There are vast constraint equations in conventional dynamics analysis of deployable structures, which lead to differential-algebraic equations (DAEs) solved hard. To reduce the difficulty of solving and the amount of equations, a new flexible multibody dynamics analysis methodology of deployable structures with scissor-like elements (SLEs) is presented. Firstly, a precise model of a flexible bar of SLE is established by the higher order shear deformable beam element based on the absolute nodal coordinate formulation (ANCF), and the master/slave freedom method is used to obtain the dynamics equations of SLEs without constraint equations. Secondly, according to features of deployable structures, the specification matrix method (SMM) is proposed to eliminate the constraint equations among SLEs in the frame of ANCF. With this method, the inner and the boundary nodal coordinates of element characteristic matrices can be separated simply and efficiently, especially on condition that there are vast nodal coordinates. So the element characteristic matrices can be added end to end circularly. Thus, the dynamic model of deployable structure reduces dimension and can be assembled without any constraint equation. Next, a new iteration procedure for the generalized-α algorithm is presented to solve the ordinary differential equations (ODEs) of deployable structure. Finally, the proposed methodology is used to analyze the flexible multi-body dynamics of a planar linear array deployable structure based on three scissor-like elements. The simulation results show that flexibility has a significant influence on the deployment motion of the deployable structure. The proposed methodology indeed reduce the difficulty of solving and the amount of equations by eliminating redundant degrees of freedom and the constraint equations in scissor-like elements and among scissor-like elements.
There are vast constraint equations in conventional dynamics analysis of deployable structures, which lead to differential-algebraic equations (DAEs) solved hard. To reduce the difficulty of solving and the amount of equations, a new flexible multibody dynamics analysis methodology of deployable structures with scissor-like elements (SLEs) is presented. Firstly, a precise model of a flexible bar of SLE is established by the higher order shear deformable beam element based on the absolute nodal coordinate formulation (ANCF), and the master/slave freedom method is used to obtain the dynamics equations of SLEs without constraint equations. Secondly, according to features of deployable structures, the specification matrix method (SMM) is proposed to eliminate the constraint equations among SLEs in the frame of ANCF. With this method, the inner and the boundary nodal coordinates of element characteristic matrices can be separated simply and efficiently, especially on condition that there are vast nodal coordinates. So the element characteristic matrices can be added end to end circularly. Thus, the dynamic model of deployable structure reduces dimension and can be assembled without any constraint equation. Next, a new iteration procedure for the generalized-α algorithm is presented to solve the ordinary differential equations (ODEs) of deployable structure. Finally, the proposed methodology is used to analyze the flexible multi-body dynamics of a planar linear array deployable structure based on three scissor-like elements. The simulation results show that flexibility has a significant influence on the deployment motion of the deployable structure. The proposed methodology indeed reduce the difficulty of solving and the amount of equations by eliminating redundant degrees of freedom and the constraint equations in scissor-like elements and among scissor-like elements.
[1] W Zhang, J Zhao. Analysis on nonlinear stiffness and vibration isolation performance of scissor-like structure with full types. Nonlinear Dynamics, 2016, 86(1):1-20.
[2] H Tagawa, N Sugiura, S Kodama. Structural analysis of deployable structure with scissor-like-element in architectural design class. IABSE Conference, Nara, Japan, May 13-15, 2015:1-8.
[3] X Sun, X Jing. Analysis and design of a nonlinear stiffness and damping system with a scissor-like structure. Mechanical Systems and Signal Processing, 2016, 66:723-742.
[4] M Takatsuka. General dynamic modeling of a scissor structure for its deployment control in space. International Journal of Space Structures, 2015, 30(3-4):245-259.
[5] T J Li, H Deng, X Ma, et al. Design and analysis of space deployable antennas composed of scissor-like hinges driven by springs. Aiaa/asme/asce/ahs/asc Structures, Structural Dynamics, and Materials Conference, Boston, Massachusetts, April 8-11, 2013:1-7.
[6] N Hua, Z Xie, L Luo, et al. Design and analysis in multiple-scissor-linkage applied to the robotics arm. 3rd International Conference on Mechatronics, Robotics and Automation, Bali, Indonesia, October 11-12, 2015:482-485.
[7] J Zhao, Z Feng, F Chu, et al. Advanced theory of constraint and motion analysis for robot mechanisms. 1st ed. Orlando:Academic Press, Inc., 2013.
[8] J Cai, X Deng, J Feng, et al. Mobility analysis of generalized angulated scissor-like elements with the reciprocal screw theory. Mechanism & Machine Theory, 2014, 82(24):256-265.
[9] Y Sun, S Wang, J Li, et al. Mobility analysis of the deployable structure of sle based on screw theory. Chinese Journal of Mechanical Engineering, 2013, 26(4):793-800.
