Kinematic Analysis of the Deployable Truss Structures for Space Applications

Transcription

1 Kinematic Analysis of the Deployable Truss Structures for Space Applications Xu Yan 1 *, Guan Fu-ling 1, Zheng Yao 1, Zhao Mengliang 1, Zhejiang University, Hangzhou China Shanghai JiangNan Architectural Design Institute China Abstract: Deployable structure technology has been used in aerospace and civil engineering structures very popularly. This paper reported on a recent development of numerical approaches for the kinematic analysis of the deployable truss structures. The dynamic equations of the constrained system and the computational procedures were summarized. The driving force vectors of the active cables considering the friction force were also formulated. Three types of macroelements used in deployable structures were described, including linear scissor-link element, multiangular scissor element, and rigid-plate element. The corresponding constraint equations and the Jacobian matrices of these macroelements were formulated. The accuracy and efciency of the proposed approach are illustrated with numerical eamples, including a double-ring deployable truss and a deployable solar array. Keywords: Kinematic analysis, Deployable truss structures, Macroelements, Deployable solar array. LIST OF SYMBOLS T X M Q i A h 1, h,, h p T() f() N(), r Kinetic energy of the system; Generalized coordinate vector; Inertia or mass matrix; First order time derivative of X; Vector that includes external and velocity dependent inertia force; Geometrical constrain equations; Jacobian matrix of constrain equations; p independent displacement modes of rigidbody movement; Row vector consists on these combined Length density of the active cable; Internal force vector; Friction force vectors in the contact point; Pressure force vectors in the contact point; Velocity and acceleration of cable length variety; Radius of the pulley; Zheda Road Xihu Hangzhou, Zhejiang China k, k 1 ij INTRODUCTION Half angle between the two active cable element nearby the point; Direction cosine of the cable element; Start and end angle according to contact region; Deployable truss structures have been applied in many applications, such as solar arrays, masts and antennas (Meguro et al., 003) that have small-stowed volumes during launch, and are deployed by certain means to assume a deployable truss structure consists of a large number of struts and kinematic pairs, which are simple, such as revolute and pantograph struts (Cherniavsky et al. deployable structure has many advantages, including lighter weight, higher precision, smaller launch volume, and higher on building the dynamic model, solving the differential J. Aerosp. Technol. Manag., São José dos Campos, Vol.4, No 4, pp , Oct.-Dec.,

2 et al only addressed simple beams and rigid bodies, which are the coordinate partitioning method of constrained Jacobian tangent space method for constrained multibody systems basis vector of the tangent space of constrained surface investigated the mechanism characteristics of deployable truss and tensegrity structures in their literatures (Calladine et al. (000) Newton chord method was employed to solve the equations approach of deployable structures based on the straight- and control methods of the hoop truss deployable antenna were systematically addressed a kinematic analysis method of deployable truss structure based on macroelements, in which EQUATIONS OF MOTION CONSIDERING DRIVING CONDITIONS Dynamic equations for the constrained system For the deployable spatial structure, the dependent Cartesian coordinates are used as generalized ones for the dynamic two revolute joints at the two ends, and the kinetic energy of 1 T T = Xo MXo X: is the generalized coordinate vector, and M T ^ d T dxh T c - - Q = 0 dt Xo m () X Q: is the vector that includes the external and velocity dependent inertia force, X T dx ^MXp - Qh = 0 (3) deployable truss structures, the vector dx, according to the types of constrains of the entire structures, the geometrical U i^xh= 0; i = 1,, g, s Since all constrains of the deployable structure are constant with time t during the deployment process, the derivative of AXo = 0 (5) A: is the corresponding Jacobian matrix of constraint Xo = ao 1h1+ ao h+ gao php = Hao h, h,, h p : are p independent displacement modes of the rigid-body movement, : is a row vector that consists of these combined 454 J. Aerosp. Technol. Manag., São José dos Campos, Vol.4, No 4, pp , Oct.-Dec., 01

3 Xp = Hap - A + AX o o Xo = Hao H MHap -H T MA AH o ao - H Q = 0 With the initial condition: Xt 0 = X, Xo = 0 t= 0 = Xo0 When the initial displacement and velocity vectors are known, the vector t=0 and acceleration of the deployable process in each time step simple and can be summarized as follows: simulation, such as the coordinates of the joints, structural topology, constraint conditions, driving mechanisms, boundary conditions and the length of time step, and so on, are provided; n, the mass matrix and driving force vectors are formed; Jacobian matrix are formulated; space of are determined; has full column, the rank will be estimated, if so go to the ninth step; displacement, velocity, and acceleration of all joints are obtained; not, go to the second step; otherwise, go to the ninth step; step, otherwise, the analysis should be stopped; Active cable driving and friction force vectors of the active cables, which forms the term Q in cables are driven by the motor, the cable length becomes the active cables will become smaller after it loops over the pulley, so the Coulomb friction law is employed to consider is assumed as T active cable and the pulley in the joints, driving forces of the cables in each deployable element are T,, T k,, T n T > > T k > > T n. and a microarc element ds in the contact point between the length density of the active cable is and the internal force vector is denoted as T( vectors in the contact point are denoted as f () and N (), (a) contact domain (b) microarc element J. Aerosp. Technol. Manag., São José dos Campos, Vol.4, No 4, pp , Oct.-Dec.,

