ANALYSIS of the FOUR-BAR LINKAGE Its Application to the Synthesis of Mechanisms a小a之
, " " . "' .. '.. "" , '0" ANALYSIS of the FOUR- BAR LINKAGE I t8 Application to the Synthesis, of Mechanisms , . - , . 1"; , ' " -, ' '" " L.. :i." .... .. ..ce"... :
ANALYSIS of the FOUR-BAR LINKAGE Its Application to the Synthesis of Mechanisms John A.Hrones George L.Nelson PROFESSOR OF ASSISTANT PROFESSOR OF MECHANICAL ENGINEERING MECHANICAL ENGINEERING THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY MANUS Published jointly by THE TECHNOLOGY PRESS of THE MASSACHUSETTS INSTITUTE of TECHNOLOGY and JOHN WILEY SONS,INC.,NEW YORK CHAPMAN HALL,LIMITED,LONDON
, .. ANALYSIS of the FOUR- BAR LINI(AGE Its Application to the Synthesis .. of Mechanisms John A. Hrones George L. Nelson PROFESSOR OF MECHANICAL ENGINEERING ASSISTANT PROFESSOR OF MECHANICAL ENGINEERING TilE MASSACHUSETTS INSTITUTE OF TECHNOLOGY Published Jointly by THE TEC~NOLOGY PRESS of THE MASSACHUSETTS INSTITUTE of . TECHNOLOGY and JOHN WILEY & SONS~ INC., NEW YORK CHAPMAN & HALL, LIMITED, LONDON .. ---..:. ~. ;.. '-, -
[vii] Classes of Linkage Operation The basie four-bar linknge is shown in figure 1.The fixed member C is the line of centers.Pinned to the extremities of C are the eranks I The Four-Bar Linkage and B.The moving ends of the cranks are joined by the fourth bar,the connecting rod A.The driving crank 1 and the follower crank B move Determinate linkage motion results when the number of independent in rotation about their fixod axes located on the.line.of centers C.The input angular motions is two less than the number of links.All links conneeting rod A in general moves in combined translation and rotation. are assumed to be rigid members and are pin-eonnected to one another. The nature of the motion of the connecting rod A and the follower Freedom of relative angular motion exists between any two members at crank B relative to the line of eenters C for a given input motion to the the pin joint.The minimum number of links which will permit relative driving crank 1 is determined by the three basie link length ratios A/1; motion between links is four.In the majority of applications one of the B/1:C/1,which will hereafter be designated by A,B,and C.The shortest links (the line of centers)is stationary while a second link (the driving link will always be designated as unity.The remaining links will be erank)is driven from an outside motion source.The motion of the remnin- labeled A,B,and C in order around the linkage.It is convenient to ing two links is a funetion of the geometry of the linkage and the motion separate the geometrically poesible motions into three main eategories of the driving crank and the line of eenters. of operation. A C Crank and Rocker Four Bar Linkage F1g.2 Fig.1 A four-bar linkage is schematieally shown in figure 1.It oonsists of the four links having pin-to-pin lengths of 1,A,B,and C.The geometry of the linkage is determined by the three ratios A/1,B/1,and C/1. Up to this point the four-bar linkage has been represented as consisting of four lines.Actually,each member is a solid body which from purely theoretical aspects can be considered as being of indefinite extent (figure 6). Manufacturing and design considerations place very real limitationa on the size of the membera.Within these limits a wide variety of motions are available.In this volume the points indicated in figure 6 on the con- necting rod included within a reetangular boundary extending a distance equal to the driving crank length in directions parallel to and at right angles to the centerline of the conneeting rod have been inveatigated and their trajeetories and velocities presented.Thus,for each series of linkago Crank and Rocker ratios the behavior of the points indicated in figure 6 has been studied and the results publisbed for ls(a)type linkage operation. F1g.3
