . . 8 2.2 Geometry of a gear tooth . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.1 Morphological matrix for transmission, Excerpt from Appendix B . . 15 3.2 Showing the differences between different transmission layouts from Figure 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3 Showing the differences between torque transfer output via shaft and torque transfer output via housing from Figure 3.1 . . . . . . . . . . 16 3.4 Kesselring matrix for transmission concepts, excerpt from Appendix E 16 3.5 Pugh matrix for transmission concepts, simplification of Appendix G 17 3.6 Final concept sketched . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.1 Bi-variate histogram of torque and speed . . . . . . . . . . . . . . . . 21 5.1 Gearbox model in KISSsys . . . . . . . . . . . . . . . . . . . . . . . . 27 5.2 Definition of stationary components as well as input and output in KISSsoft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.3 Normal load distribution on the sun gear in KISSsoft . . . . . . . . . 30 5.4 3D-models of the gear designs . . . . . . . . . . . . . . . . . . . . . . 33 5.5 3D-models of final gears, rendered in CATIA V5 . . . . . . . . . . . . 34 6.1 Illustration of normal forces at standstill for the CFS19 Car . . . . . 37 6.2 Illustration of tangential forces on one wheel at the moment of accel- eration from standstill, denoted as FT . FN denotes normal force and T the applied torque. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 6.3 Illustration of lateral forces during cornering and deceleration from top view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 6.4 Angle sensor logging the angle of the rear pushrod pivot shaft . . . . 39 6.5 Victoria, developed by CFS19 . . . . . . . . . . . . . . . . . . . . . . 40 6.6 Free body diagram from side view for hand calculations of normal forces on the front axle . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6.7 Free body diagram for hand calculations of the tangential force, de- noted FT , on one wheel . . . . . . . . . . . . . . . . . . . . . . . . . . 42 xv List of Figures 6.8 Illustration of the position of the three forces FN , FT and FL on the wheel used for dimensioning the gearbox housing and bearings . . . . 43 6.9 Imported ADC-values from the front pushrod pivot shaft angle sensors 44 6.10 Transfer function between ADC-signal and angle for both front wheels 45 6.11 All ADC-data points converted into angles on the pushrod pivot shaft 45 6.12 Graph showing the approximated normal forces for FL and FR wheel 47 6.13 Graph showing the approximated tangential forces for one wheel . . . 48 6.14 Graph showing the approximated lateral forces for both front wheels . 48 6.15 Overview of front suspension . . . . . . . . . . . . . . . . . . . . . . 49 7.1 Rendering and deformation analysis of carrier . . . . . . . . . . . . . 52 7.2 Reference points for calculating the skewing of the planet gears in carrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 7.3 3D-modelling and deformation analysis of housing . . . . . . . . . . . 54 7.4 3D-modelling and deformation analysis of final housing . . . . . . . . 55 7.5 Free body diagram of the bearing assembly in the YZ-plane . . . . . 56 7.6 Free body diagram of the bearing assembly in the XY-plane . . . . . 56 8.1 Front and section view of the gearbox . . . . . . . . . . . . . . . . . . 61 8.2 Exploded view of the carrier and its associated components . . . . . . 62 8.3 Exploded view of the housing and the ring gear, with the rotating assembly in between . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 8.4 3D-model of the final gearbox in CATIA V5 . . . . . . . . . . . . . . 63 xvi List of Tables 3.1 Requirements specification without load cases . . . . . . . . . . . . . 14 3.2 Overview of sub-functions . . . . . . . . . . . . . . . . . . . . . . . . 15 5.1 Input parameters in KISSsoft . . . . . . . . . . . . . . . . . . . . . . 27 5.2 Materials and oil choice for gears . . . . . . . . . . . . . . . . . . . . 28 5.3 Input parameters in KISSsoft step 2 . . . . . . . . . . . . . . . . . . . 28 5.4 Manufacturing process and tooth thickness tolerances for gear . . . . 28 5.5 Fine sizing in KISSsoft . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.6 Axis alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.7 Operating backlash in KISSsoft . . . . . . . . . . . . . . . . . . . . . 31 5.8 Planetary and carrier bearings in KISSsys . . . . . . . . . . . . . . . 32 5.9 Final safety factors for gears . . . . . . . . . . . . . . . . . . . . . . . 32 5.10 Micropitting and scuffing for gears . . . . . . . . . . . . . . . . . . . . 32 7.1 Specifications of the final selected bearings for the wheel axis . . . . . 59 7.2 Calculations on final selected bearings . . . . . . . . . . . . . . . . . 59 8.1 Weight comparison between CFS19 and the new gearbox . . . . . . . 63 xvii List of Tables xviii 1 Introduction Formula Student (FS) is one of the world’s largest engineering competitions for students. Teams from around the world design, build and make a business plan for a small formula style race car. Chalmers Formula Student (CFS) has participated in the competition since 2002. Initially CFS built combustion engine cars but has since 2015 built electric cars. Additionally, they have been implementing a four wheel drive system on the car since 2019. The system consists of four motors mounted on each wheel assembly, each with their respective gearbox. Since it is a racing car, it is greatly advantageous that the gearboxes are light, particularly since the gearboxes are mounted directly on the wheel which should follow the unevenness of the road as smoothly as possible. Figure 1.1: Madeleine, CFS20’s Car - Photographer: Eric Gustafsson In Sweden, there are currently eight Formula Student teams, including the team from Chalmers. The teams build a new race car every year, which they take to various competitions around the world in the summer to compete against other universities. However, Formula Student is not really about building the fastest car, it is about being able to make the optimal race car based on the conditions that the team has. It is in other words of great concern to be able to create and explain the optimal design. 1 1. Introduction 1.1 Background 1.1.1 Unsprung Mass The consequence of mounting the gearboxes in the wheel assembly is that the me- chanical powertrain follows the movement of the tyres, resulting in a high unsprung mass compared to an inboard solution [4]. Unsprung mass, in the context of vehicle dynamics, includes all mass that is not supported by a vehicle’s suspension. From a performance standpoint, it is important that the unsprung mass is as low as possible since it directly affects the dynamic normal contact force between the road and the tyre’s contact patch which is of great significance for the vehicle’s handling. 1.1.2 Formula Student Formula Student competitions take place typically during the summer. There are several different competitions around the world, and they usually last for a week. The week begins with the car being inspected to compete, then the competition itself begins. In Formula Student the teams compete in both static and dynamic events. The dynamic events are: Acceleration 75m straight. Skidpad, race in constant turning, going in a figure 8 circuit, two right hand laps and two left hand laps, fastest time wins. Autocross, one lap on a tight and winding track that is around 1 km long, fastest time wins. Endurance, the same track as autocross but you drive 20 km, one driver change, fastest time wins. Efficiency, the energy consumed during endurance is measured and the time is also taken into account. And the static events are: Engineering Design, the team presents the car to several judges and questions will be asked about their design and processes. The better the answer, the better the score. Cost and Manufacturing, Like Engineering Design but the judges focus on man- ufacturing and costs, if the judges are convinced that the work is well done, more points are obtained. Business Plan Presentation, The judges will now be investors and the team will convince the judges that their car can be monetized. How well this is done deter- mines how many points one gets. A well-developed gearbox and also a well-documented process is obviously favorable for the team, both in the static and dynamic events. A well-developed FS gearbox has two characteristic features. The first one is that it is light, Newton’s second law gives the insight that lower mass gives a higher acceleration which is advantageous for a race car. And the second one is that it is reliable and does not fail. 2 1. Introduction 1.1.3 Transmissions for Electric Machines Conventional internal combustional engines can only operate efficiently and deliver the power at a certain RPM interval. This is why transmissions with multiple stages are used, in order to maximize the torque output when accelerating and speed when the vehicle is moving. The difference when it comes to electrical motors is that they are able to offer instant power delivery at a very low RPM while easily being able to rev to 10000 RPM and more. Additionally, electrical motors do not stall, eliminating the need to have a clutch, which increases the simplicity. That is why multi-speed transmissions are not needed and can cause inefficiencies in terms of added weight [15]. 1.1.4 Previous CFS-Transmissions Since 2019 CFS has built four wheel drive cars utilizing hub motors for packaging reasons, meaning that there is one driving unit consisting of an electric motor and a transmission powering each wheel mounted directly on the wheel assembly. This assembly is illustrated in Figure 1.2. The CFS21’s gearbox solution is a 1.5 step compound planetary gearbox with a 14:1 gear ratio. A compound planetary gearbox has a pair of longitudinally connected planet gears with different radii with one set of gears meshing with the input sun and another meshing with the ring gear [18]. Figure 1.2: Exploded view of the current powertrain assembly for the CFS car In addition, CFS has decided to move towards a lower tyre profile for the 2022 season, which results in a reduced wheel circumference. Therefore, the transmission gear ratio needs to be evaluated which might enable other more effective transmission solutions. 