复杂转子耦合系统有限元建模及其动力特性研究
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摘要
转子轴承系统的动力学行为预测中,除理论分析与实验研究外,常借助于传递矩阵法和有限单元法这两种数学建模方法。国内传递矩阵法的研究水平与应用现状已接近国际领先,但有限单元法在转子动力学中的运用尚显得相对薄弱,需要提升与发展。一方面,引入有限单元法对转子轴承系统建模具有大量优势：可以考虑陀螺力矩、转动惯量、横向剪切变形、重力项等因素的影响；其运动微分方程表达清晰规范,物理意义明确；可以方便地耦合周边环境中的结构和作用力,特别适用于复杂转子耦合系统,而这一点正是传递矩阵法的不足之处。尽管有限单元法建模往往编程困难、矩阵维数大且计算耗时,但现代计算机的软件与硬件能力减小了此不利因素。另一方面,相对简单转子系统,目前线性范围内的数值计算已较为成熟,国内外研究者相当的工作量转移到考察各种非线性因素的影响,其动力学方程多基于Jeffcott转子模型；而针对复杂转子耦合系统,即使线性分析计算也不够成熟,建模方法上仍有发展空间且值得探索。
齿轮传动在机械领域广泛应用,多平行轴齿轮啮合转子系统是转子系统中重要且特殊的一类,齿轮副的存在使其动力学研究十分复杂。各平行轴转子相互耦合,系统动力学特性与非耦合的单轴转子不同,因此对各平行轴转子单独设计计算已无法满足高精度计算的要求。双转子系统在航空发动机中已有应用,内外转子反向旋转时,可抵消部分陀螺力矩,提高飞机机动性能。其转子系统由高压转子和低压转子两个子系统组成,相对单转子发动机,其结构复杂,振动特性的分析计算也更为困难。为研制先进航空发动机以提升战机的振动性能和确保可靠运行,进行双转子系统动力学研究具有重大战略意义,而在这方面国外公开的资料很少。以往国内学者常采用传递矩阵法研究双转子结构,但此方法很难计入飞行中空间运动的影响,而采用有限单元法建模可以克服这一困难。鉴于以上原因,本文以有限单元法为理论基础,重点研究多平行轴齿轮啮合转子系统和双转子系统的建模方法,通过数值计算分析了两类系统的动力特性,具体开展了以下工作：
1)针对常见的单转子轴承系统,应用有限单元法并结合Rayleigh梁理论和Timoshenko梁理论,详细地推导出了其系统运动微分方程：在此基础上理论分析了系统特征值问题、不平衡响应和瞬态响应的求解方法；基于MATLAB平台编写程序实现转子系统的临界转速、不平衡响应和瞬态计算。
2)参与了浙江省某企业某款径流式汽轮机在设计阶段的转子动力学分析,分别应用商业软件ANSYS和本文开发的程序对其进行静挠度、临界转速、振型及不平衡响应计算,两种途径得到的结果一致,验证了本文开发程序的正确与有效。同时,分析结果为径流式汽轮机转子结构的确定和轴承参数的选取提出了指导意见。
3)以浙江省重大科技专项《大功率船用齿轮箱关键技术的研究与应用》(项目代号：2009C11062)为依托,对某款三平行轴的船用齿轮箱啮合转子传动系统进行固有特征分析,考察工作转速工况下的各传动级啮合频率与系统固有频率,并通过考察各阶固有频率随齿轮啮合刚度的变化来区分弯曲振动与扭转振动。在多平行轴齿轮啮合转子系统建模方法的研究上,首先推导了在压力线坐标系下齿轮啮合单元的运动微分方程,通过坐标变换得到在全局坐标下的运动微分方程,然后给出了齿轮啮合单元刚度矩阵集成到整体转子刚度矩阵的方法以最终形成啮合转子系统运动方程。
4)在双转子系统建模方法的研究上,推导了中介轴承的力传递和内外转子的耦合过程,给出了双转子系统的动力学模型,并发展了针对双转子结构的传递矩阵法。结合数值案例,应用基于有限单元法的动力学模型和传递矩阵法模型计算双转子结构正反向旋转下的临界转速,两种方法得到的结果一致。考察了内外转子间的中介轴承刚度、转速比对系统振动特性的影响；结合直接积分法,对系统启动与停机的瞬态响应进行数值模拟。本文建立的模型为双转子系统动力学行为的预测研究提供了较为完整的基础,研究结果对于提高航空发动机转子系统的动力学性能,指导双转子系统的设计和使用具有重要的理论意义和工程应用价值。
The transfer matrix method and the finite element method are commonly applied in engineering analysis of rotor dynamics. The transfer matrix method is deeply investigated and extensively used in China, though the finite element method for rotor dynamics is insufficiently studied and immaturely developed in the domestic research. However, a finite element model of rotor-bearing systems is more powerful and advantageous, which can include the effects of gyroscopic moments, rotary inertia, transverse shear deformation and gravity, and the model features concise formula and straight forward physical meaning. This computational model can easily be coupled with the ambient structures and forces, which makes it particularly suitable for dealing with complex multi-rotor coupled systems. On the other hand, for simple rotor-bearing systems, the numerical computing is now developed maturely within the framework of linear theory, so both domestic and international scholars begin to focus their study on the influence of various nonlinear factors, which always besed on the Jeffcott rotor model. Meanwhile, for complex multi-rotor coupled systems, it is necessarily to achieve more development, modification and optimization for their computational modelings.
