运动纸带动力学及振动控制研究
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摘要
轴向运动系统横向振动和稳定性的研究有着重要的实际应用价值。但已有研究的对象主要为轴向运动的弦线、梁以及薄板。对运动纸带的研究相对较少。本文将运动纸带作为研究对象,对具有弹性边界、中间刚性支承及中间弹性支承的运动矩形纸带的横向振动和稳定性问题、加速运动矩形纸带的稳定性区域、运动矩形纸带的振动控制、纸张的振动断裁技术等问题进行了系统的研究。具体研究工作如下：
(1)研究了边界上具有弹性约束的运动矩形纸带的自由振动。将伽辽金无网格法应用于求解具有弹性约束运动矩形纸带的横向振动问题。基于弹性动力学Hamilton原理的推广,采用伽辽金无网格法建立了具有弹性约束边界的运动矩形纸带横向振动的无量纲运动方程和特征方程。通过数值计算给出了纸带的振动频率随纸带运动速度及弹性约束刚性系数的变化曲线,分析了纸带速度及弹性约束对运动矩形纸带的横向振动特性的影响。
(2)研究了具有中间刚性支承及弹性支承的运动矩形纸带的横向振动特性和稳定性问题。将Laplace变换方法应用于求解具有中间刚性支承及弹性支承的运动矩形纸带的横向振动问题。根据D'Alembert原理,导出了具有中间刚性支承及弹性支承的运动矩形纸带的运动微分方程和特征方程。用Laplace变换的方法求得了四边固支条件下运动纸带的振动频率,并获得了前3阶振动频率与纸带运动速度、纸带张力、弹性支承刚度系数、中间支承位置之间的关系。
(3)研究了线性及正弦半波型变密度运动矩形纸带的动力学特性及稳定性问题。通过引入无量纲量,得到了变密度运动矩形纸带的无量纲运动微分方程。采用微分求积法离散无量纲振型微分方程和边界条件,得到了该问题的复特征方程。通过数值计算,得到了系统的前3阶振动频率,并分析了纸带的张力比、长宽比、无量纲运动速度及纸带的密度系数对其横向振动及稳定性的影响。
(4)研究了纸带的运动速度为平均速度与简谐速度的叠加时加速运动纸带的稳定性区域的变化问题。分析了平均速度、简谐运动速度的幅值、纸带张力比、纸带长宽比对加速运动纸带的动力稳定性区域的影响。
(5)研究了变密度运动矩形纸带的横向振动控制问题。利用有限差分法,对变密度纸带的动力学方程进行离散,导出了运动矩形纸带横向振动控制系统的状态方程。应用次最优控制法,对运动矩形纸带的横向振动进行控制。仿真分析了实施控制后运动矩形纸带横向振动的响应和衰减过程。结果表明,次最优控制方法能够有效地控制运动矩形纸带的横向振动。
(6)对纸张断裁的振动裁切技术进行了研究。实现了切纸机的振动裁切新工艺,较大地降低了纸张对刀床的裁切抗力和机器受到的猛烈冲击。设计研制了振动裁切试验台,搭建了振动测试系统。通过理论计算和试验得到了最优激振频率。
The study of the vibration and stability of the axially moving systems is of great significance. The main existing subjects are mainly about the axially moving beam, the axially moving column and the axially moving plate. Few papers have been presented on the moving paper web. In this paper, the moving paper web are the object of study. The tranverse vibration and stability of the moving paper web with elastically restrained edge, intermediate rigid support and intermediate elastically support are researched. The stability region of the accelerating paper web, the control of the moving rectangular paper web and the technique of vibration cutting for paper are analyzed respectively in this paper. The main research works are as follows.
(1) The vibration problems of the moving rectangular paper web with the edge attached to distributed elastic restraint are studied. The element-free Galerkin method is proposed to solve the transverse vibration of the moving rectangular paper web with elastically restrained edge. Based on the extended Hamilton's principle for the elastic dynamics system, the dimensionless equations of motion of the moving rectangular paper web with elastically restrained edge are established by the element-free Galerkin method, and the eigenvalue equations are presented. Via numerical calculation, the curves of the vibration frequency of the moving rectangular paper web versus dimensionless moving speed and the elastic stiffness of edge elastically restrained are obtained. The effects of moving speed and edge elastically restrained on the free vibration characteristics of the moving rectangular paper web are analyzed as a conclusion.
