Introduction to Mechanism Design Introduction to Mechanism Design With Computer Applications Eric Constans and Karl B. Dyer MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software. CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2019 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-free paper International Standard Book Number-13: 978-1-138-74065-5 (Hardback) This book contains information obtained from authentic and highly regarded sources. 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Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Contents Preface ...........................................................................................................................................xiii Acknowledgments .....................................................................................................................xvii Authors .........................................................................................................................................xix 1. Introduction to Kinematics ..................................................................................................1 1.1 Introduction to Mechanical Design ............................................................................1 1.2 Fundamentals of Kinematics .......................................................................................2 1.3 Degrees of Freedom ......................................................................................................3 1.3.1 Mobility of Mechanisms .................................................................................4 1.3.2 Degrees of Freedom Example Problems .......................................................7 1.4 The Fourbar Linkage and the Grashof Condition ..................................................15 1.4.1 Classifications of the Fourbar Linkage .......................................................17 1.4.2 Fourbar Classification: The Grashof Linkages ..........................................18 1.4.3 Fourbar Classification: Non-Grashof Linkages .........................................19 1.4.4 Fourbar Classification – Special Cases........................................................20 1.4.5 Fourbar Classification – The Extreme Cases ..............................................24 1.4.6 Limiting angles for Non-Grashof Linkages ...............................................27 1.5 Practice Problems ........................................................................................................29 Acknowledgments .................................................................................................................42 Works Cited ............................................................................................................................42 2. Graphical Linkage Synthesis Using SOLIDWORKS® .................................................43 2.1 Introduction to Graphical Linkage Synthesis .........................................................43 2.2 Two Specified Positions of the Rocker .....................................................................43 2.2.1 T wo Positions of Rocker without Specified Ground Pin ..........................48 2.2.2 Q uick-Return Mechanisms ..........................................................................53 2.3 Two Specified Positions of the Coupler ....................................................................58 2.4 Three Specified Positions of the Coupler .................................................................60 2.5 Summary ......................................................................................................................71 2.6 Practice Problems ........................................................................................................74 Acknowledgments .................................................................................................................77 3. Introduction to MATLAB® .................................................................................................79 3.1 Introduction .................................................................................................................79 3.2 Simple MATLAB® – The Command Window ........................................................79 3.3 Vector Notation in MATLAB® ...................................................................................84 3.4 A First Plot....................................................................................................................86 3.5 Writing a Simple MATLAB® Script ..........................................................................86 3.6 Plotting a Filled Square ..............................................................................................90 3.7 Adding Some Structure – The for Loop ..................................................................94 3.8 A Primitive Animation ..............................................................................................97 3.9 Summary ......................................................................................................................99 Acknowledgments ...............................................................................................................100 v vi Contents 4. Position Analysis of Linkages .........................................................................................101 4.1 Introduction to Position Analysis ...........................................................................101 4.2 Review of Vectors and Matrices ..............................................................................102 4.2.1 Vector Addition ............................................................................................104 4.2.2 The Vector Loop ...........................................................................................105 4.2.3 The Dot Product ...........................................................................................107 4.2.4 The Cross Product........................................................................................109 4.2.5 Unit Vectors ..................................................................................................110 4.2.5.1 Time Derivatives of Unit Vectors ...............................................112 