Gears are widely used in the mechanical field. The multi-parallel axis gear rotor system is one kind of important and special rotor system, but the existence of gear pair makes the dynamics research complicated. The dynamic characteristics of parallel axis rotor mutual coupling system are different from those of non-coupling single axis rotor system, and therefore the separate design and calculation of each parallel axis rotor cannot meet the requirement of high-precision calculation. Dual rotor system has been applied in aviation engines. The counter-rotating of the inner and external rotor can offset some gyroscopic torque, and improve aircraft maneuverability. For these reasons, this paper took the finite element method as the theoretical theory, focused on multi-parallel shaft gear rotor system and dual rotor system modeling method, and analyzed the dynamic characteristics of these two systems by numerical calculation. Specific works were carried out as follows:
1) For the common single rotor bearing system, the finite element method combined with Rayleigh beam theory and Timoshenko beam theory was used to deduce the differential equations of motion in detail; The system eigenvalue problem and the solving method for unbalance response and transient response were theoretically analyzed based on the above; A Program written on MATLAB realized the calculation of rotor system critical speed, unbalance response and transient.
2) In the rotor dynamics analysis of a radial turbine in design stage, the commercial software ANSYS and the program developed in this paper were respectively used to carry on calculation of static deflection, critical speed, mode of vibration and unbalance response. The consistent results obtained from two ways verified that the program was correct and effective. At the same time, the analysis results were helpful to determine the radial turbine rotor structure and the selection of bearing parameters.
3) This study originated from the Key Industrial Projects founded by Zhejiang Province Science and Technology Department:"Investigation and Application of the Key Technologies for Large Marine gearboxes"(2009C11062). The research analyzed the inherent characteristics of a three parallel axis gearbox meshing rotor transmission system for marine ships, inspected each transmission level's meshing frequency and the system's frequency, and distinguished between the flexural and torsional vibrations through the inspection of each level inherent frequencies varied by the gear meshing stiffness. In the research of multi parallel axis gear meshing rotor system modeling methods, this paper derived the differential equations of motion for gear meshing unit in the line pressure coordinate firstly, which could obtain the differential equations of motion in the global coordinate by coordinate transformation; Then, this paper found a way to integrate the stiffness matrix of gear meshing unit into the whole rotor stiffness, and obtained the meshing rotor motion equation finally.
