PREFACE the analysis topics.Though this is not the "traditional"approach to the teaching of this material,the author believes that it is a superior method to that of initial concen- tration on detailed analysis of mechanisms for which the student has no concept of or- igin or purpose.Chapters I and 2 are introductory.Those instructors wishing to teach analysis before synthesis can leave Chapters 3 and 5 on linkage synthesis for later consumption.Chapters 4,6,and 7 on position,velocity,and acceleration anal- ysis are sequential and build upon each other.In fact,some of the problem sets are com- mon among these three chapters so that students can use their position solutions to find velocities and then later use both to find the accelerations in the same linkages. Chapter 8 on cams is more extensive and complete than that of other kinematics texts and takes a design approach.Chapter 9 on gear trains is introductory.The dynamic force treatment in Part II uses matrix methods for the solution of the system simultaneous equations.Graphical force analysis is not emphasized.Chapter 10 presents an intro- duction to dynamic systems modelling.Chapter 11 deals with force analysis oflinkag- es.Balancing of rotating machinery and linkages is covered in Chapter 12.Chapters 13 and 14 use the internal combustion engine as an example to pull together many dynamic concepts in a design context.Chapter 15 presents an introduction to dynamic systems modelling and uses the cam-follower system as the example.Chapters 3,8,11,13, and 14 provide open ended project problems as well as structured problem sets.The assignment and execution of unstructured project problems can greatly enhance the student's understanding of the concepts as described by the proverb in the epigraph to this preface. ACKNOWLEDGMENTS The sources of photographs and other nonoriginal art used in the text are acknowledged in the captions and opposite the title page,but the author would also like to express his thanks for the cooperation of all those individ- uals and companies who generously made these items available.The author would also like to thank those who have reviewed various sections of the first edition of the text and who made many useful suggestions for improvement.Mr.John Titus of the University of Minnesota reviewed Chapter 5 on analytical synthesis and Mr.Dennis Klipp of Klipp Engineering,Waterville,Maine,reviewed Chapter 8 on cam design. Professor William J.Crochetiere and Mr.Homer Eckhardt of Tufts University,Med- ford,Mass.,reviewed Chapter 15.Mr.Eckhardt and Professor Crochetiere of Tufts, and Professor Charles Warren of the University of Alabama taught from and re- viewed Part I.Professor Holly K.Ault of Worcester Polytechnic Institute thorough- ly reviewed the entire text while teaching from the pre-publication,class-test ver- sions of the complete book.Professor Michael Keefe of the University of Delaware provided many helpful comments.Sincere thanks also go to the large number of un- dergraduate students and graduate teaching assistants who caught many typos and errors in the text and in the programs while using the pre-publication versions.Since the book's first printing,Profs.D.Cronin,K.Gupta,P.Jensen,and Mr.R.Jantz have written to point out errors or make suggestions which I have incorporated and for which I thank them. The author takes full responsibility for any errors that may remain and invites from all readers their criticisms,suggestions for improvement,and identification of errors in the text or programs,so that both can be improved in future versions. 'R.P6ertL.:Iliprton :Mattapoisett/:Mass. 5hugs/1991
the analysis topics. Though this is not the "traditional" approach to the teaching of this material, the author believes that it is a superior method to that of initial concentration on detailed analysis of mechanisms for which the student has no concept of origin or purpose. Chapters 1 and 2 are introductory. Those instructors wishing to teach analysis before synthesis can leave Chapters 3 and 5 on linkage synthesis for later consumption. Chapters 4, 6, and 7 on position, velocity, and acceleration analysis are sequential and build upon each other. In fact, some of the problem sets are common among these three chapters so that students can use their position solutions to find velocities and then later use both to find the accelerations in the same linkages. Chapter 8 on cams is more extensive and complete than that of other kinematics texts and takes a design approach. Chapter 9 on gear trains is introductory. The dynamic force treatment in Part II uses matrix methods for the solution of the system simultaneous equations. Graphical force analysis is not emphasized. Chapter 10 presents an introduction to dynamic systems modelling. Chapter 11 deals with force analysis oflinkages. Balancing of rotating machinery and linkages is covered in Chapter 12. Chapters 13 and 14 use the internal combustion engine as an example to pull together many dynamic concepts in a design context. Chapter 15 presents an introduction to dynamic systems modelling and uses the cam-follower system as the example. Chapters 3, 8, 11, 13, and 14 provide open ended project problems as well as structured problem sets. The assignment and execution of unstructured project problems can greatly enhance the student's understanding