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FIGURE 3.19 Pole-force method for balancing a force on the coupler of a four-bar linkage.
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3. Determine lines Sab(ab) and Sac(ac); their intersection is T2. 4. Line T1T2 closes the pole-force triangle, which is transferred to Fig. 3.19b. 5. The magnitude of Fab required to balance the coupler force Fac is easily found. Many other cases, any of which you might encounter in practice, are shown by Tao [3.12].
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Most practical linkages have motion entirely in a plane or possibly in two parallel planes with duplicated mechanisms such as those in a backhoe or a front loader. Design procedures for some elementary types of spatial four-bar linkage have been created (Refs. [3.9] and [3.11]), principally for the RGGR type (Fig. 3.20). Three principal mathematical methods for writing the loop-closure equation are vectors ([3.3]), dual-number quaternions ([3.14]), and matrices ([3.13]). These techniques have evolved into general-purpose computer programs such as IMP ([3.16])
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FIGURE 3.20 An RGGR spatial linkage; R designates a revolute joint, and G designates a spherical joint.
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and ADAMS and DRAM ([3.5]); they will make your spatial linkage analysis much easier. With such tools available, you can design complex spatial mechanisms by iterative analysis.
REFERENCES
3.1 Rudolph A. Beyer, Kinematic Synthesis of Mechanisms, Herbert Kuenzel (trans.), McGraw-Hill, New York, 1964. 3.2 Ludwig Burmester, Lehrbuch der Kinematick (in German only), A. Felix, Leipzig, 1888. 3.3 Milton A. Chace, Vector Analysis of Linkages, J. Eng. Ind., ser. B, vol. 55, no. 3, August 1963, pp. 289 297. 3.4 Ferdinand Freudenstein, Approximate Synthesis of Four-Bar Linkages, Trans. ASME, vol. 77, no. 6, 1955, pp. 853 861. 3.5 Ferdinand Freudenstein and George Sandor, Kinematics of Mechanisms, in Harold A. Rothbart (ed.), Mechanical Design and Systems Handbook, 2d ed., McGraw-Hill, New York, 1985. 3.6 Richard E. Gustavson, Computer-Designed Car-Window Linkage, Mech. Eng., September 1967, pp. 45 51. 3.7 Kurt Hain, Applied Kinematics, 2d ed., Herbert Kuenzel, T. P. Goodman, et al. (trans.), McGraw-Hill, New York, 1967. 3.8 Allen S. Hall, Jr., Kinematics and Linkage Design, Prentice-Hall, Englewood Cliffs, N.J., 1961. 3.9 Richard S. Hartenberg and Jacques Denavit, Kinematic Synthesis of Linkages, McGrawHill, New York, 1964. 3.10 N. Rosenauer and A. H. Willis, Kinematics of Mechanisms, Dover, New York, 1967. 3.11 Joseph E. Shigley and John J. Uicker, Jr., Theory of Machines and Mechanisms, McGrawHill, New York, 1980. 3.12 D. C. Tao, Applied Linkage Synthesis, Addison-Wesley, Reading, Mass., 1964.
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LINKAGES 3.23
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3.13 John J. Uicker, Jr., J. Denavit, and R. S. Hartenberg, An Iterative Method for the Displacement Analysis of Spatial Linkages, J. Appl. Mech., vol. 31, ASME Trans., vol. 86, ser. E, 1964, pp. 309 314. 3.14 An T. Yang and Ferdinand Freudenstein, Application of Dual-Number and Quaternion Algebra to the Analysis of Spatial Mechanisms, J.Appl. Mech.ASME Trans., vol. 86, ser. E, 1964, pp. 300 308. 3.15 ADAMS and DRAM, Automatic Dynamic Analysis of Mechanical Systems and Dynamic Response of Articulated Machinery, developed by Chace at the University of Michigan. 3.16 IMP, the Integrated Mechanisms Program, developed by Uicker at the University of Wisconsin. 3.17 KINSYN, primarily for kinematic synthesis, developed by Kaufman while at M.I.T. (he is now at George Washington University). 3.18 LINCAGES, for kinematic synthesis and analysis, developed by Erdmann et al. at the University of Minnesota. 3.19 C. R. Mischke, Elements of Mechanical Analysis, Addison-Wesley, Reading, Mass., 1963.
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