4.2.6 A Very Brief Introduction to Matrix Algebra ..........................................114 4.2.7 Transformation of Coordinates ..................................................................118 4.3 Position Analysis of the Threebar Slider-Crank ...................................................122 4.4 Position Analysis of the Threebar Slider-Crank Using MATLAB® ...................125 4.4.1 Data Structure for the Position Calculations ...........................................128 4.4.2 The Main Loop .............................................................................................130 4.4.3 Position Calculations ...................................................................................131 4.4.4 Making a Fancy Plot and Verifying your Code.......................................136 4.4.5 Verifying Your Calculations .......................................................................137 4.4.6 Drawing the Linkage in MATLAB® ..........................................................137 4.5 Position Analysis of the Slider-Crank ....................................................................141 4.5.1 Extreme Positions of the Slider-Crank ......................................................143 4.6 Position Analysis of the Slider-Crank Using MATLAB® ....................................147 4.6.1 Verifying the Code.......................................................................................149 4.7 Position Analysis of the Fourbar Linkage .............................................................150 4.7.1 Finding the Position of Any Point on the Linkage ..................................155 4.7.2 A Digression into Trigonometric Identities .............................................161 4.7.3 Open and Crossed Configurations of the Fourbar .................................162 4.7.4 Summary .......................................................................................................162 4.8 Position Analysis of the Fourbar Linkage Using MATLAB® ..............................163 4.8.1 Data Structure for the Position Calculations ...........................................166 4.8.2 The Main Loop .............................................................................................166 4.8.3 Position Calculations ...................................................................................167 4.8.4 Making a Fancy Plot and Verifying your Code.......................................170 4.8.5 Plotting the Non-Grashof Linkage ............................................................172 4.9 Position Analysis of the Inverted Slider-Crank ...................................................174 4.9.1 Limiting Positions for the Inverted Slider-Crank ...................................177 4.10 Position Analysis of the Inverted Slider-Crank Using MATLAB® ....................180 4.10.1 P osition Analysis of the Non-Grashof Linkage .......................................183 4.11 Position Analysis of the Geared Fivebar Linkage ................................................186 4.12 Position Analysis of the Geared Fivebar Using MATLAB® ................................189 4.12.1 Verifying Your Code ....................................................................................191 4.12.2 Position of Any Point on the Linkage .......................................................192 4.13 Position Analysis of the Sixbar Linkage ................................................................193 4.13.1 Stephenson Type I Sixbar Linkage ............................................................196 4.13.2 The Remaining Sixbar Linkages................................................................198 4.13.3 The Stephenson Type II Sixbar Linkage ...................................................200 4.13.4 Summary .......................................................................................................201 4.14 Position Analysis of the Sixbar Linkage Using MATLAB® .................................201 Contents vii 4.14.1 M aking the Sixbar Plot ................................................................................204 4.14.2 T he Remaining Sixbar Linkages................................................................207 4.15 Advanced Topic: The Newton–Raphson Method ................................................207 4.15.1 T he One-Dimensional Newton-Raphson Algorithm .............................212 4.15.2 O ne Dimensional Examples .......................................................................214 4.15.3 A More Complicated Function ...................................................................215 4.15.4 N ewton–Raphson in Multidimensional Space ........................................218 4.15.5 T he Newton–Raphson Algorithm in MATLAB® ....................................220 4.15.6 Summary .......................................................................................................224 4.16 Practice Problems ......................................................................................................226 Acknowledgments ...............................................................................................................239 Works Cited ..........................................................................................................................239 5. Velocity Analysis of Linkages .........................................................................................241 5.1 Introduction to Velocity Analysis ...........................................................................241 5.1.1 Pure Rotation ................................................................................................241 5.1.2 Complex Motion ...........................................................................................243 5.1.3 Velocity of a Point Moving on a Rotating Link .......................................243 5.2 The Method of Instant Centers ...............................................................................245 5.2.1 Instant Centers of the Fourbar Linkage ....................................................246 5.2.2 SOLIDWORKS® Tutorial – Velocity Analysis of the Fourbar Linkage .....................................................................................................249 5.2.3 