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30] S. H. MacCullough, Elements of Strength of Materials, third ed, D. Van Nostrand, New York, 1949. [31] A. Pars, A Treatise on Analytical Dynamics, OX BOX Press, Woodbridge, CT, 1979. R. Cowper, The shear coefficients in Timoshenko’s beam theory, Journal of Applied Mechanics 33 (1968) 335e339. Y. Wang, On the solutions of Eringen’s micropolar plate equations and other approximate equations, International Journal of Engineering Science 28 (9) (1990) 919e935. [34] P. Zhou, Mechatronic design of flexible manipulators.
D0 is the damping matrix and we choose the linear-type Rayleigh damping [44]. As for the flexible manipulator that contains rigid motion and elastic motion, the mass matrix can be expressed as ! 142) M0 ¼ 0 0 MqW MWW 0 0 denote the mass matrices relating to elastic motion and rigid motion and Mqq where MWW 0 denotes the coupling parts of elastic and rigid motion. respectively, and MqW Likewise, the stiffness matrix can be written as 0 K ¼ 0 0 0 ! 143) 0 KWW 0 where KWW denotes the stiffness matrix relating elastic motion, and it has no elastic and rigid motion coupling.
Wang, Optimum design of vibrating cantilevers: A classical problem revisited, Journal of Optimization Theory and Applications 84 (3) (1995) 635e652. Y. Wang, On the extremal fundamental frequencies of one-link flexible manipulators, International Journal of Robotics Research 13 (2) (1994) 162e170. Y. Wang, Finding the maximum bandwidth of a flexible arm, Proceedings of 32nd IEEE Conference on Decision and Control, San Antonio, TX, 1, (1993) 619e625. Y. L. Russell, Minimum-weight robot arm for a specified fundamental frequency, Proceedings of the IEEE Conference on Robotics and Automation, Atlanta, GA, 1, (1993) 490e495.