可倾瓦轴承动力学建模及动力特性研究
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摘要
转子—轴承系统的建模是转子动力学设计的基础。动压流体轴承非线性油膜力的建模是转子—轴承系统数学模型的建立及其求解，以及系统非线性动力学行为研究的基础。
本文研究可倾瓦径向滑动轴承更为合理的动力学模型及相应的高效算法。目前绝大多数相关研究仅针对可倾瓦轴承的线性化折合刚度、阻尼系数的动力分析模型，本文在单瓦非线性油膜力及其Jacobian矩阵的计算模型、可倾瓦轴承的完整动力特性分析模型以及转子—可倾瓦轴承系统线性和非线性动力学分析模型等几个方面，进行了建模及算法的研究。本文研究工作在理论及应用上取得了一些创新成果，所获得的结论对可倾瓦轴承—转子系统运动稳定性分析和工程应用具有积极的指导意义。
具体地，本文主要的研究内容和成果有：
1．针对滑动轴承的动力学行为研究，提出了求解非线性油膜力的加权有限元互补算法。基于流体润滑的变分不等方程理论，考虑油膜厚度对动力参数的影响，选取权函数与线性插值函数乘积的形式作为油膜压力函数，建立有限元离散化的一维变分不等方程。根据雷诺方程的一维互补方程系数矩阵为三对角矩阵的特点，用修正的追赶法求解对角元素占优的三对角方程，快速且无需迭代地直接求得满足Reynolds边界条件的油膜压力分布。算例结果表明，由于采用了权函数，只需非常稀疏的一维单元划分即可获得与二维有限元算法相媲美的精度，而计算耗时仅为后者的数千分之一。
2．基于可倾瓦径向滑动轴承瓦块的扰动特性，首次提出了计算可倾瓦轴承完整动力特性系数的数学解析模型。在全局固定坐标系和局部动坐标系之间建立合适的坐标转换关系，获得作用于每块瓦和轴颈的油膜力及其Jacobian矩阵在全局坐标系下的解析表达式；其中在固结于瓦块的局部动坐标系下，单块瓦的油膜力及其Jacobian矩阵利用求解单瓦非线性油膜力的加权有限元方法求出；全局坐标系下所有子系统的油膜力向量关于广义位移和广义速度的Jacobian矩阵经过扩充后，叠加组合得到轴承油膜力的全局Jacobian矩阵。而此全局Jacobian矩阵可以用于具有快速收敛特性的Newton-Raphson方法来求解所有瓦块和轴颈的静态平衡位置，在得到可倾瓦轴承平衡位置的同时获得轴承完整的刚度和阻尼系数矩阵。在设定瓦块和轴颈的扰动频率后，此解析模型亦可以获得与传统算法一致的折合刚度和阻尼系数矩阵。本模型采用矩阵表达形式，计算过程清晰简洁，计算耗时较少，适用于转子—可倾瓦轴承系统的线性和非线性分析。
3．针对可倾瓦轴承—转子系统的动力学研究，采用完整动力分析模型，对系统进行线性和非线性分析。在线性和分线性的分析中，首次将轴颈振动和瓦块摆动的动力学行为同时纳入研究过程。通过线性化的运动微分方程，采用数值方法分析静态平衡位置的稳定性。非线性周期响应则采用打靶法直接积分求解，并通过Floquent理论判断其稳定性。通过系统线性与非线性动力行为分析结果对比以及系统非线性动力学行为的考察，得到如下结论：
1)可倾瓦轴承—转子系统并不具有‘本质稳定性’，将完整动力特性分析模型应用于系统动力学行为分析，可以计算其临界转速。
2)可倾瓦轴承的某些参数对转子系统的稳定性有重要影响，特别是几何预负荷和瓦块支点偏置比，当几何预负荷为零或者瓦块支点位于瓦块中间位置时，可倾瓦轴承的稳定性降会大为降低。
3)可倾瓦轴承—转子系统的非线性动力行为有别于固定瓦滑动轴承—转子系统。可倾瓦轴承在进入非线性分岔后，在相当长的频段内处于2倍周期运动的状态，在转速较高时，可倾瓦的扰动反馈作用对转子的影响非常明显，轴心轨迹呈现出多角盘绕的现象。
4)可倾瓦轴承的非线性失稳转速远低于线性失稳转速，随着转子不平衡量的增加，系统的非线性运动特性增强。
5)当转子的运转频率低于非线性失稳转速，且转子的不平衡量较小时，对刚性转子—可倾瓦轴承系统的不平衡响应进行非线性分析的计算结果与线性动力学所得的结果吻合较好。基于线性动力学可以用来计算转子—可倾瓦轴承系统在稳态、小不平衡量下的响应；转子转速超过分岔转速或者不平衡量较大时，线性分析的结果不能正确表现出系统的动力行为，应采用完整非线性动力学模型，研究转子—可倾瓦轴承系统的不平衡响应等非线性动力学行为。
Dynamic modeling of rotor-bearing systems is the base of rotor dynamic analysis. The most important element of the mathematical model for rotor-bearing systems is the modeling of nonlinear fluid-force in hydrodynamic bearings, which also is the precondition of nonlinear analysis for rotor dynamic behaviors.
