胶印机齿轮系统非线性模型与动力学分析

For the nonlinear dynamic characteristics of gear system for offset press, Using the lumped mass method, Newton's second law, the differential equations and series theory and method, carry out the gear system nonlinear dynamic modeling and analysis of offset printing machine algorithm research.. First of all, to offset a pair of transmission gear as object, a nonlinear dynamic model of gear system for offset printing machine with single degree of freedom was established wherein the time-varying meshing stiffness, backlash, dynamic transmission error and other nonlinear factors were considered. Secondly, to offset the single color offset printing machine gear system as object, a nonlinear dynamic model of gear system for a single color offset printing machine with three degree of freedom was established wherein the gear system backlash and rolling bearing radial clearance were considered, and assuming the shaft and bearings symmetrically on the gear center plane. Then, to offset the gear system of a multi-color offset printing machine as the object, a nonlinear dynamic model of the gear system for multi-color offset printing machine with multi degree of freedom was established. Finally, Considering the exact solution of nonlinear dynamic model for offset printing machine cannot be obtained,the right side of differential equation is transferred into Taylor expansion, and integrated directly,the any order approximate solutions can be obtained, according to the different accuracy requirement, the different algorithm can be applied. In this project, by establishing and analyzing the nonlinear dynamic model of the gear system for offset press, and the new analyzing method of nonlinear dynamic system for offset press is provided.

针对胶印机齿轮系统的非线性动态特性，运用集中质量法、牛顿第二定律、微分方程及级数理论和方法，深入开展胶印机齿轮系统非线性动力学建模及其分析算法研究：首先，以胶印机一对传动齿轮为对象，考虑变啮合刚度、齿侧间隙、动态传递误差等非线性因素，建立胶印机齿轮系统单自由度非线性动力学模型；其次，以单色胶印机齿轮系统为对象，考虑齿轮系统存在齿侧间隙和滚动轴承的径向间隙，假设传动轴和轴承均对称于齿轮中心平面，建立单色胶印机齿轮系统三自由度非线性动力学模型；然后，以多色胶印机齿轮系统为对象，建立多色胶印机齿轮系统多自由度非线性动力学模型；针对胶印机非线性动力学模型不能得到精确解的问题，通过对其微分方程的右边关于自变量进行泰勒展开，然后进行直接积分，可得到任意级近似解，根据不同的计算精度要求，可以得到不同的求解算法。通过对胶印机齿轮系统的非线性动力学建模、分析，为胶印机的非线性动力学分析提供了一种新的方法。

胶印机的主传动普遍采用齿轮传动，齿轮副的非线性模型是研究印刷机非线性动力学模型的基础。本项目以胶印机齿轮系统为研究对象，运用集中质量法、牛顿第二定律、微分方程及级数理论和方法，深入开展胶印机齿轮系统非线性动力学建模及其分析算法研究，取得了以下成果：.1）建立了单色胶印机单自由度齿轮系统非线性动力学微分方程。仅考虑轮齿的扭转振动, 忽略传动轴的横向和轴向弹性变形以及支承系统的弹性变形等因素,采用集中质量法，建立了系统的非线性动力学微分方程。.2）建立了单色胶印机多种非线性因素的三自由度齿轮系统非线性动力学微分方程。考虑齿轮传动系统存在齿侧间隙和滚动轴承的径向间隙，假设传动轴和轴承均对称于齿轮中心平面，将单色胶印机齿轮传动系统简化为具有3个自由度的齿轮振动系统，用由3个三自由度齿轮系统的非线性微分方程组来描述。.3）单色胶印机单自由度齿轮系统非线性动力学模型直接积分解。将单色胶印机单自由度齿轮系统非线性动力学微分方程，变为广义状态空间上的系统状态方程。在广义状态空间将方程的右端展开为时间的 Taylor 级数, 进一步直接积分获得非线性控制系统状态方程关于自变量时间的级数解，最终获得状态方程关于时间的无穷级数解。求解过程中除了被积分函数用Taylor级数展开之外没有用其他任何近似，所得结果十分简洁，可以对其精确解进行任意严格逼近。.4）在MATLAB软件中对单色胶印机单自由度齿轮系统非线性动力学模型直接积分解微分方程的解析解进行仿真，得到齿轮转动的角速度与角加速度与时间关系曲线。将得到的曲线与实际情况进行对比，验证了动力学模型与状态方程解析解的正确性。也证明了直接积分解法能够适合于一般的非线性系统。