4 et al * T^i+ dihcos di-t^ih- f^ih= dsl p T^i+ dihsin di = N^ih + lo ds r and : are the velocity and acceleration of cable length variety, r: is the radius of the pulley, : is the half angle between the two active cable elements the result: d and the limit is gotten by d. dt di = nt + ^prlp-nplo h provided as k and k # Tk 1 dt i - k 1 di Tk nt + rlp = - -nlo # ^ h i k and ij in a ntk 1 + ^rlp-nlo h ntk + ^rlp-nlo h - k 1 k = ni ^ - - i h e 1 T 6 ^Xi-Xjh ^Xi- Xjh@ = l Xo i lo = m^xo j- Xo ih= 6 -m m@) Xo 3 j k k =, and the relation of the internal force of cable in two adjacent deployable elements can be ntk 1 + ^rlp-n lo h ntk + ^rlp-n lo h - = nr ^ - b h e (0) = l 1 (Xj X i ) into equations of motion, being the second one employed: f^ih= N^ih cable and the pulley in the joint j fk-1 = Tk-1-Tk Q e c Q e i Q e T = " j, () Such force of the vector Q of the entire structure is obtained 456 J. Aerosp. Technol. Manag., São José dos Campos, Vol.4, No 4, pp , Oct.-Dec., 01

5 JACOBIAN MATRICES OF THE MACROELEMENTS In this section, some deployable macroelements are investigated, and the corresponding Jacobian matrices are matrices for a constant distance constraint on the members, the position constraint of the sleeve element, the angle constraint of the revolute joints and of synchronize gears have the formulations are not fully developed here, therefore see Nagaraj et al. Linear scissor-link element macroelement in the deployable truss structures, two pairs to rotate freely around the axis perpendicular to their common il and oi is uniplets ij and lk, and the Jacobian matrix is formulated as A e 1 T T mij mij 0 0 = = - G T T (3) 0 0 -mkl mkl the direction cosine of the uniplets ij is 1 ij = (Xj X Lij i ), like the kl It is assumed that the length of oi, ok are a and the length of oj, ol are k*a other at the point o relative position of the connection point o ^ a a Xi - Xjh Xi Xk Xl Xl a 1+ + = ^ - h ^ k h a ^1+ + k h kxi+ X j-kxk- Xl = 0 (5) Differentiating it with respect to X, therefore the Jacobian matrix is obtained: e A = " ki3# 3 I3# 3 -ki3# 3 -I3# 3, deployment process, the planar equation is the constraint ^rij # rikh: ril = 0 When differences are compared with respect to X, the e A = " ^ril # rjk - rij # rikh # ^rik # rilh # ^rij # rilh ^rij # rikh #, A e 3 Ae = A e 3 A e 3 Planar multiangular scissor element Planar multiangular scissor element, as illustrated in truss structures, in which the uniplets ij and lk are not aligned at an intermediate point O constraint equations of the planar multiangular scissor J. Aerosp. Technol. Manag., São José dos Campos, Vol.4, No 4, pp , Oct.-Dec.,

6 et al element include: io, jo, ko and lo of the macroelement are considered and there are four constant distance constraint equations; ij and lk are added to the macroelement; thus, there are two constant distance constraint equations; constraint, which can be formulated by the same method For the planar multiangular scissor element, the row number of the Jacobian matrix A e Rigid-plate element Planar rigid-plate element is a type of macroelement used ijkl is shown in degree of the element is analyzed: the degree of freedom of four spatial points i, j, k, l six struts are appended, the total degree of freedom becomes Six struts ij, jk, kl, li, ik and jl of the element are considered, the planar-plate element, the row number of the Jacobian matrix A e is deployed by a motor and becomes shorter, the diagonal Several planar truss elements can make a closed loop by topology is determined and the major design parameters of the NUMERICAL EXAMPLES Double-ring deployable truss A type of double-ring deployable truss based on quadrilateral elements is investigated for large-size mesh simulated by the program developed based on numerical 458 J. Aerosp. Technol. Manag., São José dos Campos, Vol.4, No 4, pp , Oct.-Dec., 01

7 between the active cables and the When the lengths of the diagonal sleeves are equal to a designed value, the locked constraint conditions of the sleeves work and (a) x axes coordinate (b) y axes coordinate J. Aerosp. Technol. Manag., São José dos Campos, Vol.4, No 4, pp , Oct.-Dec.,

8 et al which are obtained from the geometric equation of this Deployable sail arrays macroelements are dynamic machines during deployment and each kink) and, after passing over a pulley at joint, is connected = = dts is used in are equal to a designed value, the locked constraint conditions 460 J. Aerosp. Technol. Manag., São José dos Campos, Vol.4, No 4, pp , Oct.-Dec., 01

9 x-coordinate (m) y-coordinate (m) Time (s) (a) x axes coordinate Time (s) (b) y axes coordinate CONCLUSIONS forces of active cables are combined with the equations of equations and Jacobian matrices of the macroelements are deployment process and dynamic parameters at each time step can be simulated for evaluating the deployment behaviors of the capabilities of this method in the motion analysis are of error in the time step of the simulation is too large to stop works are suggested: the reliability analysis of the deployment process can be researched; and deployment control of the REFERENCES et al., 000, An explicit integration method for real time simulation of multibody vehicle models, mass-orthogonal projection method for constrained multibody for constrained multibody systems, Computer Methods In et al., 005, Large deployable space antennas based on usage of polygonal pantograph, Journal of of multibody system dynamic equations using the coordinate partitioning method in an implict Newmark scheme, J. Aerosp. Technol. Manag., São José dos Campos, Vol.4, No 4, pp , Oct.-Dec.,

10 et al for state space representation of constrained mechanical space method for constrained dynamic analysis, Journal et al., 003, Key technologies for high-accuracy et al. et al. approach for static analysis of pantograph masts, Computers decomposition for constrained dynamical systems, Journal concepts for large antenna structures or solar concentrators, method, Proceedings of the Institution of Mechanical deployable toroidal spatial truss structures for large mesh antenna, Journal of the International Association for Shell 46 J. Aerosp. Technol. Manag., São José dos Campos, Vol.4, No 4, pp , Oct.-Dec., 01