滑动轴承支承机电耦联轴系的稳定性分析
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摘要
广泛用于数控机床的电主轴具有结构紧凑、动态精度高等众多优点,已逐渐取代传统结构的变速箱。本文研究轴承-转子系统的动力学问题,主要研究内容包括:1)径向可倾瓦轴承动力系数计算;2)可倾瓦轴承稳态平衡位置搜索;3)轴承—转子系统临界转速确定;4)轴承—转子系统稳定性分析。
可倾瓦径向滑动轴承动力特性系数的计算主要基于对雷诺方程的求解:首先根据力矩平衡条件确定瓦块摆角,用偏导数法得出各瓦块稳态时的动力系数;然后,考虑瓦块摆动对系数的影响,建立静动坐标间动力系数的关系,计算出可倾瓦轴承动力特性系数。针对可倾瓦轴承稳态平衡位置搜索,提出了一种新式运动极坐标系搜索方法,该方法能有效避免搜索过程中的死循环问题,也适用于其它滑动轴承。
不平衡磁拉力的存在会改变轴承—转子系统的临界转速,运用专业软件对电主轴进行磁场分析,提取不平衡磁拉力,以更准确计算轴系的动力特性。本文介绍了两种临界转速的计算方法:1)传递矩阵法。用传递矩阵—多项式法计算临界转速,可以容易地将支承的阻尼包含在传递矩阵中,由于状态向量含有8个元素,因此特征方程阶次较高,为提高低阶根的精度,引入时间因子,借助QR法求解复数特征根;Riccati传递矩阵法将两点边值问题转变为一点初值问题,因此保持了较高精度。2)有限元法计算过程。在有限元分析软件中建立轴系的简化模型,考虑陀螺力矩影响,加载后求解,绘制出系统的CAMPBELL图,提取临界转速。本文还简单介绍了模态振型和不平衡响应的计算过程。
对数衰减率用来度量振幅衰减快慢,反应系统受扰后回复到平衡状态的能力,可以作为衡量轴系稳定性的指标,已被推广到多自由度系统。针对计算的转子系统,本文画出了复特征值对应的对数衰减率曲线,分析了轴系的稳定性。
Motorized spindle widely used in CNC machine tools has the advantages, such as compact structure and higher dynamic accuracy. It is gradually substituting the conventional transmission. Rotor dynamics is studied in this paper. The primary substance is as followings: 1) the calculation of dynamic coefficients of tilting pad journal bearing; 2) an approach searching for the steady-state equilibrium position of tilting-pad journal bearing; 3) the determination of the critical speed of the rotor-bearing system; 4) the stability analysis of a rotor system.
The calculation of dynamic coefficients is based on solving the Reynolds equation of tilting pad journal bearing. Firstly, the pad rotation angle is determined according to the balance of moment. And the method of partial derivative can be applied to calculate dynamic coefficients of tilting pad journal bearing at the equilibrium position. Then the effect of swing pad on the coefficients is not taken into account. After establishing the transformation relation to the coefficients in static coordinate system, the coefficients of tilting pad journal bearing could be calculated. A new method named dynamic polar coordinate is presented to search the steady-state equilibrium position of journal. The method can avoid effectively never-ending loop in process of search and be easily extended to other types of sliding bearings.
The existence of unbalanced magnetic pull will change the critical speed of rotor-bearing system. The electromagnetic field of motorized spindle is simulated through the application of professional software, and the unbalanced magnetic pull is calculated for better dynamic analysis. To calculate the critical speeds of rotor system, this paper covers the following aspects: 1) transfer matrix method. The matrix can easily include the damping of support by applying the transfer matrix-polynomial method to calculate the critical speed. 8 elements of state vector lead to higher order in equation inevitably. The time factor is helpful to improve the accuracy of low-order eigenvalues of the system. The shifted QR algorithm is employed for the complex roots of the characteristic equation. Riccati transfer matrix method changes two-point boundary value problem into one-point initial value one. As a result, it improves the precision. 2) the process of calculating in finite-element method. FEM analysis software is used to model the simplified shafting. The effect of gyroscopic moment is taken into account. After Loading and solving, CAMPBELL diagram is drawn to obtain the critical speeds. This paper also introduces the calculation of mode and unbalance response.
The logarithmic decrement represents the declining rate of amplitude and reflects the ability of perturbed system to return to equilibrium states. It could be used as the measure indices of the stability of spindle, which has been extended to multi-freedom degree system. The logarithmic decrement curve that corresponds to complex eigenvalues is shown in this paper. The stability analysis of the spindle is made finally.