of the concepts as described by the proverb in the epigraph to this preface. ACKNOWLEDGMENTS The sources of photographs and other nonoriginal art used in the text are acknowledged in the captions and opposite the title page, but the author would also like to express his thanks for the cooperation of all those individuals and companies who generously made these items available. The author would also like to thank those who have reviewed various sections of the first edition of the text and who made many useful suggestions for improvement. Mr. John Titus of the University of Minnesota reviewed Chapter 5 on analytical synthesis and Mr. Dennis Klipp of Klipp Engineering, Waterville, Maine, reviewed Chapter 8 on cam design. Professor William J. Crochetiere and Mr. Homer Eckhardt of Tufts University, Medford, Mass., reviewed Chapter 15. Mr. Eckhardt and Professor Crochetiere of Tufts, and Professor Charles Warren of the University of Alabama taught from and reviewed Part I. Professor Holly K. Ault of Worcester Polytechnic Institute thoroughly reviewed the entire text while teaching from the pre-publication, class-test versions of the complete book. Professor Michael Keefe of the University of Delaware provided many helpful comments. Sincere thanks also go to the large number of undergraduate students and graduate teaching assistants who caught many typos and errors in the text and in the programs while using the pre-publication versions. Since the book's first printing, Profs. D. Cronin, K. Gupta, P.Jensen, and Mr. R. Jantz have written to point out errors or make suggestions which I have incorporated and for which I thank them. The author takes full responsibility for any errors that may remain and invites from all readers their criticisms, suggestions for improvement, and identification of errors in the text or programs, so that both can be improved in future versions. 'R.P6ertL. :lI{prton :Mattapoisett/ :Mass. 5lugust/ 1991
Take to Kinematics.It will repay you.It is more fecund than geometry; it adds afourth dimension to space. PART CHEBYSCHEV TO SYLVESTER,1873 KINEMATICS OF MECHANISMS
Take to Kinematics. It will repay you. It is more fecund than geometry; it adds a fourth dimension to space. CHEBYSCHEV TO SYLVESTER, 1873
INTRODUCTION Inspiration most often strikes those who are hard at work ANONYMOUS 1.0 PURPOSE In this text we will explore the topics of kinematics and dynamics of machinery in re- spect to the synthesis of mechanisms in order to accomplish desired motions or tasks, and also the analysis of mechanisms in order to determine their rigid-body dynamic behavior.These topics are fundamental to the broader subject of machine design.On the premise that we cannot analyze anything until it has been synthesized into existence, we will first explore the topic of synthesis of mechanisms.Then we will investigate techniques of analysis of mechanisms.All this will be directed toward developing your ability to design viable mechanism solutions to real,unstructured engineering problems by using a design process.We will begin with careful definitions of the terms used in these topics 1.1 KINEMATICS AND KINETICS KINEMATICS The study of motion without regard to forces. KINETIes The study offorces on systems in motion. These two concepts are really not physically separable.We arbitrarily separate them for instructional reasons in engineering education.It is also valid in engineering design practice to first consider the desired kinematic motions and their consequences,and then subsequently investigate the kinetic forces associated with those motions.The student should realize that the division between kinematics and kinetics is quite arbitrary and is done largely for convenience.One cannot design most dynamic mechanical systems without taking both topics into thorough consideration.It is quite logical to consider them in the order listed since,from Newton's second law,F =ma,one typically needs to 3
1.0 PURPOSE In this text we will explore the topics of kinematics and dynamics of machinery in respect to the synthesis of mechanisms in order to accomplish desired motions or tasks, and also the analysis of mechanisms in order to determine their rigid-body dynamic behavior. These topics are fundamental to the broader subject of machine design. On the premise that we cannot analyze anything until it has been synthesized into existence, we will first explore the topic of synthesis of mechanisms. Then we will investigate techniques of analysis of mechanisms. All this will be directed toward developing your ability to design viable mechanism solutions to real, unstructured engineering problems by using a design process. We will begin with careful definitions of the terms used in these topics. 1.1 KINEMATICS AND KINETICS KINEMATICS The study of motion without regard to forces. KINETIcs The study of forces on systems in motion. These two concepts are really not physically separable. We arbitrarily separate them for instructional reasons in engineering education. It is also valid in engineering design practice to first consider the desired kinematic motions and their consequences, and then subsequently investigate the kinetic forces associated with those motions. The student should realize that the division between kinematics and kinetics is quite arbitrary and is done largely for convenience. One cannot design most dynamic mechanical systems without taking both topics into thorough consideration. It is quite logical to consider them in the order listed since, from Newton's second law, F = ma, one typically needs to 3
