Gearbox transmission error study using multi body analysis software MSC Adams

[1]–– Trabajo Fin de Máster Máster Universitario en Ingeniería Industrial Gearbox Transmission Error study using Multi Body Analysis software MSC Adams Author: Antonio Roldán Sánchez Tutors: Johan Wideberg Carlo Rosso Dpto. Ingeniería y Ciencia de los Materiales y del Transporte Escuela Técnica Superior de Ingeniería Universidad de Sevilla Sevilla, 2018 iii Trabajo Fin de Máster Máster Universitario en Ingeniería Industrial Gearbox Transmission Error study using Multi Body Analysis software MSC Adams Author: Antonio Roldán Sánchez Tutors: Johan Wideberg Carlo Rosso Dpto. Ingeniería y Ciencia de los Materiales y del Transporte Escuela Técnica Superior de Ingeniería Universidad de Sevilla Sevilla, 2018 v Trabajo Fin de Máster: Gearbox Transmission Error study using Multi Body Analysis software MSC Adams Author: Antonio Roldán Sánchez Tutor: Johan Wideberg El tribunal nombrado para juzgar el Proyecto arriba indicado, compuesto por los siguientes miembros: Presidente: Vocales: Secretario: Acuerdan otorgarle la calificación de: Sevilla, 2018 El Secretario del Tribunal A mi familia A mis maestros vii Agradecimientos Con este Trabajo de Fin de Máster se pone fin a una etapa muy importante de mi vida, una etapa que jamás olvidaré, una etapa llena de esfuerzos y sacrificios, una etapa de lecciones y aprendizaje, una etapa donde he cambiado, donde he madurado, una etapa que me ha convertido en la persona que soy hoy. Y me gustaría agradecer a todas las personas que han estado junto a mí en esta etapa de mi vida. En primer lugar, a mi familia, por estar siempre ahí y por apoyarme en todas mis decisiones, por ayudarme siempre que lo he necesitado y gracias a los cuales nunca me faltó nada. A mi padre, por inculcarme esa sed de conocimiento que me ha hecho llegar hasta aquí, por tener respuesta para todas las preguntas de ese niño que no podía vivir sin saber el porqué de las cosas. A mi madre, por ese amor incondicional, porque a pesar de que le discuto todo lo que me dice siempre que la necesite sé que estará ahí, porque todo lo hace pensando en mí, porque eso no tiene precio. A mis amigos de la 2.0 porque cuando estoy con ellos se me olvidan los problemas, las preocupaciones y los malos rollos. A mis Triviales, con los que he compartido esta travesía, porque ellos mejor que nadie saben lo que ha costado y el esfuerzo que hay detrás. Porque por muy lejos que estemos, cuando nos vemos es como si lleváramos juntos toda la vida. A mis Ragazzi, porque estando en un país nuevo sin conocer a nadie se convirtieron en mi familia. Porque en los momentos duros cuando no tenía a nadie más, los tenía a ellos. A mis compañeros de ARUS, con los que trabajé codo con codo horas sin descanso, noches sin dormir para poder sacar adelante un proyecto donde muchos hubieran tirado la toalla, pero ellos no. Donde aprendí mucho de ingeniería, pero mucho más de trabajo y dedicación. A todos aquellos maestros y profesores que no solo me han enseñado, sino que me han marcado a lo largo de mi vida. A don Rafael, Santi, Teacher, Sonia, Ana Cob, Brenes, Aída, Andrés, París, Pino, Lillo, y muchos más. A Johan por aceptar tutorizar este proyecto y ayudarme siempre con tanta rapidez. And special thanks to Carlo Rosso for giving me the chance to develop this project and guiding me thorough all the process helping me whenever I needed it. Antonio Roldán Sánchez Sevilla, 2018 ix xvii TABLE OF FIGURES Figure 1-1 - Gear Pair [38] 1 Figure 1-2 – Vibration limits in ANSI/AGMA 6000-B96 3 Figure 2-1 – Spur Gears [38] 6 Figure 2-2 – Helical Gears [38] 6 Figure 2-3 – Double Helix Gears [38] 7 Figure 2-4 – Normal module vs Transverse module in helical gears [39] 7 Figure 2-5- Internal Gears [38] 8 Figure 2-6 – Bevel Gears [38] 8 Figure 2-7 – Worm Gears [38] 9 Figure 2-8 – Gear teeth mating [38] 10 Figure 2-9 – Involute definition [38] 10 Figure 2-10 – Gear Geometry [39] 11 Figure 2-11 – Pitch and base circles [38] 12 Figure 2-12 – Helix angle [39] 12 Figure 3-1 – Transmission error 16 Figure 3-2 – Gear Rattle [15] 17 Figure 3-3 – Vibration excitation and transmission path [2] 18 Figure 3-4 – Harris maps for Transmission Error 19 Figure 4-1 – Multibody dynamics problema [36] 21 Figure 4-2 – Multibody Software with flexible elements [23] 22 Figure 5-1 – Complete Transmission model [1] 25 Figure 5-2 – Gearbox arquitectura [1] 27 Figure 5-3 – Insert Gear Pair wizard - Type 28 Figure 5-4 – Insert Gear Pair wizard - Method 28 Figure 5-5 – Insert Gear Pair wizard – Geometry 29 Figure 5-6 – Insert Gear Pair wizard - Material 30 Figure 5-7 – Insert Gear Pair Wizard - Connection 31 Figure 5-8 – Spur gear rigid model Adams 32 Figure 6-1 – Angular TE Comparison 36 Figure 6-2 – Linear TE Comparison 37 Figure 6-3 – Effect of torque at low speed 37 Figure 6-4 – Effect of torque at high speeds 38 Figure 6-6 – Effect of speed with light load 39 Figure 6-7 – Effect of speed with high load 39 Figure 6-8 – Effect of the type of teeth 40 Figure 7-1 – FEM Models 44 Figure 7-2 – Web Meshes 45 Figure 7-3 – 3D Model meshes 46 Figure 7-4 – Joints 1D Model 47 Figure 7-5 – Bearings Connection Details 48 Figure 7-7 – 1D Model without Bearings 49 Figure 7-7 – 3D Model with Bearings 49 Figure 8-1 – TE on 1D Model – Spur gears (1) 53 Figure 8-2 – TE on 1D Model – Spur gears (2) 53 Figure 8-3 – TE on 1D Model – Helical gears 54 Figure 8-4 – 1D Model vs 3D Model – Spur gears 54 Figure 8-5 – 1D Model vs 3D Model – Helical gears 55 xix Notation Ni Rotation speed of shaft i in rpm ωi Rotation speed of shaft i in rad/s Zi Number of teeth of gear i Ti Torque on gear i r Gear or Transmission ratio D Pitch diameter Db Base diameter α Pressure angle m Module ϑi Angular position of gear i β Helix angle TE Transmission Error MTE Manufacturing Transmission Error STE Static Transmission Error DTE Dynamic Transmission Error MBA Multibody Analysis FEM Finite Element Model MPC Multi Point constraint PPTE Peak-to-peak Transmission Error 1 INTRODUCTION ears are one of the most used components in many engineering fields, but especially in mechanical engineering. They can easily be found in almost any machine or mechanism with moving parts. We use gears in everyday household items such as a clock or a hair-drier, but they are also a key component in way more technical and complex systems such as a wind turbine. And of course, it is of utmost importance their role in the automotive industry, where they are used in many subsystems but particulary they are the essential component in the gearbox, which allows the combustion engine to be used in a wide range of speeds, and therefore make cars and motorbikes usable in everyday tasks. 1.1 Use of gears G Figure 1-1 - Gear Pair [38] Introduction 2 2 The main use of gears is to transmit rotational power, usually by applying a transmission ratio that converts speed into torque, maintaining the power. To model this, we can use the basic equation for gears: 2 2 1 2 1 1 2 1 R Z T rR Z T       (1.1) With this formula, we can see that the transmission ratio is a constant for the gear pair, which is the ratio between the number of teeth of each gear or the ratio between their radiuses and this ratio will also be the relation between the speeds of the gears and their torques. This equation derives also from the constant power condition: 1 1 2 2 P T T   (1.2) Assuming that there are no power losses, the power will be the same for both gears, so torque and speed will be inversely proportional. For this reason, we use gears when we have a shaft rotating at a high speed, but low torque and we want to transform it into low speed but high torque, or vice versa. This property is the main reason that makes gears so useful in a wide range of cases because they allow you to transform an input motion into your desired motion. For example, electric motors usually have a very high speed but low torque so we use gears to reduce the speed and make it more usable for many specific tasks. But this is not the only reason to use gears. There are many different typologies of gears that allow us to translate or change the axis of rotation, or also change the direction of the rotation. 