THIS book is the outcome of several years' experience in teaching kinematics and kinetics of machinery at the University of Illinois. For many years this subject was taught from notes prepared by Professor G. A. Goodenough, to which was added an article on the gyroscope by Professor F. B. Seely of the Department of Theoretical and Applied Mechanics. These notes were several times revised by the authors as experience showed where improvements could be made.

In the fall of 1916 the authors undertook, with the consent of Messrs. Goodenough and Seely, to rewrite these notes in textbook form. The present volume is the outcome of that undertaking. The work was interrupted by the war, which took one of the writers into the military service, and imposed on the other such a heavy burden of teaching work that further progress on the book was impossible. In the fall of 1919 the work was resumed and pushed to completion.

The introductory chapter on constraint, the chapter on plane motion, and the chapter on velocities follow closely Professor Goodenough's notes. The chapter on the gyroscope is inserted almost without change as written by Professor Seely. The chapters on accelerations, on inertia forces, on balancing and on governors have been so completely rewritten that little trace remains of the original. The chapters on toothed wheels, on cams, on wrapping connectors, and on critical speeds have been added by the authors.

It is hoped that this volume will fill a need in the curricula of our engineering schools, in that it gives systematic methods of determining velocities, accelerations, and inertia forces which can be applied to practically all mechanisms. These methods are in the main graphical; the complicated forms of the equations making analytical methods too cumbersome for practical use except in some of the simpler types of machines. If the work is done to a large scale the results should be accurate enough for all practical purposes.

The book is so arranged that it can be readily adapted to short courses as well as to more complete and detailed ones. Thus the chapters on gears, cams, and belts may be omitted where these subjects are taught in the courses in mechanism or design. The chapter on balancing can be profitably studied without the detailed analysis given in the chapters on accelerations and inertia forces. The chapter on critical speeds and parts of the chapters on governors and gyroscopes involve the use of mathematics which is perhaps beyond the range of the average undergraduate. These parts may, however, be of great value to the advanced student who intends to specialize in scientific design. For the benefit of undergraduate students a note on the solution of linear differential equations is appended.

In conclusion the authors wish to extend their thanks to Professor Goodenough for valuable suggestions and criticisms in the preparation of the work.



CONTENTS

CHAPTER I - MACHINE MOTIONS, PAIRS, LINKS, CHAINS, MECHANISMS
Scope of the Subject - Definitions - Constrained Motion. Pairs Properties of Pairs - Higher and Lower Pairs - Pairing Elements - Inversion of Pairs - Multiple Contact - Links - Chains. Unconstrained, Locked, Constrained - Mechanisms - Skeleton Links - Formation of Constrained Chains - Joints - Criterion of Constraint - Application of Criterion - Exceptional Cases - Analysis of Mechanisms - Inversion of Mechanisms - Exercises.

CHAPTER II - MOTION OF RIGID BODIES
Plane Motion of Rigid Bodies - Rotation about Fixed Center - Instantaneous Center - Properties of Instantaneous Center - Exercises - Instantaneous Center of Relative Motion - Law of Three Centers - Application of Law of Three Centers - Higher Pairs - Special Cases - Centrodes - Centrodes of Relative Motion - Exercises - Motion Produced by Rolling of Centrodes - Equivalent Mechanisms - Gears as Centrodes - Pitch Lines of Gears - Gear Teeth - Space Motion. Axodes,

CHAPTER III - VELOCITIES OF MECHANISMS
Introductory - Analytical Methods, Geared Mechanisms - Beveled Gear Trains - Steam Engine Mechanism - Method of Instantaneous Centers - Angular Velocity Ratios - Special Cases - Relative Angular Velocity - Linear Velocity Ratio - Special Cases - Method of Relative Velocities - Definitions - Special Cases - Velocity Images - Revolved Velocities - Velocity Polygons - Examples - Velocity Polygons for Linkages - Special Constructions - Three-line Construction - Exercises - Combined Method of Instantaneous Centers and Relative Velocities - Four-line Construction - Velocity Curves - Exercises.

CHAPTER IV - ACCELERATIONS IN MECHANISMS
Introductory - General Principles, Definitions - Normal and Tangential Acceleration - Translation - Rotation about a Fixed Axis - Combined Motions. Coriolis' Law - Application of Coriolis' Law - Exercises - Relative Accelerations - Notation - Graphical Constructions for Normal Acceleration - Projection of Acceleration Components - Acceleration Image - Acceleration Center - Acceleration Polygon - Selection of Scale - Ordinary Gear Train - Epicycl'c Gear Train - Four-link Chain - Steam Engine Mechanism - Klein's Construction - Sliding Pairs - Exercises - Blake Stone Crusher - Exercises - Three-line Construction - Four-line Construction - Accelerations in Cams,

CHAPTER V - INERTIA FORCES OF MACHINE PARTS
General Statement - Acceleration Produced by Single Force - Special Cases - Kinetically Equivalent System - Calculation of Line of Action of Resultant - Components of Resultant Force - Steam Engine Mechanism - Forces on Piston - Inertia Forces of Piston and Rod - Turning Effort - Atkinson Gas Engine - Inertia Forces of Piston and Rod - Forces on Piston - Forces on Flying Link - Turning Effort - Wanzer Needle-bar Mechanism.