[10] B Li, S M Wang, C J Zhi, et al. Analytical and numerical study of the buckling of planar linear array deployable structures based on scissor-like element under its own weight. Mechanical Systems & Signal Processing, 2016, 83:474-488.
[11] Y Sun, S Wang, J K Mills, et al. Kinematics and dynamics of deployable structures with scissor-like-elements based on screw theory. Chinese Journal of Mechanical Engineering, 2014, 27(4):655-662.
[12] B Li, S M Wang, R Yuan, et al. Dynamic characteristics of planar linear array deployable structure based on scissor-like element with joint clearance using a new mixed contact force model. ARCHIVE Proceedings of the Institution of Mechanical Engineers, Part C:Journal of Mechanical Engineering Science 1989-1996 (vols 203-210), 2015, 230(18):3161-3174.
[13] D Garcíd, A-Vallejo, J Valverde, et al. An internal damping model for the absolute nodal coordinate formulation. Nonlinear Dynamics, 2005, 42(4):347-369.
[14] P W Likins. Finite element appendage equations for hybrid coordinate dynamic analysis. International Journal of Solids & Structures, 1972, 8(5):709-731.
[15] J G D Jalón, E Bayo. Kinematic and dynamic simulation of multibody systems. 1st ed. New York:Springer, 1994.
[16] J C Simo, L Vuquoc. On the dynamics of flexible beams under large overall motions-the plane case:Part Ⅰ and part Ⅱ. Journal of Applied Mechanics, 1986, 53(4):849-863.
[17] A Shabana. Dynamics of multibody systems. 3rd ed. Cambridge:Cambridge University Press, 2005.
[18] M Berzeri, A A Shabana. Development of simple models for the elastic forces in the absolute nodal co-ordinate formulation. Journal of Sound & Vibration, 2000, 235(4):539-565.
[19] M A Omar, A A Shabana. A two-dimensional shear deformable beam for large rotation and deformation problems. Journal of Sound & Vibration, 2001, 243(3):565-576.
[20] R Y Yakoub, A A Shabana. Three dimensional absolute nodal coordinate formulation for beam elements:Implementation and applications. Journal of Mechanical Design, 2001, 123(4):614-621.
[21] A A Shabana, R Y Yakoub. Three dimensional absolute nodal coordinate formulation for beam elements:Theory. Journal of Mechanical Design, 2001, 123(4):606-613.
[22] A A Shabana. Definition of the slopes and the finite element absolute nodal coordinate formulation. Multibody System Dynamics, 1997, 1(3):339-348.
[23] M Berzeri, A A Shabana. Study of the centrifugal stiffening effect using the finite element absolute nodal coordinate formulation. Multibody System Dynamics, 2002, 7(4):357-387.
[24] M Langerholc, J Slavic, M Boltežar. A thick anisotropic plate element in the framework of an absolute nodal coordinate formulation. Nonlinear Dynamics, 2013, 73:183-198.
[25] A Olshevskiy, O Dmitrochenko, H-I Yang, et al. Absolute nodal coordinate formulation of tetrahedral solid element. Nonlinear Dynamics, 2017, 88(4):2457-2471.
[26] H Yu, C Zhao, B Zheng, et al. A new higher-order locking-free beam element based on the absolute nodal coordinate formulation. Proceedings of the Institution of Mechanical Engineers, Part C:Journal of Mechanical Engineering Science, 2018, 232(19):3410-3423.
[27] G Orzechowski, J Frączek, T Barczak. Physical experiment and computer simulation of the speed-up manoeuvre using the absolute nodal coordinate formulation. Proceedings of the Institution of Mechanical Engineers, Part K:Journal of Multi-body Dynamics, 2018, 232(4):473-480.
[28] P Li, C Liu, Q Tian, et al. Dynamics of a deployable mesh reflector of satellite antenna:Parallel computation and deployment simulation. Journal of Computational and Nonlinear Dynamics, 2016, 11(6):0610051-06100516.
[29] Q Tian, J Zhao, C Liu, et al. Dynamics of space deployable structures. ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Boston, USA, 2-5 August, 2015:1-8.
[30] M Shan, J Guo, E Gill. Deployment dynamics of tethered-net for space debris removal. Acta Astronautica, 2017, 132:293-302.
[31] J Gerstmayr, A A Shabana. Analysis of thin beams and cables using the absolute nodal co-ordinate formulation. Nonlinear Dynamics, 2006, 45(1-2):109-130.
[32] J Liu, J Hong. Nonlinear formulation for flexible multibody system with large deformation. Acta Mechanica Sinica, 2007, 23(1):111-119.
[33] J Chung, G M Hulbert. A time integration algorithm for structural dynamics with improved numerical dissipation:The generalized-α method. Journal of Applied Mechanics, 1993, 60(2):371-375.
[34] M Arnold, O Brüls. Convergence of the generalized-α scheme for constrained mechanical systems. Multibody System Dynamics, 2007, 18(2):185-202.