This thesis focuses on a better dynamic model of tilting-pad journal bearings and an efficient algorithm involved. Almost all of the relevant researches take the reduced stiffness and damping coefficients of tilting-pad bearings as research objects, comparatively, the full dynamic behaviors of tilting-pad bearings is studied here. This thesis studies on several aspects, such as, the efficient algorithm of nonlinear oil-film forces for a single pad, the analysis model and algorithm for the full dynamic coefficients of tilting-pad bearings, and the model of rotor dynamics of rotor-tilting-pad-bearing system. Some achievements are acquired in current research from theory and practice, and conclusions that are reached here may be beneficial to the engineering applications in stability of rotor motion.
In detail, there are the following contents and achievements in current research.
1. Based on the variational inequality theory of hydrodynamic lubrication, this thesis presents a weighted finite element algorithm to calculate the oil-film forces of journal bearings. Choosing a felicitous weighted function, two dimensional questions could be interpolated by one-dimensional elements with high accuracy. An amendatory direct-method is proposed to solve the equations whose coefficient matrices in tri-diagonal form. Then based on the interior characteristics, the solution of oil-forces is united with the solution of the Jacobian matrices. During this process, the operations are restricted within the positive pressure region, and no iteration is involved. Therefore, many redundant computations are avoided through the above measures. Numerical examples show that the results of this method agree with those of the two-dimensional finite element method very well, but significant computing time was saved.
2. A mathematical model is presented for analyzing the dynamic characteristics of a tilting-pad bearing. By means of suitable coordinate transformation, the oil-film forces acting on the journal and each pad in the global coordinate system, along with its Jacobians can be obtained in a highly concise expression. The oil-film forces and Jacobians in the pad system are computed by the weighted finite element method, which contributed by the 2nd chapter. Furthermore, the global Jacobian is used to find the equilibrium position of the journal as well as each pad in the bearing at same time by Newton-Raphson method, which is superior to the traditional methods because the later needs multi-circle of iteration steps. Once the equilibrium position is found the negative of the global Jacobians are just the complete dynamic coefficient matrices. By making an assumption that the journal and pad oscillated under the same frequency, reduced characteristics can be obtained by this analysis method.
3. The full dynamic characteristic mathematical model is used to analyze both linear and nonlinear dynamic behaviors of rotor-tilting-pad-bearing systems. The influence of pad motions is taken into account in the process of those analyses. By comparing the results of the linear analysis with that of the nonlinear analysis, conclusions as follows can be obtained:
1) Tilting-pad bearings have not 'essentially stability', when the full dynamic model was used, the critical curves of the rotor system could be decided.
2) Some parameters of tilting-pad bearing have marked influences on the stability of the system, especially the value of geometrical preload and the ratio of the offset of pad support point. When there no preload or the pad is supported in the middle, the rotor system has lower stability.
3) The nonlinear dynamic behaviors of a tilting-pad bearing differ from that of the traditional fixed pad bearings. When rotor speed exceeds the nonlinear critical-speed, the bifurcation behavior show as 2T periodic motions in a wide speed band. And at the high speed, the feedback of the pad motion is distinct.
4) The nonlinear critical-speed of tilting-pad bearings is more lower than that of linear critical-speed. With the increasing of the unbalanced mass of the rotor, the nonlinear characteristics of the system correspondingly increase.
5) When the rotary frequency is lower than nonlinear critical-speed, the difference between the linear and nonlinear results can be neglectable in case of small unbalance, whereas significant in cases of large unbalance. Once the rotor speed exceeding the critical value, the linear analysis would be useless. It is necessary to consider the influence of the whole nonlinear oil-forces in tilting-pad bearings when the dynamics performance of the rotor-tilting pad bearing system is analyzed.
