叶片—转子—轴承耦合系统的非线性动力学特性研究
摘要
随着近代旋转机械的设计向高转速、轻柔结构的趋势发展,叶片和转子的柔性在许多特定的情况下处于同一量级。高转速的转子系统常常处于超临界运动状态,低转速时可以视为刚性的旋转部件,高转速时必须考虑为弹性部件参与耦合振动。因此,在研究高转速转子系统的动力学行为时,不能再将叶片和转轴分开考虑,而是要建立更为精确的考虑叶片和转轴耦合效应的动力学模型。
另一方面,随着旋转机械机组容量的增大和设计参数的提高,非线性因素对系统动力学性能的影响也越来越显著。轴承作为旋转机械必不可少的结构之一,其中的油膜力是转子系统重要的非线性激励源。而非线性油膜力引发的油膜涡动和油膜振荡是转子系统较常见的故障。因此有必要建立由叶片、转子和轴承构成的耦合非线性动力学模型,并应用现代非线性动力学理论研究该系统的分岔、混沌等非线性动力学特性。
本文针对航空发动机、燃气轮机等大型旋转机械设计制造中存在的问题,建立了叶片-转子-轴承耦合系统的非线性动力学模型,利用该模型研究了叶片、转子和轴承的耦合动力学特性,取得了一些创新性成果。具体的研究内容和成果有:
采用Lagrange方程建立了叶片-转子-轴承耦合系统的非线性动力学模型。利用集中质量法将转子系统离散,为分析叶片对转子的惯性效应并考虑系统的时变性,将叶片模化为单摆模型。为将模型简化,首先,应用正交变换将与转子横向运动发生耦合的叶片1-节径运动和叶片其它节径运动解耦,从而将耦合的非线性系统由8 + 2 n ( n≥3)个自由度降低至12个自由度,同时得到了2n-4个描述叶片k-节径( k≠1)运动的相互独立的线性微分方程。然后,利用周期变换将耦合时变微分方程转化为常系数微分方程。在两次变换后,获得了一个含有12个自由度的常系数耦合非线性方程组和2n-4个相互独立的线性微分方程。这个降维方法即保留了叶片和转子之间的耦合效应,又降低了非线性微分方程组的维数,提高了耦合系统的求解效率,为高效准确地分析系统的非线性动力学特性奠定了基础。
当采用刚性转子假设时,叶片-刚性转子-轴承系统非线性动力学模型的自由度数进一步降至4。利用短轴承非线性油膜力模型获得非线性油膜力。对耦合模型的数值分析显示了叶片-刚性转子-轴承系统的动力学特性,与不考虑叶片振动的情况相比较,揭示了叶片振动对刚性转子-轴承系统非线性动力学特性的影响。
为分析柔性叶片、弹性转子和轴承之间的耦合动力学特性,利用对称转子假设,分别建立了叶片-单盘转子-轴承系统模型和叶片-双盘转子-轴承系统模型。由于转子的对称性,这两个模型都忽略圆盘的偏转运动和叶片的平面外运动,解耦后的模型中,系统的自由度数分别缩减至8和12。通过对模型的求解,着重分析了叶片振动和弹性转子—轴承系统之间的耦合效应。
对于非对称转子,在建立叶片-转子-轴承系统动力学模型时,将陀螺力矩对转子的影响考虑在内。此模型较全面的考察了转子与叶片的耦合效应。通过求解12个自由度的耦合非线性系统,分析了叶片振动对转子动特性的影响,并详细讨论了叶片刚度和长度等参数对转子动力学行为的影响。为分析耦合系统中叶片的振动特性,将叶片k-节径运动微分方程的解析解离散,结合耦合模型中叶片1-节径运动的数值解,获得了叶片的平面内和平面外运动的动态响应,揭示了耦合系统中叶片振动蕴含的非线性特性。
对带有外伸端结构的转子,建立了叶片-悬臂转子-挤压油膜轴承系统的动力学模型。挤压油膜阻尼器的油膜力采用Raynolds边界条件下的短轴承假设的非线性油膜力模型。该模型含有两个圆盘,其中一个圆盘位于两个轴承之间,另一个圆盘在转子的外伸端上,而且在每个圆盘上都带有若干弹性叶片。叶片-悬臂转子-挤压油膜轴承耦合系统解耦后共有24个自由度。通过对系统进行数值求解,分析了叶片振动对悬臂转子动力学行为的影响。研究表明在分析此类系统的动力学特性时,必须考虑叶片、转子和轴承之间的耦合效应。
Nowadays, in the practical design, rotating systems become lighter and more flexible with higher operating speed, and the flexibilities of the blade and rotor are close to each other in some situations. The working rotating speed of the rotor is often above the critical speed, and many rotating components, which may be assumed to be rigid at the low speed, should to be considered to be elastic components which may couple to the rotor vibration as the rotating speed is getting higher. Actually, an effective modeling strategy that addresses the interaction between the rotor and the blade is crucial in estimating system performance.
