Numerical Solution of the Vibration Suppression Problem for a Moving Web

Abstract

Mechanical processes occurring in paper production are modeled. In a papermaking machine, paper moves in the form of a thin sheet. The characteristic thickness of a sheet varies from 0.1 mm (office paper) to 1 mm (cardboard). Every papermaking machine contains open segments where a paper web passes without mechanical support in its motion from one roller to another. On such segments, the web may lose stability, performing transverse vibrations, and, as a result, might tear. The possibility of reducing these vibrations with the help of various control actuators is explored. The transverse vibrations of a moving web with nonzero bending stiffness are modeled by a fourth-order inhomogeneous partial differential equation. The action of control actuators is modeled a function on the right-hand side of the equation. The vibration amplitude is assumed to be identical across the moving web. The vibration suppression problem is reduced to the minimization of a multivariable function. The solution of the problem splits into two stages: the solution of an initial-boundary value problem with a given control and the minimization of a multivariable function. A numerical method is proposed for solving the initial-boundary value problem. The fourth-order differential equation is reduced to a system of two second-order differential ones. The latter are simplified by changing the sought functions. The resulting equations are approximated by a finite-difference scheme, which is proved to be absolutely stable. This scheme is solved using block Gaussian elimination. The multivariable function is minimized by applying the Hooke–Jeeves method. Examples of computations are given for actuators of three types, namely, point actuators, actuators acting on a web segment, and actuators acting along the entire web.