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1 MOTIVATION AND BACKGROUND 1.1 Introduction In the current literature available to engineers, planetary gear trains are given a clear treatment as far as a simple kinematic solution. Unfortunately, no publications to date present a simple, concise design and analysis technique that considers both the motion and forces present in a gear train in the general case. This thesis attempts to fill this void by presenting a technique for finding a total speed and force solution to an epicyclic gear train in the most general case possible. After developing this solution, nomographs will be used to create an intuitive design aid, allowing the designer to visualize the performance of a gear train without the need to solve equations repeatedly. Finally, the solution technique and design aids presented will be used to address the practicality of using planetary gear trains as a power coupling element in a new generation of tandem bicycles. 1.2 Motivation The research contained herein was motivated by a design effort undertaken by the Virginia Tech Human Powered Vehicle Team in 2002. During the early design of the multi-rider entry into the annual ASME competition, it was suggested that the most effective method for coupling the relatively inconsistent inputs of two human riders would be to use a planetary gear train. The concept behind the design attempted by the human powered vehicle team was to use a gear train like the one shown in figure 1 to create a system that would allow both riders to pedal at approximately the same speed and contribute approximately the same percentage of the output power. The planetary system accommodates differences in speed and power input by the two riders. The nature of the system behavior is the focus of this thesis. Figure 1 Figure 1: Gear train to be used in the Human Powered Vehicle Team’s design effort Using Willis’s [1] method for finding the kinematic solution of the gear train, it was found that the mechanism was governed by where the ω’s represent rotational speeds of each element in the gear train, and R is the basic transmission ratio of the gear train. Performing a static analysis, the torques were found to be controlled by where the T’s represent torques on each element in the train, and the N’s represent number of teeth in each gear in the train. Using these equations, it became apparent that the goal of achieving power balance at equal and opposite input speeds was impossible. If ω2 and ω5 are assumed to be equal and opposite, then to achieve a power balance, T2 and T5 must also be equal and opposite. According to equation 3, this means R must be 1. Unfortunately, this takes the denominator of equation 1 to zero, which drives ω6 to infinity. What had seemed intuitively a simple problem to solve had led to a singularity in the solution space. With deadlines for competition closing in, the design effort was abandoned in favor of a simpler solution. However, the research done in attempting to design a specific gear train became the foundation of a much broader research project. The drive of this project, rather than the design of a gear train for a specific purpose, is to create a concise design method that will allow development of planetary gear trains for any number of possible applications. By dealing with the planetary in the most general case possible, this project explores the reasons for the failure of the HPV team’s design as well as allowing engineers to define the kinematic relationships between the three branches of the planetary gear train without first selecting a physical arrangement of gears. 1.3 Background A planetary gear train is defined as any gear train containing at least one gear that orbits by rotating about its own axis and also about the axis of an arm, or carrier. The elementary planetary, or epicyclic, gear train is shown in figure 2, along with the simplified representation to be used for the remainder of this thesis. The elementary train consists of two gears, the sun (1) and planet (2) gears, and a third member, hereafter referred to as the planet carrier or arm (3). Figure 2: (a) The elementary epicyclic gear train and (b) its kinematical representation Since it is difficult to directly transmit motion to or from the planet gear, the elementary epicyclic gear train is somewhat limited in practical application. More useful, however, are the epicyclic trains referred to as the simple and complex planetary gear trains, where a second sun gear is used. These gear trains can be realized in any of the twelve arrangements set forth in figure 3, as originally presented by Lévai. The trains in quadrants I and III are classified as simple epicyclic trains, since the planet gears are in mesh with both sun gears. Those in quadrants II and IV represent the complex trains, where the planet gears are partially in mesh with each other and partially in mesh with the two sun gears. Notice that, regardless of arrangement, only one planet carrier may be used. Figure 3 Figure 3: The simple and complex epicyclic gear trains While this figure clearly shows the twelve possible arrangements of the epicyclic gear train, the notation used is difficult to grasp. To aid in the visualization of the actual trains represented, figure 4 shows a gear train of the lower arrangement in quadrant I. Figure 4 Figure 4: Epicyclic gear train of the lower arrangement of quadrant I in figure 3 The planetary gear train first appeared in ancient China, around 2600 BC, in a device referred to as the south pointing chariot. At a time when the magnetic compass was still centuries away from its birth, the Chinese faced the difficult task of navigating across the relatively featureless Gobi Desert. To surmount this difficulty, the south pointing chariot was developed. This device used a relatively complex planetary gear train attached to the two wheels of a cart to maintain a figure atop the cart pointing in the same direction, regardless of the path taken by the cart. The complexity of this device seems to indicate that the Chinese had been using differential drives for quite some time before the birth of the south pointing chariot. At this point, the planetary gear train disappears from history for quite some time. This is more likely due to a lack of writing on the subject, rather than the actual disuse of the principle. After the south pointing chariot, the next appearance of the planetary is in what has been named the Antikythera machine. Discovered by sponge divers off the coast of the Greek island ofAntikythera in 1901, it has been identified by scholars as a type of calculator used for predicting eclipses and other astrological events. This particular device has been dated back to approximately 82 BC, leaving a gap of roughly 2500 years during which the planetary gear train passed relatively unnoticed through human history [8]. The principle of the planetary gear survived Europe’s dark ages in the Far East, evidenced by the discovery of a device similar to the Antikythera machine by an Iranian savant named Al-Biz?na in the late first century AD. During the Great Renaissance, the planetary garnered wide use in astrolabes and clocks. The use and development of the mechanism continued throughout the Renaissance and on until present day. It is interesting to note at this point that, while the planetary has been successfully used since 2600 BC, it was not until the 1841 publication of Willis’s Principles of Mechanism [1] that any attempt was made to create an analytical model of the device. 