(2) The tranverse vibration and stability of the moving paper web with intermediate rigid support and intermediate elastically support are researched respectively. The Laplace transforms method is proposed to solve the transverse vibration of the moving rectangular paper web with intermediate rigid support and elastic support. According to D Alembert theory, the differential equation of motion and the eigenvalue equations of the moving rectangular paper web with intermediate rigid support and elastic support are obtained. The vibration frequencies on the condition of four edges clamped are obtained by the Laplace transforms method. The relationships of the first three order steps of vibration frequency versus velocity, tension, the stiffness of intermediate elastic support and intermediate support position of paper web are obtained.
(3) The dynamics and stability of the moving rectangular paper web with linearly and half-sinusoid varying density are investigated. The dimensionless equation of motion of the moving rectangular paper web with varying density is obtained by introducing dimensionless variables and parameters. The complex eigen-equation is derived by the discretization of the dimensionless equation of vibration modes and boundary conditions using the Differential Quadrature method. The first three order steps of vibration frequency of the system is obtained by numerical calculation and the effects of the tension ratio, aspect ratio, the dimensionless moving speed and density coefficient on the dynamic behavior and stability of the moving rectangular paper web are analyzed.
(4) The stability region of the accelerating paper web is developed when the speed is superposition by mean velocity and harmonic velocity. The influence on the stability region of the accelerating paper web exerted by the mean velocity, the amplitude of harmonic velocity, tension ratio of paper web and aspect ratio is investigated.
(5) The transverse vibration control of the moving rectangle paper web with varying density is studied. The state equation of transverse vibration control system is obtained by using finite difference calculus to separate the dynamical equation. The suboptimum method is used to control the transverse vibration of the moving rectangle paper web. Then the response and the attenuation state of the transverse vibration of the paper web are presented by simulation analysis. It shows that suboptimum method can control the transverse vibration of the moving rectangle paper web effectively.
(6) The use of vibration cutting technology in the printing machinery is studied. The new technology of vibration cutting in cutter is realized. The cutting resisting force of cutting bed and the severe shock of cutter beared are reduced significantly. The test-bed of vibration cutting is designed, and the vibration test system is built. The optimum excited frequency has been obtaind on the basis of theoretical calculation and experiments.
(1)研究了边界上具有弹性约束的运动矩形纸带的自由振动。将伽辽金无网格法应用于求解具有弹性约束运动矩形纸带的横向振动问题。基于弹性动力学Hamilton原理的推广,采用伽辽金无网格法建立了具有弹性约束边界的运动矩形纸带横向振动的无量纲运动方程和特征方程。通过数值计算给出了纸带的振动频率随纸带运动速度及弹性约束刚性系数的变化曲线,分析了纸带速度及弹性约束对运动矩形纸带的横向振动特性的影响。
(2)研究了具有中间刚性支承及弹性支承的运动矩形纸带的横向振动特性和稳定性问题。将Laplace变换方法应用于求解具有中间刚性支承及弹性支承的运动矩形纸带的横向振动问题。根据D'Alembert原理,导出了具有中间刚性支承及弹性支承的运动矩形纸带的运动微分方程和特征方程。用Laplace变换的方法求得了四边固支条件下运动纸带的振动频率,并获得了前3阶振动频率与纸带运动速度、纸带张力、弹性支承刚度系数、中间支承位置之间的关系。
(3)研究了线性及正弦半波型变密度运动矩形纸带的动力学特性及稳定性问题。通过引入无量纲量,得到了变密度运动矩形纸带的无量纲运动微分方程。采用微分求积法离散无量纲振型微分方程和边界条件,得到了该问题的复特征方程。通过数值计算,得到了系统的前3阶振动频率,并分析了纸带的张力比、长宽比、无量纲运动速度及纸带的密度系数对其横向振动及稳定性的影响。
(4)研究了纸带的运动速度为平均速度与简谐速度的叠加时加速运动纸带的稳定性区域的变化问题。分析了平均速度、简谐运动速度的幅值、纸带张力比、纸带长宽比对加速运动纸带的动力稳定性区域的影响。
(5)研究了变密度运动矩形纸带的横向振动控制问题。利用有限差分法,对变密度纸带的动力学方程进行离散,导出了运动矩形纸带横向振动控制系统的状态方程。应用次最优控制法,对运动矩形纸带的横向振动进行控制。仿真分析了实施控制后运动矩形纸带横向振动的响应和衰减过程。结果表明,次最优控制方法能够有效地控制运动矩形纸带的横向振动。
(6)对纸张断裁的振动裁切技术进行了研究。实现了切纸机的振动裁切新工艺,较大地降低了纸张对刀床的裁切抗力和机器受到的猛烈冲击。设计研制了振动裁切试验台,搭建了振动测试系统。通过理论计算和试验得到了最优激振频率。
The study of the vibration and stability of the axially moving systems is of great significance. The main existing subjects are mainly about the axially moving beam, the axially moving column and the axially moving plate. Few papers have been presented on the moving paper web. In this paper, the moving paper web are the object of study. The tranverse vibration and stability of the moving paper web with elastically restrained edge, intermediate rigid support and intermediate elastically support are researched. The stability region of the accelerating paper web, the control of the moving rectangular paper web and the technique of vibration cutting for paper are analyzed respectively in this paper. The main research works are as follows.