1.2 Gear Vibration Due to the wide use of gears in the industry, it is no wonder that gear technology is very well known and developed and that gears are the subject of many technical papers and scientific research studying different aspects inside the field of gears. One of the most important aspects when studying gears is gear vibration. Gears are a source of vibration since they are moving parts rotating at high speed with many different surfaces coming into contact at even higher speeds. The teeth of the gears engage with high-frequency excitation, for example, a gear with 30 teeth rotating at 1000 rpm will have a tooth mating frequency of 500 Hz, and this excitation will cause the whole system to vibrate. The American Gear Manufacturers Association (AGMA) has standards limits for gear vibrations, in terms of displacement, velocity, and acceleration. The standard is the ANSI/AGMA 6000-B96 which sets two different classes of gears depending on the limits they can hold, as we can see in Figure 1-2. The standard also sets procedures on how to measure vibrations, how to run the tests and what the limits of class A and B mean in terms of gear usability and life. 3 Vibration analysis is becoming nowadays a very important tool when diagnosing a gearbox, because we can detect if something is wrong with the transmission and locate a single bad tooth or even fatigue cracks growing before it becomes critical using just a vibration measurement, which is completely non-invasive and can even be carried out without disassembling the component. For this reason, vibration analysis is gaining importance in sectors where gears may play a critical role and a sudden failure can lead to significant consequences in terms of critical damage to the machine or even a safety hazard to the people such as in the automotive or aeronautic industries. But in order to use vibration analysis, we must first understand what vibrations are normally generated due to the correct working of the gearbox and which vibration are abnormal and come from some malfunction. This is where the Transmission Error comes into play. Transmission Error is the difference between the theoretical movement that the output gear should follow when we impose a movement in the input gear and the real movement of the output. And this Transmission Error, hereafter TE, is considered by many authors to be the main source of gear noise and vibration [2] [3] as we will discuss at length in 3.3. The aim of this master thesis is to study the viability of analyzing the Transmission Error of a geared transmission using commercial software MSC Adams based on Multi-Body Analysis, studying also the effect that the flexibility of different elements can have on the results of TE. This project is also a follow-up to a previous master thesis [4] developed in Politecnico di Torino that used MSC Adams too and a similar model to study the effect of profile modifications on the Transmission Error. Figure 1-2 – Vibration limits in ANSI/AGMA 6000-B96 Introduction 4 4 2 THEORY OF GEARS n this chapter we will discuss the basic theory of gears, their main parts, the different types of gears we can find, some definitions, how they work, the main parameters needed to understand them and their geometry. 2.1 Basic principles As we have already commented the main use for gears is to apply a transmission ratio to a rotating shaft to obtain some mechanical advantage. This gear ratio is defined as the relation between the speed of the input shaft and the speed of the output shaft. Although depending on the application the inverse relation might be used. 1 2 r    (2.1) Since the linear speed at the point of contact must be the same, the gear ratio can be calculated as the relation between the diameters of the gears. And since the number of teeth is directly proportional to the diameter of the gear, the ratio can also be expressed as we see in the formula (2.2) 1 2 2 1 1 2 2 2 1 1 DZ v D D r DZ          (2.2) Finally, assuming no power is lost in the transmission, which is a good assumption since the efficiency of gears is usually around 98%, we can obtain the relation between the torques and the gear ratio. The ratio of the torques is also known as Mechanical Advantage and it has the same value as the gear ratio 12 1 1 2 2 21 T P T T r MA T          (2.3) The importance of this formula should be emphasized, since what it is telling us is that, by using gears, we can obtain a mechanical advantage, and thus more force in the output than in the input, by increasing the number of rotations in the input with respect to the output. And this is the main objective when using gears. I Theory of gears 12 12 cos( ) B DD   (2.7) The addendum and dedendum are the distances between the pitch circle and the outside circle and the root circle respectively. Their values are standardized and for most common applications are the same as the module. 2 outside D D a 2 root D D d (2.8) The thickness of the tooth is measured in the pitch diameter, if it is measured along the circle it is called circular thickness, if it is measured in a straight line it is called cordal thickness. Its value is also standardized as half of the pitch, although it can be modified to tune the clearance between the gears according to the application. In helical gears, we have also another parameter which is the helix angle, that defines the twist of the helix of the gear as shown in the figure below. For two helical gears to engage they must have the same helix angle. Pitch circle Pressure angle Path of contact ≡ Line of action Base circle Figure 2-11 – Pitch and base circles [38] Figure 2-12 – Helix angle [39] 13 There are other more complex parameters such us tip and root relief, undercutting or crowning that can affect the working of the gears but those are not under the scope of this project. In conclusion, the basic parameters that define the geometry of the gear are the module, which affects the size of the gear; the pressure angle, which affects the shape of the teeth; the number of teeth; and the helix angle for helical gears; which affects the twist of the helix. Theory of gears 14 14 3 TRANSMISSION ERROR his project will be focused on studying Transmission Error and its evolution when using different operating parameters and flexible elements in the system. For this reason, we will dedicate this chapter to the TE, what it is, where it comes from, the different types there are, its relation with vibrations and the state of the art in the study of TE. 3.1 Definition The idea of the Transmission Error was first defined by Harris in [8] in 1958 and Munro, his Ph.D. student at the time. The formal definition they gave was “TE is the difference between the angular position that the output shaft of a drive would occupy if the drive were perfect and the actual position of the output”. In other words, the actual meshing of the gears is not perfect, due to the deformation of the teeth, error in the teeth profile which is not a perfect involute, and incorrect installation of the gear pair or other parameters, so the actual angular position of the output gear will not be the exact theoretical position, and the TE is defined as the difference between the theoretical one and the actual one. 1 21 2 Z TE Z   (3.1) The sign criterion is not widely stablished, some authors use this, and others use the opposite, but this is not really important because it only represents the direction of measurement of the angle. The important value is the magnitude of the angle. As defined by Mark in [9] “A pair of meshing gears with rigid, perfect, uniformly spaced involute teeth would transmit exactly uniform angular motion” and thus the transmission error would be zero. But in the real world this never happens, every gear pair has some degree of irregularity in those aspects and this translates into a not uniform angular motion transmission. Another definition of TE given by Munro in [10] is “The deviation in position of the driver gear (for any given position of the driving gear), relative to the position that the driven gear would occupy if both gears were geometrically perfect and undeformed” To measure the TE we just need to measure the angle at the input, the angle at the output, apply the gear ratio