Classical and Modern Approaches in the Theory of Mechanisms

Nicoloe Pandrea, Dinel Popa and Nicolae-Doru Stänescu University of Pitejti, Arge§, Romania

WlLEY

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Contents

Preface xi About the Companion Website xiii

1 The Structure of Mechanisms 1 1.1 Kinematic Elements 1 1.2 Kinematic Pairs 1 1.3 Kinematic Chains 2 1.4 Mobility of Mechanisms 3 1.4.1 Defmitions 3 1.4.2 Mobility Degree of Mechanisms without Common Constraints 5 1.4.3 Mobility Degree of Mechanisms with Common Constraints 5 1.4.4 Mobility of a Mechanism Written with the Aid of the Number of Loops 7 1.4.5 Familiesof Mechanisms 7 1.4.6 Actuation of Mechanisms 9 1.4.7 Passive Elements 9 1.4.8 Passive Kinematic Pairs 10 1.4.9 Redundant Degree of Mobility 10 1.4.10 Multiple Kinematic Pairs 11 1.5 Fundamental Kinematic Chains 11 1.6 Multi-pairs (Poly-pairs) 14 1.7 Modular Groups 15 1.8 Formation and Decomposition of Planar Mechanisms 16 1.9 Multi-poles and Multi-polar Schemata 18 1.10 Classification of Mechanisms 78

2 Kinematic Analysisof Planar Mechanisms with Bars 21 2.1 General Aspects 2i 2.2 Kinematic Relations 21 2.2.1 Plane-parallel Motion 21 2.2.2 Relative Motion 23 2.3 Methods for Kinematic Analysis 24 2.3.1 The Grapho-analytical Method 24 2.3.2 The Method of Projections 24 2.3.3 The Newton-Raphson Method 25 2.3.4 Determination of Velocities and Accelerations using the Finite Differences

Method 26 2.4 Kinematic Analysis of the RRR Dyad 27 2.4.1 The Grapho-analytical Method 27

Contents

2.4.2 The Analytical Method 31 2.4.3 The Assisted Analytical Method 35 2.4.4 The Assisted Graphical Method 35 2.5 Kinematic Analysis of the RRT Dyad 46 2.5.1 The Grapho-analytical Method 46 2.5.2 The Analytical Method 49 2.5.3 The Assisted Analytical Method 52 2.5.4 The Assisted Graphical Method 53 2.6 Kinematic Analysis of the RTR Dyad 60 2.6.1 The Grapho-analytical Method 60 2.6.2 The Analytical Method 63 2.6.3 The Assisted Analytical Method 66 2.6.4 The Assisted Graphical Method 66 2.7 Kinematic Analysis of the TRT Dyad 73 2.7.1 The Grapho-analytical Method 73 2.7.2 The Analytical Method 77 2.7.3 The Assisted Analytical Method 79 2.7.4 The Assisted Graphical Method 80 2.8 Kinematic Analysis of the RTT Dyad 85 2.8.1 The Grapho-analytical Method 85 2.8.2 The Analytical Method 87 2.8.3 The Assisted Analytical Method 90 2.8.4 The Assisted Graphical Method 90 2.9 Kinematic Analysis of the 6R Triad 95 2.9.1 Formulation of the Problem 95 2.9.2 Determination of the Positions 96 2.9.3 Determination of the Velocities and Accelerations 97 2.9.4 The Assisted Analytical Method 98 2.9.5 The Assisted Graphical Method 99 2.10 Kinematic Analysis of Some Planar Mechanisms 103 2.10.1 Kinematic Analysis of the Four-Bar Mechanism 103 2.10.2 Kinematic Analysis of the Crank-shaft Mechanism 109 2.10.3 Kinematic Analysis of the Crank and Slotted Lever Mechanism 113

3 Kinetostatics of Planar Mechanisms 117 3.1 General Aspects: Forces in Mechanisms 117 3.2 Forces of Inertia 118 3.2.1 The Torsor of the Inertial Forces 118 3.2.2 Concentration of Masses 118 3.3 Equilibration of the Rotors 119 3.3.1 Conditions of Equilibration 119 3.3.2 The Theorem of Equilibration 119 3.3.3 Machines for Dynamic Equilibration 121 3.4 Static Equilibration of Four-bar Mechanisms 124 3.4.1 Equilibration with Counterweights 124 3.4.2 Equilibration with Springs 126 3.5 Reactions in Frictionless Kinematic Pairs 126 3.5.1 General Aspects 126 3.5.2 Determination of the Reactions for the RRR Dyad 127 3.5.3 Determination of the Reactions for the RRT Dyad 133

