Steady Motion of a Thread Over a Rotating Roller

Dynamic equations of steady motion governing the behavior threads over rotating arbitrary-axisymmetric rollers are derived. Various types of boundary conditions resulting in initial value or boundary problems are discussed. Analytical so­lutions for the case of a circular cylinder are found. Two of the integrals obtained are exact. The third one, being a perturbation result, is thus approximate. Com­parisons of results for a circular cylinder with those for tapered and parabolic rollers are made. 1 Introduction Moving threads (strings or tapes) over rotating rollers can be found in many engineering applications, such as textile and magnetic tape recording technologies. In textile plants, threads move at speeds over 50 m/sec from spinning to winding up. Furthermore, a thread may pass over several rollers which change the direction of the thread, the thread enters and leaves a roller tangentially with some inclination. Similarly, in the magnetic tape recording process, a moving tape always executes small lateral motion along a guided roller. In general, the portion of the thread (or tape) in contact with the roller surface does not lie in a plane. Thus, the equation predicting the tension distribution in a planar thread cannot be applied. A more sophisticated model is required for the prediction of dynamic behavior of the thread over a rotating roller. Dynamic equations governing the behavior of the thread over a rotating circular cylinder were independently derived by Moustafa (1975) and Ono (1979). The following assump­tions were made in the Moustafa's derivation: (a) the motion is steady (i.e., the speed of the thread and the angular speed of the roller are constant); (b) the thread is inextensible and perfectly flexible; (c) there is Coulomb friction between the thread and roller surface with a homogeneous friction coef­ficient; and (d) the thread is in contact with the roller surface and the roller has a constant radius. His derivation ended up with a set of nonlinear ordinary differential equations. The equations were then integrated numerically for cases when the ratio of the thread speed to the tangential component of the roller surface is very close to unity. In the Ono's formulation, he used inhomogeneous friction coefficient in the derivation and ended up with a set of partial differential equations. How-