Modelling textile processes discrete models

In contrast to the continuous models, the discrete models consider the processes at the level of individual structural elements, e.g. individual fibres, threads or loops, or individual stages of the process. In these models the processes are modelled as a series of states where the transition from one state to another happens with a probability. The underpinning theories for these models are theory of Markov processes (Kemeny and Snell, 1960), queuing theory (Gross et nf, 2008), and finite automata theory (Anderson, 2006 Hopcroft et al., 2007). 

The knitted structure can be made from three basic structural elements, which are a knitted loop, a tuck loop, and a float (Spencer, 2001). Ihe formation of a knitted structure from these elements on a knitting machine is a result of a sequence of operations with the threads and loops performed 

In this model (Fig. 1.17), each cam track can be considered as an input tape containing input symbols of an algorithm written on the tape. The needle butt is the reading head of the controlling automaton with a stack memory which reads the input symbols and produces an output symbol. The output symbol, which is the direction and the distance of the needle movement, determines the operation to be performed by the needle latch represented by the operational automaton A2. The space under the hook and on the needle stem provides storage of the knitted structural elements and their transfer to the wale or to another needle. The needle can have a hook and a latch at the opposite ends these type of needles is used for the manufacture of purl structures. The structural elements therefore can be moved to the space under the hook from both sides of the needle. According to this, automaton A2 has a memory for the storage of symbols, i.e. the structural elements, in the form of a double-ended 

The finite automaton A2 can be defined by the ordered set as follows (Bratchikov, 1975)  

The basic needle positions relative to its resting position constitute the set of states Q given in Table 1.11 where superscript i = 1,2 denotes the needle bed number. The alphabet of the output symbols can be defined as = (ej, e, e ) where 4 is a float,e is a tuck loop, and 4 is a knitted loop. The alphabet of stack symbols is 17 =, d, d, e,l) where d, is a yarn being fed prior to loop formation, d is the tuck loop held by the 

---

Wang and Sun (2001) developed another numerical method to simulate textile processes and to determine the micro-geometry of textile fabrics. They called it a digital-element model. It models yams by pin-connected digital-rod-element chains. As the element length approaches zero, the chain becomes fully flexible, imitating the physical behavior of the yams. The interactions of adjacent yarns are modeled by contact elements. If the distance between two nodes on different yarns approaches the yam diameter, contact occurs between them. The yarn microstructure inside the fabric is determined by process mechanics, such as yarn tension and interyam friction and compression. The textile process is modeled as a nonlinear solid mechanics problem with boundary displacement (or motion) conditions. This numerical approach was identified as digital-element simulation rather than as finite element simulation because of a special yam discretization process. With the conventional finite element method, the element preserves... 

---