42 DESIGN OF MACHINERY CHAPTER 1 know the accelerations (a)in order to compute the dynamic forces (F)due to the mo- tion of the system's mass (m).There are also many situations in which the applied forc- es are known and the resultant accelerations are to be found. One principal aim of kinematics is to create (design)the desired motions of the sub- ject mechanical parts and then mathematically compute the positions,velocities,and ac- celerations which those motions will create on the parts.Since,for most earthbound mechanical systems,the mass remains essentially constant with time,defining the accel- erations as a function of time then also defines the dynamic forces as a function of time. Stresses,in turn,will be a function of both applied and inertial (ma)forces.Since engi- neering design is charged with creating systems which will not fail during their expected service life,the goal is to keep stresses within acceptable limits for the materials chosen and the environmental conditions encountered.This obviously requires that all system forces be defined and kept within desired limits.In machinery which moves (the only interesting kind),the largest forces encountered are often those due to the dynamics of the machine itself.These dynamic forces are proportional to acceleration,which brings us back to kinematics,the foundation of mechanical design.Very basic and early deci- sions in the design process involving kinematic principles can be crucial to the success of any mechanical design.A design which has poor kinematics will prove troublesome and perform badly 1.2 MECHANISMS AND MACHINES A mechanism is a device which transforms motion to some desirable pattern and typi- cally develops very low forces and transmits little power.A machine typically contains mechanisms which are designed to provide significant forces and transmit significant powerJI]Some examples of common mechanisms are a pencil sharpener,a camera shut- ter,an analog clock,a folding chair,an adjustable desk lamp,and an umbrella.Some examples of machines which possess motions similar to the mechanisms listed above are a food blender,a bank vault door,an automobile transmission,a bulldozer,a robot,and an amusement park ride.There is no clear-cut dividing line between mechanisms and machines.They differ in degree rather than in kind.If the forces or energy levels within A mechanism the device are significant,it is considered a machine;if not,it is considered a mechanism. A useful working definition of a mechanism is A system ofelements arranged to trans- mit motion in a predetermined fashion.This can be converted to a definition of a ma- chine by adding the words and energy after motion. Mechanisms,if lightly loaded and run at slow speeds,can sometimes be treated strictly as kinematic devices;that is,they can be analyzed kinematically without regard to forces.Machines (and mechanisms running at higher speeds),on the other hand,must first be treated as mechanisms,a kinematic analysis of their velocities and accelerations must be done,and then they must be subsequently analyzed as dynamic systems in which their static and dynamic forces due to those accelerations are analyzed using the princi- ples of kinetics.Part I of this text deals with Kinematics of Mechanisms,and Part II with Dynamics of Machinery.The techniques of mechanism synthesis presented in Part I are applicable to the design of both mechanisms and machines,since in each case some A machine collection of moveable members must be created to provide and control the desired motions and geometry
know the accelerations (a) in order to compute the dynamic forces (F) due to the motion of the system's mass (m). There are also many situations in which the applied forces are known and the resultant accelerations are to be found. One principal aim of kinematics is to create (design) the desired motions of the subject mechanical parts and then mathematically compute the positions, velocities, and accelerations which those motions will create on the parts. Since, for most earthbound mechanical systems, the mass remains essentially constant with time, defining the accelerations as a function of time then also defines the dynamic forces as a function of time. Stresses, in turn, will be a function of both applied and inertial (ma) forces. Since engineering design is charged with creating systems which will not fail during their expected service life, the goal is to keep stresses within acceptable limits for the materials chosen and the environmental conditions encountered. This obviously requires that all system forces be defined and kept within desired limits. In machinery which moves (the only interesting kind), the largest forces encountered are often those due to the dynamics of the machine itself. These dynamic forces are proportional to acceleration, which brings us back to kinematics, the foundation of mechanical design. Very basic and early decisions in the design process involving kinematic principles can be crucial to the success of any mechanical design. A design which has poor kinematics will prove troublesome and perform badly. 