轴系弯扭耦合振动及双变量分岔分析
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摘要
随着我国社会经济的发展，人们对电力的需求越来越旺盛，这使得汽轮发电机组的研制、生产向巨型化、大容量的方向发展，相应的汽轮发电机组的振动问题也越来越突出。在实际的汽轮发电机组中，由于质量不平衡和外激励等多种因素的存在，导致汽轮发电机组的弯扭耦合振动问题越来越突出。汽轮发电机组的弯扭耦合振动问题的研究极具挑战性，对该问题的研究将有助于全面的了解系统的动力学特性，提高系统故障诊断的准确性，并对该系统的参数设计具有指导意义。
首先以具有不平衡质量的双盘转子为研究对象，考虑了扭转方向主共振的情况，通过拉格朗日方程，建立了低压缸-发电机系统的弯扭耦合微分方程，非线性项是由质量不平衡引起的。采用平均法，得到了系统的解析解和分岔方程。应用奇异性理论求解出在不同保持域内的运动模式，应用龙格库塔方法对微分方程进行数值计算。不平衡质量能够引起弯扭耦合振动，质量不平衡越大，弯扭耦合越剧烈，因此减小质量不平衡有助于降低弯扭耦合振动。扭转方向主共振相当于对扭转振动的能量输入，这将会引起较大的扭转振动从而加剧转子的弯曲振动，在这种情况下应该考虑系统的弯扭耦合振动问题。
在此基础上，考虑了轴的各向异性刚度，在扭转方向主共振和组合共振(工频约等于弯曲固有频率与扭转固有频率之和)共同作用的情况下，建立了低压缸-发电机的运动微分方程。应用平均法得到系统的解析解和分岔方程，应用两个状态变量的奇异性理论分析了弯曲振动和扭转振动在不同保持域中的运动模式，应用数值算法对解析解进行验证。当在扭转方向主共振和组合共振共同作用时，系统的弯曲振动和扭转振动的幅值都大幅度增加，扭转方向主共振相当于能量输入，组合共振相当于能量交换通道，扭转方向主共振引起轴系的扭转振动大幅度增加，通过能量交换通道传递到弯曲振动上，引起弯曲振动幅值的大幅度增加。此种情况对系统的危害极大，在弯曲方向和扭转方向都有发生破坏的可能，必须要考虑弯扭耦合振动问题，在设计时尽量避免此种情况的发生。
以凸缘联轴器的平行不对中引起轴系的弯扭耦合振动问题为研究对象，考虑了由质量不平衡引起的非线性因素。在激励频率、扭转振动的固有频率、弯曲振动的固有频率之比约为1:1:2的情况时，建立了系统的弯扭耦合振动的微分方程。应用平均法和奇异性理论对该模型进行解析分析，利用龙格库塔法进行数值计算。由于组合共振关系(激励频率、扭转振动的固有频率、弯曲振动的固有频率之比约等于1:1:2)的存在，系统既有能量输入项(激励频率和弯曲固有频率之比约等于1:2)，又有能量交换通道(扭转振动的固有频率和弯曲振动的固有频率之比约等于1:2)，系统的弯曲振动和扭转振动都剧烈增加甚至有破坏的可能，所以在此种情况下必须要考虑弯扭耦合振动问题，并且在设计系统时，应该尽量避免这种组合共振关系。
在实际的工程系统中，状态变量常常是受到约束限制的，对于一个状态变量受到约束的奇异性理论研究的比较充分，而多个状态变量受到约束的奇异性理论研究到目前为止还未开展。以两个状态变量受到约束限制为研究对象，考虑七种不同的约束限制条件，分别讨论了它们的转迁集形式。在受到约束的情况下在非约束区间其分岔集和双极限点集与非约束情况下相同，滞后集不同。由于约束的存在，会产生许多新的转迁集形式。由于这些新的转迁集与边界有关，称之为由于边界引起的转迁集。以二维系统为例研究了两个状态变量的约束分岔形式，通过比较发现，由于约束条件引起许多新的转迁集，以一个例子对两个状态变量约束分岔理论进行分析说明。
As the developments of social economy, the requirement of electric isincreasing, which promotes the turbo-generator unit to large scale and large capacity.As the result, the vibration of the turbo-generator is more and more serious. Inactual system, for the existence of unbalanced mass and exciting force, thebending-torsion coupling vibration causes more and more attention. The research onthe bending-torsion coupling vibration of turbo-generator has some challenge.Further study on it will reveal the dynamical characteristics of the rotor system,increase the accuracy of fault diagnosis, and provide theoretical basis for parametricdesign.
