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LINKAGES 3.11
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FIGURE 3.8 Two positions of a plane: definition of pole P12.
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The direction in which these angles are measured is critical. For three positions, you may thus choose the fixed or the moving pivot and use this relationship to establish the location of the corresponding moving or fixed pivot, since it is also true that P12 P13 P23 = A 1 P13OA = B 1P13OB (3.14)
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The intersection of two such lines (Fig. 3.12) is the required pivot point. Note that the lines defined by the pole triangle relationships extend in both directions from the pole; thus a pivot-point angle may appear to be 180 from that defined within the triangle. This is perfectly valid. It is important to observe that arbitrary choices for pivot locations are available when three positions, or less, of the moving plane are specified.
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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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LINKAGES 3.12
MACHINE ELEMENTS IN MOTION
FIGURE 3.9 Three positions of a plane: definition of the pole triangle P12 P13P23.
3.7.3 Four Positions of a Moving Plane When four positions are required, appropriate pivot-point locations are precisely defined by theories generated by Professor Burmester in Germany during the 1880s. His work [3.2] is the next step in using the poles of motion. When you define four positions of a moving plane containing line CD as shown in Fig. 3.13, six poles are defined: P12 P13 P14 P23 P24 P34 By selecting opposite poles (P12 , P34 and P13, P24), you obtain a quadrilateral with significant geometric relationships. For practical purposes, this opposite-pole quadrilateral is best used to establish a locus of points which are the fixed pivots of links that can be attached to the moving body so that it can occupy the four prescribed positions. This locus is known as the center-point curve (Fig. 3.14) and can be found as follows: 1. Establish the perpendicular bisector of the two sides P12P24 and P13P34. 2. Determine points M and M such that P12 MQ2 P13 M Q3
3. With M as center and MP12 as radius, create circle k. With M as center and M P13 as radius, create circle k . 4. The intersections of circles k and k (shown as c0 and c in Fig. 3.14) are center 0 points with the particular property that the link whose fixed pivot is c0 or c 0 has a
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LINKAGES 3.13
LINKAGES
FIGURE 3.10 Path generation as a special case of motion generation.
total rotation angle twice the value defined by the angle(s) in step 2. The magnitude and direction of the link angle 14 are defined in the figure. Note that this construction can produce two, one, or no intersection points. Thus some link rotations are not possible. Depending on how many angles you want to investigate, there will still be plenty of choices. I have found it most convenient to solve the necessary analytic geometry and program it for the digital computer; as many accurate results as desired are easily determined. Once a center point has been established, the corresponding moving pivot (circle point) can be established. For the first position of the moving body, you need to use the pole triangle P12 P13 P23 angles to establish two lines whose intersection will be the circle point. In Fig. 3.15, the particular angles are P13 P12 P23 c1 P12 c0
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 3.14
MACHINE ELEMENTS IN MOTION
FIGURE 3.11 Geometric relationship between pole triangle angle(s) and location of link fixed and moving pivot points.
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