Advanced Mathematical Modeling of Woolen Knitting Dynamics Using Laplace Transform and Fourth-Order Runge–Kutta Method

The knitting of wool is a complicated textile process that is driven by nonlinear interactions of yarn tension, loop creation, and needle dynamics. These interactions have a considerable impact on the quality of the fabric as well as the efficiency with which it is produced. The traditional empirical and linearized models often fail to reflect the genuine dynamic behavior of knitting systems, which results in a significant gap in the optimization of the textile process. In order to overcome this constraint, the current research endeavors to establish a sophisticated mathematical framework for modeling the dynamics of wool knitting by using the fourth-order Runge–Kutta (RK4) technique for numerical simulations. The key goals are to build a reliable model that can forecast yarn tension, loop mechanics, and subsystem reliability, as well as to quantify the influence that different subsystem failure and repair rates have on the availability of the system over the long run. The technique incorporates probabilistic reliability modeling using differential equations, which are then solved by means of Laplace transformation and RK4 integration on the basis of boundary conditions. This is then followed by lengthy simulations that are carried out over a period of three hundred and 60 days. The findings indicate that the model achieves an accuracy of 92% when forecasting yarn tension in comparison to experimental benchmarks. Furthermore, the model displays steady performance up to 1.2 m per second for the yarn speed, and it identifies the tumbler subsystem as the most crucial aspect affecting overall dependability and availability. Additionally, it was shown that differences in the failure and repair rates of winders and tumblers had a substantial impact on mean time between failures (MTBFs), which highlighted regions that need focused maintenance. In addition to advancing theoretical knowledge of knitting mechanics, the model that was built also offers a practical decision-support tool that can be used to optimize machine design, predictive maintenance, and intelligent control in the textile manufacturing industry. Its incorporation into industrial settings is in accordance with the concepts of Industry 4.0, which makes it possible to manufacture fabric that is both environmentally friendly and of high quality.