and subtract one from the other. As we can see this TE has a unit of angle, however, the results are rarely given as an angular measurement, but it is multiplied by the pitch radius of the gear to obtain a linear measurement since this is much more informative (usually below 5μm). The great advantage of specifying TE as a linear measurement is that it is a direct indication of the quality of the gears, two gears of the same quality will have the same TE regardless of their size, module or diameter, so we can compare different gears very easily [2] T Transmission Error 16 16 In literature we can find the typical values of peak-to-peak TE (PPTE) obtained in gears with different levels of quality. For very large gears and low speed machinery we can see values of up to 20μm but this type of gears is only used in situations where vibrations and noise is not a concern at all. Values of 10μm are common on very rough and low-quality gears not suitable for many applications where vibrations must be kept under control. For medium and small, which are the most used ones in the industry, 3 to 5 μm of PPTE are typical of good quality gears. And for high precision gearing we can find values of PPTE around 1 μm which would be representative of extremely good quality and very precise gears [2].   21 2 1 2 2 1 1 2 1 22 DZ TE D D Z            (3.2) Even though TE is a variable in time, it ussually follows a wave form, we do not need the whole evolution of the TE in time, but the single value of peak-to-peak TE is often reported since it gives information on the amplitude of the TE as Houser explains in [11]. If some further information of the vibration is needed Fourier analysis is also used to identify the most important frequencies of the signal in order to detect any problem in the transmission. 3.2 Types and origins There are three main types of TE that are often referred to in literature, and these types are related to the source and origin of the TE [12]. 3.2.1 Manufacturing Transmission Error (MTE) The simplest type of TE is the MTE which is caused by an error during the manufacturing of the gear that generally makes the tooth profile deviate from the perfect involute profile that it should follow. If the teeth don’t have a perfect involute profile, this will make the transmission of torque not perfectly uniform. But MTE also accounts for the uneven spacing of the teeth along the pitch circle, errors during the assembly of the gearbox Figure 3-1 – Transmission error 𝜃1 𝜃1∗ 𝐷 2 𝐷1 𝜃2 T E 𝑅1 𝑅2 17 which may lead to the distance between gears not be the correct one, or even to a misalignment in the shafts which will make the axes not parallel to each other. This MTE is directly linked to the quality of the gears and its manufacturing process. To measure the MTE we measure the TE of a gear pair under no load or a light load, in order to avoid any deformations that may occur under load, and at a low speed, to avoid any possible dynamic effects [3]. 3.2.2 Static Transmission Error (STE) Another source of TE can be found in the deflection and deformation of the system when it is under load. This deformation can be found in the gear teeth, but also in the bearings, shafts and even casing of the gearbox. Static Transmission Error (STE) accounts mainly for this source of TE which depends basically on the stiffness of the system. In order to measure STE, we measure the TE of a gear pair under load, but in static conditions of low speed, to avoid dynamic effects. It should be noted that since the STE is measured by measuring the TE, it is impossible to get rid of the effect of the MTE, so the STE will include the MTE. 3.2.3 Dynamic Transmission Error (DTE) The last kind of TE and probably the most important one is the Dynamic Transmission Error. It is the most important one because it considers all the variables that affect the system during its working life since it includes all manufacturing error, the static deformation of the system, but also the effects of the masses and inertial forces of the system, that generate also dynamic deformations. To measure DTE, the TE is measured under load and at a high-speed condition, which is generally the working condition of many gears. So, it accounts for all the possible reasons of transmission error 3.3 Relation with vibrations Vibrations and noise are an intrinsic response of almost any mechanical system with moving parts, and gears are not an exception to this rule. On the contrary, gears are more susceptible to generate vibrations and noise than many other mechanical components because they consist of many different teeth meshing and coming into contact at very high speeds. When the surfaces of these teeth contact with each other a contact force appears. This contact force is the responsible of the movement and working of the gears, but at the same time if this contact force is not constant it will become an excitation to the system leading to vibrations and noise. Thanks to the involute profile described in 2.3 we make sure that the direction of the contact force does not change and remains the same during the movement of the point of contact along the whole line of action. However, it is the variation in the amplitude of the force that gives the excitation for the vibration. According to Smith in [13] the source of this force variation is a variation in the smoothness in the contact and it is due to a combination of small variations of the form of the tooth and a varying elastic deflection of the teeth. There are two main types of gear noise, [15] gear whine and gear rattle. The first one, Gear Whine, is a purely tonal noise, whose frequency changes in proportion to the speed. It is generated by the torsional vibration of the Figure 3-2 – Gear Rattle [15] Transmission Error 18 18 loaded gears due to the meshing stiffness variation during the teeth engagement and thus, the frequency of this noise is the meshing frequency of the teeth. And the second kind of noise, Gear Rattle, is not provoked by the deformation of the teeth but due to the clearance necessary for gears to work there is certain backlash when two gears are engaged. So, when the gears are not loaded the driven gear might vibrate back and forth within this backlash generating noise. The main reason why Transmission Error is so important, and it is the subject of many studies nowadays, this one among them, is because there have been numerous studies that point out the correlation between Transmission Error and gear whine, and TE is widely considered as one of the main causes of gear noise, if not the most important one. In Figure 3-3 – Vibration excitation and transmission pathwe can see the overall path to noise, and how all the small deflections of the gears, distortions and deviations from the involute profile translate into Transmission Error and this TE will later derive into noise. 3.4 State of the art As we commented before the formal definition of the term Transmission Error was given by G. Harris in [8] in the late 50s, even though the basic idea of TE had been previously used. However, it wasn’t until the main bulk of the classic studies on the subject had been carried out by Harris, Munro and Gregory [8, 10, 14, 15, 16, 17] that the term spread out and became widely accepted. In [8] Harris also explained why tip relief is a good option to reduce TE. If the gear teeth have a perfect involute Figure 3-3 – Vibration excitation and transmission path [2] 19 profile, to have the smoothest transmission possible the tips of the next pair of teeth must be coming into contact when the tips of the current pair are disengaging, so the distance between teeth is set with this rule in mind. The problem is that the teeth are flexible, and under load they will deflect, and this distance will change. If the distance is set to be exactly the needed one under deflection the teeth could not engage smoothly, so by adding tip relief this distance is increased and we ensure a smoother transition between teeth. To explain this and other phenomena he also developed what we know as “Harris Maps” which are graphs that plot the theoretic evolution of the TE along the roll distance for various loads. We can see an example of Harris maps