1.2 MECHANISMS AND MACHINES A mechanism is a device which transforms motion to some desirable pattern and typically develops very low forces and transmits little power. A machine typically contains mechanisms which are designed to provide significant forces and transmit significant powerJI] Some examples of common mechanisms are a pencil sharpener, a camera shutter, an analog clock, a folding chair, an adjustable desk lamp, and an umbrella. Some examples of machines which possess motions similar to the mechanisms listed above are a food blender, a bank vault door, an automobile transmission, a bulldozer, a robot, and an amusement park ride. There is no clear-cut dividing line between mechanisms and machines. They differ in degree rather than in kind. If the forces or energy levels within the device are significant, it is considered a machine; if not, it is considered a mechanism. A useful working definition of a mechanism is A system of elements arranged to transmit motion in a predetermined fashion. This can be converted to a definition of a machine by adding the words and energy after motion. Mechanisms, if lightly loaded and run at slow speeds, can sometimes be treated strictly as kinematic devices; that is, they can be analyzed kinematically without regard to forces. Machines (and mechanisms running at higher speeds), on the other hand, must first be treated as mechanisms, a kinematic analysis of their velocities and accelerations must be done, and then they must be subsequently analyzed as dynamic systems in which their static and dynamic forces due to those accelerations are analyzed using the principles of kinetics. Part I of this text deals with Kinematics of Mechanisms, and Part II with Dynamics of Machinery. The techniques of mechanism synthesis presented in Part I are applicable to the design of both mechanisms and machines, since in each case some collection of moveable members must be created to provide and control the desired motions and geometry
INTRODUCTION 1.3 A BRIEFHISTORY OF KINEMATICS Machines and mechanisms have been devised by people since the dawn of history.The ancient Egyptians devised primitive machines to accomplish the building of the pyra- mids and other monuments.Though the wheel and pulley (on an axle)were not known to the Old Kingdom Egyptians,they made use of the lever,the inclined plane (or wedge), and probably the log roller.The origin of the wheel and axle is not definitively known. Its first appearance seems to have been in Mesopotamia about 3000 to 4000 B.C A great deal of design effort was spent from early times on the problem of timekeep- ing as more sophisticated clockworks were devised.Much early machine design was directed toward military applications (catapults,wall scaling apparatus,etc.).The term civil engineering was later coined to differentiate civilian from military applications of technology.Mechanical engineering had its beginnings in machine design as the in- ventions of the industrial revolution required more complicated and sophisticated solu- tions to motion control problems.James Watt (1736-1819)probably deserves the title of first kinematician for his synthesis of a straight-line linkage (see Figure 3-29a on p. 121)to guide the very long stroke pistons in the then new steam engines.Since the plan- er was yet to be invented (in 1817),no means then existed to machine a long,straight guide to serve as a crosshead in the steam engine.Watt was certainly the first on record to recognize the value of the motions of the coupler link in the fourbar linkage.Oliver Evans (1755-1819)an early American inventor,also designed a straight-line linkage for a steam engine.Euler (1707-1783)was a contemporary of Watt,though they apparent- ly never met.Euler presented an analytical treatment of mechanisms in his Mechanica sive Motus Scienta Analytice Exposita (1736-1742),which included the concept that pla- nar motion is composed of two independent components,namely,translation of a point and rotation of the body about that point.Euler also suggested the separation of the prob- lem of dynamic analysis into the "geometrical"and the "mechanical"in order to simpli- fy the determination of the system's dynamics.Two of his contemporaries,d'Alembert and Kant,also proposed similar ideas.This is the origin of our division of the topic into kinematics and kinetics as described above. *Ampere is quoted as writing "(The science of In the early 1800s,L'Ecole Polytechnic in Paris,France,was the repository of engi- mechanisms)must neering expertise.Lagrange and Fourier were among its faculty.One of its founders therefore not define a was Gaspard Monge (1746-1818),inventor of descriptive geometry (which incidental- machine,as has usually ly was kept as a military secret by the French government for 30 years because of its been done,as an instru- ment by the help of which value in planning fortifications).Monge created a course in elements of machines and the direction and intensity set about the task of classifying all mechanisms and machines known to mankind!His of a given force can be colleague,Hachette,completed the work in 1806 and published it as what was probably altered,but.as an the first mechanism text in 1811.Andre Marie Ampere (1775-1836),also a professor instrument by the help of at L'Ecole Polytechnic,set about the formidable task of classifying "all human knowl- which the direction and edge."In his Essai sur la Philosophie des Sciences,he was the first to use the term "ein- velocity of a given motion ematique,"from the Greek word for motion,*to describe the study of motion without can be altered.To this science ..Ihave given the regard to forces,and suggested that "this science ought to include all that can be said with name Kinematics from respect to motion in its different kinds,independently of the forces by which it is pro- KtVll<x-motion." in duced."His term was later anglicized to kinematics and germanized to kinematik. Maunder,L.