On the other hand, improving the parameters of the rotating machinery, the effect of the nonlinear factor on the behavior of the rotor is more significantly. The journal bearing is one of the necessary components of the rotating machine, and the nonlinear oil film force of the journal bearing is one of the nonlinear factors in the rotor system and can cause the occurrence of the oil whirl and oil whip which may cause the failure of the machine. Therefore, the precise nonlinear model should be developed to describe the interaction between the rotor, blade and bearing, and to be used to analyze the nonlinear behaviors such as the bifurcation and chaos by the theory of the nonlinear dynamics.
In the design and manufacture of large-scale rotating machinery such as aircraft engines and steam turbines, the coupling dynamics of rotor-bearing systems with blades is a crucial problem. In this dissertation, a coupling nonlinear dynamical model of the blade-rotor- bearing system is developed to analyze the interaction between the rotor, blade and bearing. Research achievements on the coupling dynamics of blade-rotor-baering model are obtained, and this may be beneficial to the engineering applications. The main contents and achievements are listed here.
The nonlinear coupling model of the blade-rotor-bearing system is established by the Lagrange approach. The lumped mass method is used to discrete the rotor, and the blades are modeled as pendulums to emphasize the inertia coupling between blades and rotor. To reduce the coupling model, first, orthogonal transformations are employed to decouple the one nodal diameter equations of motion, which are coupled with the equations of motion of the rotor, from k-nodal diameter ( k≠1) equations of motion for an array of elastic blades. The coupling equations with 8 + 2n( n≥3)-degree-of-freedom are divided into two parts: one is the coupling nonlinear model with 12-degree-of- freedom which describe the interaction between the motion of the rotor and the 1-nodal diameter motion of blades, the others are the independent linear system with (2n-4)-degree-of-freedom which describe the k-nodal diameter ( k≠1) motion of blades. Then, the coupling equations of the blade-rotor-bearing system are transferred to a time-invariant model in terms of periodic transformations. After the two sets of transformations, the coupling nonlinear model is reduced to a lower dimensional system which can be easily solved numericaly in comparison with the solving of the original coupling system.
With the assumption of a rigid rotor, the degree-of-freedom of the blade-rigid rotor-bearing system is reduced to 4. The nonlinear oil film force model based on the short bearing theory is adopted here. The system is numericaly solved and the dynamical responses of the system are used to analyze the nonlinear dynamical behaviors of the coupling model. The bifurcation diagrams for the corresponding rigid rotor-bearing model, in which the effect of the vibration of blade is neglected, are given as a comparison of the results obtained from the blade-rigid rotor-bearing model.
To analyze the interaction between the elastic blade, flexible rotor and journal bearing, the model of blade-rotor-bearing systems with both single and double disks are proposed respectively based on the assumption of the symmetrical rotor, in which the deflection motions of the rotor and the out-of-plane motion of blades are neglected. After decoupling, the degree-of-freedom of these two models are reduced to 8 and 12, respectively. Then the coupling interaction between the elastic blade and rotor-bearing system is investigated.
With the assumption of the asymmetrical rotor, the model of the blade-rotor-bearing system with the gyroscopic moment is developed. Obtaining the numerical solutions of the coupling model with the 12-degree-of-freedom, the nonlinear dynamical behavior of the coupling system is investegated. And the effect of the different stiffnesses and lengths of the blade on the behavior of the rotor is discussed. By analyzing the responses of the in-plane and out-of-plane motion of blades, the nonlinear characteristic of the vibration of blades is shown.