in Figure 3-4, where the top curve is the unloaded case, and then as load is applied the double contact regime steadily expands around the changeover point, which is the point where the contact passes from a teeth pair to the next one. Since there is such a strong connection between transmission error and noise, most of the research carried out in the field of gear noise has also been related with TE. There are many good books and publications on gear noise that can be used as an introduction to the subject and explain the basic fundaments and more complicated aspects. Three of the best books on the subject of gear noise are [2] by J. Derek Smith, [18] by C.M Harris and [4] by Dudley. Welbourn [3] collected an extensive research on the field of gear noise with many experiments in the field run by different researchers in the 70s. In its research he stablished the relation between transmission error and noise. The general consensus of the experiments pointed out that when PPTE of the gear pair doubled the noise generated by the system increased by 6 dB. The main factors that affect TE are power, tooth load and speed, when doubling these factors, they observed an increase of 3 dB in the sound generated and 6 dB for the case of speed. Welbourn also remarked the importance of tip relief for reducing the total TE. Thanks to the investigations of Mark [9] and Munro [17] we can have a deeper understanding of the main sources of the TE which are the geometric errors in the tooth profile, the deflection of the teeth under load and the dynamics of the gears at high speeds. With their research they also formulated a general classification of the different types of TE depending on their origin and divided the TE into the three main categories seen in 3.2, Manufacturing TE (MTE), Static TE (STE) and Dynamic TE (DTE). All of the above can be considered as the classical research work carried out in the field of transmission error, and this knowledge has been generally accepted and has formed the basic theory of the field. But the most recent works in this field are focusing on two main trends, the first and most important is related with the development of computer models that can simulate the gear contact and predict TE on a system without the need of real world tests, and the second trend is more related with TE and noise measurements in real world tests. Figure 3-4 – Harris maps for Transmission Error Transmission Error 20 20 In [19] Fernandez del Rincon develops a model for studying the meshing stiffness of a gear pair. The meshing stiffness of the gear pair is a very important parameter in when studying TE because it is directly linked to the deflection of the teeth and thus to the STE, and this plays a fundamental role in the DTE. He uses FEM in order to test his model but the actual model is an analytical one, and it allows us to determine the STE as well as the meshing stiffness taking into account factors such as the deformation of the gear teeth, modifications on the working distance due to deformations of the supports, friction and the possibility of contacts on both flanks is also included. The deformation of the tooth has been approached by its decomposition in to local and global deformations, in this way the nonlinearity becomes linked only to the local contact model reducing considerably the computational cost With [20] Lin and He present a method for analyzing the transmission error of helical gears with machining errors, assembly errors and tooth modifications. Just like Fernandez del Rincon they first use a FEM model the simulate the gear transmission system and then they establish the bending-torsional-axial coupling dynamic model of the transmission system based on the lumped mass method. With this method they try to evaluate the results of their model in terms of STE and DTE and compare them to the results obtained with the first FEM model. Belingardi [1] uses a Multibody approach for the dynamic analysis of a gear transmission of an electric vehicle. He presents a transmission model, which is the same used in this very project and simulates it using an MBA program called RecurDyn to obtain measures of the contact forces and DTE values, and then analyzes the frequency response of the system. The results he obtains are very helpful in the evaluation of the dynamic loads applied to the shafts, the bearings, and the housing of the gearbox, in addition those results allow to make considerations on vibration and transmission noise and evaluate the presence of high loads that are generated under dynamic conditions. There are many papers that study how TE can be reduced by introducing intentional geometric modifications to the gear teeth. One of these studies is [21] which is the base for this project, and it follows a MBA approach very similar to the one followed in this project. Using the Advanced 3D contact toolkit in Adams d’Addato generates gearbox model and run several tests to verify that teeth modifications such as tip and root relief have an effect on TE and the results obtained from Adams meet the expectations of the theory. Also, Tharmakulasingam [12] for his Ph.D. dissertation developed a model to simulate and analyze the effects of tooth profile modifications on TE but following a FEM approach using Abaqus. For that he has created a Phyton script that makes the gear profile generation process automatic inside Abaqus, generates a FEM model which is able to account for the many nonlinearities of this particular problem, and with this model evaluates the STE as well as the DTE comparing them and checking the results. Finally, he uses a hybrid numerical/analytical method to optimize the gear profile that gives the best results of STE. 4 MULTIBODY ANALYSIS he aim of this project is to study the feasibility of using a multibody dynamics software to try to model a gearbox, particularly focusing on the results of transmission error. Analyzing the effect of changing different values on the input, such as torque, speed, type of gear as well as different models introducing flexible elements to try to see if the results obtained are valid and correspond to what we could expect in a real situation. 4.1 Multibody dynamics Multibody dynamics is a very powerful tool used in many engineering and science fields with the objective of analyzing the kinematic and dynamic behavior of a system composed of different interconnected bodies. It has many applications in various fields such as mechanical engineering, aerospace, biomechanics, robotics, particle simulation, vehicle dynamics and so on. T Figure 4-1 – Multibody dynamics problema [36] Description of the Rigid Model 28 28 5.2.1 Advanced 3D Contact parameters Once we have all the dimension we need to define the gears we open the Create Gear Pair in Adams inside the Machinery module. A wizard box will open where we will have to introduce all the parameters we need. The first step is the type of gear, where we will select Spur or Helical depending on which model we are preparing. In the second step, we must select the method for modeling the gear pair that Adams will use. Here we select Advanced 3D Contact as we have explained before. The third step refers to the geometry of the gear pair, where we must introduce the module and the pressure angle, helix angle, center location, number of teeth and gear width of each diameter. It should be noted that for helical gears, the module that should be introduced is the Transverse module, not the normal one, and the same goes for the pressure angle. Rim/Bore Diameter refers to the internal diameter of the gear since Adams models only the crown of the gear as will be shown later, and this diameter must be within some limits. Here we can also select if one of the gears is internal, but in our case, all of them are external gears. There are also different Figure 5-3 – Insert Gear Pair wizard - Type Figure 5-4 – Insert Gear Pair wizard - Method 29 tabs where we can modify parameters related to Tooth Profile and Tooth Modification, but those are not the object of this project so will be left as default. For a deeper understanding of those options and their effect on the Transmission Error, the thesis [21] can be consulted. There