2012 10 31 10 Mechanical Science and Technology for Aerosace Engineering October Vol. 31 2012 No. 10 1 2 1 2 1 2 1 2 1 300387 2 300387 Matlab /Simulink Simulink TH112 A 1003-8728 2012 10-1664-06 Dynamics Simulation of Metamorhic Mechanism Based on the Simultaneous Constraint Method Liang Dong 1 2 Jin Guoguang 1 2 Chang Boyan 1 2 Wei Zhan 1 2 1 School of Mechanical Engineering Tianjin Polytechnic University Tianjin 300387 2 Tianjin Key Laboratory of Advanced Mechatronics Equiment Technology Tianjin 300387 Abstract Acting as a new kind of mechanism metamorhic mechanisms are widely used in engineering fields. In this aer a dynamics model is established for a metamorhic mechanism containing closed-loo constraint by using simultaneous constraint method and the simulation analysis is imlemented using Matlab /Simulink. The results of numerical simulation show that its kinematic variables and force have saltation when the configuration changes from one to another and a method is roosed to solve this roblem. Key words metamorhic mechanism dynamics simulink simulation 3 4 Huston 1 2 Kane 5 6 2011-03-15 51275352 09JCYBJC04600 1985 - ld850725@ sina. com jinguoguang@ tju. edu. cn
10 1665 1 2 f 2 f 2 f q 1 q 1 q 1 q 2 q 1 q n 2 f 2 f 2 f H = q 2 q 1 q 2 q 2 q 2 q n 9 7 2 f 2 f 2 f q n q 1 q n q 2 q n q n H R n n n Hessian 8 1. 1 n q 1 q 2 q n f i q 1 q 2 q n = 0 i = 1 2 n 1 t d f i dt = n f i d q j q j dt i = 1 2 n = 0 1 2 J q = 0 3 J Jacobian f 1 f 1 f 1 q 1 q 2 q n f 2 f 2 f 1 J = q 1 q 2 q n f n f n f n q 1 q 2 q n q = q 1 q n T n 1 4 5 q - 1 q - 2 q - n Newton-Rahson 7 i C i 1 t n i = 1 n n k = 1 2 f q i q j q i q j + f qk = 0 6 q k q T H q + J q = 0 q = q1 q n T n 1 7 8 H ij = 2 f q i q j = [ ] 2 f 1 2 f 2 2 f n q i q j q i q j q i q j T 10 H 8 n 6 n q T H k q + J q k = 0 k = 1 2 n 11 1. 2 N 1 1 q i = i -1 j = 0 N r j + r Ci i = 1 2 N 12 r j j = 0 1 N j j r Ci O i - 1 i 12 t i a Ci = i -1 j = 0 C i rj + rci i = 1 2 N 13
1666 31 1. 3 2 i M W M M F xj F yj - m i a Ci x = 0 14 - m i a Ci y = 0 15 M F yj x j - x Ci - F xj y j - y Ci W + T zk - J zi αi = 0 16 k = 1 F xj F yj i x y T zk i 1 2 x j y j F xj F yj x Ci y Ci 3 rad/s t 1 =2. 094 s i m i J i i a Ci x a Ci y i αi i 11 ~ 15 A R = B 17 A R R B 3 4 t 4 1 3 2 4 1 2 AE AB = DE = 40 cm BC = CD = 200 cm AE =150 cm r =2. 5 cm ρ = 7 800 kg /m 3 1 = 0 2 =1. 186 4 rad 3 =1. 955 2 rad 4 =0 1 1 =3 rad/s 4 4 = - 9t +3 rad/s 2π E 4 t 3 t = t 1 2 π rad 2 4 3 2 4 2 2 1 4 = 0 4 = π rad 4
