The problem of synthesizing a planar four-bar linkage with two given fixed pivots such that the coupler curve passes through five given points is considered with the Groebner-Sylvester hybrid approach. First, closed-form equations of a single point are constructed. The reduced Groebner basis in degree lexicographic ordering for the closed-form equations is then obtained using computer algebra. A 23 × 23 Sylvester’s matrix can be constructed by selecting 23 out of 89 Groebner bases. A 36th degree univariate equation is obtained directly from the determinate of the matrix. The same result can be obtained with a continuation method. A numerical example is given and verifies that the problem has at most 36 solutions in the complex field.
HUANG Xiguang, LIAO Qizheng, WEI Shimin, XU Qiang
. Five precision point-path synthesis of planar
four-bar linkage using algebraic method[J]. Frontiers of Electrical and Electronic Engineering, 2008
, 3(4)
: 470
-474
.
DOI: 10.1007/s11460-008-0063-x
1. Li Baoqi . Classified Graphic Spectrum of Space Model for Curve of Linkage. Beijing: China Machine Press, 1993 (in Chinese)
2. Huo Jingping, Cao Weiqing, Ruan Zhongtang . Synthesis of path generating four-bar linkage with useof fuzzy mathematics. Chinese Journal ofMechanical Engineering, 1989, 25(2): 6l–66 (in Chinese)
3. Li Zhen, Gui Changlin . Morphologicaly analyzingmethod for the characteristics of coupler curves of linkage mechanism. Chinese Journal of Mechanical Engineering, 2002, 38(2): 27–30 (in Chinese)
4. Hoeltzel D A, Chieng W H . Pattern matching synthesisas an automated approach to mechanism design. ASME Journal of Mechanical Design, 1990, 112(2): l90–l99. doi:10.1115/1.2912592
5. Chu Jinkui, Cao Weiqing . Synthesis of coupler curvesof planar four-bar linkages through fast Fourier transform. Chinese Journal of Mechanical Engineering, 1993, 29(5): 117–122 (in Chinese)
6. Sun Xiaobin, Xiao Renbin, Wang Xue . New approach to rapid searching path generating mechanisms. Chinese Journal of Mechanical Engineering, 2000, 36(11): 98–l05 (in Chinese)
7. Wang Chengzhi, Ji Yuebo, Sun Daoheng . Research on applying wavelet analysis to path synthesisfor coupler curves of planar four-bar linkages. Chinese Journal of Mechanical Engineering, 2004, 40(8): 34–39 (in Chinese)
8. Wang Delun, Wang Shufen, Li Tao . New approach for mechanisms synthesis by adaptive saddlefitting. Chinese Journal of MechanicalEngineering, 2001, 37(12): 21–26 (in Chinese)
9. Peng Qiang, Zhang Tongzhuang, Liu Huilin . Path generation synthesis of planar 4-bar linkage withfive precision points using Wu's method. Machinery Design & Manufacture, 2004, (4): 66–67 (in Chinese)
10. Wampler C W, Morgan A P, Sommese A J . Complete solution of the nine-point path synthesis problemfor four-bar linkages. ASME Journal ofMechanical Design, 1992, 114(1): 153–159.. doi: 10.1115/1.2916909
11. Morgan A P, Wampler C W . Solving a planar four-bardesign problem using continuation. ASMEJournal of Mechanical Design, 1990, 112(4): 544–550. doi:10.1115/1.2912644
12. Morgan A P, Sommese A J, Watson L T . Finding all isolated solutions to polynomial systemsusing HOMPACK. ACM Transactions on MathematicalSoftware, 1989, 15(2): 93–122. doi:10.1145/63522.64124
13. Verschelde J . PHCpack:a general purpose solver for polynomial systems by homotopy continuation. ACM Transactions on Mathematical Software, 1999, 25(2): 251–276. doi:10.1145/317275.317286