is a final tab to modify Mesh Properties, for the mesh of the FEM that Adams runs. This tab can be useful to tune in order to modify computational time and precision of the results. Figure 5-5 – Insert Gear Pair wizard – Geometry Description of the Rigid Model 30 30 Next step is related to the Material. Here we can select the material type of the gear, which will be steel, and the program will calculate its mass and inertia matrix according to the geometry previously defined, or we can also input the matrix number by number if we knew its values. In this step, we can also define the Contact Settings such as friction, oil damping and more, but these options will be left as default. Finally, we must select the Modelling option, among three choices RunTime, PreComputed and Rigid, as we explained in 4.3. For external reasons of this project, due to the fact that this one was a very recent release, there was a bug in the software which made impossible to use the RunTime option to model the gear contact, so all the studies have been carried out with the Rigid option, which may not be the best option in terms of precision but is quite a step forward compared with the previous option thanks to the Finite Element meshing of the gears when calculating the reaction forces. The final step is Connection, where we have to define the constraints connecting the gears to the rest of the bodies of the system. In this first model since we are only modeling the meshing of the gears and not considering any deformation of other components the connection will be a perfect rotational constraint of the gear to the ground of the model which will allow them to rotate freely around their axis of rotation while restraining the rest of degrees of freedom. But for the gear 3, we must introduce a fix constraint to the gear 2, since both gears are mounted on the same shaft in the real gearbox, and they must have the same rotational speed. Figure 5-6 – Insert Gear Pair wizard - Material 31 This process must be done for both gear pairs and the program will create the corresponding bodies for each gear as well as a Gear System for each gear pair where a lot of information about the gear contact will be calculated, such as contact force, distance and misalignment, friction, contact pressures, transmission error and many more variables. With all the gears defined and generated by the program, there are still some details left to be defined so the model can be completed. 5.2.2 Measures One of the strongest points of Multibody Simulation software is that once you have your model generated you can add Measures to monitor almost any variable you can imagine in the results, position, relative distance, speed, acceleration, force, torque, angular displacement, anything. Even though by creating the gear pairs with the Advanced 3D Contact tool we can see the Transmission Error of each pair in the results, we have decided to monitor the value of TE with measures, since the one given by the program is not clearly defined, the units are not specified, and can only measure the TE of each pair, not the total TE of the whole system. For these reasons, measures for the TE will need to be created. First of all, we need to create markers, which are construction points of reference that will be attached to the bodies and we will measure the movement of those markers. So, we create one marker for each gear in the center of the gear connected to each gear and another one in each center connected to the ground. Then with those markers, we create 3 measures for the angle of rotation of each axle. Once we have the angle of rotation of each axle we can calculate the TE as we have seen previously. 1 12 2 1 2 ang Z TE Z   3 34 4 3 4 ang Z TE Z   3 1 41 34 ang total Z Z TE ZZ   (5.2) But those measures are in degrees, and we are interested in the linear TE so we need to convert them into first into radians and then into μm. 2 12 12 1000 2 180 m ang D TE TE     4 34 34 1000 2 180 m ang D TE TE     41000 2 180 m total ang total D TE TE     (5.3) Figure 5-7 – Insert Gear Pair Wizard - Connection Description of the Rigid Model 32 32 5.2.3 Forces There is only one force we need to introduce as an input in our model, and that force is a resistant torque that will be applied on the output gear (gear 4) opposing to the direction of motion. The value of this resistant torque will vary depending on the test we are running. We will have two different conditions, loaded and unloaded. On the loaded condition a torque of 10Nm will be applied which is the load resistance torque of the vehicle calculated in [1]. And for the unloaded condition, we will use a torque of 0.1Nm. Even the unloaded condition must have some small load because if the output wasn’t loaded the gears would rattle too much and the values measured would not be a result of the transmission error but due to the excessive rattle of the gears. 5.2.4 Motion Another condition we must impose in the model is the motion of the input gear. So, we go to the Motions tab in Adams, add a Rotational Joint Motion and we select the rotational joint between the first gear and the ground. The speed of this motion will also vary for the different simulations. We have two speed conditions, a low speed of 0.2 rad/s (1.9 rpm) that will be used to simulate static conditions, and a high speed of 945 rad/s (9024 rpm) to simulate dynamic conditions. These speed conditions are the same as the ones used in the previous thesis [4]. 5.2.5 Simulation controls Finally, we have the simulation controls, since this is a dynamic simulation of a system that it is evolving in time we need to specify the duration of the simulation as well as the number of steps or the size of the step. These values will vary from simulation to simulation, making sure that the End Time of the simulation is long enough so we have reached a permanent state of the system, and that the Step size is small enough that the sampling frequency can capture the real evolution of the variable without any antialiasing problems. Having defined all the elements necessary and all the inputs required our model is completed and we can proceed with the analysis. Figure 5-8 – Spur gear rigid model Adams 33 6 RESULTS FOR THE RIGID MODEL his chapter will be dedicated to describing the analyses carried out on the first model as well as presenting and discussing the results obtained from those analyses. The first model is an ideal model, we are not accounting for any flexibilities in the model, all the elements are completely rigid and the connections of the gears to the ground are also perfectly rigid. This means that no deformations will occur during the simulation, and the distance between the centers of engaged gears will remain constant and with its design value. These assumptions may make the model not suitable for real applications where a high precision is required but is more than enough for most applications trying to simulate gears and it will help us understand the effect of certain parameters, more precisely speed torque, and type of gears will have on the TE. 6.1 Parameters under study As we have commented on this model we will run analyses to investigate the effect of the input speed, the resistant torque and the type of teeth on the value of TE. For this, we will run analyses with helical and spur gears, loaded and unloaded on the output, and with low speed and high speed on the input, as explained in 5.2.3 and 5.2.4. We have three different variables with two different values each, what gives us a total of 8 different analyses. In the table below, we can see the different analyses carried out with their corresponding conditions and the code they receive to named them. Table 6-1 – Rigid model Analyses Code Teeth type Input Speed (rad/s) Resistant Torque (Nm) Spur _S02_L100 Spur 0.2 0.1 Spur_S02_L10000 Spur 0.2 10 Spur_S945_L100 Spur 945 0.1 Spur_S945_L10000 Spur 945 10 Hel _S02_L100 Helical 0.2 0.1 Hel_S02_L10000 Helical 0.2 10 Hel_S945_L100 Helical 945 0.1 Hel_S945_L10000 Helical 945 10 T Results for the Rigid Model 36 36 The values obtained in the analyses with low speed will represent what we have called Static Transmission Error (STE) in chapter 3.2, and the values obtained from the analyses with high speed will give us the value of Dynamic Transmission Error (DTE) since the speed is high enough for dynamic effects to play an important role. For the Manufacturing Transmission Error (MTE) we can use the tab Tooth profile in the wizard when creating the geometry of the gears to modify some factors of the tooth flank, cutter rack or involute to tune the profile of the teeth, but the problem is that since the MTE is originated by manufacturing errors that are very difficult to measure in the real life, these parameters will be very difficult to tune perfectly to our exact gear. Anyways these modifications are out of the scope of this thesis and will be left as default. Another source for MTE in our model would be the FEM mesh that Adams generates to solve the contact. This error could be reduced by using a finer mesh if it was necessary. 