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
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9 20 B= 30 C=25 Fig.12 SIMPLE COMPUTER LINKAGE [xvJ
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