3 1. Introduction 1.1.5 Gear Ratio The gear ratio from the electric machine to the wheel mainly depends on two pa- rameters, the gearbox gear ratio and the roll diameter of the wheel. CFS intends to change from the 18” used in recent years to 16” in wheel diameter. This will affect the total gear ratio, which will be demonstrated in a calculation example below. CFS wants an equally optimal gear ratio, with a lighter gearbox, which should result in a faster car. To do this, simulations are used, below is a simulation that shows different lap times depending on the gear ratio. It is also known that the gear ratio on the CFS19 car was 14:1 [12]. Figure 1.3: Lap time simulation - Created by: Daniel Persson Ilonen, CFS19 [12] The conclusion from Figure 1.3 is that a gear ratio between approximately 13:1 and 14:1 gives the same lap time. An equation for the total gear ratio is generated in Equation 1.1. itot = igearbox · iwheel (1.1) Where igearbox is the gear ratio for the gearbox, which is what this project should determine and design a gearbox that can deliver. iwheel is the ratio that the wheel diameter change contributes with. If the wheel diameter changes this gives a ratio that needs to be taken into consideration if the same lap time wants to be achieved. Then the value of the limits that still gives an equally fast car are recalculated with the new smaller wheel (18”→ 16”) is evaluated in equation 1.2 and 1.3. The lower limit: itotNewLow = 13 · 16 18 = 11.555... (1.2) The upper limit: itotNewLow = 14 · 16 18 = 12.444... (1.3) The values are rounded and made to an interval within which the new gearbox must be [11.5 : 1− 13 : 1]. 4 1. Introduction 1.2 Purpose The purpose of this project is to develop a gearbox for the upcoming CFS car that is lighter than the most recent manufactured CFS gearbox. This is done with the overall objective to lower the unsprung mass which should result in a better performing car. The gearbox design should be durable enough to last a full season of running. The secondary purpose is that the outcome of this project should result in a development for a design methodology that can be used by future CFS teams. 1.3 Methodology The structure of this report is presented in a specific way. The report is constituted of different sections beginning with concept generation, transmission load cases, gear design, bearings and housing load cases, housing design and bearing selection and final design. Each section has its own method, results and discussion. This is due to the fact that each section is dependent on the previous section. Before beginning with the gear design for instance, the concept generation and transmission load cases have to be finished, to know what type of gearbox is going to be used and the loads affecting it. The first step is to generate a concept. This concept is subjected to loads meaning a load case spectrum has to be defined using MATLAB scripts provided by previous CFS teams. A model geometry is then created in KISSsys and the spectrum is uploaded to the software. The loads and stresses are analyzed in KISSsoft and the gears are optimized to get acceptable safety factors. Simultaneously, using MATLAB scripts that compute normal, tangential and lateral forces during the course of a lap, load cases are defined for the housing and bearings in order to dimension them correctly. As a next step, a CAD model is generated in CATIA V5, and finally, the housing is analyzed using a FEM software called ANSYS to study the stresses affecting it. 1.4 Limitations As with many projects, there exists a few delimitations related to this project as well. The budget for all four gearboxes should not exceed a market value of 100 000 Swedish crowns. This will have implications on the design of the gearbox when it comes to the choice of materials, manufacturing processes, how advanced the gearbox can be made and which outsourcing that can be done. It is also important to note that the time for developing the gearbox is limited to the spring of 2021 (2021-01-18 to 2021-05-14). Moreover, it is worth mentioning that the new design of the gearbox may be bigger than the current one, which implies that the upright (see Figure 1.2) has then to be bigger. Thus it will have a larger volume and consequently a higher mass. However this is not an area which will be studied or analysed in detail in this project but estimating calculations will be done to this matter. 5 1. Introduction 6 2 Theory 2.1 Gear Geometry and Gearbox Kinematics There exist myriads of different gearbox designs and a multitude of different gear tooth geometries. The ones that are found to be relevant to this project are presented in this chapter to give some background for the reader. Perhaps the out most simple gear transmission is the spur gear transmission (see Figure 2.1 a) which in its most basic form consists of only two cylindrical gears with external teeth on parallel shafts. The gear ratio of a spur gear transmission is calculated as [17]: i = z2 z1 (2.1) Where i is the transmission ratio defined as the number of turns of the input for every turn of the output1, z1 is the number of teeth on the driven gear (input gear) and z2 is the number of teeth on the output gear. Another geared transmission is the internal gear transmission (see Figure 2.1 b) this consists of a smaller spur gear (red) enclosed in a larger internally toothed gear (green). The transmission ratio calculation is the same as for the spur gear transmission. The internal gear transmission has a larger contact ratio than that of two externally toothed gears working together. This means that more teeth on average are in contact with each other which is beneficial as the load is distributed over more teeth. [17] A third common gear transmission is the planetary transmission (see Figure 2.1 c) (also known as a epicyclic gear train), this is a very compact way to achieve a comparatively high transmission ratio. The input and output may be any of the three: sun gear (yellow), planet carrier (green) or ring gear (red). The transmission ratio of a planetary gearbox is dependant on which parts are made to be input, output and which are stationary. If the sun gear is the input, the carrier is stationary and the ring gear is the output the ratio will be: i = −zr zs (2.2) Where i again is the transmission ratio as defined under equation 2.1, zs is the number of teeth of the sun gear and zr is the number of teeth of the ring gear. If the sun gear is the input, the ring gear is stationary and the carrier is the 1Sometimes defined as the inverse but this is how it is used in this report. 7 2. Theory output the ratio will be [17]: i = zr zs + 1 (2.3) This is the highest ratio configuration possible with a this type of planetary gearbox, with one input. Other configurations are possible as well but will not be discussed in this report. a) Spur gears b) Internal gear c) Planetary gearbox (epicyclic gearing) Figure 2.1: Three examples of geared transmissions Source: Wikimedia Commons One of the most used gear geometries is the straight cut involute gear as it has favourable transmission characteristics as well as being relatively easy to produce. This is the type of gear that will be used in all later examples of gearboxes. The involute part can be found on the flank of the gear tooth which represents the side of the tooth (see Figure 2.2). The gear tooth root is the base of the tooth from the flank to the lowest part of the valley. The gear being straight cut refers to the teeth being parallel to the shaft on which the gear is mounted, as opposed to helical gears on which the teeth are cut at an angle. [17] Figure 2.2: Geometry of a gear tooth 8 2. Theory The gears themselves are defined by a set of parameters which are standardised, a few of these will be very central to the performance of the gearbox, others will be forced by outside influences (not all parameters will be discussed in this part). One of the first parameters to decide is the number of teeth on each gear to achieve the desired transmission ratio. The kinematics of this looks very different for different types of gearboxes but in the end it is always the number of teeth that in some way decides the gear ratio. Many (theoretically infinite) sets of teeth numbers will yield the desired ratio but only a few will be practical. Here the tooth size needs to be determined, known as the module. The module together with the approximate desired size of the gearbox will determine which of the theoretically working sets of number of teeth will be used. Interacting gears, known as meshing gears, need to have the same module2 to be able to mesh correctly. The module may in theory be any value (size) but since for most methods of producing gears a special tool is needed for each module, the sizes are standardised. This means that one should strive to use a standard modulus to save a significant amount of machining cost. A set of gears might have restrictions on the exact distance between the shafts, in this case profile shifting might be needed to keep the gears meshing properly. This is a process where the tool is shifted radially from “normal” cutting depth to create teeth with slightly different geometry. [17] 2.2 Load Cases A load case denotes a group of loads, supports or displacements that are applied to a model at a specific time. A model can be subjected to different load cases at different times [2]. 2Actually, the base pitch needs to be the same but in practice this means that the module will have to match. 9 2. Theory 10 3 Concept Generation 3.1 Method The process of defining possible concepts to the decision of final concept/concepts to further develop is described in this chapter. First, the fundamental boundaries are defined which sets the rules for the generation of concepts. The requirements needed to ensure the function of the final concept are also defined. The boundaries combined with studies of earlier FS-teams and literature results in a number of possible solutions which are then compared with decision matrices based on the requirements. These methods results in a ranking of most promising concepts, of which the final concept/concepts can be chosen. 3.1.1 Boundaries Since the time frame of the project is narrow the concept generation needs to be very effective. Therefore the method of introducing reasonable boundaries is used to minimize the risk of unnecessary work before the concept generation starts. The boundaries are defined with this purpose in mind and chosen carefully to not exclude any useful and innovative solutions. The literature used to support the boundaries are found at the Chalmers library as well as on web sites on the internet. Members from CFS-teams and experts at Chalmers and from the industry are then contacted to verify that the boundaries are reasonable. 