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.

中文翻译：

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运动腹板振动抑制问题的数值解

摘要

对造纸过程中发生的机械过程进行了建模。在造纸机中，纸张以薄片形式移动。纸张的特征厚度从0.1毫米（办公纸）到1毫米（纸板）不等。每台造纸机都包含开放段，纸幅在没有机械支撑的情况下从一个辊移动到另一个辊而通过。在这样的段上，幅材可能失去稳定性，进行横向振动，结果可能会撕裂。探索了借助各种控制执行器来减少这些振动的可能性。具有非零弯曲刚度的活动腹板的横向振动通过四阶不均匀偏微分方程建模。控制执行器的作用在方程的右侧建模为一个函数。假设整个移动的幅材的振动幅度相同。减振问题减少到了多元函数的最小化。该问题的解决方案分为两个阶段：使用给定控制的初始边界值问题的解决方案和多元函数的最小化。提出了一种数值方法来求解初边值问题。四阶微分方程被简化为两个二阶微分方程的系统。后者通过更改所需功能来简化。通过有限差分方案对所得方程进行逼近，事实证明该方程是绝对稳定的。使用块高斯消除来解决该方案。通过应用Hooke-Jeeves方法将多变量函数最小化。