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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)CA-B1+1). Ciasa ()(drag link mechanism). (1)Line of eenters must be the shortest link (1). 2②)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
Mathematical Relationships Motion f Points on Connecting Rod.The relationships for displacement, Dricer-Follower Crank.(See figure 9.)The follower crank angular velocity,and acceleration of a point located anywhere on the connecting position is given by 4. rod are considerably more cumbersome than the above equations and are 业=a+a given in reference 1. gin 0 Illustrative Examples.(See reference 2.) a1-tan C+c080 1.Duell Period from Straigh-Line Path.Figure1 how the path a4=0gr1R+2C6089 of a point on the conneeting rod of a linkage which is approximately straight ·2BL between points a and b.A rotstion of 55 of the drive erank produces this K -1+B+C-A* straight portion.This is indieated by the eleven dashes which constitute L-1+Ci +2C cos 0 the path lying between a and b. =K*+2C cos 6 If the point having this displacement path drives a member with a S=√4BF-M radial slot constrained to rotate about the fixed point e,an angular motion -mce。+ow+Co0 sin 6 2BL ) of 279 will result.In terms of drive crank angle there will be a 55 dwell, a 220 forward stroke,and an 85 return stroke.These figures are obtained directly from the trajectory by counting the number of dashes in each phase Differentiating equation I with respeet to time yields the following equation for the velocity of the follower crank: of the motion and multiplying by 5. If the pivot of the rotating link is loeated at d the total angular motion 整-[cm0+)+c如+】] of the link is 20.At a uniform drive erank speed approximately one-half of each cyele is used for the forward stroke,one-third for the return stroke, A second differentintion yields an expression for the angular accelera- and one-sixth of ench eyele for the dwell. tion of the follower crank: An alternative method of obtaining a dwell is to drive s"sootch yoke," a member sotted parallel to the straight portion ab but constrained to 器-[品cm+)+0收+】 move in the direetion ef perpendicular to ab.In this case the ratio of +[+9）(2Ce-如+9 forward return stroke is not adjustable and is equal to 51/21 or 2.4.The ratio of the length of stroke to drive erank length is 1.37. -9-(-2+〗僧 2.Duell Period from Circular-Line Path.Figure 11 shows the path ofa point on the conneeting rod of a four-bar linknge whose basic ratios are 4;2.5;3.5./Between points a and b an approximate circular are exists over90 drive crank ange displneement.Link ae pinned to the midpoint of the conneeting rod drives a bell crank rotating about the same fixed axis as the drive erank.Link proportions are selected such that e is the center of curvature of the are ab.The bell erank has a dwell period of one-fourth the total cycle and a total angle of travel of 34 with approxi- mately oqual times for advance and return. The time ratio of forward to return stroke and the angle of oseillation ean be adjuated by choosing other locations for the bell crank fixed axis, with corresponding changes in the length of the bell crank arm so that c remains the center of curvature of the are a.For instance,it is poesible using this linkage to locate the bell crank axis at the fixed axis of the follower crank.Care must be taken to avoid a dead center position between the link ac and the bell crank arm to which it is pinned. Cos 6 The sme fundamental linkage can be employed to produce a straight F1g.9 line reciprocating motion with the same dwell period at the end of the 剪
Mathematical Relationships Driver-Follower Crank. (See figure 9.) The follower crank ,angular position is given by 4. 1ft = alsin 8 al C+cos8 K2 +2C cos 8 a2 = cos 2BL K2 = 1 + B2 C2 = 1 + C2 2C cos 8 M'i K2 +2C cos 8 S2 ...J 4B2V M4 1. sin (j K2 2C cosO 'Y =, an cos + cos 8 2BL Differentiating equation 1 with respect to time yields the following equation for the velocity of the follower crank: (1) Motion of Points on Connecting Rod. The relationships for displacemen~, velocity, and acceleration of a point located anywhere on the connecting" r()d are considerably more cu~bersome than the above equations and are given in reference 1. IUustrative Exam les, (See reference 2. 1. DwellPeriod from Straight-Line Path. Figure 10 shows the path of a point on the connecting rod of a linkage which is approximately straight between points and b. A rotation of 550 of the drive crank produces this straight portion, This is indicated by the eleven dashes which constitute the path lying between and If the point having this displacement path drives a member with a radial slot constrained to rotate about the fixed point l;, an angular motion of 27P will result. In terms of drive crank angle there will be a 550 dwell a 2200 forward stroke, and an 850 return stroke. These figures are obtained directly from the trajectory by counting the number of dashes in each phase of the motion and multiplying by 50 If the pivot of the rotating link is located at the total angular motion of the link is 200. At a uniform drive crank speed approximately one-half of each cycle is used for the forward stroke, one~third for the return stroke and one-sixth of each cycle for the dwell. All alternative method of obtaining a dwell is to drive a "scotch yoke a member slotted parallel to the straight portion ab but constrained to move in the direction ef perpendicular to ab, In this case the ratio of forward return stroke is not adjustable" and is equal to 51/21 or 2,4. The ratio of the length of stroke to drive crank length is 1.37. d1ft (C cos 8 + 1) sin 8 2 + dt dt V S2 A second differentiation yields an expression for the angularaccelera- tion of the follower crank: " 1ft d28 l(Ccos 8 + 1) + sin 8 2 + dt2 dt2 V S2 202 sin2 8 (2B2 - M2) C cos 8 2C2 sin2 8 1 - sin 8 1 - 2(C cos 8 + I VS2 L2 OIC': , , "2 f/I ' -~ - ' ~ - - 6r~9) - ' - - Fig, 2. Dwell Period from Circular-Line Path. Figure 11 shows the path of a point on tbF connecting rod of a four-bar linkage whose basic ratios are 4; 2,5;3. 