CHAPTER VI - BALANCING OF ENGINES
Introductory - Kinetic Load Due to Unbalanced Mass - Centrifugal Force - Centrifugal Couple - Masses in Same Plane of Revolution - Masses in Different Transverse Planes - Balancing of Rotating Masses - Example - General Relations - Conditions for Balance - Analytic Method - Inertia Effect of Reciprocating Masses. Harmonic Motion - Balancing Conditions - Engines with Finite Rods - Acceleration of Piston - Secondary Balance - Conditions for Primary and Secondary Balance - Partial Balance - Single-crank Engine - Counterbalancing - Two-crank Engine - Three-crank Engine, Complete Balance, Partial Balance - Engines with More than Three Cranks - Four-cylinder Automobile-type Engine - Six-cylinder Automobile-type Engine - Eight-cylinder Automobile-type Engine - Twelve-cylinder Automobile-type Engine - Radial Engine - Opposed Engine - Rotary Engine - Offset Engine.

CHAPTER VII - GOVERNORS
Purpose - Classification, Fly-ball Governor, Shaft Governor - Force Reduction - Mass Reduction - Moment of Centrifugal Force - Elementary Fly-ball Governor - Weighted Fly-ball Governor - Horizontal Spring-controlled Governor - Vertical Spring-controlled Governor - Conditions of Stability - Oscillation of Fly-ball Governor - Conditions of Stability - Effect of Change in Speed of Engine - Damping-out of Oscillations - Elementary Centrifugal Shaft Governor - Elementary Inertia Governor - Inertia Forces in Shaft Governor - Moment about Governor Pivot - Oscillations in Governor - Mathematical Analysis of Governor Action - Damping-out Oscillations.

CHAPTER VIII - THE MECHANICS OF THE GYROSCOPE
Introductory - Gyroscopic Couple - Surging - Line of Action of Resultant - Wheel on Circular Path - Angular Momentum - Illustrations - Analysis of Gyroscope by Method of Angular Momentum - Precession - Loss of Energy in Setting up Precession.

CHAPTER IX - CRITICAL SPEEDS AND VIBRATIONS
Introductory - Revolving Shaft Loaded at Middle, Critical Speed - Natural Period of Vibration - Uniformly Loaded Shaft - Critical Speed - Natural Period of Vibration - Shaft Rotating in Fixed Bearings, Uniformly Loaded - Critical Speed - Natural Period of Vibration - Inclination of Rotating Disc - Effect on Critical Speed - Effect on Period of Vibration - Other Systems of Loading - References.

CHAPTER X - TOOTHED WHEELS
Introductory - Rolling Contact - Parallel Axes, Rolling Cylinders - Intersecting Axes, Rolling Cones - Axes neither Parallel nor Intersecting, Hyperboloid of One Sheet - Friction Gearing - Spur Friction Gearing - Grooved Wheels - Beveled Friction Gearing - Crown Friction Gearing - Definitions - Relation between Circular and Diametral Pitch - Rectification of Circular Arcs - Generation of Cycloidal Curves - Teeth of Wheels - Relation between Pitch and Arcs of Approach and Recess - Size of Describing Circle - Interchangeable Wheels - Rack and Pinion - Annular Gears - Approximate Method for Laying Out Cycloidal Teeth - Grant's Odontograph for Cycloidal Teeth - Involute System - Involutes in Sliding Contact - Path of Point of Contact - Teeth of Wheels - Relation between Height of Tooth and Arcs of Approach and Recess - Annular or Internal Gears - Rack - Interference, Limit of Addendum - Approximate Layout for Involute Tooth - Grant's Odontograph for Involute Tooth - Stub Teeth - Fellows' System - Nuttall System - Comparison between Standard and Stub Tooth Gears - Proportions of Cast Teeth Proportions of Cut Teeth - Beveled Gears - Form of Teeth - Layout of Beveled Gears - Transmission between Non-intersecting Shafts - Screw Gears, Helical Gears, Worm Gears - Helical Gears for Parallel Shafts, Herringbone Gears - Helical Gears for Non-intersecting Shafts - Velocity Ratio - Worm Gearing, Cylindrical Worm, Hindley Worm - Circular Pitch, Axial Pitch of Worm - Velocity Ratio Comparison of Hindley and Cylindrical Worm Gears - Skew Beveled Gears - Comparison of Systems.

CHAPTER XI - CAMS
General Principles - Layout of Cams - Definitions - Base Curves - Layout of Base Curves - Pressure Angle, Cam Factors - Size of Cams - Rotary Cams, Radial Follower - Working Surface of Cams - Two-step Cams - Size of Follower Roller - Rotary Cam, Offset Follower - Rotary Cam, Swinging Follower - Positive Motion Cams - Rotary Cams, Tangential Follower - Rotary Cams, Sliding Tangential Follower - Plane Sliding Cams, Sliding Follower - Plane Sliding Cams, Pivoted Follower - Cylindrical Cam - Cylindrical Cam, Sliding Follower.

CHAPTER XII - WRAPPING CONNECTORS
Introductory, Belts, Ropes, Chains - Transmission of Motion, Equivalent Link - Friction between Belt and Pulley - Power Transmitted by Belt - Length of Belts, Cone Pulleys - Open and Crossed Belts - Arrangements of Twisted Belts, Reversibility - Idlers - Crowned Pulleys - Materials of Belts - Rope Drives, Multiple System, Continuous System - Friction between Rope and Pulley - Chain Drives - Hoisting Tackle - Differential Pulley.

NOTE A - IRREGULAR GEARS
NOTE B - PROPOSITIONS ON VELOCITY POLYGONS
NOTE C – LOCUS OF THE CENTER OF ACCELERATION
NOTE D - SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS
Definitions - Types of Solutions - Linear Differential Equations, Second Member Zero - Character of the Roots - Second Member not Zero - Complimentary Function - Particular Integral - Summary.
NOTE E - INVESTIGATION OF FORCES IN GASOLINE ENGINE 354
Data - Timing Diagram - Indicator Card - Total Pressure Card - Wrist-pin Pressure - Side Thrust - Turning Effort - Combined Turning Effort - Shaking Forces - Counterbalancing.