UNIVERSITY OF CINCINNATI Date:___________________ I, _________________________________________________________, hereby submit this work as part of the requirements for the degree of: in: It is entitled: This work and its defense approved by: Chair: _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ Gear Modeling By Simulating The Fabrication Process A thesis submitted to the Office of the Graduate School of the University of Cincinnati in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in the Department of Mechanical, Industrial and Nuclear Engineering of the College of Engineering 2005 by Deepa Nair B.E., Osmania University, India, 2001 Committee Chair : Dr. Ronald L. Huston Abstract One of the major impediments facing designers and engineers of power transmission and gearing systems is the geometry of the gears themselves. Even for simple involute spur gear pairs, the geometry is hardly trivial, and the geometric complexity increases dramatically with helical, bevel and hypoid gears. Now it appears it may not be necessary to master all the geometric details of involutes, evolutes, envelopes, spirals, tip relief and crowning. With recent and current advances in computer graphics, CAD/CAM and computer hardware, it has become possible to develop models of gears by simulating the fabricating processes. In this thesis, a mathematical basis for gear tooth modeling is presented, along with computer graphics simulations of spur, helical, bevel and spiral bevel gears. With models like these, a designer can now readily predict the responses of gears in field applications and determine expected stresses, strains, deformation displacements and predict fatigue, life and failure. But perhaps of even more importance is that non-standard tooth forms can also be developed. Specifically, non-standard tooth forms can be developed such that ideal tooth geometry is approached when the gears are under load. Such developments can then also be used to determine the fabrication processes for these non-standard forms. Acknowledgement I would like to express my sincere thanks to Dr. Ronald L. Huston for being the guiding force behind this thesis. His encouragement and guidance at all times has been invaluable. My sincere thanks to Dr. David F. Thompson, my academic advisor, who has always supported, guided and encouraged me to achieve my professional goals. I would also like to thank Dr. Richard L. Shell for serving on my thesis committee and for his support and encouragement. My thanks to Mechanical, Industrial and Nuclear Engineering department staff at the University of Cincinnati for their timely assistance whenever I needed it. I would also like to thank the staff at the Engineering Library at the University of Cincinnati for their assistance in procuring reference books and papers. Finally I would like to thank my parents who have provided me with opportunities to seek knowledge. Without their blessing I would not have been able to pursue my academic goals. TABLE OF CONTENTS Sr. No. Title Pg. No. A List of Figures 3 1 Computer Aided Design 4 1.1 Introduction 4 1.2 Definition of a CAD System 4 1.3 The Design Process 5 1.4 CAD and The Design Process 8 2 Computer Graphics 13 2.1 Introduction 13 2.2 CAD Graphics Software 17 2.3 Functions of a CAD Graphics Software 19 3 Gear Modeling By Simulating The Fabrication Process 22 3.1 Introduction 22 3.2 Preliminary Concepts 23 3.3 Envelopes 24 3.4 Involutes 25 3.5 Evolutes 29 1 Sr.No. Title Pg.No. 3.6 Envelopes/Involute Geometry of a Gear Blank 31 Rolling Over a Reciprocating Trapezoidal Cutter 3.7 Envelope of a Gear Blank Rolling Over a Wheel 33 With an Involute Tooth Form as Cutter 3.8 Computer Graphics Simulation 42 4 Summary and Conclusions 66 References 69 2 LIST OF FIGURES Sr.No. Title Pg.No 1.1 The General Design Process 7 1.2 Application of Computers to the Design Process 9 3.1 Gear Blank Meshing With a Reciprocating Rack Cutter 23 3.2 Involute Teeth on Gear Blank 23 3.3 A Plane Curve 24 3.4 A Family of Plane Curves and Its Envelope 24 3.5 Involute of a Circle Formed as a Locus 26 3.6 An Involute of a Circle 27 3.7 A Computer Generated Graph of an Involute 28 3.8 Center of Curvature of a Plane Curve 29 3.9 A Plastic Wheel Rolling Over a Rigid Step 31 3.10 A Gear Blank Rolling on a Wheel With a Reciprocating Cutter 34 3.11 Axes Generated by Rolling Disks 35 3.12 A Gear Blank Rolling Over a Rack Cutter (Front view) 43 3.13 A Gear Blank Rolling Over a Rack Cutter (Isometric view) 44 3.14-3.17 Spur Gear 45 3.18-3.21 Helical Gear 49 3.22-3.28 Bevel Gear 53 3.29-3.34 Spiral Bevel Gear 60 3 Chapter 1 COMPUTER AIDED DESIGN 1.1 Introduction: In recent years the computer has become a powerful tool in design and manufacture. CAD/CAM systems (Computer Aided Designing and Computer Aided Manufacturing) have revolutionized the design and manufacturing industry by increasing design accuracy, reducing lead times and improving overall engineering productivity. 