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robotics Designing the Mechanisms for Automated Machinery Part 12

Tham khảo tài liệu 'robotics designing the mechanisms for automated machinery part 12', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 9.3 Kinematics of Manipulators 345 The general solution of the dynamic equations then is B11cos fc1t Ju1 B12cos fc2t Ju2 2 11 21 cos it jq B12 22 cos fc2í jU2 . The parameters Bn B12 Mị M2 are determined through the initial conditions t 0 Pi Pw p2 P20 P1 P1O and Ộ2 P20- Thus the free vibrations of the arm close to the average position consist of two oscillation processes with two different and usually not commensurable frequencies fci and k2 which in turn means that in general these oscillations are not of a periodic nature. For further analyses it is convenient to use the so-called main or normal coordinates zm which are defined in the following form 2 Pr ỵ rmZm r m l 2 . m ỉ For any initial conditions these new variables change periodically monoharmonically zm Dmcos kmt ym 771 1 2 where the constants Dm and ym are determined by the initial conditions. An example follows. Supposing mỉ m2-m and Cl c2 c we rewrite 9.46c in the form . 2 11 . o_ . an mr 3cosợ a22 ml2 3 4 3_ cosq . L3 2 ứ12 ml2 For q 0 this becomes __i2 20. aỵỵ - mr 3 a22 mỉ2 4 3 ứ 2 mĩ2 . 6 For the natural frequencies in this case we obtain 9.46k 9.461 fci2 er.iAMl RNnZ2 fc22 9.164c mZ2 and Kt S0356ỈỊẶ K2 3.027 346 Manipulators For q ĨIỈ2-. - 7211. an mr 3 22 ml2 9.46m 12 -ml For the natural frequencies in this case we obtain kị 0.2341 dml2 kị 1.373dml2. The coefficients describing the shape of the oscillations are correspondingly 12 - 1 ệ22 -2.203 s2 511- 1 21 0.4537 It is possible to obtain an approximate estimation for the initial deformations appearing in the manipulator under discussion which determine the amplitudes of the free oscillations after the motors are stopped. For this purpose we write kinetostatical equations of acting torques Qj and ọ2. _ __ I I ml2 _ 3Z 3Z ml2 _ __ 20_j2_ . Qi meQ n mE0 Eo mE0 2121 - mrEn 22 12 2 2 12 3 9.46n _ 31 I ml2 17 2 ọ2 mE0 - E0 mE02ZZ ml q . 2 2 12 6 This is done with an assumption that the angle q 0 and that angular acceleration before the stop was Eo .