Classical and Modern Approaches in the Theory of Mechanisms

Nicolae Pandrea, University of Pitesti, Romania Prof. Pandrea teaches at the Faculty of Mechanics and Technology, University of Pitesti, Romania.  He has served as President of the Scientific and Technical Committee in Romania, and is a member of the Romanian Academy of Technical Sciences. Professor Pandrea is the author of 250 papers and six books.Dinel Popa, University of Pitesti, Romania Prof. Dinel Popa has taught at the University of Pitesti for over 25 years. His areas of expertise are: Mechanics, Mechanisms, Vibrations, Acoustics, and Numerical Methods. Nicolae-Doru Stãnescu, University of Pitesti, RomaniaProf. Stãnescu teaches at the Faculty of Mechanics and Technology, University of Pitesti, Romania. He is a member of the International Institute of Acoustic and Vibration, in USA, and Société des Ingénieurs de l’Automobile, France. He has written 200 papers and 10 books, and his areas of expertise include non-linear vibrations, dynamical systems, stability, chaos, and numerical analysis.

• Preface xiAbout the Companion Website xiii1 The Structure of Mechanisms 11.1 Kinematic Elements 11.2 Kinematic Pairs 11.3 Kinematic Chains 21.4 Mobility of Mechanisms 31.4.1 Definitions 31.4.2 Mobility Degree of Mechanisms without Common Constraints 51.4.3 Mobility Degree of Mechanisms with Common Constraints 51.4.4 Mobility of a MechanismWritten with the Aid of the Number of Loops 71.4.5 Families of Mechanisms 71.4.6 Actuation of Mechanisms 91.4.7 Passive Elements 91.4.8 Passive Kinematic Pairs 101.4.9 Redundant Degree of Mobility 101.4.10 Multiple Kinematic Pairs 111.5 Fundamental Kinematic Chains 111.6 Multi-pairs (Poly-pairs) 141.7 Modular Groups 151.8 Formation and Decomposition of PlanarMechanisms 161.9 Multi-poles and Multi-polar Schemata 181.10 Classification of Mechanisms 182 Kinematic Analysis of Planar Mechanisms with Bars 212.1 General Aspects 212.2 Kinematic Relations 212.2.1 Plane-parallel Motion 212.2.2 Relative Motion 232.3 Methods for Kinematic Analysis 242.3.1 The Grapho-analytical Method 242.3.2 The Method of Projections 242.3.3 The Newton–Raphson Method 252.3.4 Determination of Velocities and Accelerations using the Finite Differences Method 262.4 Kinematic Analysis of the RRR Dyad 272.4.1 The Grapho-analytical Method 272.4.2 The Analytical Method 312.4.3 The Assisted Analytical Method 352.4.4 The Assisted Graphical Method 352.5 Kinematic Analysis of the RRT Dyad 462.5.1 The Grapho-analytical Method 462.5.2 The Analytical Method 492.5.3 The Assisted Analytical Method 522.5.4 The Assisted Graphical Method 532.6 Kinematic Analysis of the RTR Dyad 602.6.1 The Grapho-analytical Method 602.6.2 The Analytical Method 632.6.3 The Assisted Analytical Method 662.6.4 The Assisted Graphical Method 662.7 Kinematic Analysis of the TRT Dyad 732.7.1 The Grapho-analytical Method 732.7.2 The Analytical Method 772.7.3 The Assisted Analytical Method 792.7.4 The Assisted Graphical Method 802.8 Kinematic Analysis of the RTT Dyad 852.8.1 The Grapho-analytical Method 852.8.2 The Analytical Method 872.8.3 The Assisted Analytical Method 902.8.4 The Assisted Graphical Method 902.9 Kinematic Analysis of the 6R Triad 952.9.1 Formulation of the Problem 952.9.2 Determination of the Positions 962.9.3 Determination of the Velocities and Accelerations 972.9.4 The Assisted Analytical Method 982.9.5 The Assisted Graphical Method 992.10 Kinematic Analysis of Some Planar Mechanisms 1032.10.1 Kinematic Analysis of the Four-Bar Mechanism 1032.10.2 Kinematic Analysis of the Crank-shaft Mechanism 1092.10.3 Kinematic Analysis