3.1.2 Requirements To eliminate insufficient concepts and to determine which concepts are most suitable for the project, a list of requirements is compiled to a requirements specification. The goal is not to make a heavy product-oriented requirements specification, instead only the most important requirements for function and competition compliance are included to support an effective concept generation. The functions are defined by studying Mechanical Powertrain (MPT) reports from earlier CFS-teams to see what external features (brake disc/caliper, upright mounting etc) are needed to be taken into consideration when developing the transmission housing. For comparing the concepts, quantifiable wishes are also defined, and these can be used as support for those decisions. There are also a few rules regarding the gearbox. These rules are defined by Formula Student Germany which is considered to be the leader in FS. 11 3. Concept Generation 3.1.3 Sub-Functions A number of sub-functions should be developed by analyzing which functions the gearbox solves, as well as the functions that the gearbox indirectly affects. This is partly done by controlling which requirements the function must maintain [23]. To ensure that the final concept have all these functions, they are called sub-functions of the concept and will be the foundation of the concept generation. The idea is that each of the sub-functions will be assigned alternative solutions which then will be combined into multiple complete concepts to compare and choose the final concept from. 3.1.4 Concept Matrices The sub-functions are divided into two separate parts: transmission and accessories. Transmission includes the sub-functions for the rotating assembly as well as the housing. Accessories includes the sub-functions that will be mounted on the housing, for example the break caliper and upright seen in Figure 1.2. The reason for this division is to simplify the concept generation process since time is limited. The division means that the number of concepts in the first step decreases, this is because the number of concepts is the number of the solution for each sub-function multiplied by each other. By looking at fewer sub-functions at a time, the number of concepts to be compared at the same time is reduced, which eliminates unrealizable and less favorable concepts earlier in the process. There are still theoretically many concepts to be implemented, but the process is narrowed down. The risk with this method is that a “heavy” transmission-concept that is eliminated could be lighter when combined with the Accessories. The risk of this being the case is considered very low. First, the transmission sub-concept is concept generated with a Morphological matrix [23]. All combinations are evaluated but the physically impossible combina- tions are ignored. The remaining concepts are evaluated in Pugh [23] and Kesselring matrices. The criteria in both matrices are based on the wishes from the require- ment list. In the first iterations of the Pugh matrix for the transmission-concept, the gearbox from CFS21 (explained in Section 1.1.4 and can be seen in Figure 1.2) is used as a reference. From the last iterations a final ranking from the Pugh matrix method is achieved from which the lowest ranked concepts could be eliminated leaving a fewer number of concepts to continue choosing from. Before these set of concepts are carried over to the Kesselring matrix, the criteria must be defined and quantified. These criteria are based on the wishes from the requirement list as stated before, but are more specific than in the Pugh matrix. This is because each criteria must be a measurable factor based on which all concepts can be evaluated. Therefore the criteria in the Pugh- and Kesselring matrices might differ. From the Kesselring matrix, a final ranking of the transmission-concepts is achieved. From this ranking, one or two final concepts can be chosen. Next, the chosen transmission concept/concepts are combined with the acces- sories in a morphological matrix, which results in a concept of the whole gearbox with accessories. All possible solutions are evaluated in the above mentioned matri- 12 3. Concept Generation ces again, resulting in one or two final complete concepts. 3.2 Results 3.2.1 Boundaries First different gearbox types are benchmarked to use as basis for the concept selec- tion. To investigate what solutions were suitable a literature study was done. This helped exclude irrelevant solutions. In the study it was found that a friction gear, a V-belt or flat-belt solution would have a lower efficiency than a gear solution [17]. It was also found that they had a lower precision, which however, does not have a huge significance but is still disadvantageous [17]. When it comes to chains and sprockets, they perform in a similar manner as a gear to gear solution, but with the possibility to move the shafts further apart, which in this case is the opposite of what is wanted to be achieved. FS requires a compact solution that fits inside the rim. The chain also adds mass. The result of this study is that gear driven transmissions are superior to friction gears, belt and chain drives in terms of packaging, strength and efficiency. Therefore all friction gears, belt and chain drives are excluded from this project since they both are difficult to package in the space given and have low efficiency in terms of friction losses. Therefore only gear-driven transmissions will be evaluated in this project. Bearings are central components in a transmission and there are many pos- sible types of bearings both in terms of models and locations in the housing. To determine what type of bearing solution is most suitable, the load cases need to be defined in order to know what characteristics are needed (axial movement, etc.). The bearings will not be included as a sub-function in the concept generation. This boundary simplifies the concept generation but also means that the concepts need to be thought through so that there are space for bearings. Concerning the type of gears to be used, it is concluded that only spur gears would be used. The reason to this choice lies in their simple design, making them easy and cheap to produce. Add to this they produce no axial forces [20]. Moreover, in planetary gearboxes, a different number of planets can be used and each has its own advantages. In this project however, only 3 planets will be used because the axial forces developed between the mesh of the gears cancel each others in a 3 planet-system [3]. 3.2.2 Requirements A requirements specification (Table 3.1) was established after both discussions with those who had previously been responsible for MPT within CFS and by analyzing the adaptations required for a new tyre diameter. The requirements specification was made as simple as possible and as quantifiable as possible to simplify the concept generation. It was also affected by a number of rules, Formula Student Germany (FSG) defines the rules regarding FS. There are some specific rules for the gearbox which are available as an extract in the Appendix A. The conclusion drawn from 13 3. Concept Generation the rule book was that, primarily, the rotating assembly needs to be protected with 2mm steel or 3mm aluminium alloy 6061-T6, secondly the lubrication needs to be sealed so that no leakage exists when the car is tilted 60°. The details of the load case specific requirements can be found in Chapter 4 and 6. Table 3.1: Requirements specification without load cases 3.2.3 Sub-Functions The requirements from Table 3.1 are used as a base to analyse which sub-functions the concepts need to solve. The main function of the gearbox is of course to transmit torque. This has been divided into two sub-functions, the first is transmission layout and the second is torque transfer output. With transmission layout it is meant the type of planetary gear that is intended and with torque transfer output it is meant how the torque is transmitted to the rim. The gearbox also has some secondary functions regarding the braking and mounting. Since the brakes must be mounted somewhere between the motor and the wheel, a complete solution must be designed taking this into account. The gearbox must also be mounted together with the wheel suspension, this is usually done with a type of upright solution. The secondary functions are also divided into sub-functions, these are mounting brake disc, mounting brake caliper and upright. These three sub-functions will be called accessories. All functions are compiled in 14 3. Concept Generation Table 3.2 to provide a better overview. Table 3.2: Overview of sub-functions # Sub-functions Transmission Sub-functions 1 Transmisson Layout 2 Torque Transfer Output Solution Accessories Sub-functions 3 Mounting Solution Brake Disc 4 Mounting Solution Brake Caliper 5 Upright The reason for the division between transmission- and accessories sub-functions is discussed in Section 3.1.4. 3.2.4 Concept Matrices The concept generation began with a morphological matrix (see Figure 3.1), based on the transmission sub-functions from Table 3.2, to generate 18 concepts to solve the transmissions part of the gearbox. Figure 3.1: Morphological matrix for transmission, Excerpt from Appendix B In Figure 3.1 different types of solutions for the transmission layout output are stated, some of which are explained in Figure 3.2. Figure 3.2: Showing the differences between different transmission layouts from Figure 3.1 15 3. Concept Generation In Figure 3.1 different types of solutions for the torque transfer output are stated, these are explained in Figure 3.3. Figure 3.3: Showing the differences between torque transfer output via shaft and torque transfer output via housing from Figure 3.1 All combinations are evaluated but the physically impossible combinations are ignored, this was done through evaluating against elimination criteria from the requirements specification (Table 3.1). The evaluation was done using an elimination matrix (Appendix C), this resulted in 15 concepts remaining. The 15 concepts were then evaluated using two iterations of a Pugh Matrix (Appendix D), the criteria for the Pugh matrix are based on the wishes from the requirements specification (Table 3.1). In the first iteration the CFS21-solution (explained in Section 1.1.4 and can be seen in Figure 1.2) was used as a reference, this was done to ensure that the concepts that have been generated at least have some advantages over the current solution. The second time the highest ranked concept from the first iteration was used as reference, concept-1 (declared in Appendix B). Then the lowest ranked concepts were eliminated leaving eight concepts to choose from. Before these set of concepts were carried over to the Kesselring matrix (Ap- pendix E), the criteria was defined and quantified. These criteria are based on the wishes from the requirement list as stated before. Figure 3.4: Kesselring matrix for transmission concepts, excerpt from Appendix E 16 3. Concept Generation Two final concepts were then chosen by using a Kesselering Matrix, Figure 3.4. The two final concepts for solving the transmission part were concept number one and twelve. The concepts are described in text form as: 1: 1-stage planetary with locked ring in a stationary housing with torque trans- fer via the output shaft. The brake disc is mounted on the transmission output shaft. 