In this paper, the primary resonance of the rotor system with double disks andunbalanced mass is studied. The bending-torsional coupling motion equations of lowpressure cylinder-engine system are derived by using Lagrange equation, where thenonlinear factor is caused by unbalanced mass. By using average method, theanalytical solutions and bifurcation equations are obtained. And the differentbifurcation modals in persistent regions are obtained by singularity theory. After that,the motion equations are simulated by Runge-Kutta method. It is found that theunbalanced mass can arouse bending-torsion coupling vibration and the balancemore large the vibration more serious. Therefore, decreasing the unbalanced masscan reduce the bending-torsional coupling vibration. For the system, primaryresonance may be considered as the input energy of the torsional vibration, whichcan intensify the bending vibration. Therefore, for the actual system,bending-torsion coupling vibration should be considered.
In subsequence, motion equations of low pressure cylinder-engine system underprimary resonance and combined resonance (the frequency of exciting force is thesum of natural frequency of bending vibration and the one of torsinal vibration) intosional direction are constructed, where anisotropic stiffness of shaft is considered.Similarly, the analytical solutions and bifurcation equations are obtained by usingaverage method. And the different bifurcation modals in persistent regions areobtained by singularity theory. The analytical solutions are tested by numericalcalculation. It is found that under the primary resonance and combined resonancethe amplitudes of bending vibration and torsional vibration both increase largely. Inthis case, primary resonance can be considered as the input energy of the torsionalvibration, and combined resonance can be considered as energy switch path. Whenthe primary resonance causes the torsional vibration of shaft, the energy istransmitted to bending vibration through switch path. Then the amplitude of bending vibration increases. For this case, the system can be destroyed both in bendingdirection and torsional direction. Therefore during the dynamical analysis of systemthe bending-torsion coupling vibration should be considered, and in the design phaseof the system this case is best to avoid.
The bending-torsion coupling vibration of shaft caused by parallel misalignmentof flange coupling is analyzed, where the nonlinear factor is caused by unbalancedmass. The bending-torsion coupling motion equations are constructed for the case ofcombined resonance, i.e. the ratio of the frequency of exciting force, naturalfrequency of bending vibration and natural frequency of torsinal vibration is1:1:2.By using average method and singularity theory, the analytical solutions andbifurcation equations are obtained. And the numerical solutions are simulated byRunge-Kutta method. For the combined resonance (the ratio of the frequency ofexciting force, natural frequency of bending vibration and natural frequency oftorsinal vibration is1:1:2), there is not only input energy (the ratio of the frequencyof exciting force and natural frequency of bending vibration is1:1), but also energyswitch path (the ratio of natural frequency of bending vibration and naturalfrequency of torsinal vibration is1:2), which can cause the damage of the system.Therefore, in this case the bending-torsion vibration should be considered. Duringthe design of the system, the combined vibration is best to avoid.
For the actual system, the state variables are usually constrained. There aremany studies on the one state variable singularity theory with constraints, whilelittle study on the two state variables singularity theory with constraints. In thispaper, the two state variables system with constraints is studied, where sevendifferent constraints are considered. The transition sets are obtained. In theunconstrained region, the bifurcation set and double limit set are confirmed with theones without constraints, but hysteresis set is different. The constraints arouse somenew transition sets. These new transition sets are associated with boundary.Therefore, they are called boundary induced transition sets. As an example, thebifurcation of a two dimensional system with constraints is analyzed. It is found thatthere are some new transition sets are induced by the constraints, which is a test ofthe bifurcation theory of two state variables system with constraints.