4) In the research on modeling method for rotor system, this paper derived the transfer of medium bearing force and the coupling process of internal and external rotor, provided the dynamic model of dual rotor system, and developed the transfer matrix method of dual rotor structure. In the numerical case, the critical speed of the dual rotor structure with normal and reverse rotation calculated by the finite element method dynamics model and the transfer matrix method model was agreed. This paper inspected the effects of the intermediate bearing's rigidity and speed ratio to the system's vibration characteristics; Combined with the direct integral method, the numerical simulations were carried on the system's start and stop transient response. The model established in this paper provided the relatively complete basis for the prediction of dual rotor system's dynamic behavior, and the results of this research have important theoretical significance and engineering application value for improving the dynamic performance of aero engine's rotor system and guiding the design of dual rotor system.
齿轮传动在机械领域广泛应用,多平行轴齿轮啮合转子系统是转子系统中重要且特殊的一类,齿轮副的存在使其动力学研究十分复杂。各平行轴转子相互耦合,系统动力学特性与非耦合的单轴转子不同,因此对各平行轴转子单独设计计算已无法满足高精度计算的要求。双转子系统在航空发动机中已有应用,内外转子反向旋转时,可抵消部分陀螺力矩,提高飞机机动性能。其转子系统由高压转子和低压转子两个子系统组成,相对单转子发动机,其结构复杂,振动特性的分析计算也更为困难。为研制先进航空发动机以提升战机的振动性能和确保可靠运行,进行双转子系统动力学研究具有重大战略意义,而在这方面国外公开的资料很少。以往国内学者常采用传递矩阵法研究双转子结构,但此方法很难计入飞行中空间运动的影响,而采用有限单元法建模可以克服这一困难。鉴于以上原因,本文以有限单元法为理论基础,重点研究多平行轴齿轮啮合转子系统和双转子系统的建模方法,通过数值计算分析了两类系统的动力特性,具体开展了以下工作：
1)针对常见的单转子轴承系统,应用有限单元法并结合Rayleigh梁理论和Timoshenko梁理论,详细地推导出了其系统运动微分方程：在此基础上理论分析了系统特征值问题、不平衡响应和瞬态响应的求解方法；基于MATLAB平台编写程序实现转子系统的临界转速、不平衡响应和瞬态计算。
2)参与了浙江省某企业某款径流式汽轮机在设计阶段的转子动力学分析,分别应用商业软件ANSYS和本文开发的程序对其进行静挠度、临界转速、振型及不平衡响应计算,两种途径得到的结果一致,验证了本文开发程序的正确与有效。同时,分析结果为径流式汽轮机转子结构的确定和轴承参数的选取提出了指导意见。
3)以浙江省重大科技专项《大功率船用齿轮箱关键技术的研究与应用》(项目代号：2009C11062)为依托,对某款三平行轴的船用齿轮箱啮合转子传动系统进行固有特征分析,考察工作转速工况下的各传动级啮合频率与系统固有频率,并通过考察各阶固有频率随齿轮啮合刚度的变化来区分弯曲振动与扭转振动。在多平行轴齿轮啮合转子系统建模方法的研究上,首先推导了在压力线坐标系下齿轮啮合单元的运动微分方程,通过坐标变换得到在全局坐标下的运动微分方程,然后给出了齿轮啮合单元刚度矩阵集成到整体转子刚度矩阵的方法以最终形成啮合转子系统运动方程。
4)在双转子系统建模方法的研究上,推导了中介轴承的力传递和内外转子的耦合过程,给出了双转子系统的动力学模型,并发展了针对双转子结构的传递矩阵法。结合数值案例,应用基于有限单元法的动力学模型和传递矩阵法模型计算双转子结构正反向旋转下的临界转速,两种方法得到的结果一致。考察了内外转子间的中介轴承刚度、转速比对系统振动特性的影响；结合直接积分法,对系统启动与停机的瞬态响应进行数值模拟。本文建立的模型为双转子系统动力学行为的预测研究提供了较为完整的基础,研究结果对于提高航空发动机转子系统的动力学性能,指导双转子系统的设计和使用具有重要的理论意义和工程应用价值。
The transfer matrix method and the finite element method are commonly applied in engineering analysis of rotor dynamics. The transfer matrix method is deeply investigated and extensively used in China, though the finite element method for rotor dynamics is insufficiently studied and immaturely developed in the domestic research. However, a finite element model of rotor-bearing systems is more powerful and advantageous, which can include the effects of gyroscopic moments, rotary inertia, transverse shear deformation and gravity, and the model features concise formula and straight forward physical meaning. This computational model can easily be coupled with the ambient structures and forces, which makes it particularly suitable for dealing with complex multi-rotor coupled systems. On the other hand, for simple rotor-bearing systems, the numerical computing is now developed maturely within the framework of linear theory, so both domestic and international scholars begin to focus their study on the influence of various nonlinear factors, which always besed on the Jeffcott rotor model. Meanwhile, for complex multi-rotor coupled systems, it is necessarily to achieve more development, modification and optimization for their computational modelings.