12: Compound planetary in a rotating housing with torque transfer via housing. The brake disc is mounted on the housing, same as CFS21 (explained in Section 1.1.4 and can be seen in Figure 1.2). The concepts were chosen even though they were not the two highest ranked con- cepts. This was because the group considered it too risky to proceed with two single stage gearboxes in case it turned out that it was not physically possible to construct a single stage gearbox according to the existing requirements. Hence the decision to keep the best single stage and the subsequent best concept that was not single stage. Next, the chosen transmission concepts were combined with the accessories in a morphological matrix, see Appendix B. Concept 1 and 12 were further developed into eight concepts with accessories, four from each. An elimination matrix was created (see Appendix F) and used in the same way as before by using the requirements specification from Table 3.1. Two concepts did not reach the requirements of the elimination matrix, thus six concepts remained. Figure 3.5: Pugh matrix for transmission concepts, simplification of Appendix G Then two pugh matrices were made (next to each other in Appendix G ), the result can be seen in Figure 3.5. On the left hand side, concepts 1.2, 1.3, 1.4 were evaluated against concept 1.1 and on the right hand side, concept 12.1 was evaluated against concept 12.2. This was done because the group considered that there was a risk in proceeding with only a single stage planetary gearbox if it later turns out that it did not meet the high requirements. So as a backup, it was also analyzed which was the best compound planetary gearbox. The conclusion from the pugh matrices was that 1.3 was the best single stage planetary gearbox concept and 12.1 was the best compound planetary gearbox concept. 17 3. Concept Generation 3.2.5 Final Concept Primarily, concept 1.3 has been chosen as the main concept to design. The concept can be described with words as: 1.3: Single stage planetary gearbox with a locked ring in a stationary housing with torque transfer via the output shaft. The brake disc is mounted on the transmission output shaft and with the brake caliper mounted on the housing. The upright is separate from housing. The concept has also been sketched for visual understanding in Figure 3.6. Figure 3.6: Final concept sketched Secondarily, Concept 12.2 has been chosen as a backup concept for design if the primarily concept turns out to not be realizable. Concept 12.2 can be described as: 12.1: Compound planetary gearbox in a rotating housing with torque transfer via the housing. The brake disc is mounted on the housing and with the brake caliper mounted on the upright. The upright is separate from the housing. 3.3 Discussion The division of sub-functions that are then concept generated in rounds may have resulted in a heavier total solution. However, the risk that the solution is not the lightest is low. With that said, the final concept in that case should only be slightly heavier than the optimal solution and that is a trade off that needed to be taken since the time is limited and the project is extensive. Moving forward with two concepts is of great importance because there is a great uncertainty in the best concept. There is a risk when working outside of the normal gear ratio and the conclusion could be that it might not be robust enough. 18 4 Transmission Load Cases 4.1 Method To create the transmission load cases, data are gathered from the car on the track as this should give a more accurate depiction of the forces acting on the gearbox than computer simulations would. The real world data should contain unpredictable forces that are missed by the computer simulations. Another reason for choosing measured data is that it can easily be applied to future generation of cars, all using data from the previous year without having to update or create new computer models. Here in this part the transmission will refer only to the internal parts of the gearbox e.g. the actual gears. These are assumed to be unaffected by outside forces acting on the gearbox such as those transferred from the wheel hitting bumps on the road. The only forces assumed to act on the transmission are the moments on the input and output shafts. Even if the torque from the motor is assumed to be the only load acting on the transmission, other data is also important for the analysis. These are: the speed from the motor and GPS data for calculating distance traveled. The speed and torque values are used from all four wheels and the worst value from these four at any given point in time is sorted into one theoretical “worst case” wheel, this “wheel” will include all extraordinary loads experienced by any of the wheels. How this is achieved is discussed in closer detail in the result section. A bi-variate histogram is created from this “worst case wheel” with a grid of speed and torque intervals respectively and the frequency of these scenarios occur- ring making up the third dimension. This load spectrum is then used by the gear designing program KISSsoft for an accurate dimensioning goal. The distance trav- eled during the event can be derived from the GPS data, this in conjunction with the speed of the vehicle determines the expected service life when scaled to the life time distance specified in the requirement specifications (see Table 3.1). 19 4. Transmission Load Cases 4.2 Results 4.2.1 Data Sourcing Five consecutive laps from the autocross event of a celebratory unofficial competition at Achen in Germany were used as a basis for all data in the analysis. The race in Achen was used since it is the only one where all four motors were working without issues. The extracted data was: torque and speed for all four wheels as well as position data from GPS. The torque and speed were sampled at 33.3 Hz and the GPS data was sampled at 10 Hz. The speed was measured directly with a sensor while the torque was (from beforehand) calculated from the current driving the motors. The data was evaluated in MATLAB. To properly import the data, which was stored in .mdf format in the log it was imported into a MATLAB applica- tion (.mlapp) and exported as a MATLAB struct. The MATLAB application was developed by CFS for this purpose. 4.2.2 Load Spectrum A MATLAB program (see Appendix H) was developed to analyse the data from the race and generate a load spectrum to be used in KISSsoft for the dimensioning of the gears. The torque and speed for all wheels are imported, some sign errors are corrected and the vectors are cut to start at the same time stamp as well as to be the same length. Here the sample rate is also checked so that it is the same for all vectors as some sensors use a different sample rate and it all varies among different races. The data is also put through a low pass filter (using MATLAB’s “lowpass” function) to weed out some noise. To make sure to represent a worst case scenario for dimensioning, the torque and speed from the wheel that is experiencing the most extreme of either of those at any given point in time is sorted into a new vector representing a fictional “worst case” wheel. The most extreme here is defined as the largest, positive or negative. This fictional wheel is the basis for all later calculations in the program. When the vehicle is stationary the gearbox is not experiencing any significant load, these data point are thus filtered out, with a small margin around zero to cut out any data points close to zero. If they were not filtered out, a majority of the load case exported to KISSsoft would consist of no load which would lower the resolution of the important data points as well as waste computing power. From these vectors -torque and speed, a bi-variate histogram (see Figure 4.1) is created with a specified number of bins (load levels) in each of the axes. The number of different cases is therefore Ntorque bins · Nspeed bins (N = number of bins). The number of negative values is very low but might be important for the analysis. If these values were to be removed a potentially critical part of the analysis might be missed but if they were to be left untouched they would just increase the frequency of the lowest bins and therefore artificially make the load spectrum less demanding. A separate bin is therefore created for all negative values and the remaining positive 20 4. Transmission Load Cases values are sorted into N − 1 number of bins. Figure 4.1: Bi-variate histogram of torque and speed Now that the data is broken up into cases comprising the combination of a certain torque and speed, some of such cases never occur e.g. very high speed and no torque. These cases with a frequency equaling zero are removed to not waste computing power and time when the KISSsoft analysis is made. The histogram data is then restructured such that the Ntorque bins · Nspeed bins cases are put as a row in a matrix with the columns: frequency, torque and speed. These columns are then normalized such that the highest value in torque equals 1, the highest value in speed equals 1 and the sum of the values in frequency equals 100 (representing 100%). The load case matrix is then written to an Excel spreadsheet (see Appendix I). Since the data is from a car with another gear ratio some modifications are needed for the implementation in KISSsoft. Assuming the same weight of the new car as the CFS19 car as well as the same overall movement on the track, the speed and torque needs to be adjusted according to the new gear ratio. The load spectrum is normalized as described above, such that the maximum value of the torque and speed is 1, this enables KISSsoft easy control of the load values as the maximum value is simply entered and other values are scaled according to this. The maximum value used in KISSsoft is the maximum value of torque and speed bins respectively when scaled to the new gear ratio. The speed is adjusted as: ωnew = ω19 13/1 14/1 (4.1) Where ωnew is the new speed and ω19 is the old speed (from CFS19). 13/1 is the new “gear ratio” (note that this is not the actual gear ratio of the new gearbox but the gear ratio that would be used if the same tyre is kept, it represents a new value 21 4. Transmission Load Cases chosen from Figure 1.3) and 14/1 is the old gear ratio. The torque is adjusted as: Tnew = T19 14/1 13/1 (4.2) Where Tnew is the new torque and T19 is the old torque (from CFS19). The fractions are the gear ratios as described for the speed above. When the change in effective gear ratio is accounted for (note again that this is not the actual gear ratio of the transmission but rather only a movement in Figure 1.3) the new and adjusted speed and torque values can be entered in KISSsoft. 