Accurate and consistent FE modelling of soft docking of micro/nano paired-satellites using variational inequalities

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Accurate and consistent FE modelling of soft docking of micro/nano paired-satellites using variational inequalities Xiang Zhang . S. A. Meguid . Yiyong Huang . Xiaoqian Chen

Received: 11 August 2015 / Accepted: 25 August 2015 Ó Springer Science+Business Media Dordrecht 2015

Abstract Time-dependent variational inequalitybased formulations describing the behavior of elastodynamic contact problems have been used to model the soft docking of micro/nano paired-satellites. The solution strategy is based upon the iterative use of two subproblems. In the first subproblem, the technique of Quadratic Programming was used to predict the contact surface and the stresses acting on it. In the second subproblem, the technique of Lagrange multipliers was used to impose the boundary conditions and contact constraint obtained in step I. The solution accounts for the effect of friction through the use of an appropriate regularization technique in the virtual work expressions. The accuracy of the proposed variational inequality (VI) model of the soft docking process is validated by comparing the VI results with commercially available software. Finally, the effects of the bending stiffness of the docking probe, the friction between the candidate contact surfaces are investigated and analyzed using the newly developed model.

X. Zhang  Y. Huang  X. Chen College of Aerospace Science and Engineering, National University of Defense Technology, Changsha 410073, China X. Zhang  S. A. Meguid (&) Mechanics and Aerospace Design Laboratory, Department of Mechanical and Industrial Engineering, University of Toronto, Toronto ON M5S 3G8, Canada e-mail: [email protected]

Keywords Soft docking  Contact  Variational inequality  Micro/nano satellite  Dynamic  Friction  Quadratic programming  Lagrange multiplier Notations b CT c

dp E1, Ep F FN FT Frc b F g I2 k b k l1, l2 l3 lp M m2

Length from the initial contact point to the inner edge of the docking cone Damping matrix Length from mass center of target satellite to the inner edge of docking cone Diameter of the docking probe Young’s modulus of the chaser satellite and the docking probe External nodal force vector Normal contact force Tangential friction force Resultant contact force Effective load vector Contact gap Moment of inertia of the target satellite Global stiffness matrix Equivalent stiffness matrix Geometry size of the chaser satellite and the target satellite Length of the generatrix of the docking cone Length of the docking probe Global mass matrix Mass of the target satellite

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m, n N O2 Op R2 Rp S T t U u uTb u2, V2 u_ v_ x–y–z xb  yb  zb aH, aB B, c Cc CD Ct e h K l m1, mp q1, qp r X

Number of active contact constraints and freedom Normal vector Mass center of the target satellite Center of the capture ball mounted on the top of the docking probe Radius of the inner edge Radius of the capture ball Point on the candidate contact surface Tangential velocity matrix External applied surface load Global nodal displacement vector Displacement vect