of the Crank and Slotted Lever Mechanism 1133 Kinetostatics of Planar Mechanisms 1173.1 General Aspects: Forces in Mechanisms 1173.2 Forces of Inertia 1183.2.1 The Torsor of the Inertial Forces 1183.2.2 Concentration of Masses 1183.3 Equilibration of the Rotors 1193.3.1 Conditions of Equilibration 1193.3.2 The Theorem of Equilibration 1193.3.3 Machines for Dynamic Equilibration 1213.4 Static Equilibration of Four-bar Mechanisms 1243.4.1 Equilibration with Counterweights 1243.4.2 Equilibration with Springs 1263.5 Reactions in Frictionless Kinematic Pairs 1263.5.1 General Aspects 1263.5.2 Determination of the Reactions for the RRR Dyad 1273.5.3 Determination of the Reactions for the RRT Dyad 1333.5.4 Determination of the Reactions for the RTR Dyad 1393.5.5 Determination of the Reactions for the TRT Dyad 1453.5.6 Determination of the Reactions for the RTT Dyad 1503.5.7 Determination of the Reactions at the Driving Element 1553.5.8 Determination of the Equilibration Force (Moment) using the Virtual Velocity Principle 1563.6 Reactions in Kinematic Pairs with Friction 1573.6.1 Friction Forces and Moments 1573.6.2 Determination of the Reactions with Friction 1603.7 Kinetostatic Analysis of some Planar Mechanisms 1613.7.1 Kinetostatic Analysis of Four-bar Mechanism 1613.7.2 Kinetostatic Analysis of Crank-shaft Mechanism 1643.7.3 Kinetostatic Analysis of Crank and Slotted Lever Mechanism 1664 Dynamics of Machines 1694.1 Dynamic Model: Reduction of Forces and Masses 1694.1.1 Dynamic Model 1694.1.2 Reduction of Forces 1694.1.3 Reduction of Masses 1714.2 Phases of Motion of a Machine 1734.3 Efficiency of Machines 1744.4 Mechanical Characteristics of Machines 1754.5 Equation of Motion of a Machine 1764.6 Integration of the Equation of Motion 1774.6.1 General Case 1774.6.2 The Regime Phase 1784.7 Flywheels 1814.7.1 Formulation of the Problem: Definitions 1814.7.2 Approximate Calculation 1824.7.3 Exact Calculation 1834.8 Adjustment of Motion Regulators 1864.9 Dynamics of Multi-mobile Machines 1895 Synthesis of Planar Mechanisms with Bars 1955.1 Synthesis of Path-generating Four-bar Mechanism 1955.1.1 Conditions for Existence of the Crank 1955.1.2 Equation of the Coupler Curve 1965.1.3 Triple Generation of the Coupler Curve 1985.1.4 Analytic Synthesis 1995.1.5 Mechanisms for which Coupler Curves Approximate Circular Arcs and Segments of Straight Lines 2015.1.6 Method of Reduced Positions 2015.2 Positional Synthesis 2045.2.1 Formulation of the Problem 2045.2.2 Poles of Finite Rotation 2055.2.3 Bipositional Synthesis 2065.2.4 Three-positional Synthesis 2075.2.5 Four-positional Synthesis 2105.2.6 Five-positional Synthesis 2145.3 Function-generating Mechanisms 2156 Cam Mechanisms 2196.1 Generalities. Classification 2196.2 Analysis of Displacement of Follower 2236.2.1 Formulation of the Problem 2236.2.2 The Analytical Method 2246.2.3 The Graphical Method 2336.2.4 Analysis of Displacement of Follower using Auto Lisp 2366.3 Analysis of Velocities and Accelerations 2376.3.1 Analytical Method 2376.3.2 Graphical Method: Graphical Derivation 2416.4 Dynamical Analysis 2436.4.1 Pre-load in the Spring 2436.4.2 The Work of Friction 2456.4.3 Pressure Angle, Transmission Angle 2456.4.4 Determination of the Base Circle’s Radius 2476.5 Fundamental Laws of the Follower’s Motion 2486.5.1 General Aspects: Phases of Motion of the Follower 2486.5.2 The Linear Law 2496.5.3 The Parabolic Law 2506.5.4 The Harmonic Law 2526.5.5 The Polynomial Law: Polydyne Cams 2546.6 Synthesis of Cam Mechanisms 2566.6.1 Formulation of the Problem 2566.6.2 The Equation of Synthesis 2576.6.3 Synthesis of Mechanism with