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A PC Implemented Kinematic Synthesis System for Planar Linkages A PC Implemented Kinematic Synthesis System for Planar Linkages by Christodoulos S. Demetriou Thesis submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering APPROVED: Charles F. Reinholtz, Chairman Michael P. Deisenroth John B. Kosmatka December, 1987 Blacksburg, Virginia A PC Implemented Kinematic Synthesis System for Planar Linkages by Christodoulos S. Demetriou Charles F. Reinholtz, Chairman Mechanical Engineering (ABSTRACT) The purpose of this thesis is to develop a PC implemented kinematic synthe- sis system for four-bar and six-bar planar linkages using Turbo Pascal. CYPRUS is an interactive program that calculates and displays graphically the designed four-bar and six-bar linkages. This package can be used for three and four position synthesis of path generation, path generation with input timing, body guid- ance, and body guidance with input timing linkages. The package can also be used for function generation linkages where the user may enter a set of angle pairs or chooce one of the following functions: tangent, cosine, sine, exponential, logarithmic, and natural logarithmic. The above syntheses can be combined to design linkages that produce more complex motion. For each kinematic synthesis case the code calculates a certain number of solutions. Then the designer chooses the most suitable solution for the particular application at hand. After a mechanism is synthesized, it can be animated for a check of the mechanical action. Watching this animation allows the designer to judge cri- teria such as clearances, forces, velocities and acceleration of the moving links. The software operates on an IBM PC or any other PC compatible. The lan- guage used is Turbo Pascal, an extremely effective tool and one of the fastest high level languages in compilation and execution time. Acknowledgements I would like to extend my deepest gratitude to the chairman of my committee, Dr. Charles F. Reinholtz, for his guidance, encourangement and support during the course of this study. I would like also to thank Dr. Michael P. Deisenroth and Dr. John B. Kosmatka for their guidance, service on the graduate committee and support. For their support, my thanks to Mitchel Keil, Ashit Gandhi, Steve Wagner, Jayaram Sankar and generally all my fellow students, who provided technical help and moral support. A special thanks to Richard Cobb, a University instructor, for his help and guidance in Pascal Programming. Finally, I am indebted to my wife Nitsa and all my family members, especially my uncle Suhel Turjman, for their continuous love and support, which made my studies in the United States possible. Acknowledgements iii Table of Contents Introduction . 1 Literature review . 4 Design Considerations and Programming Guides . 9 3.1 Planar Linkages . 9 3.2 Mechanism Defects . 13 3.3 Pascal Programming Language . 16 Body Guidance Synthesis . 22 4.1 Example Problem . 26 Path Generation with Input Timing . 30 5.1 Cognate Linkages . 31 5.2 Example Problem . 36 Function Generation Synthesis . 40 Table of Contents iv 6.1 Chebyshev Spacing . 40 6.2 Freudenstein's Method . 44 6.3 Example Problem . 46 Body Guidance Synthesis with Input Timing ........... , . 49 7.1 Six-Bar Linkage .................................................... 49 7.2 Example Problem . 51 System Structure . 55 8.1 Main Procedure . 56 8.2 Body Guidance . 56 8.3 Body Guidance with Input Timing . 58 8.4 Path Generation with Input Timing . 58 8.5 Function Generation . 60 8.6 Cognates . 60 8.7 Animation Routine . 63 Concluding Remarks and Recommendations . 67 References ............................... , ..... , . 69 Body Guidance Synthesis Equations . 72 A.1 Equations Used for Three Position Synthesis . 72 A.2 Equations Used for Four Position Synthesis . 74 Pascal Code . 77 Table of Contents V Procedures and Functions Used . • . 190 Vita ................................................................ 208 Table of Contents vi List of Illustrations Figure 1. Four-Bar linkage ......................................... 5 Figure 2. Stephenson's I, II, and Ill six-bar linkages ..................... 11 Figure 3. Watt's I and II six-bar linkages ............................. 12 Figure 4. Function generation mechanism. .......................... 14 Figure 5. A typical body guidance problem. .......................... 15 Figure 6. The two possible branches of a four-bar linkage. .............. 17 Figure 7. Body guidance solution using complex numbers. .............. 23 Figure 8. Four-bar body guidance linkage. ........................... 29 Figure 9. Original four-bar linkage. ................................ 32 Figure 10. Cognate linkages. ...................................... 