Instant Centers of the Slider-Crank Linkage ...........................................251 5.2.4 Instant Centers of the Inverted Slider-Crank ..........................................252 5.2.5 Instant Center Example Problems .............................................................256 5.2.6 Velocity Ratios ..............................................................................................261 5.2.7 Mechanical Advantage ...............................................................................263 5.2.7.1 Mechanical Advantage in the Slider-Crank .............................266 5.3 Velocity Analysis of the Threebar Slider-Crank ...................................................267 5.3.1 Velocity of Any Point on the Linkage .......................................................270 5.3.2 Velocity Analysis of the Threebar Slider-Crank Using MATLAB® ......271 5.3.2.1 Verifying the Code .......................................................................273 5.3.2.2 Verifying the Code – An Alternative Approach ......................274 5.4 Velocity Analysis of the Slider-Crank ....................................................................281 5.4.1 Example Slider-Crank .................................................................................282 5.5 Velocity Analysis of the Fourbar Linkage .............................................................283 5.5.1 Velocity of Any Point on the Linkage .......................................................285 5.5.2 Fourbar Velocity Analysis Using MATLAB® ...........................................287 5.5.3 Verifying the Code.......................................................................................289 5.6 Velocity Analysis of the Inverted Slider-Crank ....................................................293 5.7 Velocity Analysis of the Geared Fivebar Linkage ................................................296 5.7.1 Example Fivebar Linkage ...........................................................................298 5.8 Velocity Analysis of the Sixbar Linkage ................................................................301 5.8.1 Some Example Solutions for the Sixbar Linkage ....................................303 5.9 Introduction to Electric Motors ...............................................................................304 5.9.1 AC Motors .....................................................................................................306 5.9.2 DC Motors .....................................................................................................312 5.9.3 Brushless Motors ..........................................................................................313 viii Contents 5.9.4 Servo Motors .................................................................................................317 5.9.5 Stepper Motors .............................................................................................318 5.10 Practice Problems ......................................................................................................318 Acknowledgments ...............................................................................................................328 6. Acceleration Analysis of Linkages .................................................................................329 6.1 Introduction to Acceleration Analysis ...................................................................329 6.1.1 A cceleration of a Moving Point on a Moving Link .................................331 6.2 Acceleration Analysis of the Threebar Slider-Crank ...........................................333 6.2.1 C omputing the Accelerations Using MATLAB® .....................................335 6.2.2 Acceleration at the Pins ...............................................................................337 6.3 Acceleration Analysis of the Slider-Crank ............................................................340 6.3.1 Slider-Crank with Constant Crank Angular Velocity ............................341 6.3.2 A Note on the Angular Acceleration of the Crank .................................343 6.4 Acceleration Analysis of the Fourbar Linkage .....................................................350 6.4.1 Computing the Accelerations Using MATLAB® .....................................351 6.5 Acceleration Analysis of the Inverted Slider-Crank ............................................356 6.5.1 Computing the Accelerations Using MATLAB® .....................................357 6.6 Acceleration Analysis of the Geared Fivebar Linkage ........................................357 6.6.1 Computing the accelerations using MATLAB® .......................................359 6.7 Acceleration Analysis of the Sixbar Linkage ........................................................362 6.7.1 Some Example Solutions for the Sixbar Linkage ....................................365 6.8 Summary ....................................................................................................................365 6.9 Practice Problems ......................................................................................................374 Acknowledgments ...............................................................................................................381 7. Force Analysis on Linkages ..............................................................................................383 7.1 Fundamentals of Dynamics ....................................................................................383 7.1.1 Dynamic Models ..........................................................................................383 7.1.1.1 Mass ...............................................................................................384 7.1.1.2 Center of Mass ..............................................................................384 7.1.1.3 Mass Moment of Inertia ..............................................................387 7.1.2 The Parallel Axis Theorem .........................................................................389 7.1.3 Using SOLIDWORKS® to Calculate Moment of Inertia .........................393 7.2 Newtonian Kinetics of a Rigid Body ......................................................................395 7.2.1 Equations of Motion for the Rigid