For the overhung rotor, the model of the blade-cantilever rotor- squeeze film bearing model is established. In the model, there are two disks on the rotor, one is fixed between two bearing, and the other is fixed at the end of the overhung rotor. The many elastic blades are fixed to each disk. The oil film force of the squeeze film damper based on the theory of short bearing with the Raynolds boundary is adopted. After decoupling, there exist 24 equations in the coupling nonlinear system. Using the numerical solutions of the coupling system, the effect of the vibration of blades on the nonlinear dynamical behavior of the overhung rotor is shown.
另一方面,随着旋转机械机组容量的增大和设计参数的提高,非线性因素对系统动力学性能的影响也越来越显著。轴承作为旋转机械必不可少的结构之一,其中的油膜力是转子系统重要的非线性激励源。而非线性油膜力引发的油膜涡动和油膜振荡是转子系统较常见的故障。因此有必要建立由叶片、转子和轴承构成的耦合非线性动力学模型,并应用现代非线性动力学理论研究该系统的分岔、混沌等非线性动力学特性。
本文针对航空发动机、燃气轮机等大型旋转机械设计制造中存在的问题,建立了叶片-转子-轴承耦合系统的非线性动力学模型,利用该模型研究了叶片、转子和轴承的耦合动力学特性,取得了一些创新性成果。具体的研究内容和成果有:
采用Lagrange方程建立了叶片-转子-轴承耦合系统的非线性动力学模型。利用集中质量法将转子系统离散,为分析叶片对转子的惯性效应并考虑系统的时变性,将叶片模化为单摆模型。为将模型简化,首先,应用正交变换将与转子横向运动发生耦合的叶片1-节径运动和叶片其它节径运动解耦,从而将耦合的非线性系统由8 + 2 n ( n≥3)个自由度降低至12个自由度,同时得到了2n-4个描述叶片k-节径( k≠1)运动的相互独立的线性微分方程。然后,利用周期变换将耦合时变微分方程转化为常系数微分方程。在两次变换后,获得了一个含有12个自由度的常系数耦合非线性方程组和2n-4个相互独立的线性微分方程。这个降维方法即保留了叶片和转子之间的耦合效应,又降低了非线性微分方程组的维数,提高了耦合系统的求解效率,为高效准确地分析系统的非线性动力学特性奠定了基础。
当采用刚性转子假设时,叶片-刚性转子-轴承系统非线性动力学模型的自由度数进一步降至4。利用短轴承非线性油膜力模型获得非线性油膜力。对耦合模型的数值分析显示了叶片-刚性转子-轴承系统的动力学特性,与不考虑叶片振动的情况相比较,揭示了叶片振动对刚性转子-轴承系统非线性动力学特性的影响。
为分析柔性叶片、弹性转子和轴承之间的耦合动力学特性,利用对称转子假设,分别建立了叶片-单盘转子-轴承系统模型和叶片-双盘转子-轴承系统模型。由于转子的对称性,这两个模型都忽略圆盘的偏转运动和叶片的平面外运动,解耦后的模型中,系统的自由度数分别缩减至8和12。通过对模型的求解,着重分析了叶片振动和弹性转子—轴承系统之间的耦合效应。
对于非对称转子,在建立叶片-转子-轴承系统动力学模型时,将陀螺力矩对转子的影响考虑在内。此模型较全面的考察了转子与叶片的耦合效应。通过求解12个自由度的耦合非线性系统,分析了叶片振动对转子动特性的影响,并详细讨论了叶片刚度和长度等参数对转子动力学行为的影响。为分析耦合系统中叶片的振动特性,将叶片k-节径运动微分方程的解析解离散,结合耦合模型中叶片1-节径运动的数值解,获得了叶片的平面内和平面外运动的动态响应,揭示了耦合系统中叶片振动蕴含的非线性特性。
对带有外伸端结构的转子,建立了叶片-悬臂转子-挤压油膜轴承系统的动力学模型。挤压油膜阻尼器的油膜力采用Raynolds边界条件下的短轴承假设的非线性油膜力模型。该模型含有两个圆盘,其中一个圆盘位于两个轴承之间,另一个圆盘在转子的外伸端上,而且在每个圆盘上都带有若干弹性叶片。叶片-悬臂转子-挤压油膜轴承耦合系统解耦后共有24个自由度。通过对系统进行数值求解,分析了叶片振动对悬臂转子动力学行为的影响。研究表明在分析此类系统的动力学特性时,必须考虑叶片、转子和轴承之间的耦合效应。
Nowadays, in the practical design, rotating systems become lighter and more flexible with higher operating speed, and the flexibilities of the blade and rotor are close to each other in some situations. The working rotating speed of the rotor is often above the critical speed, and many rotating components, which may be assumed to be rigid at the low speed, should to be considered to be elastic components which may couple to the rotor vibration as the rotating speed is getting higher. Actually, an effective modeling strategy that addresses the interaction between the rotor and the blade is crucial in estimating system performance.
