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The second equality could also be written P23 P13 P12 c1P13 c0 180
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A locus of points thus defined can be created as shown in Fig. 3.16. Each point on the circle-point curve corresponds to a particular point on the center-point curve. Some possible links are defined in Fig. 3.16; each has a known first-to-fourth-position rotation angle. Only those links whose length and/or pivot locations are within prescribed limits need to be retained. The two intermediate positions of the link can be determined by establishing the location of the moving pivot (circle point) in the second and third positions of the moving body. Since the positions lie on the arc with center at the fixed pivot (center point) a and radius aa , it is easy to determine the link rotation angles as 12 = A1OA A2 13 = A1OA A 3
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Linkages need to be actuated or driven by one of the links. Knowing the three rotation angles allows you to choose a drive link which has the desired proportions
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LINKAGES
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FIGURE 3.12 Determining the moving or fixed pivot by using the pole triangle.
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FIGURE 3.13 Four positions of a plane: definition of the opposite-pole quadrilateral formed by lines P13 P24 and P12 P34.
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3.15 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
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FIGURE 3.14 Determination of points on the center-point curve.
FIGURE 3.15 Determination of a circle point corresponding to a particular center point.
3.16 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
LINKAGES 3.17
LINKAGES
FIGURE 3.16 Some of the links which can be attached to the plane containing CD.
of motion. Proper care in selection of the two links will result in a smooth-running four-bar linkage.
3.7.4 Five Positions of a Plane It would seem desirable to establish as many precision positions as possible. You can choose two sets of four positions (for example, 1235 and 1245) from which the Burmester curves can be created. The intersections (up to six) of those two centerpoint curves are the only fixed pivots which can be used to guide the moving body through the five positions. Since those pivots and/or link lengths have virtually always been outside the prescribed limits, I never use five-position synthesis.
3.7.5 Available Computer Programs Two general-purpose planar linkage synthesis programs have been created: KINSYN ([3.17]) and LINCAGES ([3.18]). They involve the fundamentals described in
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LINKAGES 3.18
MACHINE ELEMENTS IN MOTION
this section and can be valuable when time is limited. I have found it more advantageous to create my own design and analysis programs, since the general programs almost always need to be supplemented by routines that define the particular problem at hand.
3.8 DIMENSIONAL SYNTHESIS OF THE PLANAR FOUR-BAR LINKAGE: CRANK-ANGLE COORDINATION
Many mechanical movements in linkages depend on the angular position of the output crank. In general, you will have to design the four-bar linkage so that a prescribed input crank rotation will produce the desired output crank rotation. Significant work was performed in an attempt to generate functions ([3.9]) using the four-bar linkage until the advent of the microcomputer. Although it is seldom necessary to utilize the function capability, you will find many applications for crankangle coordination. Two methods are possible: geometric and analytical.
3.8.1 Geometric Synthesis In a manner similar to that for motion generation (Sec. 3.7), the concept of the pole is once again fundamental. Here, however, it is a relative pole, since it defines relative motions. Suppose that you need to coordinate the rotation angles 12 for the crank (input) and 12 for the follower (output). Refer to Fig. 3.17, where the following steps have been drawn: 1. Establish convenient locations for the fixed pivots OA and OB. 2. Draw an extended fixed link OAOB. 3. With OA as vertex, set off a line at angle 12 /2 (half rotation angle, opposite direction).
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