Body ...................................................396 7.2.2 Rotational Equations of Motion .................................................................398 7.2.3 A Digression on Moments, Torques, and Couples ..................................403 7.3 Force Analysis on a Single Link ..............................................................................405 7.3.1 Another Useful MATLAB® Function ........................................................409 7.3.2 Force Analysis of a Threebar Linkage using MATLAB® .......................413 7.4 Force Analysis of the Threebar Slider-Crank .......................................................417 7.4.1 Code Verification ..........................................................................................420 7.4.1.1 Static Verification ..........................................................................420 7.4.1.2 Verifying the Code using the Energy Method .........................420 7.4.2 Summary .......................................................................................................424 7.5 Force Analysis Example 1 – The Threebar Door Closing Mechanism .............424 7.5.1 The Problem Statement ...............................................................................426 7.5.1.1 Critical Dimensions of the Linkage ...........................................427 Contents ix 7.5.1.2 Inertial Properties of the Mechanism .......................................427 7.5.1.3 External Forces Acting on the Mechanism ..............................428 7.5.1.4 Free-Body Diagrams of each Link in the Mechanism ............428 7.5.1.5 Motion of the Crank .....................................................................431 7.5.1.6 Solving for the Pin Forces and Plotting Results ......................432 7.5.2 Verification of the Code ...............................................................................433 7.5.3 Summary .......................................................................................................435 7.6 Force Analysis of the Slider-Crank .........................................................................436 7.6.1 Force Analysis of the Example Linkage ...................................................438 7.7 Force Analysis Example 2 – The Air Compressor Mechanism ..........................444 7.7.1 First, a Simple Model ...................................................................................445 7.7.2 Inertial Properties of the Links ..................................................................446 7.7.3 Driving Torque without Pressure Force ...................................................447 7.7.4 And Now, a Little Thermo ..........................................................................448 7.7.5 Adding Friction to the Model ....................................................................452 7.7.6 Potential Energy of Air Inside the Cylinder ............................................455 7.8 Force Analysis of the Fourbar Linkage ..................................................................456 7.8.1 Force Analysis of the Sample Linkage ......................................................458 7.9 Force Analysis Example 3 – The Grill Lid Lifting Mechanism ..........................464 7.9.1 Designing the Fourbar Mechanism ..........................................................466 7.9.2 D etermine the Critical Dimensions of the Linkage ................................468 7.9.3 D etermine the Inertial Properties of Each Body in the Mechanism .....470 7.9.4 D etermine the Nature of the External Forces Acting on the Linkage ......472 7.9.5 Draw Free-body Diagrams of Each Link in the Mechanism ................473 7.9.6 Determine the Nature of the Motion of the Crank .................................475 7.9.7 Solve the Equations of Motion and Plot the Desired Results ................476 7.9.8 Using the Code to Improve the Design ....................................................477 7.9.9 Summary .......................................................................................................481 7.10 Force Analysis of the Inverted Slider-Crank.........................................................481 7.11 Force Analysis Example 4 – The Bicycle Air Pump .............................................488 7.11.1 Determine the Critical Dimensions of the Linkage ................................490 7.11.2 Calculate the Inertial Properties of Each Body in the Mechanism ......490 7.11.3 Determine the External Forces ..................................................................491 7.11.4 Draw Free-Body Diagrams of Each Link in the Mechanism ................493 7.11.5 Determine the Nature of the Movement of Crank ..................................495 7.11.6 Solve for the Pin Forces and Driving Force ..............................................496 7.12 Force Analysis of the Geared Fivebar Linkage .....................................................499 7.12.1 Some Gear Geometry ..................................................................................500 7.13 Force Analysis of the Sixbar Linkage .....................................................................511 7.13.1 F orce Matrices for Sixbar Linkages ..........................................................521 7.14 Practice Problems ......................................................................................................527 Acknowledgments ...............................................................................................................536 Work Cited ............................................................................................................................536 8. Gears and Gear Trains .......................................................................................................537 8.1 Introduction to Gears ...............................................................................................537 8.1.1 Spur Gears ....................................................................................................537 8.1.2 H elical Gears ................................................................................................537 8.1.3 B evel Gears ...................................................................................................537