5. IBetween points andb an approximate circular arc exists over a 900 drive crank angle displacement, Link ac pinned to the midpoint of the connecting rod drives a bell crank rotating about the same fixed axis as the drive crank, Link proportions are selected such that is the center of curvature' of the arc ab. The bell crank has a dwell period of one-fourth the total cycle and a total angle of travel of 340 with approximately equal times for advance and return. The "time ratio of forward to return stroke and the angle of oscillation can be adjusted by choosing other locations for the bell crank fixed axis with corresponding changesin the length of the bell crank arm so that remains the center of curvature of the arc ab. For instance, it is possible using this linkage to locate the bell crank axis at the fixed axis of the follower crank, Care must be taken to avoid a dead center position between the iink ac and the" bell crank arm to which it is pillned. The same fundamental linkage can be employed to produce a straight line reciprocating motion with the same dwell period at the end of the (xi)
stroke.Substitute for the bell crank a slider constrained to move in a driven by a point on the conneeting rod whose trajeetory exhibits an inter- fixed straight slot passing through point c.Adjust the direction of this seetion.The two output oscillations can differ in amplitude and time slot to obtain the desired time ratio of forward to return stroke.For Figure 14 illustrates the special case where each oscillation is of the same example,using the line cd as the axis of the slot,the slider will dwell at amplitude.The pivot of the output member is located at point a.The time required for each part of the cyele is obtained by eounting the dashes point c for 90,advanee to point d in 150,and return to point e in 120 between the points of tangeney b,c,d,and e and is given in the table below. rotation of the drive crank. The linkage has the basie ratios 2,2.5,and 2.The drive point is at the 3.Computer Linbages.Four-bar linkages are often used as computers. coordinate location (+1,-1). Beeause of the infinite number of output-input relationships available a wide variety of functions can be represented over limited ranges of the Stroke Path Dashss Degr件 variables appearing in the desired funetions.Where a high degree of l威forward BC 10 50 18 90 accuraey is required more than one four-bar linkage is often necessary. Ist return 135 In this event the primary linkage approximates the desired relationship 2d forward E E 17 85 while additional linkages apply corrections to bring the maximum errors within the tolerance limits required.(See reference 3.) Figureowlin hich ceely satisfie thefunei 5.Symmetrical-Motion Paths.In a number of applications it is 2L5」 desirable to obtain a path which is symmetrical with respeet to some over a range of from 0 to 55,when used as indicated below.In finding referenoe line.Linkages where the connecting rod and follower crank are this mechanism the procedure was as follows.The drive crank angle was of equal length (A B)have points on the connecting rod whose trajec- assumed to be the variable 0.The value of eorresponding to values of tories meet this condition.The loeus of such points is a eirele of radius A from 0 to 75 at 5>intervals was caleulated.The caleulated angular on the connecting rod with its center at the moving end of the follower erank. positions were then accurately laid out on transparent paper.Repeated Figure 15 shows the paths of twelve pointa on the conneeting rod of superposition of this layout on various charts resulted in finding a trajectory four-bar linkage with the basic ratios 2;2;2.5.The pointe are equally on which the lines representing the angular position(of the overlay)fell spaced on the dashed cirele.The trajectory of each point is symmetrieal on successive dash terminals.The point on this particular linkage giving about the straight line passing through the follower crank fixed axis and this desired result can be used to drive a radially slotted member pivoted the sero position of the trajectory.The twelve trajectories illustrate s at the intersection of the lines on the overlay sheet. typical set of aymmetrical-motion paths and show the great variety of The seleeted linknge has the basie ratios 2;3;2.5.The drive point on curves available for e where symmetry of forward and return stroke is the conneeting rod has the coordinate location +1.5,+1.The slotted essential. member is pivoted at Within the range offromto55a good repre- sentation of d is obtained. Figure 13 shows a linkage in which the output position is the logarithm of the input position over a limited range.As in the previous problem an overlay was construeted and the charts searched for satisfactory matehing of the overlay over the desired range.In the mechanism shown,the radially slotted output member pivoted at a has angular displacements proportional to the logarithm of the drive crank displacement in the range of positions 1 to 10.Except at position 1 the aeeuracy is good.Two linkages of this type feeding a differential unit could be usedasmultiplier. The basie linkage ratios are 2.5,2.5,and 1.5.The coordinate location of the point on the connecting rod is +1.5,+1. 4.Dowble Oscillating Crank.A slotted erank whoee frequeney of oscillation is twice that of the drive erank is shown in figure 14.It is