本文研究可倾瓦径向滑动轴承更为合理的动力学模型及相应的高效算法。目前绝大多数相关研究仅针对可倾瓦轴承的线性化折合刚度、阻尼系数的动力分析模型，本文在单瓦非线性油膜力及其Jacobian矩阵的计算模型、可倾瓦轴承的完整动力特性分析模型以及转子—可倾瓦轴承系统线性和非线性动力学分析模型等几个方面，进行了建模及算法的研究。本文研究工作在理论及应用上取得了一些创新成果，所获得的结论对可倾瓦轴承—转子系统运动稳定性分析和工程应用具有积极的指导意义。
具体地，本文主要的研究内容和成果有：
1．针对滑动轴承的动力学行为研究，提出了求解非线性油膜力的加权有限元互补算法。基于流体润滑的变分不等方程理论，考虑油膜厚度对动力参数的影响，选取权函数与线性插值函数乘积的形式作为油膜压力函数，建立有限元离散化的一维变分不等方程。根据雷诺方程的一维互补方程系数矩阵为三对角矩阵的特点，用修正的追赶法求解对角元素占优的三对角方程，快速且无需迭代地直接求得满足Reynolds边界条件的油膜压力分布。算例结果表明，由于采用了权函数，只需非常稀疏的一维单元划分即可获得与二维有限元算法相媲美的精度，而计算耗时仅为后者的数千分之一。
2．基于可倾瓦径向滑动轴承瓦块的扰动特性，首次提出了计算可倾瓦轴承完整动力特性系数的数学解析模型。在全局固定坐标系和局部动坐标系之间建立合适的坐标转换关系，获得作用于每块瓦和轴颈的油膜力及其Jacobian矩阵在全局坐标系下的解析表达式；其中在固结于瓦块的局部动坐标系下，单块瓦的油膜力及其Jacobian矩阵利用求解单瓦非线性油膜力的加权有限元方法求出；全局坐标系下所有子系统的油膜力向量关于广义位移和广义速度的Jacobian矩阵经过扩充后，叠加组合得到轴承油膜力的全局Jacobian矩阵。而此全局Jacobian矩阵可以用于具有快速收敛特性的Newton-Raphson方法来求解所有瓦块和轴颈的静态平衡位置，在得到可倾瓦轴承平衡位置的同时获得轴承完整的刚度和阻尼系数矩阵。在设定瓦块和轴颈的扰动频率后，此解析模型亦可以获得与传统算法一致的折合刚度和阻尼系数矩阵。本模型采用矩阵表达形式，计算过程清晰简洁，计算耗时较少，适用于转子—可倾瓦轴承系统的线性和非线性分析。
3．针对可倾瓦轴承—转子系统的动力学研究，采用完整动力分析模型，对系统进行线性和非线性分析。在线性和分线性的分析中，首次将轴颈振动和瓦块摆动的动力学行为同时纳入研究过程。通过线性化的运动微分方程，采用数值方法分析静态平衡位置的稳定性。非线性周期响应则采用打靶法直接积分求解，并通过Floquent理论判断其稳定性。通过系统线性与非线性动力行为分析结果对比以及系统非线性动力学行为的考察，得到如下结论：
1)可倾瓦轴承—转子系统并不具有‘本质稳定性’，将完整动力特性分析模型应用于系统动力学行为分析，可以计算其临界转速。
2)可倾瓦轴承的某些参数对转子系统的稳定性有重要影响，特别是几何预负荷和瓦块支点偏置比，当几何预负荷为零或者瓦块支点位于瓦块中间位置时，可倾瓦轴承的稳定性降会大为降低。
3)可倾瓦轴承—转子系统的非线性动力行为有别于固定瓦滑动轴承—转子系统。可倾瓦轴承在进入非线性分岔后，在相当长的频段内处于2倍周期运动的状态，在转速较高时，可倾瓦的扰动反馈作用对转子的影响非常明显，轴心轨迹呈现出多角盘绕的现象。
4)可倾瓦轴承的非线性失稳转速远低于线性失稳转速，随着转子不平衡量的增加，系统的非线性运动特性增强。
5)当转子的运转频率低于非线性失稳转速，且转子的不平衡量较小时，对刚性转子—可倾瓦轴承系统的不平衡响应进行非线性分析的计算结果与线性动力学所得的结果吻合较好。基于线性动力学可以用来计算转子—可倾瓦轴承系统在稳态、小不平衡量下的响应；转子转速超过分岔转速或者不平衡量较大时，线性分析的结果不能正确表现出系统的动力行为，应采用完整非线性动力学模型，研究转子—可倾瓦轴承系统的不平衡响应等非线性动力学行为。
Dynamic modeling of rotor-bearing systems is the base of rotor dynamic analysis. The most important element of the mathematical model for rotor-bearing systems is the modeling of nonlinear fluid-force in hydrodynamic bearings, which also is the precondition of nonlinear analysis for rotor dynamic behaviors.