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3.5.4 Determination of the Reactions for the RTR Dyad 139 3.5.5 Determination of the Reactions for the TRT Dyad 145 3.5.6 Determination ofthe Reactions for the RTT Dyad ISO 3.5.7 Determination ofthe Reactions at the Driving Element 1SS 3.5.8 Determination ofthe Equilibration Force (Moment) using the Virtual Velocity

Principle 156 3.6 Reactions in Kinematic Pairs with Friction 757 3.6.1 Friction Forces and Moments 157 3.6.2 Determination of the Reactions with Friction 160 3.7 Kinetostatic Analysis of some Planar Mechanisms 161 3.7.1 Kinetostatic Analysis of Four-bar Mechanism 161 3.7.2 Kinetostatic Analysis of Crank-shaft Mechanism 164 3.7.3 Kinetostatic Analysis of Crank and Slotted Lever Mechanism 166

4 Dynamics of Machines 169 4.1 Dynamic Model: Reduction of Forces and Masses 169 4.1.1 Dynamic Model 169 4.1.2 Reduction of Forces 169 4.1.3 Reduction of Masses 171 4.2 Phases of Motion of a Machine 173 4.3 Efficiencyof Machines 174 4.4 Mechanical Characteristics of Machines 175 4.5 Equation of Motion of a Machine 176 4.6 Integration of the Equation of Motion 177 4.6.1 General Case 177 4.6.2 The Regime Phase 178 4.7 Flywheels 181 4.7.1 Formulation of the Problem: Definitions 181 4.7.2 Approximate Calculation 182 4.7.3 Exact Calculation 183 4.8 Adjustment of Motion Regulators 186 4.9 Dynamics of Multi-mobile Machines 189

5 Synthesis of Planar Mechanisms with Bars 195 5.1 Synthesis of Path-generating Four-bar Mechanism 195 5.1.1 Conditions for Existence of the Crank 195 5.1.2 Equation ofthe Coupler Curve 196 5.1.3 Triple Generation of the Coupler Curve 198 5.1.4 Analytic Synthesis 199 5.1.5 Mechanisms for which Coupler Curves Approximate Circular Ares and Segments of

Straight Lines 201 5.1.6 MethodofReducedPositions 201 5.2 Positional Synthesis 204 5.2.1 Formulation ofthe Problem 204 5.2.2 Poles ofFinite Rotation 205 5.2.3 Bipositional Synthesis 206 5.2.4 Three-positional Synthesis 207 5.2.5 Four-positional Synthesis 210 5.2.6 Five-positional Synthesis 214 5.3 Function-generating Mechanisms 215

Cam Mechanisms 219 1 Generalities. Classification 219 2 Analysis of Displacement of Follower 223 2.1 Formulation of the Problem 223 2.2 The Analytical Method 224 2.3 The Graphical Method 233 2.4 Analysis of Displacement of Follower using AutoLisp 236 3 Analysis ofVelocities and Accelerations 237 3.1 Analytical Method 237 3.2 Graphical Method: Graphical Derivation 241 4 Dynamical Analysis 243 4.1 Pre-load in the Spring 243 4.2 The Work ofFriction 245 4.3 Pressure Angle, Transmission Angle 245 4.4 Determination ofthe Base Circle's Radius 247 5 Fundamental Laws ofthe Follower's Motion 248 5.1 General Aspects: Phasesof Motion ofthe Follower 248 5.2 The Linear Law 249 5.3 The Parabolic Law 250 5.4 The Harmonie Law 252 5.5 The Polynomial Law: Polydyne Cams 254 6 Synthesis of Cam Mechanisms 256 6.1 Formulation of the Problem 256 6.2 The Equation of Synthesis 257 6.3 Synthesis of Mechanism with Rotational Cam and Translational Follower 258 6.4 Synthesis of Mechanism with Rotational Cam and Rotational Follower 260 6.5 Cam Synthesis using AutoLisp Functions 262 6.6 Examples 263