6.2 Results First of all, we should comment on the results of all the Measures taken for a single analysis and the relation between the variables. On Figure 6-1, we can see a comparison of the different values of the angular TE for a single analysis which corresponds to the code Spur_S02_L100, but the conclusion obtained from this particular analysis can be extrapolated to the rest of cases. It is important to point out that the values obtained are negative, so the vertical axis of the graph is negative too, but we do not care about the sign of the TE but its absolute value, so when we compare two different variables we will be speaking about their absolute values. The first thing we see in this graph is that the TE of the Pair12 is lower than the Pair34 in mean value due to the lower torque it is been subjected to, but the peak-to-peak value (PPTE) is higher, this is because it is rotating at a higher speed and has a higher meshing frequency than the Pair34. And the value of TE of the whole system, Total, seems to be almost like the summation of the two pairs but it is not the exact value. This makes sense since in a gear train the total TE between the input and the output should be an accumulation of all the TE of each pair it is made of, but not exactly the same value of the summation since the different gear ratios must affect the impact of each TE on the total one. On Figure 6-2, we have the values for the linear TE, which are same graphs as for the angular ones but each of them is scaled by the diameter of the corresponding gear to transform them from angular to linear displacements. Figure 6-1 – Angular TE Comparison 37 Once we know the relation between the different TE inside a model, which is the same for all analyses, it is easier and gives more information to study only one of them when comparing two different analyses, for that reason from now on only the linear total TE will be studied and hereon when we speak about TE we will be referring to the same. In addition to that, the variable of TE is a continuous variable in time, which takes the form of a wave with the frequency corresponding to the meshing frequency of the specific gear pair, so it is fixed. Then to characterize these waves we only need two scalar variables which give us all the information we need about the curve. Those variables are: first the peak-to-peak TE (PPTE), the amplitude between the maximum and minimum values of the wave, and the mean value of the wave. Both variables are measured in μm. It is also worth remarking that the values we are obtaining are always negative, this is due to the signs when we defined the TE, but it doesn’t really matter, only the absolute value is important. So, when reporting the values of mean TE, the absolute value will be used to help with clarity. 6.2.1 Effect of torque Figure 6-2 – Linear TE Comparison Figure 6-3 – Effect of torque at low speed 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 PPTE (μm) 0 10 20 30 40 50 60 70 80 90 Mean TE (μm) Description of the Flexible Model 44 44 The program used to generate the FEM models will be Simulia Abaqus by Dassault Systemes which is one the most powerful FEM software available nowadays and it is widely used in the industry. In addition, Abaqus has the advantage that allows us to export the models into an MNF file that can be later imported into Adams and used in our MBA model. 7.1.1 1D Model For the 1D model, we will actually model the shafts and the webs as individual parts and then join them in Adams with constraints, this will allow us to make each individual part rigid or flexible for different analyses without having to create a new FEM model. The first step when creating a FEM model is the geometry, which in this case is extremely simple. For the shafts, we only need to use the option Create Part select wire shape and draw a line with the length of our shaft. We also have divided the shaft in the points where the webs and the bearings will be, so when we mesh the part a node will be generated in that point. To do that we use Create Datum Point and enter the coordinates of those points, and then use Partition Edge to divide the geometry. For the webs, we use again Create Part but select shell shape instead and draw a circle with the dimensions of the webs. And then we use Partition face: Sketch to divide the circle into 4 sectors so the mesh is uniform, and CAD Part 1D FEM Model 3D FEM Model Figure 7-1 – FEM Models 45 it generates a node in the center of the circle. With the geometries generated and divided we just have to go to the mesh module select the approximate size for the elements using Seed part with a size from 3 to 5 mm and then use Mesh Part. The meshed parts are presented below. The bigger parts will obviously have more elements. The next step would be to define the material, which will be steel. It is important to remember the units we are using, in this case, we are using Newtons (N) for the force and millimeters (mm) for distances, this gives us the stresses in Megapascals (MPa) and the mass must be in tonnes (T) to be consistent. So, the values needed to define the material are the Young’s Modulus E=210e3 MPa, Poisson ratio υ=0.3 and density ρ=7.85e-9 T/mm3. Then the sections are created for the webs as homogeneous and specify the material and the thickness of each web, after that those sections must be assigned to their corresponding webs. For the shaft, we must create a profile that will be circular and assign it to the shaft part. To export the model into an MNF file we need to create a substructure in Abaqus and run a frequency analysis to obtain the modal shapes of vibrations, so to do that, we must create two different steps in Abaqus. Create Step -> Linear perturbation -> Substructure generation and Create Step -> Linear perturbation -> Frequency The last step is the boundary conditions. For the step substructure we need to create a boundary condition called Retained nodal DOFs, this boundary condition retains the selected DOFs in the nodes that we choose, so when the substructure is created the only nodes that we will see are those retained ones, and those are the only nodes that will appear on Adams. So, we need to retain the nodes of the shaft where the webs and the bearings will be and also the ends of the shafts, and for the webs the nodes in the center of each web, and we will retain also 4 nodes distributed in the perimeter of the web to join them to the crown of the gears in Adams. And for the step frequency, we need to encastrate the parts in a single point, so they are not constrained to vibrate but we eliminate the movements as rigid bodies. Once the whole model has been defined, the only thing left to do is create a job for each model and run the analyses. When we run the jobs the program will create some files that will be necessary to generate the MNF file. Web 1 Web 2 Web 3 Web 4 Figure 7-2 – Web Meshes Description of the Flexible Model 46 46 7.1.2 3D Model For the 3D model, the process is very similar. However, the geometry for the 3D model is more complex to generate in Abaqus, so it will be generated in CAD software, specifically Catia, and then imported to Abaqus. Once imported the geometry files, we have to divide the parts using Partition Cell in smaller, simpler cells that can be meshed with a uniform mesh and generating nodes in the points that we want as we did for the 1D model. Then the parts are meshed, and we obtain something the following meshes. The material, the steps, and the boundary conditions are generated exactly as for the 1D model. The only thing that must be done in this model is that we have to generate Multi Point Constraints (MPC). An MPC is a kind of constraint that links the displacements and strains of a region or a number of nodes to a single node. This is necessary because when we use the Retained Nodal DOFs we select single nodes, and a node is a point in the part, it does not have dimensions, so if we apply a force in Adams to that dimensionless point it becomes a concentrated force creating a huge stress riser and the stresses that force produces are too high for the material. So, by using an MPC we connect that point to a section of the shaft and with the same force, the stresses are much lower. In addition, it is necessary to create MPCs linking the outer surfaces of the webs to the central point of the web, for reasons that will be discussed in chapter 7.2. 