1.2 Definition of a CAD System: Computer Aided Design can be defined as any type of design activity which makes use of the computer to develop, analyze, or modify an engineering design. [1] Hardware for a CAD system typically consists of a computer, graphics display unit, keyboards and other peripherals. CAD software consists of computer programs to implement computer graphics on the system plus application programs to facilitate the engineering functions of the user company.[1] Most of the application programs are generally stress-strain analyses of components, dynamic response of mechanisms, part programming for CNC machines or heat transfer calculations. There are several fundamental reasons for implementing a computer-aided design system: [1] 1. To improve legibility: A CAD system permits more standardization in the drawings, better documentation of the design, greater legibility and portability. 4 2. To increase the productivity of the designer: The productivity of the designer can be enhanced by helping him visualize the product and its component subassemblies and parts; and by reducing the time required to create the design, analyze and document it. Improving productivity can in turn reduce design costs and shorten project completion times. 3. To improve the quality of design: Performing a thorough engineering analysis and investigation of several alternative designs is comparatively easy when a CAD system is used. The greater accuracy provided by the system also ensures that design errors are reduced. 4. To create a database for manufacturing: A large part of the database required to manufacture the product is created in the process of documenting the product design, for example the dimensions and geometries of the product and its components, material specifications of the parts, bill of materials etc. 1.3 The Design Process: The process of designing begins when there is a need. It may be the need for a new product, an improvement over an existing product or correction of a defect in an existing product. According to Shigley, [2] the design process is an iterative process which consists of six steps: 1 Recognition of need 2 Definition of problem 3 Synthesis 4 Analysis and optimization 5 Evaluation 5 6 Presentation As mentioned earlier, the entire designing process begins when there is a recognized need, such as the need for a new product, an improvement over an existing design or correction of defects in components. Identification of need is the very first step of the design process. Problem definition is the detailed specification of all aspects of the design to be implemented, which can include physical features, functional characteristics, performance criteria etc. In the synthesis phase the designer creates the model as per the specifications. He then proceeds to analyze it, make any necessary corrections and improvements over the original model, and redesign it. This process is repeated until an optimum design is achieved within the constraints. In the evaluation phase the design is checked to see if it matches the original specifications mentioned in the problem definition stage. Often a prototype is built to assess various performance criteria. Presentation is the final phase of the design process. It includes the creation of a design database containing design drawings, material specifications, assembly lists etc. [1] 6 Recognition of need Definition of problem Synthesis Analysis and Optimization Evaluation Presentation Figure 1.1 The general design process according to Shigley [2] 7 1.4 CAD And The Design Process: The accuracy of several phases of the conventional design process can be enhanced by the use of a CAD system. According to Groover and Zimmers [1] the various design related tasks which are performed by a modern CAD system can be classified into four functional areas mentioned below: [1] 1. Geometric modeling 2. Engineering Analysis 3. Design review and evaluation 4. Automated drafting These four functional areas compare with the last four stages of Shigley’s model of general design. Geometric modeling is comparable to the synthesis stage in which the actual physical design is created on the graphics unit. Engineering analysis can be compared to the analysis and optimization stage. Design review and evaluation is similar to the evaluation stage in Shigley’s model. Automated drafting corresponds to the presentation stage in which engineering drawings are created directly from the CAD database. 8 The Design Process Computer Aided Design Recognition of need Problem definition Synthesis Geometric modeling Analysis and Engineering Optimization analysis Evaluation Design review and evaluation Presentation Automated drafting Figure 1.2 Application of computers to the design process 9 Geometric Modeling: In CAD, geometric modeling alludes to the physical generation of the design model on the ICG (Integrated Computer Graphics) unit.