On the other hand, improving the parameters of the rotating machinery, the effect of the nonlinear factor on the behavior of the rotor is more significantly. The journal bearing is one of the necessary components of the rotating machine, and the nonlinear oil film force of the journal bearing is one of the nonlinear factors in the rotor system and can cause the occurrence of the oil whirl and oil whip which may cause the failure of the machine. Therefore, the precise nonlinear model should be developed to describe the interaction between the rotor, blade and bearing, and to be used to analyze the nonlinear behaviors such as the bifurcation and chaos by the theory of the nonlinear dynamics.
In the design and manufacture of large-scale rotating machinery such as aircraft engines and steam turbines, the coupling dynamics of rotor-bearing systems with blades is a crucial problem. In this dissertation, a coupling nonlinear dynamical model of the blade-rotor- bearing system is developed to analyze the interaction between the rotor, blade and bearing. Research achievements on the coupling dynamics of blade-rotor-baering model are obtained, and this may be beneficial to the engineering applications. The main contents and achievements are listed here.
The nonlinear coupling model of the blade-rotor-bearing system is established by the Lagrange approach. The lumped mass method is used to discrete the rotor, and the blades are modeled as pendulums to emphasize the inertia coupling between blades and rotor. To reduce the coupling model, first, orthogonal transformations are employed to decouple the one nodal diameter equations of motion, which are coupled with the equations of motion of the rotor, from k-nodal diameter ( k≠1) equations of motion for an array of elastic blades. The coupling equations with 8 + 2n( n≥3)-degree-of-freedom are divided into two parts: one is the coupling nonlinear model with 12-degree-of- freedom which describe the interaction between the motion of the rotor and the 1-nodal diameter motion of blades, the others are the independent linear system with (2n-4)-degree-of-freedom which describe the k-nodal diameter ( k≠1) motion of blades. Then, the coupling equations of the blade-rotor-bearing system are transferred to a time-invariant model in terms of periodic transformations. After the two sets of transformations, the coupling nonlinear model is reduced to a lower dimensional system which can be easily solved numericaly in comparison with the solving of the original coupling system.
With the assumption of a rigid rotor, the degree-of-freedom of the blade-rigid rotor-bearing system is reduced to 4. The nonlinear oil film force model based on the short bearing theory is adopted here. The system is numericaly solved and the dynamical responses of the system are used to analyze the nonlinear dynamical behaviors of the coupling model. The bifurcation diagrams for the corresponding rigid rotor-bearing model, in which the effect of the vibration of blade is neglected, are given as a comparison of the results obtained from the blade-rigid rotor-bearing model.
To analyze the interaction between the elastic blade, flexible rotor and journal bearing, the model of blade-rotor-bearing systems with both single and double disks are proposed respectively based on the assumption of the symmetrical rotor, in which the deflection motions of the rotor and the out-of-plane motion of blades are neglected. After decoupling, the degree-of-freedom of these two models are reduced to 8 and 12, respectively. Then the coupling interaction between the elastic blade and rotor-bearing system is investigated.
With the assumption of the asymmetrical rotor, the model of the blade-rotor-bearing system with the gyroscopic moment is developed. Obtaining the numerical solutions of the coupling model with the 12-degree-of-freedom, the nonlinear dynamical behavior of the coupling system is investegated. And the effect of the different stiffnesses and lengths of the blade on the behavior of the rotor is discussed. By analyzing the responses of the in-plane and out-of-plane motion of blades, the nonlinear characteristic of the vibration of blades is shown.
For the overhung rotor, the model of the blade-cantilever rotor- squeeze film bearing model is established. In the model, there are two disks on the rotor, one is fixed between two bearing, and the other is fixed at the end of the overhung rotor. The many elastic blades are fixed to each disk. The oil film force of the squeeze film damper based on the theory of short bearing with the Raynolds boundary is adopted. After decoupling, there exist 24 equations in the coupling nonlinear system. Using the numerical solutions of the coupling system, the effect of the vibration of blades on the nonlinear dynamical behavior of the overhung rotor is shown.
引文
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