7.1.3 Abaqus/Adams Interface With the FEM models completed, they now need to be exported into an MNF file so they can be used in Adams. This step was complicated to reach because even though Abaqus is capable of translating a substructure file into MNF thanks to an Abaqus/Adams Interface, such Interface cannot be found inside the main program, but must be executed through the command window, and the reference page inside Abaqus Help does not really explain the process very clearly. So, the process to transform the files obtained from Abaqus into an MNF will be described step by step. The first thing to do is modify the INP file that Abaqus generates. The INP file is the input file that Abaqus sends to the solver which contains all the information from the model, but nothing has been calculated yet. The INP file is a text file that can even be written with a text editor such as Notepad. After generating our model in Abaqus we have to open the INP file it generates with a text editor and add the following lines at the end of the file inside the options for the step generating the substructure: *SUBSTRUCTURE GENERATE, MASS MATRIX=YES, RECOVERY MATRIX=YES *FLEXIBLE BODY, TYPE=ADAMS Shaft 1 Shaft 2 Shaft 3 Figure 7-3 – 3D Model meshes 47 After adding that to the model we save the modified file. Now we must run that modified file in Abaqus solver, to do that we can open Abaqus and run the job or we can do it with the Windows command line, here we will explain the second option since the next steps involve also the command line. To run the analysis, we need to open the Windows command line in the folder that contains the files, the easiest way to do that is to right-click the folder in question while holding the Shift key and choose Run Command Prompt here… and in the command line we must run the following commands: ABAQUS JOB=job_name ABAQUS ADAMS JOB=substructure_name UNITS=mmks MASS=Tonne The first command will run the Abaqus analysis on the modified file. And the second command generates the MNF file, we have to input the name of the substructure file, which is usually the same as for the INP but followed by “_Z1”, the units of our model are MMKS (mm, kg, N,s), but the mass is actually in tonnes as we saw in 7.1.1 so it must be modified with the last option. After running those commands, the MNF file should be ready to use and be imported into Adams. 7.2 Adams Model The process of generating the Adams model will be exactly the same as followed in chapter 5.2 with the only exception of the last step when defining the gears when we have to choose the connection to the rest of the models we previously chose a rotational joint with the ground and now will be left unconnected. Then we need to import the flexible MNFs generated, for that, we use the option Flexible Bodies  Adams Flex: Create a flexible body we specify the name of the part we are importing and chose the MNF file we want to import. And the part should appear in the Adams environment. With the parts in Adams, they need to be positioned with the Rotate and Move tools. The next step is to connect all the parts using joints. For the 1D Model, the webs must be connected to the shafts with Fixed Joint at the center of each web, and they will also be connected to the crown of the gears with 4 Fixed Joints in the retained nodes that we selected in the substructure generation, we can see those joints in the figure below. For the 3D model, we only need to connect the crowns to the retained node in the center of the webs that was linked with an MPC to the outer surface of the webs. Crown-web joints Web-Shaft joints Figure 7-4 – Joints 1D Model Description of the Flexible Model 48 48 The connection to the ground will depend on the model we are trying to study. If the model does not account for bearings we just apply a Revolute Joint between the shaft and the ground, two joints per shaft in the corresponding locations. If we want to study a model with bearings it is useful for us again the Machinery module inside Adams. We go to Machinery tab and open the tool Create Bearing. For this project we will use the method Compliant that allows us to generate a bearing with linear behavior and lets us specify the stiffness in each direction, but it should be mentioned that there is a more complex option, Detailed¸ that can be used if we need more precision when modelling the bearings you can specify the precise type of bearing and all its properties. In the geometry tab, we need to specify the point where the bearing will be created as well as its axis of rotation. In the connection tab, we must choose which part will be the shaft and which will be the housing (the ground in our case), we can also use the option Impose Motion… to impose the input motion to one of the gears of the first shaft. And finally, in this tab we can configure all the parameters of the bearing related to the stiffness and damping of the gears, those parameters will be left as default, except for the case of helical gears, where the gears in addition to the radial force also generate an axial force on the bearings, so the value of the axial stiffness must be increased. Since we are studying a general case we don’t really care about the specific values, assuming that we are within a sensible range, if we were studying a specific case with specific bearings we would have to use the values provided by the supplier. After creating all the bearings necessary we have completed our model and the only thing left is to run the analyses. In the figures below, we can see some examples of the completed models. Figure 7-5 – Bearings Connection Details 49 7.3 Limitations of this approach This approach for simulating the dynamics of a gearbox taking into account the flexibility of some parts it is very versatile and has a huge potential, but however it also has certain limits that will be discussed in this section. The first limit has already been discussed and it is related with the bug in the code that makes impossible the use of the RunTime model, so we have only been able to model the gears using the Rigid Gear option, and thus not considering the flexibility of the teeth. This is certainly not a limitation of this approach but some bug in our specific version of the program that we hope will be solved in future versions, however, it has been a limitation for this very project. And another limitation when considering the deformations of the flexible parts is that Adams does not really solve the FEM problem for each part, but it tries to solve the deformed state by using Modal Superposition. Modal Superposition is the method used to analyze flexible parts in dynamic situations, and it uses the shape modes of the firsts natural frequencies of the part in a linear combination to try to obtain the actual deformed shape of the part. However, if the shape of the load is not simple, it may be difficult to calculate the deformed state using Modal Superposition. The more complicated the load the higher the number of modes that we will have to use to represent the deformed state. The number of modes we are using is a variable that can be selected when running the Abaqus