( viiiJ The F our- Bar Linkage Classes of Linkage Operation The basic four-bar linkage is shown in figure 1. The fixed member is the line of centers. . Pinned to the extremities of are the cranks 1 and B. The moving ends of the cranks are joined by the fourth bar, the coIWecting rod A. The driving crank 1 "and the follower crank move in rotation about their fixed axes located on the .line of centers G. The connecting rod in general moves jn combined translation and rotation. The nature of the motion of the conhecting rod and the follower crank relative to the line of centers fora given input motion to the driving crank 1 is determined by the three basic link length ratios All; BIl; Gl1 which will hereaJterbe designated by , B and G. The shortest link will always be designated as unity, The remaining links will be labeled , B ahd in order around the linkage. It is convenient to separate the geometrically possible motions into three main categories of operation. Determi~ate linkage motion results when the number of independent input angular motions is two less than the number of links, All links are assumed to be rigid members and are pin-connected to one another. Freedom of relative angular motion exists between any two members at the pin joint. The minimum number of links which will permit relative motion between links is four. In the majority of applications one of the links (the line of centers) is stationary while a second link (the driving crank) is driven from an outside motion source. The motion of the remain- ' . ing two links isa function of the geometry of the linkage and the motion of the driving crank and the line of centers. Crank and Rocke Four Bar Linkage Fig. 2 Fig. 1 A four-bar linkage isschema.tically shown in figure 1. It consists of the four links having pin-to-pin lengths of 1 , B and G. The geometry of the linkage is determined by the three ratios All, Bl1 and GI1. Up ,to this point the four-bar linkage has been represented as consisting of four lines, Actually, each member is a solid body which from purely theoretical aspects can be considered as being of indefinite extent (figure 6). Manufacturing and design considerations place very real limitations on the size of the members. Within these limits a wide variety of motions are available. In this volume the points indicated in figure 6 on the connecting rod included.. within a rectangular boundary extending a distance equal to the driving crank length in directions parallel to and at right angles to the centerline of the connecting rod have been investigated and, their trajectories and velocities presented, Thus, for each series of linkage ratios the behavior of the points indicated in figure 6 has been studied and the results published for class (a) type linkage operation. Crank and Rocker Fig. 3 -..,,- ~c-'" " "",' T..- ~"'--
Class (a)One erank is eapable of rotation through a complete revo- lution while the second crank can only oseillate. Class (b)Both cranks are capable of rotating through 360. Class (e)Both cranks oscillate,but neither can rotate through a complete revolution. The above classifieations arise from a consideration of the motions of the various links relative to the fixed link.As any link ean be fixed arbitrarily the classification of a given linkage is dependent upon the choice of s fixed member.In figure 2,link C is fixed.Crank I ean make a complete revolution while the erank B can only oscillate.The linkage is operating as a class (a)unit commonly known as a erank-and-rocker linkage.Similarly,if link A is fixed (fgure 3),Class (a)operation results. Drag Link If link I is fixed (figure 4),both eranks are capable of full 360 rotation and class (b)operation results.This linkage is often referred to as Fig.4 drag link mechanism.If link B is fixed (figure 5),the two eranks A and C ean only oscillate,hence class (c)operation takes plaoe Though the above classifications are helpful in design,it is important to realize that the relative motion of any link to the remaining members of the linkage is the same,regardless of which member is fixed. Criteria for determining the class of operation when link ratios are known are listed below. Class (a)(crank and rocker mechanism). (1)Drive crank must be the shortest link (1). (2)C<(A+B-1). ()C>A-B1+1). Ciasa ()(drag link mechanism). (1)Line of eenters must be the shortest link (1). 2②)C<(A+B-1). (3)C>(A-B+1) Same as class (a). Ciass (e)(double rocker mechaniam). Two (1)All cases where the conneeting rod is the shortest link (1). Rockers (2)All linkages in which (2)and (3)for classes (a)and (b)are not satisfied. Fig.5 The above conditions have been graphically expressed in figure 7. The ratio C is the ordinate;ratio B is the abscissa;and values of ratio A are plotted as 45 lines.For a given value of A,if the point specifiod by Fig.6 the coordinates B and C ialls inside the oblique reetangular space bounded by the lines of constant A,the linkage will operate in class (a)or (b): class (a)if the shortest link is made a crank;class (b)if the shortest link is made the line of centers.If the point determined by the coordinates 4-B]sbsolute value of (A -B). [