1.4 Literature Review Robert Willis’s 1857 publication, Principles of Mechanism, is widely regarded as the first publication dedicated solely to the field now called kinematics. In his work, Willis discusses for the first time in published literature the analytical modeling of an epicyclic gear train. As this work is a study purely in mechanism, Willis presents only a solution for the rotational speeds in the gear train. After developing this solution, the author spends the remainder of the work dedicated to epicyclic gear trains in discussing applications of the mechanism. While this discussion is well conceived, it covers four remarkably obscure applications of the epicyclic gear train, owing to the age of the work. As stated previously, this work studied only the pure kinematics of the gear train, without any discussion of the torques present in the mechanism. In his doctoral dissertation for The Technical University of Building, Civil and Transport Engineering in Hungary, Theory of Epicyclic Gears and Epicyclic Change-Speed Gears, Dr Z. Lévai attempts to unify all of the previously written literature on epicyclic trains and what he calls “epicyclic change speed gears”, which appear to simply be multiple speed transmissions. In explaining to the reader exactly what constitutes an epicyclic train Lévai identifies, for the first time, the twelve possible variations on the epicyclic train. It is also stated that these twelve variations can be neatly divided into those with and without auxiliary planets or planet pairs. This is the first publication where any attempt was made to clearly and concisely define all possible arrangements of the planetary train. After defining the epicyclic train, Lévai turns his attention to its solution. After briefly discussing the solution method laid out by Willis, and the graphical method of Kutzbach [5] as it applies to trains without auxiliary planets, he discusses at length two different modifications that can be performed to apply the Kutzbach method to a train with auxiliary planets. Again, he offers no treatment of the torques present in the system. Deane Lent, professor of Mechanical Engineering at Massachusetts Institute of Technology, published his work, Analysis and Design of Mechanisms, in 1961. In this work Lent again presents in detail the methodology of Willis for finding the rotational speeds of each branch of the epicyclic gear train, along with specific methods for the design of three and four gear trains. While these techniques are well written and simple to follow, there is again no discussion of torques present in the system. Also included in this publication are several applications of the planetary gear train, all significantly more relevant than those discussed by Willis. Joseph Shigley and John Uicker published their kinematics text, Theory of Machines and Mechanisms, in 1980. Within this work are not only a treatment of Willis’s methodology, but also a more complete definition of the epicyclic gear train. Not only do they dedicate a significant amount of discussion to this definition, but they also reproduce Lévai’s figure demonstrating the twelve possible variations of the planetary gear train. Most importantly, however, they present a solution technique for the torques present in the gear train. Unfortunately they do not approach the static force analysis for the general case; rather they present the solution in terms of free body diagrams for a specific arrangement of the planetary. While this method is relatively simple, it limits the designer to a single arrangement early in the design process. Mechanisms and the Dynamics of Machinery, the publication of Hamilton Mabie and Charles Reinholtz, presents largely the same information as Shigley and Uicker. While the treatment of the kinematics and static forces of the mechanism are nearly identical, Mabie and Reinholtz also present a brief section considering circulating power flow in controlled planetary gear systems. While this discussion has no direct application to this thesis, it does hint at the methods used herein to solve for the static forces in the gear train for the general case. John Molnar published his Nomographs in 1981. This work presents an excellent introduction to nomographs, as well as discussing at length their use and construction. This work was instrumental in the construction of the nomographs presented herein. While the bulk of this publication is dedicated to the reproduction of nomographs covering the broad general category of problems dealing with air, water, and related mechanical devices, the introduction provides more than enough information for a novice to completely understand the construction and use of nomographs for the solution of nearly any problem. 