A General Energy Analysis of Fabric Mechanics Using Optimal Control Theory

Abstract

A general energy analysis has been formulated and applied to the study of fabric mechanics with the aid of computer programs. The analysis is chiefly concerned with the elastic or recoverable mechanisms of fabric rheology. The computer programs can be readily adapted to provide a solution for various fabric structures and deformations simply changing the boundary conditions.The energy analysis is based on the fundamental principle that elastic structures alway s assume a confguration of minimum strain energy, regardless of the deformation applied. The resulting minimization problem is solved with the use of a specific optimization technique -i.e., it is treated as an optimal control problem. In particular, the total yarn strain encrgy (consisting of the sum of the individual yarn bending. torsion, lateral compression, and longitudinal extension strain energies) is minimized, subject to certain constraints which give rise to internal forces and couples acting within the fabric structure itself. The constraints lead to the necessity of optimizing the "total" energy, or the sum of the potential energy due to the internal forces of constraint, and the yarn strain energy comprising the bending, torsion, and extension energies. < , at UQ Library on June 20, 2015 trj.sagepub.com Downloaded from

A Model for Predicting Set in Helices

“…The derivation of the equations of equilibrium for the helix constrained to lie on the surface of a cylinder is based on the principle of minimum energy, which we have discussed in detail elsewhere [3,4]. The neotation we used previously [3,4] is also employed here, with a number of exceptions indicated below.…”

“…The neotation we used previously [3,4] is also employed here, with a number of exceptions indicated below.…”

“…If the fiber axis when simply unbent has a pretwist m (m here refers to m3 used previously [3,4]), the torsion of the fiber axis is The bending strain energy Ek and the torsional strain energy El are given by B, and C, are the bending rigidity and torsional rigidity, respectively, of the fiber axis (B and G used previously [3,4]). The pammeters % and l~ are the initial curvature and torsion of the helix prior to wrapping it around the mandrel.…”

“…Second, the equations of equilibrium for a helix forced to lie on the surface of a mandrel by a constraining couple are established by means of the principle of minimum energy [3,4]. Third, the effect of inserting twist into the helical fiber axis, prior to wrapping it around the mandrel, is included in the analysis.…”

“…CONSTRAINED HELIX Suppose a fiber (or yam) is held in a helical shape 1 by wrapping it around the surface of a circular cylinder by means of a couple K, and a helix axial force 7!t. The appendix shows that using minimum energy principles [3,4], the equation of equilibrium of the helix in state 1 is now In contrast to Equations 1 and 2, in this case there is only one equation of equilibrium since only the angle a, is unspecified, the radius r, being fixed. This equation is equivalent to one derived by Konopasek and Hearle [11] using a force method of analysis.…”