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FIGURE 3.1 Kinematic pairs useful in linkage design. The quantity f denotes the number of degrees of freedom.
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interest in most linkages is to provide a particular output for a prescribed input, we deal with closed kinematic chains, examples of which are depicted in Fig. 3.2. Considerable work is now under way on robotics, which are basically open chains. Here we restrict ourselves to the closed-loop type. Note that many complex linkages can be created by compounding the simple four-bar linkage. This may not always be necessary once the design concepts of this chapter are applied.
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3.1.2 Freedom of Motion The degree of freedom for a mechanism is expressed by the formula
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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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FIGURE 3.2 Closed kinematic chains. (a) Planar four-bar linkage; (b) planar six-bar linkage; (c) spherical four-bar linkage; (d) spatial RCCR four-bar linkage.
where
l j fi
= = = = = =
number of links (fixed link included) number of joints f of ith joint integer 3 for plane, spherical, or particular spatial linkages 6 for most spatial linkages
Since the majority of linkages used in machines are planar, the particular case for plane mechanisms with one degree of freedom is found to be 2j 3l + 4 = 0 (3.2)
Thus, in a four-bar linkage, there are four joints (either revolute or prismatic). For a six-bar linkage, we need seven such joints. A peculiar special case occurs when a sufficient number of links in a plane linkage are parallel, which leads to such special devices as the pantograph. Considerable theory has evolved over the years about numerous aspects of linkages. It is often of little help in creating usable designs. Among the best references available are Hartenberg and Denavit [3.9], Hall [3.8], Beyer [3.1], Hain [3.7], Rosenauer and Willis [3.10], Shigley and Uicker [3.11], and Tao [3.12]. 3.1.3 Number Synthesis Before you can dimensionally synthesize a linkage, you may need to use number synthesis, which establishes the number of links and the number of joints that are
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required to obtain the necessary mobility. An excellent description of this subject appears in Hartenberg and Denavit [3.9]. The four-bar linkage is emphasized here because of its wide applicability.
3.2 MOBILITY CRITERION
In any given four-bar linkage, selection of any link to be the crank may result in its inability to fully rotate. This is not always necessary in practical mechanisms. A criterion for determining whether any link might be able to rotate 360 exists. Refer to Fig. 3.3, where l, s, p, and q are defined. Grubler s criterion states that l+s<p+q (3.3)
If the criterion is not satisfied, only double-rocker linkages are possible. When it is satisfied, choice of the shortest link as driver will result in a crank-rocker linkage; choice of any of the other three links as driver will result in a drag link or a doublerocker mechanism. A significant majority of the mechanisms that I have designed in industry are the double-rocker type. Although they do not possess some theoretically desirable characteristics, they are useful for various types of equipment.
3.3 ESTABLISHING PRECISION POSITIONS
In designing a mechanism with a certain number of required precision positions, you will be faced with the problem of how to space them. In many practical situations, there will be no choice, since particular conditions must be satisfied. If you do have a choice, Chebychev spacing should be used to reduce the structural error. Figure 3.4 shows how to space four positions within a prescribed interval [3.9]. I have found that the end-of-interval points can be used instead of those just inside with good results.
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