4.2.3 Expected Service Life The gearbox must last 1500 km as specified in the requirements specification (see Table 3.1), this distance had to be translated into time for the KISSsoft analysis. Since the Autocross event from Achen was used for the load spectrum this data set was used to calculate the time as well as this ensures that the time is representative of the load case. The calculation was made in the same MATLAB program (see Appendix H) used to generate the load spectrum as they must match at various attributes. The GPS data is imported and cleaned since it contains a few errors. First the origin of the data is moved closer to the track as it was found to be placed about 12 km away, this is not strictly necessary but it helps when performing the subsequent processing of the data. A handful of data points were found to be a many hundreds of meters off the track, to the human eye these were an obvious error of the GPS. The faulty data points are removed entirely by the program as no obvious way was found to restore their position short of guessing. This was however not a great loss as the lost data points were few and a straight line drawn between the gaps roughly followed the track (as laid out by other laps where the section in question remained intact). The distance is found by taking the euclidean norm of two subsequent data points and adding them all together. The time driving is found by taking the time between two subsequent data points, making sure to only add the pairs where the distance between them is non-zero as waiting time would otherwise be included. Thus the time and distance for the autocross event (consisting of five laps) is known. This time is computed with the equation 4.3 Expected service life = time of 5 laps · dimensioning distance distance of 5 laps = 193 · 1500 1.76 ≈ 164480 s ≈ 46 h (4.3) Note again that “time of 5 laps” is time spent moving. The time it would take to drive 1500 km if the distance is covered in the autocross event was found to be 46 hours. 22 4. Transmission Load Cases 4.3 Discussion It is far from guaranteed that the autocross event in Achen 2019 represents the toughest of load cases for the gears. The autocross event is generally considered to be the toughest event of the competition, which is why it was chosen for the load case. Yet this very race in Achen might not be that demanding compared to another race at a different time and place. If more races would be analysed the load cases could be set with more confidence. The load level for each case in the load spectrum exported to KISSsoft is the middle value of the bin edges in the bi-variate histogram. This means that the load spectrum do not actually contain the maximum loads (nor the minimum), this might be a risk in the analysis. Experimentation in KISSsoft showed that the actual maximum load did not influence the expected service life significantly as the analysis concerned fatigue failure. It should be noted that by taking the middle value of a bin, not only is the values above the middle lowered to the middle but the values below the middle are raised to the middle, this somewhat cancels out in the fatigue calculation. 23 4. Transmission Load Cases 24 5 Gear Design 5.1 Method To be able to determine if the concept will be durable enough and not fail during the season, a model has been created in a commercial software called KISSsys, and then analyzed in KISSsoft. The analysis is done based on the load spectrum created earlier, that is uploaded to KISSsoft’s database tool. The model comes included with the shafts, bearings, couplings and all other necessary components. The next step is to add the boundary conditions and define some key parameters. The maximum torque and speed of the motor is then defined and a gear ratio is chosen. When the model is updated, only the size of the gears is changed. Thus, a resizing of all the other elements needs to be done. Moreover, to be able to effectively develop a gearbox that is dimensioned cor- rectly, a load case report has to be generated. The report analyses the durability, fatigue of all the components in the transmission as well as giving safety factors for the root, flank, micropitting and scuffing. Micropitting is the phenomena where pits (holes) begin to form on the tooth of the gear as a result of excessive stress or the lubrication film not succeeding to protect it well enough [14]. Scuffing describes the phenomena when the lubrication film fails to separate two metal parts, increasing friction and causing material to transfer from one part to the other. This is revealed by scratches on the surface of the metal [19]. When the model is finished and the kinematics are checked to see if the gearbox works properly, the work with KISSsoft begins. The focus in this software is to obtain flank and root safety factors higher than 1 for all gears while not exceeding the maximum allowable outer diameter for the gearbox. The first step is to choose the appropriate materials for the sun gear, planet gears and the ring gear as well as a proper oil for the transmission. The next step is to set the maximum input torque, motor revolutions, service life and the application factor Ka that indicates what type of the shock, the driven and driving machines will be subjected to. The calculation method comes to play afterwards as it needs to be chosen properly so that it suits the type of module designed. After defining the previous parameters, fine sizing comes to play. Here a combination of teeth for the sun, planets and ring are chosen by defining a range for different parameters. The first one is defining a range for the module. The next parameter is the normal pressure angle as it affects the tooth strength and the contact ratio. A range for the center distance should be decided next while staying within the maximum allowable dimensions of the gearbox. The number of teeth on 25 5. Gear Design the gears can be left for KISSsoft to decide by itself or can be partly decided by the user by having a fix number of teeth on only one gear and letting KISSsoft generate different combinations based on that. The last parameter to adjust is the face width of the gears which has a strong effect on the outcome of the safety factors and face load distribution. The manufacturing process is determined in the next step to indicate how the gears are going to be machined. A range of different manufacturing and modification processes are available to choose from. The gear reference profile has to be defined as well to determine how the gears are going to be machined and which tool to select for that purpose. The following step is to do a contact analysis to see the load distribution along the width of the gears and the stresses that the teeth are subjected to. To be able to do that correctly values for the planet carrier tilting relative to the gear axis and the tilting of the planet gears relative to the planet pin axis have to be defined. Furthermore, it is important to define again that the input torque drives the sun gear and the carrier provides the output torque, so that the calculations are done correctly. After the calculation is finished, graphs that show the normal load distribution along the width of the gears, tooth root stress and Hertzian stress should be checked to see how evenly the load is distributed. Hertzian stress denotes a type of stress that is formed when a load is applied between two surfaces with different radii that are in contact [6]. The final step in KISSsoft is to carry out an operating backlash calculation. What it does is that it calculates how much play there exists in between the teeth of the gears in contact as well as the angular play of the sun gear. Last but not least, a choice for the bearings inside the planet gears and on the carrier shafts has to be done. Different types of bearings can be suitable for the same application so the choice can be made based on the previous years’ decisions and on what seems to be reasonable. 5.2 Results 5.2.1 Configurating Fundamental Geometry Starting with KISSsys, a template for a single-stage planetary gearbox was chosen (since this was the conclusion from Section 3.2.5). The input shaft was chosen to drive the sun gear and the carrier was chosen as the output. A value of 19.89 Nm was set for the input torque and 15 282 RPM for the speed of the motor, these values are the highest values in Figure 4.1 scaled with equations 4.1 and 4.2. They are intended to represent scale factors for the load spectrum that is uploaded into the software. Section 1.1.4 states that the gear ratio should be in the range [11.5 : 1− 12.5 : 1]. Since a single-stage planetary gearbox has a recommended gear ratio between [1 : 3 − 1 : 10] [10] it was therefore chosen to design for a 11.5 : 1 gear ratio. It is acceptable to work outside the recommended range as the software will calculate and controls that it lasts. The chosen gear ratio was then used as an input parameter(see Table 5.1). 26 5. Gear Design The output torque and speed obtained were 255.73 Nm and 1328.9 rpm respec- tively. The shafts, bearings, carrier and planet pins were then resized accordingly. The resulting model is shown in Figure 5.1. Table 5.1: Input parameters in KISSsoft Ratio 11.5:1 Input torque (Nm) 19.89 Input motor speed (RPM) 15282 Output torque (Nm) 255.73 Output speed (RPM) 1328.9 Figure 5.1: Gearbox model in KISSsys 5.2.2 Defining parameters Regarding the material choice, by relying partially on the choice of the material done by the CFS team of the previous year and partially on the recommendations from Per Forsberg, an R&D engineer who works at Atlas Copco Industrial Technique, the following choices shown in Table 5.2 were made. A hardened steel means that carbon has been incorporated into the steel to make it harder. Case hardened steel is a steel where just the surface has been hardened to make it more resistant to fractures while retaining a softer core, whereas through hardened steel is a steel where both the surface and the core are hardened, making it very hard but more brittle [24]. Thus, Case hardened steel was chosen for the sun and planet gears as it is commonly used in the automotive industry for powertrain components and offers high tensile strength, impact resistance, fatigue strength and hardenability. For the ring gear nitrided through hardened steel was chosen as this gear has a thin wall section, thus it should be stiff as it can be subjected to distortion [24]. The oil was also chosen to a consistency of 75W-90 since this is a common oil used in transmissions according to Forsberg. Important to note that since the transmission has a very short service life the choice of oil will not strongly affect how the gearbox performs and how long it lasts. 