(1) The vibration problems of the moving rectangular paper web with the edge attached to distributed elastic restraint are studied. The element-free Galerkin method is proposed to solve the transverse vibration of the moving rectangular paper web with elastically restrained edge. Based on the extended Hamilton's principle for the elastic dynamics system, the dimensionless equations of motion of the moving rectangular paper web with elastically restrained edge are established by the element-free Galerkin method, and the eigenvalue equations are presented. Via numerical calculation, the curves of the vibration frequency of the moving rectangular paper web versus dimensionless moving speed and the elastic stiffness of edge elastically restrained are obtained. The effects of moving speed and edge elastically restrained on the free vibration characteristics of the moving rectangular paper web are analyzed as a conclusion.
(2) The tranverse vibration and stability of the moving paper web with intermediate rigid support and intermediate elastically support are researched respectively. The Laplace transforms method is proposed to solve the transverse vibration of the moving rectangular paper web with intermediate rigid support and elastic support. According to D Alembert theory, the differential equation of motion and the eigenvalue equations of the moving rectangular paper web with intermediate rigid support and elastic support are obtained. The vibration frequencies on the condition of four edges clamped are obtained by the Laplace transforms method. The relationships of the first three order steps of vibration frequency versus velocity, tension, the stiffness of intermediate elastic support and intermediate support position of paper web are obtained.
(3) The dynamics and stability of the moving rectangular paper web with linearly and half-sinusoid varying density are investigated. The dimensionless equation of motion of the moving rectangular paper web with varying density is obtained by introducing dimensionless variables and parameters. The complex eigen-equation is derived by the discretization of the dimensionless equation of vibration modes and boundary conditions using the Differential Quadrature method. The first three order steps of vibration frequency of the system is obtained by numerical calculation and the effects of the tension ratio, aspect ratio, the dimensionless moving speed and density coefficient on the dynamic behavior and stability of the moving rectangular paper web are analyzed.
(4) The stability region of the accelerating paper web is developed when the speed is superposition by mean velocity and harmonic velocity. The influence on the stability region of the accelerating paper web exerted by the mean velocity, the amplitude of harmonic velocity, tension ratio of paper web and aspect ratio is investigated.
(5) The transverse vibration control of the moving rectangle paper web with varying density is studied. The state equation of transverse vibration control system is obtained by using finite difference calculus to separate the dynamical equation. The suboptimum method is used to control the transverse vibration of the moving rectangle paper web. Then the response and the attenuation state of the transverse vibration of the paper web are presented by simulation analysis. It shows that suboptimum method can control the transverse vibration of the moving rectangle paper web effectively.
(6) The use of vibration cutting technology in the printing machinery is studied. The new technology of vibration cutting in cutter is realized. The cutting resisting force of cutting bed and the severe shock of cutter beared are reduced significantly. The test-bed of vibration cutting is designed, and the vibration test system is built. The optimum excited frequency has been obtaind on the basis of theoretical calculation and experiments.
引文
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