The Problem of Damping the Transverse Oscillations on a Longitudinally Moving String

Abstract

Introduction. The problem under consideration is relevant to production processes associated with the longitudinal movement of materials, for example, for producing paper webs. For these processes transverse disturbances, which in the vertical section are described by the hyperbolic equation of a longitudinally moving string, are extremely undesirable. That gives the problem of damping these oscillations within a finite time. Materials and Methods. To solve the problem of damping the oscillations, the authors suggest reducing it to the trigonometric problem of the moments at an arbitrary time interval. When considering moving materials, the construction of the basis systems forming the moment problem is a special challenge, since the hyperbolic equation contains a mixed derivative (Coriolis acceleration). Therefore, the classical method of separating variables is not applicable in this case. Instead, a new method is used to find self-similar solutions of non-stationary equations, which makes it possible to find the basis systems explicitly. Results. In the case of paper web, it is necessary to find a minimal in the whole class of admissible perturbations time interval, within which the trigonometric system forming the problem of moments is the Riesz basis, that make it possible through using the system conjugate with it to find the optimal control way in the form of a series and, therefore, to build a so-called optimal damper. Conclusions. As a result of the study, a generalized solution of the problem of transverse oscillations is constructed. For the problem of damping oscillations, the exact damping time is obtained, namely, a time T0 at which the total energy of the system is zero. Optimum control is found in the form of a Fourier series. Keywords: damping oscillations, hyperbolic equation, Coriolis acceleration, trigonometric moment problem, Riesz base For citation: Muravey L. A., Petrov V. M., Romanenkov A. M. The Problem of Damping the Transverse Oscillations on a Longitudinally Moving String. Vestnik Mordovskogo universiteta = Mordovia University Bulletin. 2018; 28(4):472–485. DOI: https://doi.org/10.15507/0236-2910.028.201804.472-485 Acknowledgements: The work was supported by grant No. 16-01-00425 A from the Russian Foundation for Basic Research.