:;,,~':;""'~ """" --.. ,~,- " ',- (xii) , stroke, Substitute for the bell crank ~ slider constrained to move in a fixed" straight slot passing through point c, Adjust the direction of this slot to obtain the desired time ratio of forward to return stroke. For example, using the line cd as the axis of the slot, the slider will dwell at point cfor 900, advance to point d in 1500, and return to point c in 1200 rotation of the drive crank, 3, Computer Linkages. Four-bar linkages are often used as computers. Because of the infinite number of output-input relationships available a wide variety of functions can be represented over limited ranges of the variables appearing in the desired functions. Where a high degree of accuracy is required more than one four-bar linkage is often necessary, In this event the primary linkage approximates the desired relationship while additional linkages apply corrections to bring the maximum errors within the tolerance limits required, (See reference 3. Figure 12 shows a linkage which closely satisfies the function4'=~(~l5 over a range of (Hrom 0 to 550, when used as indicated below, In finding this mechanism the procedure was as follows. The drive crank angle was assumed to be the variable (), The value of 4'corresponding to values of () from 0 to 750 at 50 intervals was calculated. The calculated angular positions 4' were then accurately laid out on transparent paper. Repeated superposition of this layout on various charts resulted in finding a trajectory on which the lines representing the angular position cp (of the overlay) fell on successive dash terminals, The point on this particular linkage giving this desired result can be used to drive a radially slotted member pivoted a~ the intersection of the lines on the overlay sheet. The selected linkage has the basic ratios 2; 3; 2,5, The drive point the conne~ting rod has the coordinate location + 1.5 , + 1. The slotted member is pivoted ata. Within the range of () from 0 to 550 a good representation of cp is obtained. Figure 13 shows a linkage in which the output position is the logarithm of the input position over a limited range. As in the previous problem an overlay was constructed and the charts searched for satisfactory matching of the overlay over the desired range. In the mechanism shown, the radially slotted output member pivoted at a has angular displacements proportional to the logarithm of the drive crank displacement in the range of positions 1 to 10. Except at position 1 the accuracy is good, Two linkages of this type feeding a differential unit could be used as a multiplier, , The basic linkage ratios are 2, , 2,5, and 1.5. The coordinate location of the point on the connecting rod is + 1.5, + 1. 4, Double Oscillating Crank, A slotted crank whose frequency of oscillation is twice that of the drive crank is shown in figure 14, It -:-~~t.. :"- driven by a point on the connecting rod whose trajectory exhibits an intersection, The two output oscillations can differ in amplitude and time, Figure 14 illustrates the special case where each oscillation is of the sa~e amplitude: The pivot of the output member is located at point a. The time required for each part of the cycle is obtained by counting the dashes between the points of tangency b, c, d, and e and is given in the table below, The linkage has the basic ratios 2 , and 2, The drive point is at the coordinate location (+1 , - 1). Drit'e Crank Stroke Path Dashes Degrees 1st forward 1st return 2d forward 135 2d return 5. Symmetrical-Motion Pat~s. In a number of applications it is desirable to obtain a path which is symmetrical with respect to some reference line. Linkages where the connecting rod and follower crank are of equal length (A = B) have points on the connecting rod whosetrajectories meet this condition, The locus of such points is a circle of radius A on the connecting rod with its center at the moving end of 'the follower crank. Figure 15 shows the paths of twelve points on the connecting rod a four-bar linkage with the basic ratios 2; 2; 2,5. The points are equally , spaced on the dashed circle. The trajectory of each point is symmetrical about the straight line passing through the follower crank fixed axis and the zero position of the trajectory. The twelve trajectories illustrate a typical set of symmetrical-motion paths and show the great variety of curves available for use where symmetry of forward and return stroke is essential. . J "---'" ", ~_...... c--- -,_ "-- "". 'C,=, " " ,..w "","".~.........-~~-~"",",",.""" ...." ...?~~~_.._,." ""."....~...~ "" """':~ , ":""- ......." _
Analyuis of the Four Bar Linkage 20° A2.0 B 2.5 C=20 @ a e Fig.10 b DWELL PERIOD FROM 22 STRAIGHT LINE PATH [xii] 回
"-- "-- '-- .... "-- .... "-- ' .... "-- "-- 'y \;-~,, , / !/"g = V 8' , 0. V"I' f)prmBaUD.lH . "'J' UOB'JilN aBu:J(Uf, .mg a:m :Jo8Jdl1ruv .t110~. .LH9IV~..1~ H-LVd 3N/7 773MG aOIC/3d I/VO~..::I 0/:1 0/ ' ", I ,,- ,," (mxJ - . c j , /
B=25 O 吸 Fig.I 湖 DWELL PERIOD FROM CIRCULAR ARC PATH
,~, 0 ;"'o '~ ;
9 20 B= 30 C=25 Fig.12 SIMPLE COMPUTER LINKAGE [xvJ
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