Gear Mechanisms 273 1 General Aspects: Classifications 273 2 Relative Motion of Gears: Rolling Surfaces 273 3 Reciprocal Wrapped Surfaces 278 4 Fundamental Law of Toothing 280 5 Parallel Gears with Spur Teeth 281 5.1 Generalities. Notations 281 5.2 Determination of the Conjugate Profile and Toothing Curve 281 5.3 The Involute of a Circle 283 5.4 Involute Conjugate Profile and Toothing Line 283 5.5 The Main Dimensions of Involute Gears 284 5.6 Thickness of a Tooth on a Circle of Arbitrary Radius 286 5.7 Building-up of Gear Trains 287 5.8 The Contact Ratio 288 5.9 Interference of Generation 289 6 Parallel Gears with Inclined Teeth 290 6.1 Generation ofthe Flanks 290 6.2 The Equivalent Planar Gear 291 7 Conical Concurrent Gears with Spur Teeth 293 8 Crossing Gears 295 8.1 Helical Gears 295

Contents ix

7.8.2 Cylindrical Worm and Wheel Toothing 296 7.9 Generation ofthe Gears usinga CAD Soft 297 7.9.1 Gear Tooth Manufacture 297 7.9.2 Algorithm and AutoLisp Functions for Creating Gears from Solids 297 7.9.3 Generation ofthe Cylindrical Gears with Spur and Inclined Teeth 298 7.9.4 The Generation ofthe Cylindrical Gears with Curvilinear Teeth 302 7.9.5 The Generation ofConical Gears with Spur Teeth 305 7.10 Kinematics of Gear Mechanisms with Parallel Axes 311 7.10.1 Gear Mechanisms with Fixed Parallel Axes 311 7.10.2 The Willis Method 311 7.10.3 Planetary Gear Mechanisms with Four Elements 312 7.10.4 Planetary Gear Mechanisms with Six Mobile Elements 313 7.11 Kinematics of Mechanisms with Conical Gears 314 7.11.1 Planetary Transmission with Three Elements 314 7.11.2 Planetary Transmission with Four Elements 314 7.11.3 Automotive Differentials 315

8 Spatial Mechanisms 317 8.1 Kinematics of Spatial Mechanisms: Generalities 317 8.1.1 Kinematics ofthe RSSR Mechanism 317 8.1.2 Kinematics ofthe RSST Mechanism 321 8.1.3 Spatial Mechanism Generating Oscillatory Motion 323 8.2 Hydrostatic Pumps with Axial Pistons 325 8.3 Cardan Transmissions 328 8.4 Tripod Transmissions 330 8.4.1 General Aspects 330 8.4.2 The C2-C Tripod Kinematic Pair 332 8.4.3 The C l - C Tripod Kinematic Pair 335 8.4.4 The Sl-P Tripod Kinematic Pair 338 8.4.5 The S2-P Tripod Kinematic Pair 338 8.4.6 Simple Mechanisms with Tripod Joints 340 8.4.7 Tripod Joint Transmissions 344 8.5 Animation of the Mechanisms 349 8.5.1 The Need for an Animation 349 8.5.2 The Animation Algorithm 349 8.5.3 Positional Analysis 349 8.5.4 Modelling the Elements of a Mechanism 357 8.5.5 Creation of the Animation Frames 361 8.5.6 Creation of Animation File for the Mechanism 364 8.5.7 Conclusions 366

9 Industrial Robots 369 9.1 General Aspects 369 9.2 Mechanical Systems of Industrial Robots 370 9.2.1 Structure 370 9.2.2 The Path-generating Mechanism 371 9.2.3 The Orientation Mechanism 373 9.2.4 The Grip Device 377 9.3 Actuation Systems of Industrial Robots 380 9.3.1 Electrical Actuation 380

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9.3.2 Hydraulic Actuation 381 9.4 Control Systems of Industrial Robots 382 9.5 Walking Machines 384 9.5.1 The Mechanical Model ofthe Walking Mechanism 385 9.5.2 Animation ofthe Walking Machine 386

10 Variators of Angular Velocity with Bars 391 10.1 Generalities 391 10.2 Mono-loop Mechanisms Used in the Construction of the Variators of Angular

Velocity with Bars 391 10.2.1 Kinematic Schemata 391 10.2.2 Kinematic Aspects 393 10.2.3 Numerical Example 394 10.3 Bi-loop Mechanisms in Variators of Angular Velocity with Bars 398 10.3.1 Kinematic Schemata 398 10.3.2 Kinematic Analysis 400

Further Reading 411

Index 417