Gears are widely used in the mechanical field. The multi-parallel axis gear rotor system is one kind of important and special rotor system, but the existence of gear pair makes the dynamics research complicated. The dynamic characteristics of parallel axis rotor mutual coupling system are different from those of non-coupling single axis rotor system, and therefore the separate design and calculation of each parallel axis rotor cannot meet the requirement of high-precision calculation. Dual rotor system has been applied in aviation engines. The counter-rotating of the inner and external rotor can offset some gyroscopic torque, and improve aircraft maneuverability. For these reasons, this paper took the finite element method as the theoretical theory, focused on multi-parallel shaft gear rotor system and dual rotor system modeling method, and analyzed the dynamic characteristics of these two systems by numerical calculation. Specific works were carried out as follows:
1) For the common single rotor bearing system, the finite element method combined with Rayleigh beam theory and Timoshenko beam theory was used to deduce the differential equations of motion in detail; The system eigenvalue problem and the solving method for unbalance response and transient response were theoretically analyzed based on the above; A Program written on MATLAB realized the calculation of rotor system critical speed, unbalance response and transient.
2) In the rotor dynamics analysis of a radial turbine in design stage, the commercial software ANSYS and the program developed in this paper were respectively used to carry on calculation of static deflection, critical speed, mode of vibration and unbalance response. The consistent results obtained from two ways verified that the program was correct and effective. At the same time, the analysis results were helpful to determine the radial turbine rotor structure and the selection of bearing parameters.
3) This study originated from the Key Industrial Projects founded by Zhejiang Province Science and Technology Department:"Investigation and Application of the Key Technologies for Large Marine gearboxes"(2009C11062). The research analyzed the inherent characteristics of a three parallel axis gearbox meshing rotor transmission system for marine ships, inspected each transmission level's meshing frequency and the system's frequency, and distinguished between the flexural and torsional vibrations through the inspection of each level inherent frequencies varied by the gear meshing stiffness. In the research of multi parallel axis gear meshing rotor system modeling methods, this paper derived the differential equations of motion for gear meshing unit in the line pressure coordinate firstly, which could obtain the differential equations of motion in the global coordinate by coordinate transformation; Then, this paper found a way to integrate the stiffness matrix of gear meshing unit into the whole rotor stiffness, and obtained the meshing rotor motion equation finally.
4) In the research on modeling method for rotor system, this paper derived the transfer of medium bearing force and the coupling process of internal and external rotor, provided the dynamic model of dual rotor system, and developed the transfer matrix method of dual rotor structure. In the numerical case, the critical speed of the dual rotor structure with normal and reverse rotation calculated by the finite element method dynamics model and the transfer matrix method model was agreed. This paper inspected the effects of the intermediate bearing's rigidity and speed ratio to the system's vibration characteristics; Combined with the direct integral method, the numerical simulations were carried on the system's start and stop transient response. The model established in this paper provided the relatively complete basis for the prediction of dual rotor system's dynamic behavior, and the results of this research have important theoretical significance and engineering application value for improving the dynamic performance of aero engine's rotor system and guiding the design of dual rotor system.
引文
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