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3.4 PLANE FOUR-BAR LINKAGE
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3.4.1 Basic Parameters The apparently simple four-bar linkage is actually an incredibly sophisticated device which can perform wonders once proper design techniques are known and used. Figure 3.5 shows the parameters required to define the general case. Such a linkage can be used for three types of motion: 1. Crank-angle coordination. Motion of driver link b causes prescribed motion of link d. 2. Path generation. Motion of driver link b causes point C to move along a prescribed path. 3. Motion generation. Movement of driver link b causes line CD to move in a prescribed planar motion.
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FIGURE 3.3 Mobility characteristics. (a) Closed four-link kinematic chain: l = longest link, s = shortest link, p, q = intermediate-length links; (b) crank-rocker linkage; (c) double-rocker linkage.
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FIGURE 3.4 Four-precision-point spacing (Chebychev) x1 = xA + 0.0381(xB xA) x2 = xA + 0.3087(xB xA) x3 = xA + 0.6913(xB xA) x4 = xA + 0.9619(xB xA) In general, for n precision points 1 xj = (xA + xB) 2 (2j 1) 1 (xB xA) cos 2n 2
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j = 1, 2, . . . ,n
3.4.2 Kinematic Inversion A very useful concept in mechanism design is that by inverting the motion, new interesting characteristics become evident. By imagining yourself attached to what is actually a moving body, you can determine various properties, such as the location of a joint which connects that body to its neighbor. This technique has been found useful in many industrial applications, such as the design of the four-bar automobile window regulator ([3.6]). 3.4.3 Velocity Ratio At times the velocity of the output will need to be controlled as well as the corresponding position. When the motion of the input crank and the output crank is coordinated, it is an easy matter to establish the velocity ratio d/ b. When you extend line AB in Fig. 3.5 until it intersects the line through the fixed pivots OA and OB in a point S, you find that OA S d = b OAOB + OA S (3.4)
Finding the linear velocity of a point on the coupler is not nearly as straightforward. A very good approximation is to determine the travel distance along the path of the point during a particular motion of the crank. 3.4.4 Torque Ratio Because of the conservation of energy, the following relationship holds: Tbd = Td d (3.5)
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LINKAGES 3.7
LINKAGES
FIGURE 3.5 General four-bar linkage in a plane.
Since both sides of (3.5) can be divided by dt, we have, after some rearranging, n= Tb d /dt d = = Td d /dt b (3.6)
The torque ratio n is thus the inverse of the velocity ratio. Quite a few mechanisms that I have designed have made significant use of torque ratios. 3.4.5 Transmission Angle For the four-bar linkage of Fig. 3.5, the transmission angle occurs between the coupler and the driven link. This angle should be as close to 90 as possible. Useful linkages for motion generation have been created with approaching 20 . When a crank rocker is being designed, you should try to keep 45 < < 135 . Double-rocker or drag link mechanisms usually have other criteria which are more significant than the transmission angle.
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