Action Principle and Ritz Method for Yarn Dynamics in Ring Spinning

• Published in: Proceedings in applied mathematics and mechanics Vol. 21; no. 1
• Main Authors: Lenz, Michael, Beitelschmidt, Michael
• Format: Journal Article
• Language: English
• Published: Berlin Wiley-VCH GmbH 01.12.2021
• ISSN: 1617-7061; 1617-7061
• DOI: 10.1002/pamm.202100227

Ring spinning is the most relevant method of producing high quality staple yarns. The rotating yarn forms one or multiple balloon shapes governed by inertial, eyelet friction and drag forces. Traditionally, the stationary yarn path has been computed by direct integration of an equation of motion, and stability and natural oscillations have been assessed by application of Galerkin's method to these ([1], [4], and others). Recent technological advances by using frictionless superconducting magnetic bearings allow for increased process speeds and thus motivate a renewed interest in the dynamics of ring spinning. An integral formulation of the yarn dynamics problem is presented using Hamilton's principle for the one‐dimensional continuum and the magnetic bearing. While therefrom the known equations of motions can be deduced elegantly, here Ritz's method is employed to achieve direct discretization of the problem, with cases to be made for both local and global shape functions. While applicable to the classical study of stationary balloon shapes and their stability, the resulting models in particular enable the study of the instationary yarn path and natural and driven yarn oscillations, which both have received limited attention in literature, but are known to affect yarn tension and thus the process viability.