27 5. Gear Design Table 5.2: Materials and oil choice for gears Materials choice Oil choice Sun gear Case hardening steel-18CrNiMo7-6 Klubersynth GE 4 75 W 90 Planet gears Case hardening steel-18CrNiMo7-6 Klubersynth GE 4 75 W 90 Internal gear Through hardening steel-34CrNiMo 6 Klubersynth GE 4 75 W 90 The lifetime of the transmission was set to 46 hours which corresponds to the amount of time that the vehicle is competing and subjected to the load cases defined. Important to point out that when using a load case spectrum in KISSsoft the application factor should be set to 1, otherwise the software will send a warning. However a factor of 1.25 was chosen to ensure a higher safety margin. This specific number means that the car will be subjected to light shocks which should be more than enough since it will be driven on a slick track. The calculation method was chosen as DIN 3990:1987 Method B, which according to KISSsoft is recommended for planetary stage gearboxes. All of those parameters are shown in Table 5.3. Table 5.3: Input parameters in KISSsoft step 2 Service life (h) 46 Application factor (KA) 1.25 Calculation method DIN 3990:1987 Method B The choices made in the manufacturing process here did not affect the dura- bility of the gearbox, however they were still important as it had to be checked if the steel maker had the possibility to manufacture the gears as described in KISS- soft. Consequently, hobbing was chosen for the sun and planet gears since it is a relatively fast process as well as being precise. For the ring gear hobbing was not possible because of the shape of the hobbing machine, shaping was thus chosen [8]. For the modifications process the option “not defined” was chosen as the other op- tions only included grinding. According to Forsberg it was not advised to grind the gears considering that the module is so small as the process could eat away from the hardened surface of the teeth, thus making the surface less wear resistant. The optimal solution was consequently to tumble the gears. For that same reason, the tooth thickness tolerances were set to DIN 58405 8e standard for the sun and planet gears and DIN58405 9e for the ring gear as shown in Table 5.4. This meant that the gears did not need to be ground [7]. Table 5.4: Manufacturing process and tooth thickness tolerances for gear Sun gear Planet gears Ring gear Manufacturing process Hobbing Hobbing Shaping Tooth thickness tolerance DIN 58405 8e DIN 58405 8e DIN 58405 9e For fine sizing a module of 0.6 mm was first chosen however the safety factors were not met as the flank safety for the sun for example was lower than one. By setting a range from 0.4 and to 1, it was found that a module of 0.8 mm met 28 5. Gear Design the requirements for the safety factors. A module of 1 mm rendered the gearbox dimensions too big to fit inside the rim. Normal pressure angle was set to 20°since this is the standard [17]. A range of 20 to 40 mm was chosen for the center distance and the goal here was to remain in the allowable dimensions at the same time as creating a large range for KISSsoft to generate many results and see which ones that fulfill the requirements better. The number of teeth was also noticed to play a big role when it comes to safety factors. The higher the amount on the gears the worse the safety factors as the teeth get smaller. Thus, the number was reduced to obtain the best achievable durability and was the following: 14 teeth for the sun, 74 for each planet gear and 166 for the ring gear. The last parameter to adjust was the facewidth of the gears. It affected the safety factors the most and after starting with small width of 13 mm, it was noticed that the safety factors were not met. Thus, the value was quickly risen to 22 mm for the planet gears and 23 mm for the sun and ring gear. Values bigger than that did not bring a significant improvement and rendered the gearbox rather wide, which also had a negative effect on the load distribution on the teeth as it became uneven. The sun and ring gear were chosen to be 1 mm wider than the planet gears after consulting Per Forsberg in order to reduce the bending stress in the sun gear. Table 5.5 illustrates all of the chosen parameters. Table 5.5: Fine sizing in KISSsoft Module (mm) 0.8 Normal pressure angle (deg) 20 Center distance (mm) 36 Number of teeth for the sun 14 Number of teeth for the planets 74 Number of teeth for the ring gear 166 Face width of the sun (mm) 23 Face width of the planets (mm) 22 Face width of the ring gear (mm) 23 5.2.3 Contact, Stress and Load Analysis The tilting values in the axis alignment window were set to 10 and 20 µm consecu- tively and are shown in Table 5.6. Moreover, these values were obtained by creating a model of the planet carrier in ANSYS and doing a FEM analysis. Additionally, the sun gear had to be chosen to come from I, as shown in Figure 5.2, in other words from the input side. That means that KISSsoft will take into consideration that the sun gear is driven by the motor axis, thus taking into account the torsion of this gear caused by the torque of the motor. That step was done based on the recommendation of Forsberg. Per Forsberg also gave the recommendation to have a floating sun gear, which means that the sun is allowed some radial movement, since this allows more skewing and which greatly improves the safety factors. 29 5. Gear Design Table 5.6: Axis alignment Tilting of the planet carrier relative to the gear axis (µm) 10 Tilting of the planet pin relative to the planet carrier axis (µm) 20 Figure 5.2: Definition of stationary components as well as input and output in KISSsoft After doing the contact analysis, the next step was to analyse the load distri- bution along the width of the gears. Ideally one would want the distribution to be even, in this case though, the load seem to peak at the edge of the face width of the sun as shown in Figure 5.3. It begins to subside moving along the width with a minimal load in the middle. That has a negative effect on the gear performance as there will be variations, it also increases the bending stresses in the gear and causes micropitting and increased vibrations. Figure 5.3: Normal load distribution on the sun gear in KISSsoft 30 5. Gear Design Taking a look at Appendix J showing the tooth root stress of the sun gear, it can be seen that the stresses are more or less evenly distributed. The hertzian stress between the sun and planets however, is higher at the edges of the tooth as shown in Appendix K. Moreover, when running the analysis on the gears, a message from KISSsoft kept appearing saying the following: “Mesh gear 2-3: Pitch point C is outside the path of contact. The calculation of scuffing or micropitting can be inaccurate!”. With gear 2 representing the planet gear and gear 3 representing the ring gear. After contacting KISSsoft support, it was found that it was a special case and gears having the pitch point outside the path of contact could even have a better performance in scuffing or micropitting and that it was not a cause for concern. This was also confirmed by Forsberg. For the operating backlash, the material for the planet carrier was chosen to be billet aluminium considering it is lighter than steel and following last year’s team’s choice. After running the calculation, the results in Table 5.7 showed reasonable values for the play with none of the values being negative meaning that there was no clash occurring in the teeth. Table 5.7: Operating backlash in KISSsoft Planet Carrier’s angel of rotation for a fixed sun/fixed gear rim Without planet carrier pitch deviation(degrees) 0.092 / 0.340 With planet carrier pitch deviation(degrees) 0.030 / 0.277 Pair 1 Pair 2 Min. circumferential backlash (mm) 0.031 0.044 Max. circumferential backlash (mm) 0.181 0.211 Pitch error due to dilatation (µm) 0.000 0.000 Minimum tip clearance (mm) 0.170 0.268 Maximum tip clearance (mm) 0.393 0.617 Min. transverse contact ratio 1.246 1.337 Max. transverse contact ratio 1.371 1.609 Comment: The calculation is performed with the working pressure angle 5.2.4 Planetary Bearings Going back to KISSsys, Figure 5.8 a) shows that two bearings were inserted into each planet gear. These were chosen to be needle roller bearings. The bearings for the carrier shaft on the input side was chosen to be a deep groove ball bearing as shown in the cross-sectional view in Figure 5.8 b). For the bearing on the output side the same type of bearing was chosen as the one on the input side, these are shown in Figure 5.8 b). Important to note, however, that the choice for the carrier bearings was done only to have a complete model in KISSsys to be able to generate a load case report later on. The carrier bearings that were actually chosen are shown in Chapter 7. 31 5. Gear Design a) Planetary bearings in KISSsys b) Carrier bearings in KISSsys Table 5.8: Planetary and carrier bearings in KISSsys The final results displayed in Tables 5.9 and 5.10 show all the safety factors for the gears. It is obvious that the ring gear has more than acceptable flank and root safety factors of 1.971 and 1.730, which means that the possibility of a failure occurring is little. For the sun gear the safety factors are 1.261 and 2.119 for the flank and root respectively. Being over 1, the flank should be fine for the loads that they are dimensioned against during the course of the competition. Concerning the planet gears, the obtained flank and root safety factors were 1.374 and 1.891 respectively. In other words, while having an application factor of 1.25 over the load case, all gears still got a minimum flank and root safety factors of over 1. Regarding the safety against micropitting and scuffing, it was noticed that all factors were above 1 except for the micropitting factor between the sun and the planets, which was at 0.539. Table 5.9: Final safety factors for gears Flank safety Root safety Sun 1.261 2.119 Planets 1.374 1.891 Ring gear 1.971 1.730 Table 5.10: Micropitting and scuffing for gears Safety factors Micropitting between sun and planets 0.539 Micropitting between planets and ring gear 1.042 Scuffing between sun and planets 4.972 Scuffing between planets and ring gear 7.725 32 5. Gear Design 5.2.5 Gear Modelling With all parameters defined resulting in a gear geometry with acceptable safety factors, the model generated with KISSsys is exported for the final modelling. The KISSsys model is represented in Figure 5.4 a). a) The sun, ring and planet gears generated by b) Section view of one planet KISSsys, rendered in CATIA V5 gear in CATIA V5 Figure 5.4: 3D-models of the gear designs As seen in the figure, KISSsys exports a model with the gear geometries as shells. Therefore, the gear walls must be further 3D-modelled in CATIA V5. In order to decide a appropirate wall thickness for the planet gears, CFS19 planet gears are studied. These gears have a wall thickness of 5 mm and since CFS19 used a 1.5 step compound planetary gearbox, the planet wall is subjected to twisting forces from the secondary planet gear which is not present in a 1 stage planetary gearbox. Therefore, a wall thickness of 4 mm is assumed to be adequate for this application. Two bearing seats divided by a lip for the planetary bearings are also designed. See figure 5.4 b). As described in Section 5.2.2, Per Forsberg recommended a floating sun gear. Therefore, a splined input shaft is added to the sun gear geometry as well as a hole for a M4 bolt for locking sun gears axial movement, see Figure 5.5 a). The ring gear remains the same as the KISSsys model except for the addition of flanges for mounting. See figure 5.5 b) for the gear design including planetary bearings. All wall thicknesses generated from KISSsys have remained unchanged since this have great effect on the gear strength. 