Rotational Cam and Translational Follower 2586.6.4 Synthesis of Mechanism with Rotational Cam and Rotational Follower 2606.6.5 Cam Synthesis using Auto Lisp Functions 2626.6.6 Examples 2637 Gear Mechanisms 2737.1 General Aspects: Classifications 2737.2 Relative Motion of Gears: Rolling Surfaces 2737.3 Reciprocal Wrapped Surfaces 2787.4 Fundamental Law of Toothing 2807.5 Parallel Gears with Spur Teeth 2817.5.1 Generalities. Notations 2817.5.2 Determination of the Conjugate Profile and Toothing Curve 2817.5.3 The Involute of a Circle 2837.5.4 Involute Conjugate Profile and Toothing Line 2837.5.5 The Main Dimensions of Involute Gears 2847.5.6 Thickness of a Tooth on a Circle of Arbitrary Radius 2867.5.7 Building-up of Gear Trains 2877.5.8 The Contact Ratio 2887.5.9 Interference of Generation 2897.6 Parallel Gears with Inclined Teeth 2907.6.1 Generation of the Flanks 2907.6.2 The Equivalent Planar Gear 2917.7 Conical Concurrent Gears with Spur Teeth 2937.8 Crossing Gears 2957.8.1 Helical Gears 2957.8.2 Cylindrical Worm and Wheel Toothing 2967.9 Generation of the Gears using a CAD Soft 2977.9.1 Gear Tooth Manufacture 2977.9.2 Algorithm and Auto Lisp Functions for Creating Gears from Solids 2977.9.3 Generation of the Cylindrical Gears with Spur and Inclined Teeth 2987.9.4 The Generation of the Cylindrical Gears with Curvilinear Teeth 3027.9.5 The Generation of Conical Gears with Spur Teeth 3057.10 Kinematics of Gear Mechanisms with Parallel Axes 3117.10.1 Gear Mechanisms with Fixed Parallel Axes 3117.10.2 The Willis Method 3117.10.3 Planetary Gear Mechanisms with Four Elements 3127.10.4 Planetary Gear Mechanisms with Six Mobile Elements 3137.11 Kinematics of Mechanisms with Conical Gears 3147.11.1 Planetary Transmission with Three Elements 3147.11.2 Planetary Transmission with Four Elements 3147.11.3 Automotive Differentials 3158 Spatial Mechanisms 3178.1 Kinematics of Spatial Mechanisms: Generalities 3178.1.1 Kinematics of the RSSR Mechanism 3178.1.2 Kinematics of the RSST Mechanism 3218.1.3 Spatial Mechanism Generating Oscillatory Motion 3238.2 Hydrostatic Pumps with Axial Pistons 3258.3 Cardan Transmissions 3288.4 Tripod Transmissions 3308.4.1 General Aspects 3308.4.2 The C2–C Tripod Kinematic Pair 3328.4.3 The C1–C Tripod Kinematic Pair 3358.4.4 The S1–P Tripod Kinematic Pair 3388.4.5 The S2–P Tripod Kinematic Pair 3388.4.6 Simple Mechanisms with Tripod Joints 3408.4.7 Tripod Joint Transmissions 3448.5 Animation of the Mechanisms 3498.5.1 The Need for an Animation 3498.5.2 The Animation Algorithm 3498.5.3 Positional Analysis 3498.5.4 Modelling the Elements of a Mechanism 3578.5.5 Creation of the Animation Frames 3618.5.6 Creation of Animation File for the Mechanism 3648.5.7 Conclusions 3669 Industrial Robots 3699.1 General Aspects 3699.2 Mechanical Systems of Industrial Robots 3709.2.1 Structure 3709.2.2 The Path-generating Mechanism 3719.2.3 The Orientation Mechanism 3739.2.4 The Grip Device 3779.3 Actuation Systems of Industrial Robots 3809.3.1 Electrical Actuation 3809.3.2 Hydraulic Actuation 3819.4 Control Systems of Industrial Robots 3829.5 Walking Machines 3849.5.1 The Mechanical Model of the Walking Mechanism 3859.5.2 Animation of the Walking Machine 38610 Variators of Angular Velocity with Bars 39110.1 Generalities 39110.2 Mono-loop Mechanisms Used in the Construction of the Variators of Angular Velocity with Bars 39110.2.1 Kinematic Schemata 39110.2.2 Kinematic Aspects 39310.2.3 Numerical Example 39410.3 Bi-loop Mechanisms in Variators of Angular Velocity with Bars 39810.3.1 Kinematic Schemata 39810.3.2 Kinematic Analysis 400Further Reading 411Index 417