Class (a) One crank is capable of rotation through a complete revolution while the second crank can only oscillate. Both cranks are capable of rotating through 3600 Both cranks oscillate, but neither can rotate through a complete revolution. ' Class (b) Class (c) 1: ' t c ~he above classifications arise from a consideration of the motions of the variolls links relative to the fixed link. As any link can be fixed , arbitrarily the classification of a given linkage is dependent upon the choice of a fixed member. In figure 2, link is fixed. Crank 1 can make a complete revolution while the crank can only oscillate. The linkage operating as a class (a) unit commonly known as a crank-and-rocker linkage, Similarly, if link A is fixed (figure 3), Class (a) operation results, If link 1 is fixed (figure 4), both cranks are capable of full 3600 rotation and class (b) operation results, ' This linkage is often referred to as a drag link mechanism, If link is fixed (figure 5), the two cranks A and can only oscillate, hence class (c) operation takes place. Though the above clas.sifications are helpful in design, it is important to realize that the relative. motion of any link to the remaining members of the linkage is the same, regardless of which member is fixed. Criteria for determining the class of operation when link ratios are known are listed below. Class (a) (crank and rocker mechanism), (1) Drive crank mustbe the shortest link (1). (2) C.c (A 1). (3) C;:. (IA BI* +1). Class (b) (drag link mechanism), (1), Line of centers must be the shortest link (1). (2) C.c (A B- 1). " (3) C;:. (IA BI + I), Same as class (a). Class (c) (double rocker mechanism). (1) All cases where the connecting rod is the shortest link (1). (2) All linkages in ~hich (2) and (3) for classes (a) and (b) are not satisfied; The above conditions have been graphically expressed in figure 7. The ratio is the ordinate; ratio is the abscissa; and values of ratio A are plotted as 450 lines, For a given value of A , if the point specified by the coordinates Band falls inside the oblique rectangular spa~e bounded by the lines of constant A, the ' linkage will operate in ' class (a) or (b): class (a) if the shortest link is made a crank; class (b) if the shortest liI~lk is made the line of centers, If Jhe point determined by the coordinates IA- BI = absolute value of (A B). Two Rockers Fig, / . --- (ixJ . . . ' . .,......... , " :.. . . , Fig. 6
冈 10 60 P B and Cfalls outside the reetangular area bounded by the lines of constant 7.0 A,class(c)operation is indiented.Crank and rocker linkages [elnss (a) operation]included in henvily outlined area are covered in this book. 2.0 Example: Data:Drive erank =4";Connecting Rod -10" 6.0 Follower Crank =8";Line of Centers 12" Pind:(1)Class of operation. 3.0 (2)Variation in link ratios possible without changing class of operstion. 5.0 Solution (see figure 8): 4-碧-26；B=g-2:c-是-8 4 4.0 Drive crank is the shortest link.Refer to figure 7.Point (B -2. 4.0 C=3)falls within oblique reetangular area bounded by lines A -2.5. Therefore s()operation is indieated 5.0 1,A -2.5,B =2 C may be varied from 1.5 to 3.5 1,A -2.5,C=3 B may be varied from 1.5 to 4.5 3.0 1,B -2,C=3 A may be varied from 2 to4 6.0 2 2.0 7.0 1.0 1.0 2.0 3.0 40 5.0 6.0 7.0 B Fig.7 POUR-BAR LINKAGE CLASSIFICATION CHART B Linkage whose ratios B and C determine a point which lies in the oblique reetangular area bounded by lines of eonatant ratio A operates Crank and Rocker or as a Drag F1g.8 Link
Ex) '1. Fig; 7 FOUR~BAR LINKAGE CLASSIFICATION CHART Linkage whose ratios Band determine a point which lies in the oblique rectangular area bounded by lines of constant ratio Link, operates as a Crank and Rocker or as a Drag :",,-:..,' ~" .;.. . "' ;~,;.it1i'&"~;,:J!f;;t. y+'",,,;,,;, - iii Band C falls outside the rectangular area bounded by the lines of constant class (c) operation is indicated, Crank and rocker linkages (class (a) operation) included in heavily outlined area are covered in this book. '1; Example: Data: Drive crank = 4" ; Connecting Rod = 10" Follower Crank = 8"; Line of Centers = 12" Find: (1) Class of operation. (2) Variation in link ratios possible without changing class of operation. Solution (see figure 8): - = 2, 10 5; = - = 2; C = - = Drive crank is the shortest link, Refer to figure 7. Point (B = 2 C = 3) falls within oblique rectangular area- bounded by lines =2. Therefore class (a) operation is indicated = 2, = 2 C may be varied from 1.5 to 3. = 2. , C = 3 may be varied from 1.5 to 4. = 2, C = may be varied from 2 to 4 bs. I\~ bs. '-' Fig, III ..~.. "'. ~~- ...c