33 5. Gear Design a) Sun gear with splined shaft b) Final design of the gears including the planetary bearings Figure 5.5: 3D-models of final gears, rendered in CATIA V5 5.3 Discussion As mentioned before, the main goal of the work in KISSsoft was to ensure that the gearbox would last through the whole season by making sure the safety factors were over 1. The question after that would become how much over 1 the factors needed to be; here a delimitation had to be made as the primary goal of the project was to make the gearbox lighter than the years before. With that being said, and taking into consideration that it is a new concept that has not been implemented before, it was deemed to be reasonable to have a bit of a safety margin. That, to ensure that the gears would be more than capable to take the loads, thus reducing the chance of a failure in the middle of the competition as much as possible. Furthermore, it would have been possible to increase the width of the gears in order to obtain a minimum factor of 1.5 for all gears, following Peter Wittke and Tommie Hall’s recommendation, two engineers that work with developing transmissions at Volvo Cars. Their point was that this factor would then consider the dynamic loads that can arise if there is not enough traction and the wheels slip for example. However, this would have made it harder to reach the goal of obtaining a lighter gearbox. Concerning the uneven load distribution that was obtained, it was deemed to not affect the service life of the gearbox, as the flank and root safety factors achieved were obtained taking it into account. In other words if the load distribution was even the safety factors would have been even higher. As seen in the results, there were no risk of the gears scuffing. Nevertheless, when it came to micropitting between the sun and planet gears the low safety factor obtained meant that pits would probably form on the surface of the teeth during the lifetime of the gearbox. Be that as it may, this did not play as much role in 34 5. Gear Design predicting failure as the flank and root safety factors did. The micropitting safety factor would not bring much concern in terms of a catastrophic failure happening on the short term but would cause a problem on the long term [13]. In other words, the gearbox would still be functional even with micropitting symptoms on the gears. An investigation to see if the gear has been subjected to micropitting would however be interesting to do after the runtime to understand how it has affected the teeth. The reason to why a double bearing setup was used was that using only one bearing resulted in having to choose a large one. This in its turn meant that the minimal bearing load was not achieved. Using two small bearings meant that the load was equally distributed between these and that they were appropriately loaded. When generating the load case report, it was noticed that the bearings sitting on the carrier shaft were not exposed to the minimal load required. After being in contact with KISSsoft support and Forsberg, it was assured that for the bearings on the carrier it was normal for the minimal load not to be reached when the gravity force is deactivated and with no extra radial forces or moments. There would be no resulting extra force as a result. When setting the service life to 10000 hours, it was noticed that the safety factors obtained were not much lower than before. The flank safety for the sun went down from 1.261 to 0.880 which came as a surprise. It was expected that the factors would be considerably lower, as they would be subjected to fatigue stress. 35 5. Gear Design 36 6 Bearings & Housing Load Cases 6.1 Theory The gearbox housing and its bearings are components that experience great forces during acceleration, retardation and cornering. To design a light gearbox housing while being able to handle the stresses it is important to properly define the forces that cause the stresses. At standstill, a static normal force is acting on the wheels from the ground as a result of the vehicle’s mass. The force is acting radially on the tyre perpendicular to the ground, see Figure 6.1. During acceleration, retardation and cornering the normal forces acting on each wheel vary depending on the acceleration of the mass centre of the vehicle. This is one of two radial forces acting on the centre of the wheel. Figure 6.1: Illustration of normal forces at standstill for the CFS19 Car The second radial force is the tangential force acting on the gearbox transmis- sion shaft during acceleration and retardation as a result of the torque transmitted from the electric machine and the brakes at each wheel, see Figure 6.2. 37 6. Bearings & Housing Load Cases Figure 6.2: Illustration of tangential forces on one wheel at the moment of acceleration from standstill, denoted as FT . FN denotes normal force and T the applied torque. The third and last force is the lateral force acting axially on the wheel centre during cornering. This axial force is pushing the centre of the wheel inwards the vehicle on one side and pulling the centre of the wheel outwards the vehicle one the other side during cornering. This force is the reason the gearbox needs to house a guiding bearing in order to prevent the inner wheel to tear the gearbox apart in a corner, see Figure 6.3. Figure 6.3: Illustration of lateral forces during cornering and deceleration from top view 38 6. Bearings & Housing Load Cases 6.2 Method The gearbox housing and its bearings are subjected to the forces defined in Section 6.1 via the wheels. These forces are therefore used for dimensioning of the gearbox housing. The more trustworthy approximation of the forces, the less safety margin is needed during dimensioning, resulting in a lighter design with sufficient strength to handle the stresses. Therefore, much effort is put into finding reliable data defining the forces corresponding to the maximum loads during competition which can be used to analyse strength and deformation as well as average repetitive forces for fatigue analysis on the carrier and the housing. From the maximum loads, equivalent forces can be computed which is used to dimension the bearings. 6.2.1 Data Sourcing The data needs to contain information that can be translated to how much force the wheels or suspension endures during its most critical use, which in the case of this project is during the competition. This type of information can be found from multiple types of sources. In order to find the most suitable source for this project, the available information was found by consulting CFS-members and the project supervisor Björn Pålsson. Historically, the load cases for the gearboxes have been derived from software computations in MATLAB, simulation models in ADAMS and OptimumLap. It is also concluded that the cars have a set of sensors logging various information from the car each time it is tested or competed with. After inspecting the logged sensor data from the competitions it seemed like a promising data source. Acceleration in three dimensions, angles of the pushrod pivot shaft and the torque output from each motor, see Figure 6.4, are logged. This information is considered enough to construct a good estimation of all loads on each wheel during competition which can be translated into forces on the housing and its bearings. Figure 6.4: Angle sensor logging the angle of the rear pushrod pivot shaft 39 6. Bearings & Housing Load Cases Since the alternative sources are based on software simulations including as- sumptions and simplifications, the data from physical testing is considered more accurate. Furthermore, the use of physical sensor data is considered more interesting from a CFS methodology perspective. If this project results in a method for defining accurate load cases for the gearboxes, the logged data can be updated with the data from the most recent competition each year. Because of these reasons, the logged data from the accelerometer and the angle sensor on the pushrod pivot shaft are chosen as the data source of the gearbox housing and bearing load cases. This choice is further motivated by the fact that the hypothesis of the project is that earlier CFS gearboxes have been over dimensioned when designed utilizing these simulations. CFS19 was the most recent team to compete with their car named Victoria, see Figure 6.5. Therefore, the logged data from Victoria is chosen as the data source for developing load cases for the gearbox housing and bearings. More specifically, the data from autocross competition in Achen 2019. Figure 6.5: Victoria, developed by CFS19 6.2.2 Deriving Load Cases from Data All sensor data is loaded into MATLAB with tools from CFS into a new script. Then, the plots of the data are studied and interpreted with the help of former CFS member Erik Lund. Lund was a member of the group responsible for the low voltage-system in CFS19 which installed the sensors on Victoria. Erik Lund provided information about the specifications of the angle sensor and assisted during measurement of the output of the angle sensor in relation to the wheel motion. This data is then loaded into a SIMULINK model that defines the transfer function between the logged data and the angle of the pushrod pivot shaft angle. Because of problematic data logs, this methodology deviated from the project purpose. It was also concluded that only the pushrod pivot shaft angle sensors on the front wheels gave a reliable output. For further