# Introduction to Mechanisms

Yi Zhang
with
Susan Finger
Stephannie Behrens

1 Physical Principles
1.1 Force and Torque
1.1.1 Force
1.1.2 Torque
1.2 Motion
1.2.1 Motion Along a Straight Path
1.2.2 Linear Motion in Space
1.2.3 Motion of a Rigid Body in a Plane
1.3 Newton's Law of Motion
1.3.1 Newton's First Law
1.3.2 Newton's Second Law
1.4 Momentum and Conservation of Momentum
1.4.1 Impulse
1.4.2 Momentum
1.4.3 Conservation of Momentum
1.5 Work, Power and Energy
1.5.1 Work
1.5.2 Power
1.5.3 Energy
2 Mechanisms and Simple Machines
2.1 The Inclined Plane
2.1.1 Screw Jack
2.2 Gears
2.2.1 Gear Trains
2.2.2 Gear Ratios
2.3 Belts and Pulleys
2.4 Lever
2.6 Wedge
2.7 Efficiency of Machines
3 More on Machines and Mechanisms
3.1 Planar and Spatial Mechanisms
3.2 Kinematics and Dynamics of Mechanisms
3.3 Links, Frames and Kinematic Chains
3.4 Skeleton Outline
3.5 Pairs, Higher Pairs, Lower Pairs and Linkages
3.6 Kinematic Analysis and Synthesis
4 Basic Kinematics of Constrained Rigid Bodies
4.1 Degrees of Freedom of a Rigid Body
4.1.1Degrees of Freedom of a Rigid Body in a Plane
4.1.2 Degrees of Freedom of a Rigid Body in Space
4.2 Kinematic Constraints
4.2.1 Lower Pairs in Planar Mechanisms
4.2.2 Lower Pairs in Spatial Mechanisms
4.3 Constrained Rigid Bodies
4.4 Degrees of Freedom of Planar Mechanisms
4.4.1 Gruebler's Equation
4.2.2 4.4.2 Kutzbach Criterion
4.5 4.5 Finite Transformation
4.5.1 Finite Planar Rotational Transformation
4.5.2 Finite Planar Translational Transformation
4.5.3 Concatenation of Finite Planar Displacements
4.5.4 Planar Rigid-Body Transformation
4.5.5 Spatial Rotational Transformation
4.5.6 Spatial Translational Transformation
4.5.7 Spatial Translation and Rotation Matrix for Axis Through the Origin
4.6 Transformation Matrix Between Rigid Bodies
4.6.1 Transformation Matrix Between Two Arbitray Rigid Bodies
4.6.2 Kinematic Constraints Between Two Rigid Bodies
4.6.3 Denavit-Hartenberg Notation
4.6.4 Application of Transformation Matrices to Linkages
5.1 Introduction
5.1.1 What are Linkage Mechanisms?
5.1.2 Functions of Linkages
5.2 Four Link Mechanisms
5.2.1 Examples
5.2.2 Definitions
5.2.3 Classification
5.2.4 Transmission Angle
5.2.6 Slider-Crank Mechanism
5.2.7 Inversion of the Slider-Crank Mechanism
6 Cams
7 Gears
7.1 Gear Classification
7.2 Gear-Tooth Action
7.2.1 Fundamental Law of Gear-Tooth Action
7.2.2 Constant Velocity Ratio
7.2.3 Conjugate Profiles
7.3 Involute Curve
7.3.1 Generation of the Involute Curve
7.3.2 Properties of Involute Curves
7.4 Terminology for Spur Gears
7.5 Condition for Correct Meshing
7.6 Ordinary Gear Trains
7.6.1 Velocity Ratio
7.7 Planetary gear trains
7.7.1 Velocity Ratio
7.7.2 Example
8 Other Mechanisms
8.1 Ratchet Mechanisms
8.2 Overrunning Clutch
8.3 Intermittent Gearing
8.4 The Geneva Wheel
8.5 The Universal Joint
8.5.1 Analysis of a Universal Joint
8.5.2 Double Universal Joint
Index
References

## Figures

Chapter 1
1-1 A lever with balanced forces
1-2 Position vector and displacement vector
1-3 Motion of a rigid body in a plane
1-4 Collision of billiard balls Chapter 2
2-1 Cross section of a power cylinder in a diesel engine
2-2 Skeleton outline
2-3 Inclined plane
2-4 The screw jack
2-5 Gears
2-6 Gear train
2-7 Compound gears
2-8 Belts and pulleys
Chapter 3
3-1 Skeleton outline
3-2 Turning pair
3-3 Prismatic pair
3-4 Higher pairs
Chapter 4
4-1 Degrees of freedom of a rigid body in a plane
4-2 Degrees of freedom of a rigid body in space
4-3 A planar revolute pair (R-pair)
4-4 A planar prismatic pair (P-pair)
4-5 A spherical pair (S-pair)
4-6 A planar pair (E-pair)
4-7 A cylindrical pair (C-pair)
4-8 A revolute pair (R-pair)
4-9 A prismatic pair (P-pair)
4-10 A screw pair (H-pair)
4-11 Rigid bodies constrained by different kinds of planar pairs
4-12 Kinematic Pairs in Planar Mechanisms
4-13 Transom mechanism
4-14 Dump truck
4-15 Degrees of freedom calculation
4-16 Point on a planar rigid body rotated through an angle
4-17 Point on a planar rigid body translated through a distance
4-18 Concatenation of finite planar displacements in space
4-19 Relative position of points on constrained bodies
4-20 Denavit-Hartenberg Notation
Chapter 5
5-1 Windshield Wiper
5-2 Do-it-yourself four link mechanism
5-3 Four bar linkage in SimDesign
5-4 Front loader mechanism
5-5 Crank and Slider Mechanism
5-6 Crank and Piston
5-7 Block Feeder
5-8 Four bar linkage
5-9 Transmission angle
5-11 Work fixture
5-12 Overcoming the dead point by asymmetrical deployment (V engine)
5-13 Crank-slide mechanism
5-14 The inversion of crank-slide mechanisms
5-15 A pump device
6-1 A simple experiment
6-2 Classification of cam mechanisms
6-3 Translating cam - translating follower
6-4 Grooved cam
6-5 Cylindrical cam and end cam
6-6 Constant diameter cam
6-7 Dual cam
6-8 SimDesign translating cam
6-9 SimDesign oscillating cam
6-10 Cam nomenclature
6-11 Motion events
6-12 A Skeleton Diagram of disk cam with knife-edge translation
6-13 Profile design of translating cam follower
6-14 Disk cam with knife-edge oscillating follower
6-15 Cam profile design for a rotating follower
6-16 The trace point of the follower on a disk cam
6-17 The tangent point, P, of a roller to the disk cam

Chapter 7
7-1 Spur, helical, rack and pinion, spur and worm gears
7-2 Two gearing tooth profiles
7-3 Involute curve
7-4 Spur Gear
7-5 Two meshing gears
7-6 Ordinary gear trains
7-7 Planetary gear trains

Chapter 8
8-1 Ratchet
8-2 Overrunning clutch
8-3 Intermittent gearing
8-4 Geneva wheel
8-5 Universal joint
8-6 General form for a universal joint
8-7 Analysis of a universal joint