This thesis focuses on a better dynamic model of tilting-pad journal bearings and an efficient algorithm involved. Almost all of the relevant researches take the reduced stiffness and damping coefficients of tilting-pad bearings as research objects, comparatively, the full dynamic behaviors of tilting-pad bearings is studied here. This thesis studies on several aspects, such as, the efficient algorithm of nonlinear oil-film forces for a single pad, the analysis model and algorithm for the full dynamic coefficients of tilting-pad bearings, and the model of rotor dynamics of rotor-tilting-pad-bearing system. Some achievements are acquired in current research from theory and practice, and conclusions that are reached here may be beneficial to the engineering applications in stability of rotor motion.
In detail, there are the following contents and achievements in current research.
1. Based on the variational inequality theory of hydrodynamic lubrication, this thesis presents a weighted finite element algorithm to calculate the oil-film forces of journal bearings. Choosing a felicitous weighted function, two dimensional questions could be interpolated by one-dimensional elements with high accuracy. An amendatory direct-method is proposed to solve the equations whose coefficient matrices in tri-diagonal form. Then based on the interior characteristics, the solution of oil-forces is united with the solution of the Jacobian matrices. During this process, the operations are restricted within the positive pressure region, and no iteration is involved. Therefore, many redundant computations are avoided through the above measures. Numerical examples show that the results of this method agree with those of the two-dimensional finite element method very well, but significant computing time was saved.
2. A mathematical model is presented for analyzing the dynamic characteristics of a tilting-pad bearing. By means of suitable coordinate transformation, the oil-film forces acting on the journal and each pad in the global coordinate system, along with its Jacobians can be obtained in a highly concise expression. The oil-film forces and Jacobians in the pad system are computed by the weighted finite element method, which contributed by the 2nd chapter. Furthermore, the global Jacobian is used to find the equilibrium position of the journal as well as each pad in the bearing at same time by Newton-Raphson method, which is superior to the traditional methods because the later needs multi-circle of iteration steps. Once the equilibrium position is found the negative of the global Jacobians are just the complete dynamic coefficient matrices. By making an assumption that the journal and pad oscillated under the same frequency, reduced characteristics can be obtained by this analysis method.
3. The full dynamic characteristic mathematical model is used to analyze both linear and nonlinear dynamic behaviors of rotor-tilting-pad-bearing systems. The influence of pad motions is taken into account in the process of those analyses. By comparing the results of the linear analysis with that of the nonlinear analysis, conclusions as follows can be obtained:
1) Tilting-pad bearings have not 'essentially stability', when the full dynamic model was used, the critical curves of the rotor system could be decided.
2) Some parameters of tilting-pad bearing have marked influences on the stability of the system, especially the value of geometrical preload and the ratio of the offset of pad support point. When there no preload or the pad is supported in the middle, the rotor system has lower stability.
3) The nonlinear dynamic behaviors of a tilting-pad bearing differ from that of the traditional fixed pad bearings. When rotor speed exceeds the nonlinear critical-speed, the bifurcation behavior show as 2T periodic motions in a wide speed band. And at the high speed, the feedback of the pad motion is distinct.
4) The nonlinear critical-speed of tilting-pad bearings is more lower than that of linear critical-speed. With the increasing of the unbalanced mass of the rotor, the nonlinear characteristics of the system correspondingly increase.
5) When the rotary frequency is lower than nonlinear critical-speed, the difference between the linear and nonlinear results can be neglectable in case of small unbalance, whereas significant in cases of large unbalance. Once the rotor speed exceeding the critical value, the linear analysis would be useless. It is necessary to consider the influence of the whole nonlinear oil-forces in tilting-pad bearings when the dynamics performance of the rotor-tilting pad bearing system is analyzed.
引文
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