A New Strategy for Form Finding and Optimal Design of Space Cable Network Structures

Abstract

Cable network structures, which are a class of nonlinear flexible structures, have been widely used in infrastructures and spacecrafts. In this work, a new form-finding method, namely, the fixed nodal position method (FNPM), is developed for optimal design of geometric configuration and internal force distribution for cable network structures, to meet the operation requirement of high shape/surface accuracy. Different from conventional methods, which usually adopts a stress-first-and-displacement-later procedure in form finding, the FNPM first assigns nodal coordinates for a cable network structure and then determines the internal force distribution of the structure by a nonlinear optimization process. The highlight of the FNPM is that the prescribed nodal coordinates are unchanged during the form-finding process. This unique feature of fixed nodal positions makes it possible to place the nodes of a cable network structure at desired locations, satisfying complicated structural constrains and yielding high shape/surface accuracy as required. As another advantage, the FNPM in form finding undertakes the assignment of geometric configuration (nodal coordinates) and the determination of internal force distribution separately. This translates into significant savings in computational effort, compared with conventional form-finding methods. The new form-finding method is applied to the optimal design of a large deployable mesh reflector of 865 nodes.

Appendix

1.1 Global Minimizer and Solution of the Optimization Problem (6.12)

This appendix shows that the solution of internal forces given by Eq. (6.13) and (6.14) is a global minimizer of the optimization problem (6.12).

The general solution of Eq. (6.2) for truss structures of types III and IV with consistent external forces is given by

$$ T=-{M}^{+}{F}_c+{M}_{\mathrm{null}}\alpha $$

(A1)

where M _null is the null space of matrix M and α is an arbitrary s by 1 vector, with s being the states of self-stress. Substituting the solution T into the quadratic objective function in the optimization problem (6.12) yields

$$ f={\left\Vert T-{T}_{\mathrm{des}}\right\Vert}^2={\alpha}^T\left({M}_{\mathrm{null}}^T{M}_{\mathrm{null}}\right)\alpha -2{C}^T{M}_{\mathrm{null}}\alpha $$

(A2)

where C is a constant vector given by

$$ C={M}^{+}{F}_c+{T}_{\mathrm{des}} $$

(A3)

Consider the following lemma, which is proved in [55].

Lemma

Let f _q be a quadratic function given by

$$ {f}_q={\alpha}^T B\alpha +2{g}^T\alpha $$

(A4)

where α is a vector of variables, and B is any symmetric matrix. The following two statements hold true:

• (1) Function f _q attains a minimum if and only if B is positive semi-definite and g is in the range of B. If B is positive semi-definite, then every α satisfying Bα = −g is a global minimizer of f _q.

• (2) Function q has a unique minimizer if and only if B is positive definite.

Note that function f in Eq. (A2) has the same form as function f _q in (A4), with g =  − C ^T M _null and $B={M}_{\mathrm{null}}^T{M}_{\mathrm{null}}$, which is positive definite because all the column vectors M _null are linearly independent. By the lemma, function f has the unique global minimizer given by

$$ {\alpha}^{\ast }={B}^{-1}g={\left({M}_{\mathrm{null}}^T{M}_{\mathrm{null}}\right)}^{-1}\left({C}^T{M}_{\mathrm{null}}\right)={M}_{\mathrm{null}}^T{T}_{\mathrm{des}} $$

(A5)

With α ^*, the internal force distribution T ^*, which is the minimum deviation from the desired internal force distribution T _des, is then obtained as

$$ {T}^{\ast }=-{M}^{+}{F}_c+{M}_{\mathrm{null}}{\alpha}^{\ast } $$

(A6)