2 CAD/CAM 2.1 Introduction to CAD/CAM Thoughout the history of our industrial society,many inventions have been patented and whole new technologies have evolved.Perhaps the single development that has impacted manufacturing more quickly and significantly than any previous technology is the digital computer.Computer are being used increasingly for both design and detailing of engineering components in the drawing office. Computer-aided design (CAD) is defined as the application of computers and graphics software to aid or enhance the product design from conceptualization to documentation.CAD is most commonly associated with the use of an interactive computer graphics system,referred to as a CAD system.Computer-aided design systems are powerful tools and are used in the mechanical design and geometric modeling of products and components. There are several good reasons for using a CAD system to support the engineering design function: l To increase the productivity l To improve the quality of the design l To uniform design standards l To eliminate inaccuracies caused by hand-copying of drawings and inconsistency between drawings Computer-aided manufacturing (CAM) is defined as the effective use of computer technology in manufacturing planning and control.CAM is most closely associated with functions in manufacturing engineering,such as process and production planning, machining, scheduling, management, quality control, and numerical control (NC) part programming, Computer-aided design and computer-aided manufacturing are often combined into CAD/CAM systems. This combination allows the transfer of information from the design stage into the stage of planning for the manufacturing of a product,without the need reenter the data on part geometry manually.The database developed during CAD is stored; then it is processed further,by CAM,into the necessary data and instructions for operating and controlling production machinery,material-handling equipment,and automated testing and inspection for product quality. 2.2 Rationale for CAD/CAM The rationale for CAD/CAM is similar to that used to justify any technology-based omprovement in manufacturing.It grows out of a need to continually improve productivity,quality and competitiveness.There are also other reasons why a company might make a conversion from manual processes to CAD/CAM: l Increased productivity l Better quality l Better communication l Common database with manufacturing l Reduced prototype construction costs l Faster response to customers 2.3 CAD/CAM Hardware The hardware part of a CAD/CAM system consists of the following components:(1)oneor more design workstaions,(2)digital computer,(3)plotters,printers and other output devices,and (4)storage devices. The relationship among the components is illustrated in Fig.10.1.In addition,the CAD/CAM system would have a communication interface to permit transmission of data to and from other computer systems,thus enabling some of the benefits of computer integration. The workstation is the interface between computer and user in the CAD system.The design of the CAD workstation and its available features have an important influence on the convenience,productivity,and quality of the user’s output.The workstation must include a graphics display terminal and a set of user input devices.CAD/CAM applications require a digital computer with a high-speed control processing unit(CPU).It contains the main memory and logic/arithmetic section for the system.The most widely used secondary storage medium in CAD/CAM is the hard disk,floppy diskette,or a combination of both. The typical I/O devices used in a CAD system are shown in Fig.10.2. Input devices are generally used to transfer information from a human or storage medium to a computer where “CAD functions”are carried out.There are two basic approaches to input an existing drawing:model the object on a drawing or drawing or digitize the drawing.The standard output device for CAD/CAM is a CRT display.There are two major types of CRT displays:random-scan-line-drawing displays and aster-scan displays.In addition to CRT,there are also plasma panel displays and liquid-crystal displays. 2.4 CAD/CAM Software Software allows the human user to turn a hardware configuration into a powerful design and manufacturing system.CAD/CAM software falls into two broad categories,2-D and 3-D, based on the number of dimensions visible in the finished geometry.CAD packages that represent objects in two dimensions visible in the finished geometrey.CAD packages that represent objects in two dimensions are called 2-D software.Early systems were limited to 2-D.This was a serious shortcoming because 2-D representations of 3-D objects is inherently confusing.Equally problem has been the inability of manufacturing personnel to properly read and interpret complicated 2-D representations of objects.3-D software permits the parts to be viewed with the three-dimensional planes-height,width,and depth-visible.The trend in CAD/CAM is toward 3-D representation of graphic images.Such representations approximate the actual shape and appearance of the object to be produced;therefore,they are easier to read and understand. 2.5 Applications of CAD/CAM The emergence of CAD/CAM has had a major impact on manufacturing,by standardizing product development and by reducing design effort,tryout,and prototype work;it has made possible significantly reduced costs and improved productivity. Some typical applications of CAD/CAM are as follows: l Programming for NC,CNC,and industrial robots; l Design of dies and molds for casting,in which,for example,shrinkage allowances are preprogrammed; l Design of tools and fixtures and EDM(electrical-discharge machining)electrodes; l Quality control and inspection---for instance,coordinate-measuring machines programmed on a CAD/CAM workstation; l Process planning and scheduling. 2.6 Advantage of CAD/CAM There are many reasons for using CAD;the most potent driving force is competition.In order to win business,companies used CAD to produce better designs more quickly and more cheaply than their competitors.Productivity is much improved by a CAD program enabling you to easily draw polygons,ellipses,multiple parallel lines and multiple parallel curves.Copy,rotate and mirror facilities are also very handy when drawing symmetrical parts.Many hatch patterns are supplied with CAD programs. Filling areas in various colors is a requirement in artwork and presentations.Different style fonts for text are always supplies with any CAD programs.The possibility of importing different graphic file formats and scanning of material(photographs)into a CAD program is also an asset especially as the image can be manipulated,retouched and animated. Another advantage of CAD system is its ability to store entities,which are frequently used on drawings.Libraries of regularly used parts can be purchased separately or can be created by the draughtsman.For repetitive use on a drawing,a typical item may be retrieved and positioned in seconds,also oriented at any angle to suit particular circumstances. Using CAD products,assembly drawings can be constructed by inserting existing component drawings into the assembly drawing and positioning them as required. Clearance between different components can be measured directly from the drawing,and if required,additional components designed using assembly as reference. CAD is very suitable for fast doc