可倾瓦径向滑动轴承动力特性系数的计算主要基于对雷诺方程的求解:首先根据力矩平衡条件确定瓦块摆角,用偏导数法得出各瓦块稳态时的动力系数;然后,考虑瓦块摆动对系数的影响,建立静动坐标间动力系数的关系,计算出可倾瓦轴承动力特性系数。针对可倾瓦轴承稳态平衡位置搜索,提出了一种新式运动极坐标系搜索方法,该方法能有效避免搜索过程中的死循环问题,也适用于其它滑动轴承。
不平衡磁拉力的存在会改变轴承—转子系统的临界转速,运用专业软件对电主轴进行磁场分析,提取不平衡磁拉力,以更准确计算轴系的动力特性。本文介绍了两种临界转速的计算方法:1)传递矩阵法。用传递矩阵—多项式法计算临界转速,可以容易地将支承的阻尼包含在传递矩阵中,由于状态向量含有8个元素,因此特征方程阶次较高,为提高低阶根的精度,引入时间因子,借助QR法求解复数特征根;Riccati传递矩阵法将两点边值问题转变为一点初值问题,因此保持了较高精度。2)有限元法计算过程。在有限元分析软件中建立轴系的简化模型,考虑陀螺力矩影响,加载后求解,绘制出系统的CAMPBELL图,提取临界转速。本文还简单介绍了模态振型和不平衡响应的计算过程。
对数衰减率用来度量振幅衰减快慢,反应系统受扰后回复到平衡状态的能力,可以作为衡量轴系稳定性的指标,已被推广到多自由度系统。针对计算的转子系统,本文画出了复特征值对应的对数衰减率曲线,分析了轴系的稳定性。
Motorized spindle widely used in CNC machine tools has the advantages, such as compact structure and higher dynamic accuracy. It is gradually substituting the conventional transmission. Rotor dynamics is studied in this paper. The primary substance is as followings: 1) the calculation of dynamic coefficients of tilting pad journal bearing; 2) an approach searching for the steady-state equilibrium position of tilting-pad journal bearing; 3) the determination of the critical speed of the rotor-bearing system; 4) the stability analysis of a rotor system.
The calculation of dynamic coefficients is based on solving the Reynolds equation of tilting pad journal bearing. Firstly, the pad rotation angle is determined according to the balance of moment. And the method of partial derivative can be applied to calculate dynamic coefficients of tilting pad journal bearing at the equilibrium position. Then the effect of swing pad on the coefficients is not taken into account. After establishing the transformation relation to the coefficients in static coordinate system, the coefficients of tilting pad journal bearing could be calculated. A new method named dynamic polar coordinate is presented to search the steady-state equilibrium position of journal. The method can avoid effectively never-ending loop in process of search and be easily extended to other types of sliding bearings.
The existence of unbalanced magnetic pull will change the critical speed of rotor-bearing system. The electromagnetic field of motorized spindle is simulated through the application of professional software, and the unbalanced magnetic pull is calculated for better dynamic analysis. To calculate the critical speeds of rotor system, this paper covers the following aspects: 1) transfer matrix method. The matrix can easily include the damping of support by applying the transfer matrix-polynomial method to calculate the critical speed. 8 elements of state vector lead to higher order in equation inevitably. The time factor is helpful to improve the accuracy of low-order eigenvalues of the system. The shifted QR algorithm is employed for the complex roots of the characteristic equation. Riccati transfer matrix method changes two-point boundary value problem into one-point initial value one. As a result, it improves the precision. 2) the process of calculating in finite-element method. FEM analysis software is used to model the simplified shafting. The effect of gyroscopic moment is taken into account. After Loading and solving, CAMPBELL diagram is drawn to obtain the critical speeds. This paper also introduces the calculation of mode and unbalance response.
The logarithmic decrement represents the declining rate of amplitude and reflects the ability of perturbed system to return to equilibrium states. It could be used as the measure indices of the stability of spindle, which has been extended to multi-freedom degree system. The logarithmic decrement curve that corresponds to complex eigenvalues is shown in this paper. The stability analysis of the spindle is made finally.
引文
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[2]孟庆国,陈予恕.基金重大项目“大型旋转机械非线性动力学问题”中期成果总结.力学进展,2001(2):304~306.
[3] N. N. Bhujappa, M.G. Basanagouda. Dynamic Reynolds equation for micropolar fluid lubrication of porous slider bearings. Journal of Marine Science and Technology, 2008, 16(3): 182~190.
[4]魏民祥,马利峰.建立可倾瓦轴承动态雷诺方程的一种新方法.机械科学与技术,1998,17(6):933~934.
[5] Ladislav P?st, Jan Kozánek. Evolutive and nonlinear vibrations of rotor on aerodynamic bearings. Nonlinear Dynamics, 2007, 50(4): 829–840.
[6] R. Tiwari, A.W. Lees, M.I. Friswell. Identification of dynamic bearing parameters: a review. Shock and Vibration Digest, 2004, 36(2): 99~124.
[7] Christian Brecher, Christoph Baum, Markus Winterschladen et al. Simulation of dynamic effects on hydrostatic bearings and membrane restrictors. Prod. Eng. Res. Devel, 2007, 1(4): 415~420.
[8]朱均.滑动轴承油膜非线性动力学分析.太原重型机械学院学报,2004,25(增刊):4~7.
[9]陈峻华,袁慧,陈凌珊等.径向可倾瓦轴承润滑参数的分析方法及应用.广东工业大学学报,1999,16(1):35~38.
[10]陆永忠,廖道训,黄其柏.多层油膜阻尼转子系统动力学建模方法的研究.华中理工大学学报,2000,28(1):18~19,22.
[11]王正.Riccati传递矩阵法的奇点及其消除方法.振动与冲击,1987 (2):74~78.
[12]顾家柳.传递矩阵一直接积分法及其应用.航空学报,1983,4(4):48~56.
[13]黄太平.转子动力学中传递矩阵阻抗耦合法.航空动力学报.1988,3(4):315~318.
[14]黄太平.多转子系统振动的子系统分析方法——阻抗耦合法与分振型综合法.振动工程学报.1988,3(l):30~40.
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