(1979). "Theory and Practice." Robert Willis (1800-1875)wrote the text Principles of Mechanism in 1841 while a Proc.5th World Congo on professor of natural philosophy at the University of Cambridge,England.He attempted Theory of Mechamisms and to systematize the task of mechanism synthesis.He counted five ways of obtaining rel- Machines,Montreal,p.I
1.3 A BRIEFHISTORY OF KINEMATICS Machines and mechanisms have been devised by people since the dawn of history. The ancient Egyptians devised primitive machines to accomplish the building of the pyramids and other monuments. Though the wheel and pulley (on an axle) were not known to the Old Kingdom Egyptians, they made use of the lever, the inclined plane (or wedge), and probably the log roller. The origin of the wheel and axle is not definitively known. Its first appearance seems to have been in Mesopotamia about 3000 to 4000 B.C. A great deal of design effort was spent from early times on the problem of timekeeping as more sophisticated clockworks were devised. Much early machine design was directed toward military applications (catapults, wall scaling apparatus, etc.). The term civil engineering was later coined to differentiate civilian from military applications of technology. Mechanical engineering had its beginnings in machine design as the inventions of the industrial revolution required more complicated and sophisticated solutions to motion control problems. James Watt (1736-1819) probably deserves the title of first kinematician for his synthesis of a straight-line linkage (see Figure 3-29a on p. 121) to guide the very long stroke pistons in the then new steam engines. Since the planer was yet to be invented (in 1817), no means then existed to machine a long, straight guide to serve as a crosshead in the steam engine. Watt was certainly the first on record to recognize the value of the motions of the coupler link in the fourbar linkage. Oliver Evans (1755-1819) an early American inventor, also designed a straight-line linkage for a steam engine. Euler (1707-1783) was a contemporary of Watt, though they apparently never met. Euler presented an analytical treatment of mechanisms in his Mechanica sive Motus Scienta Analytice Exposita (1736-1742), which included the concept that planar motion is composed of two independent components, namely, translation of a point and rotation of the body about that point. Euler also suggested the separation of the problem of dynamic analysis into the "geometrical" and the "mechanical" in order to simplify the determination of the system's dynamics. Two of his contemporaries, d' Alembert and Kant, also proposed similar ideas. This is the origin of our division of the topic into kinematics and kinetics as described above. In the early 1800s, L'Ecole Polytechnic in Paris, France, was the repository of engineering expertise. Lagrange and Fourier were among its faculty. One of its founders was Gaspard Monge (1746-1818), inventor of descriptive geometry (which incidentally was kept as a military secret by the French government for 30 years because of its value in planning fortifications). Monge created a course in elements of machines and set about the task of classifying all mechanisms and machines known to mankind! His colleague, Hachette, completed the work in 1806 and published it as what was probably the first mechanism text in 1811. Andre Marie Ampere (1775-1836), also a professor at L'Ecole Polytechnic, set about the formidable task of classifying "all human knowledge." In his Essai sur la Philosophie des Sciences, he was the first to use the term "einematique," from the Greek word for motion,* to describe the study of motion without regard to forces, and suggested that "this science ought to include all that can be said with respect to motion in its different kinds, independently of the forces by which it is produced." His term was later anglicized to kinematics and germanized to kinematik. Robert Willis (1800-1875) wrote the text Principles of Mechanism in 1841 while a professor of natural philosophy at the University of Cambridge, England. He attempted to systematize the task of mechanism synthesis. He counted five ways of obtaining rel- * Ampere is quoted as writing "(The science of mechanisms) must therefore not define a machine, as has usually been done, as an instrument by the help of which the direction and intensity of a given force can be altered, but as an instrument by the help of which the direction and velocity of a given motion can be altered. To this science ... Ihave given the name Kinematics from KtVIl<x-motion." in Maunder, L. (1979). "Theory and Practice." Proc. 5th World Congo on Theory of Mechanisms and Machines, Montreal, p. I