33 Figure 11. Cayley's diagram. ...................................... 35 Figure 12. Four-bar path generator with input timing. ................... 39 Figure 13. Synthesis structural error.................................. 42 Figure 14. Chebyshev spacing for four precision spacing. ................ 43 Figure 15. Four-bar linkage ........................................ 45 Figure 16. Four-bar function generator. .............................. 48 Figure 17. Stephenson Ill six-bar linkage. ............................ 50 Figure 18. Cognate of the six-bar linkage. ............................ 52 Figure 19. Six-bar body guidance linkage with input timing. .............. 54 List of Illustrations vii Figure 20. Logic diagram of the main program. ........................ 57 Figure 21. Logic diagram of dyad presentation. ........................ 59 Figure 22. Description of the function generation routine. ................ 61 Figure 23. Logic diagram of the cognates routine. ...................... 62 Figure 24. Description of the animation routine. ....................... 64 Figure 25. Logic diagram of the iterative routine ......................... 66 List of Illustrations viii List of Tables Table 1. Software available for mechanism synthesis ..................... 8 Table 2. Possible Solutions for Body Guidance Problem .................. 25 Table 3. Body guidance linkage specifications ......................... 28 Table 4. Body guidance linkage links' angles ........................... 28 Table 5. Resulting cognates ....................................... 37 Table 6. Path generation linkage link's angles. ........................ 38 Table 7. Function generator specifications. ........................... 47 Table 8. Six-bar linkage specifications ...... : ........................ 53 Table 9. Six-bar linkage links' angles ................................. 53 List of Tables Ix Chapter 1 Introduction Linkages are often the simplest and most economical way to generate motion in a wide range of industrial and consumer goods. Linkages are found in automo- biles, aircraft, office equipment, robots, prosthetic devices, production equipment and toys. However, designing a linkage that works can be difficult. Often, linkage design is done by trial and error. Traditional techniques involve making drawings and building models of proposed linkages. But even after extensive trials, an optimized design may be elusive. In many cases, the difficulty of finding an effective linkage results in the use of more complicated mechanisms including cams, gears, belts and cables. Erdman [ 1] insists that this problem no longer exists since improved computer design procedures have been developed that simplify the design of proper linkage. According to Barker [2] successful mechanism design implies that: • all the links of the mechanism are of manageable length, Introduction 1 • the required positions are obtained in the right order, • the mechanism has the desired mobility, (ie. it does not become locked within the range of movement required), and • acute transmission angles requiring large driving torques are avoided. In some applications where gear sectors are used, four-bar linkages can pro- vide an acceptable, sometimes better, alternative. Linkages tend to have less inertia than gears, and they are also capable of transmitting higher loads. Another advan- tage of linkages over gears is their lower manufacturing cost. The essential differ- ence between the two alternatives is that the gears can provide perfect geometric accuracy over as long a range as required, whereas linkage provide only an approx- imation and over a limited range [3]. Although robotics is an important component of computer-aided manufactur- ing, many associated problems are best solved by devices simpler than robots [4]. Inexpensive planar and spatial mechanisms are frequently best suited for the solution of these problems. Planar mechanisms, being purely mechanical single-input de- vices, tend to be more reliable and more energy efficient than electronically- controlled multiple-input devices. Many experts, among them Sandor and Erdman [5]. believe that closed loop devices, such as planar mechanisms are known to be capable of running at higher speeds and carrying greater loads with more precision than open loop devices, such as robotic manipulators. Pascal is a general purpose high level language originally designed by pro- fessor Niklaus Wirth of the Technical University of Zurich, Switzerland and named it in honor of Blaise Pascal, the famous French seventeen century philosopher and mathematician. According to professor Wirth, the Pascal language was intended to Introduction 2 aid the teaching of a systematic approach to computer programming, specifically
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