reading, see Appendix L. The logged data is converted to represent the angles of the pushrod pivot shaft with the SIMULINK transfer function. Because of above mentioned uncertainties, the data needs to be calibrated in order to ensure reliable results. The angle data is normalized and scaled with approximated normal forces. These approximations are done with hand calculations supported by the logged accelerations and vehicle 40 6. Bearings & Housing Load Cases specifications from CFS reports. See Figure 6.6 followed by the equations 6.1 and 6.2. Figure 6.6: Free body diagram from side view for hand calculations of normal forces on the front axle y A : FN · L3 −m · g · (L3 − L2)−m · a · L1 = 0 (6.1) =⇒ FN = m(a · L1 + g · (L3 − L2)) L3 (6.2) The accelerometer data log is supplemented by a video recording from the competi- tion and an arbitrary time stamp in the data log is used as a reference. The chosen timestamp is at acceleration from standstill. At standstill, the downforce generated from the aerodynamic features of the vehicle can be neglected which provides a more accurate reading. The longitudinal acceleration at this timestamp is inserted into equation 6.2 which computes the normal force on the front axle. This normal force is evenly distributed between the two front wheels and compared to the angle read- ing at the same timestamp. All other data points are then calibrated by comparing their angle to the angle of this timestamp. This results in a data log of normal forces suitable to use for dimensioning. The tangential force, is computed with the logged torque output-data and the radius of the wheel when acceleration see Figure 6.7 followed by the equation 6.3 for computation of tangential forces for one wheel. 41 6. Bearings & Housing Load Cases Figure 6.7: Free body diagram for hand calculations of the tangential force, denoted FT , on one wheel FT = T R (6.3) When the car decelerates the tangential force is derived from Newton’s second law using the car’s accelerometer data and mass: FT = m · a (6.4) The reason for using both equation 6.3 and equation 6.4 is that when the car accel- erates the motors draws current which is then logged. But when the car decelerates this type of data becomes unreliable since the car uses both generative breaking and traditional disc brakes. Therefore, during deceleration, the car’s accelerom- eter is used instead. The accelerometer could be used for both acceleration and deceleration, however, this is not done because the current data is a more direct source. With the tangential and normal force defined, the lateral forces are next. These are computed with the lateral acceleration data from the accelerometer and the normal forces. According to CFS report [16], the weight distribution between the front and rear axle is even during cornering. In order to define the weight distribution between the left and right wheels, an assumption that the distribution is directly proportional to the distribution between left and right normal forces is done. See equations 6.5 to 6.7 below for the calculation of the FR wheels lateral forces. FL,tot = m · a (6.5) FL,front = FL,tot 2 (6.6) FL,FR = FL,front · FN,FR FN,tot = m · a 2 · FN,FR FN,tot (6.7) 42 6. Bearings & Housing Load Cases Now all forces affecting the wheel are defined from which the load cases can be constructed, see Figure 6.8. The maximum values of the forces can be used as a reference for finite analysis and when calculating the service life of the bearings the whole driving cycle can be used. Figure 6.8: Illustration of the position of the three forces FN , FT and FL on the wheel used for dimensioning the gearbox housing and bearings 43 6. Bearings & Housing Load Cases 6.3 Results 6.3.1 Data Interpretation The complete data log from the final autocross lap in Achen 2019 is loaded into MATLAB and plotted. The data log includes five laps with pauses in between. To concentrate the data to relevant data points, only the data from the final lap is chosen. The raw Analog to Digital Converter-values (ADC) from the pushrod pivot shaft angle sensor are presented in Figure 6.9 Figure 6.9: Imported ADC-values from the front pushrod pivot shaft angle sensors The SIMULINK-model described in Appendix L generates a transfer function that is used to convert the ADC-values into angles on the pushrod pivot shafts. Figure 6.10 illustrates the transfer function with ADC-values on the y-axis and corresponding angles on the x-axis. 44 6. Bearings & Housing Load Cases Figure 6.10: Transfer function between ADC-signal and angle for both front wheels As seen in the figure above there are intervals where the transfer function for the front left wheel can output two different angles for one ADC-signal. For example, if the ADC-value is 200 the angle could either be circa 32°or 48°. Since there are non- continuous values in the data log for the front left wheel, the angle sensor must cross the threshold at circa 40.5°. Furthermore, the wheel motion must be continuous. This knowledge enables a limited interval of [33◦, 48◦] to be assumed. All data points from the Achen 2019 data log are converted into angles with the transfer function. This results in a new data set plotted in Figure 6.11. Figure 6.11: All ADC-data points converted into angles on the pushrod pivot shaft 45 6. Bearings & Housing Load Cases 6.3.2 Deriving Loads From Data To calibrate the angles to their corresponding normal force, the theoretical stationary normal force is used as a reference. The normal force is calculated with equations 6.8, 6.9 and 6.10 and data from Carl Larsson’s CFS-report [16]. FN,tot = m(a · L1 + g · (L3 − L2)) L3 = 990.6 N (6.8) FN,FL = FN,tot 2 = 495.3 N (6.9) FN,FR = FN,tot 2 = 495.3 N (6.10) The pushrod pivot shaft angle data points are normalized and scaled in such way that the angles at standstill corresponds to above calculated stationary normal forces. In order to scale the amplitude of the angles to corresponding loads, the angles during a specific time stamp in the data log is used as a reference against the theoretical normal force amplitude. The moment when the car accelerates from standstill is used as the reference time stamp since the aerodynamic impact of the angles are considered low during this moment. During the chosen time stamp the longitudinal acceleration is 9.42 m/s2 according to the acceleration data log. The normal force amplitude is calculated according to equations 6.11, 6.12 and 6.13 with the data from Carl Larsson’s CFS-report [16]. Acceleration is noted with negative values and retardation is noted with positive values since the front axle is evaluated. This notation simplifies calculation of normal forces since the front axle lifts during acceleration which results in less normal forces. FN,tot,amp = m(a · L1) L3 = −169.0 N (6.11) FN,FL,amp = FN,tot,amp 2 = −84.5 N (6.12) FN,FR,amp = FN,tot,amp 2 = −84.5 N (6.13) Similarly to the calibration of the stationary normal forces, the normal force am- plitudes are computed by scaling the angle amplitude to the corresponding normal force amplitude. All other angle data points are then scaled with the same factor which results in the data set of normal forces in Figure 6.12. 46 6. Bearings & Housing Load Cases Figure 6.12: Graph showing the approximated normal forces for FL and FR wheel The tangential forces are computed with the resulting forces from the torque output of the electric machines when the car accelerates. When the car decelerates, the accelerometer is used in combination with Newton’s second law. If the car accelerates: FT = Ttot · igearbox R · 1 4 (6.14) In equation 6.14 a factor of 1 4 is used, this is to divide the forces evenly between all four wheels. Ttot is the total torque from all four motors combined. When the car decelerates Newtons second law is used to calculate the tangential forces. FT = m · a · 1 4 (6.15) The resulting tangential forces for each data point from all wheels are calculated in MATLAB. To simplify the calculation, all wheels are assumed to have equal tangential forces, see Figure 6.13 for one wheel’s tangential force. Negative values means that the car is retarding which gives the forces the opposite direction to the forces during acceleration. 47 6. Bearings & Housing Load Cases Figure 6.13: Graph showing the approximated tangential forces for one wheel The lateral forces are computed with the data from the lateral accelerometer, the total mass of the vehicle and the normal forces. The total lateral force from the lateral acceleration of the mass centre is distributed evenly between the front and rear axle because of the evenly distributed weight distribution of the vehicle. The left and right lateral force distribution is approximated to correspond to the left and right normal force distribution. Each lateral acceleration data point is computed with its corresponding normal force data into a lateral force according to equation 6.16, 6.17 and 6.18. This results in the lateral force data set plotted in Figure 6.14. FL,tot = m · aL (6.16) FL,FL = FL,tot · FN,FL FN,tot (6.17) FL,FR = FL,tot · FN,FR FN,tot (6.18) Figure 6.14: Graph showing the approximated lateral forces for both front wheels 48 6. Bearings & Housing Load Cases 6.3.3 Final Load Cases The load vectors shown as graphs in Figure 6.12, 6.13 and 6.14 defines the final load cases. The gearbox housing is only stress- and deformation analyzed, therefore the maximum normal-, tangential- and lateral forces defines the gearbox housing load case. For the bearings the complete load vectors combined with the motors’ rotational speeds are used as the load case for computing the bearing service life. 6.4 Discussion Initially, the idea was to use the pivot shaft angle sensor data combined with a kinematic model for the springs and dampers to approximate the normal forces. Because of complications described in Appendix L this was not possible, which is the reason that the data was scaled with hand calculations. This method requires the simplification that the shock absorbers are fully linear but since CFS19 utilized a heave-roll suspension system it is highly likely that the suspension system is non- linear, see figure 6.15. This is because the heave-roll system compresses a roll- spring during cornering (lateral acceleration) and a heave-spring during longitudinal acceleration or unevenness in the road surface. Since Victoria have a stiff suspension and low center of gravity, it is assumed that the roll of the vehicle is negligible. CFS did also explain that the damper and spring rates might have been adjusted during competition and can therefore be considered as unknown variables. For the intended kinematic model, this would have meant assumptions which could have affected the final approximated normal force cons