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 repre￾sentation 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 inter￾section, 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 whosetrajec￾tories 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 "","".~.........-~~-~"",",",.""" ...." ...?~~~_.._,." ""."....~...~ "" """':~ , ":""- ......." _