analysis, and that information is later translated in the MNF file. Finally, the last important limitation is that Adams is not designed to work with flexible parts but with rigid bodies, and this can be seen in the different Joints available within the program. Most joints are point joints, this means that they join two different bodies in one specific location. This is not a problem for rigid bodies, but for flexible parts, if we join two parts in a single point, all the forces will be transmitted from one body to the other through that single point. Resulting in a huge stress riser and much higher strains that they should be. There isn’t a way to connect two surfaces in Adams as we could in a FEM program. So, in order to try to solve this problem, two approaches have been used. For the 1D model instead of connecting the gears to the webs in a single point, they have been connected through 4 different points and thus distributing the load at least in 4 points instead of only one. For the 3D model this approach could not be followed, because if we connect a flexible 3D part to a rigid body in more than one point the part becomes completely rigid and thus we lose all the information the flexible part can give us. So, to work around this problem what we did was modifying the FEM model, and inserting an MPC constraint that connects the outer surface of the web to the center point, and then in Adams connect that single point to the gear with a Fixed Joint, as we explained in 7.1.2. Figure 7-7 – 1D Model without Bearings Figure 7-7 – 3D Model with Bearings Description of the Flexible Model 50 50 8 RESULTS FOR THE FLEXIBLE MODEL ust like chapter 6 was dedicated to describing the analyses and present the results obtained from the Rigid Model, this chapter will be dedicated to explaining the analyses carried out on the Flexible model, as well as their corresponding results 8.1 Parameters under study On the Rigid Model we already examined the effect of the input speed and the resistant torque on the gearbox, so for this model, those parameters will be out of the scope of the study and will be set to a fixed value. More precisely all the analyses will be run under conditions of high speed (945 rad/s) and high load (10000 Nmm). Instead of that, we will be analyzing the effects of introducing the flexibility of the different parts. To do that there is a very useful option in Adams, where we can select if a flexible body can behave as flexible or completely rigid. So, we will choose the specific flexible elements for each analysis. We will also compare the results of the 1D Model against the 3D Model and those at the same time against the values obtained from the Rigid Model. And all those analyses will be carried out with both spur and helical gears. In the table below, we can see an overview of the different analyses that will be run. J Results for the Flexible Model 52 52 Table 8-1 – Flexible Model Analyses Code Teeth type Flexible Bearings Flexible Shafts Flexible Webs Flex1D_AllRigid Spur No No No Flex1D_Webs Spur No No Yes Flex1D_Shafts Spur No Yes No Flex1D_ShaftsWebs Spur No Yes Yes Flex1D_Bearings Spur Yes No No Flex1D_BearingsWebs Spur Yes No Yes Flex1D_BearingsShafts Spur Yes Yes No Flex1D_AllFlexible Spur Yes Yes Yes Flex3D_AllRigid Spur No No No Flex3D_ShaftsWebs Spur No Yes Yes Flex3D_Bearings Spur Yes No No Flex3D_AllFlexible Spur Yes Yes Yes Flex1DHel_AllRigid Helical No No No Flex1DHel_Webs Helical No No Yes Flex1DHel_Shafts Helical No Yes No Flex1DHel_ShaftsWebs Helical No Yes Yes Flex1DHel_Bearings Helical Yes No No Flex1DHel_BearingsWebs Helical Yes No Yes Flex1DHel_BearingsShafts Helical Yes Yes No Flex1DHel_AllFlexible Helical Yes Yes Yes Flex3DHel_AllRigid Helical No No No Flex3DHel_ShaftsWebs Helical No Yes Yes Flex3DHel_Bearings Helical Yes No No Flex3DHel_AllFlexible Helical Yes Yes Yes 8.2 Results After running all the simulations, we analyze the results obtained, starting with the 1D Model with spur gears. In Figure 8-1, we have the evolution of the total TE plotted against time, while in Figure 8-2 we can see the values of mean TE and PPTE for the different analyses so they can be compared. Both figures give us the same information, increasing the number of flexible elements in the system has two different effects on the total TE, it reduces the value of PPTE but increases the mean value of TE. Those were the expected effects since by introducing more flexible elements means that the total deformations will be higher, so the mean TE is higher, but at the same time those higher deformations reduce the peaks in the contact forces and thus reduce the amplitude of the vibrations and the PPTE. The impact of the flexibility of each part is not the same, introducing flexible bearings has the highest impact, followed by the impact of the flexible shaft, and in last place, we have the impact of the flexible webs, which has almost no noticeable effect. This is because the webs are 2D elements with the loads on the plane of the element which is much more rigid than the shafts which are 1D elements with the loads on the transversal plane. 53 Figure 8-1 – TE on 1D Model – Spur gears (1) 0 0,2 0,4 0,6 0,8 1 1,2 1,4 PPTE (μm) 53,5 54 54,5 55 55,5 56 56,5 57 57,5 58 Mean TE (μm) Figure 8-2 – TE on 1D Model – Spur gears (2) References 60 60 [18] C. Harris, Handbook of Acoustical Measurements and Noise Control, McGraw-Hill Inc., 1991. [19] A. F. d. Rincon, F. Viadero, M. Iglesias, P. García, A. de-Juan and R. Sancibrian, "A model for the study of meshing stiffness in spur gear transmission," Mechanism and Machine Theory, vol. 61, pp. 30-58, 2013. [20] T. Lin and Z. He, "Analytical Method for coupled transmission error of helical gear sysstem with machining error, assembly erros and tooth modification," Mechanical Systems and Signal procesing, vol. 91, pp. 167-182, 2017. [21] F. d'Addato, Dynamic Simulation of power transmissions, Torino: Master thesis, Politecnico di Torino, 2016. [22] MSC Software, Adams Machinery Gear Advanced 3D Contact Solution Brief, MSC Software, 2017. [23] MSC Software, "Adams 2017.2 Online Help," 2017. [24] H. Kohler, A. Pratt and A. Thompson, "Dynamics and noise of parallel axis gearing," Institute of Mechanical Engineers Conference in Gearing, pp. 111-121, 1970. [25] R. Tharmakulasingam, G. Alfano and M. Atherton, "Static and dynamic transmission error of spur gear pair," Journal of Sound and Vibration, 2010. [26] P. Velex and M. Ajmi, "On the modelling of excitations in geared systems by transmission errors," Journal of Sound and Vibration, no. 290, pp. 882-909, 2006. [27] F. Kayama, The Dynamics of Parallel Axis Gears in an Automotive Transmission, PhD Thesis, University of Leeds, 2005. [28] R. Parker, "Progress and problems in gear vibration and noise," in Second International Conference on Damping Technologies, Stellenbosch, South Africa, 2006. [29] R. Kumar, N. Tiwari, D. Kunwar, R. V. Lakshmi and M. Chhetri, "Transimission Error on Spur Gears," International Journal of Advanced Engineering Research and Studies, vol. III, pp. 122-125, 2012. [30] P. Davoli, C. Gorla, F. rosa, F. Rossi and G. Boni, "Transmission Eror and Noise Emission of Spur Gears," Gear Technology, 2007. [31] M. Henriksson, On noise generation and dynamic transmission error of gears, Stockholm: Universitetsservice, 2009. [32] E. Hiroaki and S. Nader, Gearbox Simulation Models with Gear and Bearing Faults, Mechanical Engineering, Dr. Murat Gokcek (Ed.), ISBN: 978-953-51-0505-3, InTech, Available from:, 2012. [33] D. Lee and K. M. W. Lee, "Characteristics of Transmission Error and Vibration of broken tooth contact," Journal of Mechanical Science and Technology, vol. 12, no. 30, pp. 5547-5553, 2016. [34] O. Bauchau, Flexible Multibody Dynamics, Springer, 2011. [35] F. Amirouche, Fundamentals of Multibyody Dynamics. Theory and Applications, Brikhäuser, 2006. [36] K. Arczewski, W. Blajer, J. Fraczek and M. Wojtyra, "Multibody Dynamics," Computational Methods 61 and Applications, vol. 23, pp. 195-215, 2001. [37] M. Akerblom, Gear Noise and Vibrations - A Literature Survey, Eskilstuna, Sweden, 2016. [38] J. Dominguez Abascal, Teoría de Máquinas y Mecanismos, Sevilla: Universidad de Sevilla, 2016. References 62 62