10 1667 2. 1 1 1 r 1 e j 1 + r 2 e j 2 - r 3 e j 3 - r 4 e j 4 - r 0 e j0 = 0 18 17 Jacobian - r 2 sin 2 r 3 sin 3 J = [ r 2 cos 2 - r 3 cos ] 3 19 17 - r 2 2sin 2 + r 3 3sin 3 = r 1 2 1 cos 1 + r 1 1sin 1 + r 2 2 2 cos 2 - r 3 2 3 cos 3 - r 4 2 4 cos 4 - r 4 4sin 4 20 a C1 y = r C1 1cos 1 - r C1 2 1 sin 1 23 a C2 x = - r 1 1sin 1 - r 1 2 1 cos 1 - r C2 2sin 2 - r C2 2 2 cos 2 24 a C2 y = r 1 1cos 1 - r 1 2 1 sin 1 + r C2 2cos 2 - r C2 2 2 sin 2 25 a C3 x = - r 4 4sin 4 - r 4 2 4 cos 4 - r C3 3sin 3 - r C3 2 3 cos 3 26 a C3 y = r 4 4cos 4 - r 4 2 4 sin 4 + r C3 3cos 3 - r C3 2 3 sin 3 27 a C4 x = - r C4 4sin 4 - r C4 2 4 cos 4 28 5 1 F 01 x + F 21 x - m 1 a C1 x = 0 30 F 01 y + F 21 y - G 1 - m 1 a C1 y = 0 31 r 2 2cos 2 - r 3 3cos 3 = M 01 + F 01 x r C1 sin 1 - F 01 y r C1 cos 1 - F 21 x r 1 - r C1 sin 1 + r 1 2 1 sin 1 - r 1 1cos 1 + r 2 2 2 sin 2 - F 21 y r 1 - r C1 cos 1 - J z1 1 = 0 32 2 r 3 2 3 sin 3 - r 4 2 4 sin 4 + r 4 4cos 4 21 - F 2 21 x + F 32 x - m 2 a C2 x = 0 33 - F 21 y + F 32 y - G 2 - m 2 a C2 y = 0 34 - F 21 x r C2 sin 2 + F 21 y r C2 cos 2 + F 32 y r 2 - r C2 cos 2 - F 32 x r 2 - r C2 sin 2 - J z2 2 = 0 35 a C1 x = - r C1 1sin 1 - r C1 2 1 cos 1 22 3 - F 32 x + F 43 x - m 3 a C3 x = 0 36 - F 32 y + F 43 y - G 3 - m 3 a C3 y = 0 37 F 32 x r 3 - r C3 sin 3 - F 32 y r 3 - r C3 cos 3 + F 43 x r C3 sin 3 - F 43 y r C3 cos 3 - J z3 3 = 0 38 4 - F 43 x + F 04 x - m 4 a C4 x = 0 39 - F 43 y + F 04 y - G 4 - m 4 a C4 y = 0 40 M 04 + F 04 x r C4 sin 4 - F 04 y r C4 cos 4 + F 43 x r 4 - r C4 sin 4 - F 43 y r 4 - r C4 cos 4 - J z4 4 = 0 41 30 ~ 41 F 01 x F 01 y F 21 x F 21 y F 32 x F 32 y F 43 x F 43 y F 04 x F 04 y M 01 M 04 a C1 x a C1 y a C2 x a C2 y a C4 y = r C4 4cos 4 - r C4 2 4 sin 4 29 a C3 x a C3 y a C4 x a C4 y 1 2 3 4 20 21 3 22 ~ 29 5
1668 31 4 Simulink 7 20 ~ 41 A 22 22 R 22 1 = B 22 1 42 1 4 2. 2 Matlab /Simulink 1 7 Matlab /Simulink 9 6 6 7 Simulink 2 t = 5 s 8a ~ 8d
10 1669 8 2 3 t 1 4 4 t 1 1 1 2 3 1 4 A D - 0. 818 rad /s 2-0. 818 rad /s 2 104. 076 N m 41. 946 N m 44. 786 N - 106. 989 N 3 1 3 4 J. 2004 2 Matlab /Simulink J 1 Dai J S Jones J Rees. Mobility in metamorhic mechanism of foldable /erectable kinds J. Transaction of the ASME Journal of Mechanical Design 1999 121 3 375 ~ 382 2 Dai J S Zhang Q X. Metamorhic mechanism and their configuration models J. Chinese Journal of Mechanical Engineering English Edition 2000 13 3 212 ~ 218 3 ~ 223 7. 4 8 M 9 F. R. 2003. 25 4 401 ~ 405 5.. 2006 25 9 1092 ~ 1905 6 Li D L Ding X L Zhang Q X et al. Mechanism theory of metamorhic mechanism and alication in sace technology A. In Proceedings of 2002 International Symosium on Dee Sace Exloration Technology and Alication C 2002 229 Matlab /Simulink M. 2007.. 2006. Matlab Simulink M. 2002