首先以具有不平衡质量的双盘转子为研究对象，考虑了扭转方向主共振的情况，通过拉格朗日方程，建立了低压缸-发电机系统的弯扭耦合微分方程，非线性项是由质量不平衡引起的。采用平均法，得到了系统的解析解和分岔方程。应用奇异性理论求解出在不同保持域内的运动模式，应用龙格库塔方法对微分方程进行数值计算。不平衡质量能够引起弯扭耦合振动，质量不平衡越大，弯扭耦合越剧烈，因此减小质量不平衡有助于降低弯扭耦合振动。扭转方向主共振相当于对扭转振动的能量输入，这将会引起较大的扭转振动从而加剧转子的弯曲振动，在这种情况下应该考虑系统的弯扭耦合振动问题。
在此基础上，考虑了轴的各向异性刚度，在扭转方向主共振和组合共振(工频约等于弯曲固有频率与扭转固有频率之和)共同作用的情况下，建立了低压缸-发电机的运动微分方程。应用平均法得到系统的解析解和分岔方程，应用两个状态变量的奇异性理论分析了弯曲振动和扭转振动在不同保持域中的运动模式，应用数值算法对解析解进行验证。当在扭转方向主共振和组合共振共同作用时，系统的弯曲振动和扭转振动的幅值都大幅度增加，扭转方向主共振相当于能量输入，组合共振相当于能量交换通道，扭转方向主共振引起轴系的扭转振动大幅度增加，通过能量交换通道传递到弯曲振动上，引起弯曲振动幅值的大幅度增加。此种情况对系统的危害极大，在弯曲方向和扭转方向都有发生破坏的可能，必须要考虑弯扭耦合振动问题，在设计时尽量避免此种情况的发生。
以凸缘联轴器的平行不对中引起轴系的弯扭耦合振动问题为研究对象，考虑了由质量不平衡引起的非线性因素。在激励频率、扭转振动的固有频率、弯曲振动的固有频率之比约为1:1:2的情况时，建立了系统的弯扭耦合振动的微分方程。应用平均法和奇异性理论对该模型进行解析分析，利用龙格库塔法进行数值计算。由于组合共振关系(激励频率、扭转振动的固有频率、弯曲振动的固有频率之比约等于1:1:2)的存在，系统既有能量输入项(激励频率和弯曲固有频率之比约等于1:2)，又有能量交换通道(扭转振动的固有频率和弯曲振动的固有频率之比约等于1:2)，系统的弯曲振动和扭转振动都剧烈增加甚至有破坏的可能，所以在此种情况下必须要考虑弯扭耦合振动问题，并且在设计系统时，应该尽量避免这种组合共振关系。
在实际的工程系统中，状态变量常常是受到约束限制的，对于一个状态变量受到约束的奇异性理论研究的比较充分，而多个状态变量受到约束的奇异性理论研究到目前为止还未开展。以两个状态变量受到约束限制为研究对象，考虑七种不同的约束限制条件，分别讨论了它们的转迁集形式。在受到约束的情况下在非约束区间其分岔集和双极限点集与非约束情况下相同，滞后集不同。由于约束的存在，会产生许多新的转迁集形式。由于这些新的转迁集与边界有关，称之为由于边界引起的转迁集。以二维系统为例研究了两个状态变量的约束分岔形式，通过比较发现，由于约束条件引起许多新的转迁集，以一个例子对两个状态变量约束分岔理论进行分析说明。
As the developments of social economy, the requirement of electric isincreasing, which promotes the turbo-generator unit to large scale and large capacity.As the result, the vibration of the turbo-generator is more and more serious. Inactual system, for the existence of unbalanced mass and exciting force, thebending-torsion coupling vibration causes more and more attention. The research onthe bending-torsion coupling vibration of turbo-generator has some challenge.Further study on it will reveal the dynamical characteristics of the rotor system,increase the accuracy of fault diagnosis, and provide theoretical basis for parametricdesign.
In this paper, the primary resonance of the rotor system with double disks andunbalanced mass is studied. The bending-torsional coupling motion equations of lowpressure cylinder-engine system are derived by using Lagrange equation, where thenonlinear factor is caused by unbalanced mass. By using average method, theanalytical solutions and bifurcation equations are obtained. And the differentbifurcation modals in persistent regions are obtained by singularity theory. After that,the motion equations are simulated by Runge-Kutta method. It is found that theunbalanced mass can arouse bending-torsion coupling vibration and the balancemore large the vibration more serious. Therefore, decreasing the unbalanced masscan reduce the bending-torsional coupling vibration. For the system, primaryresonance may be considered as the input energy of the torsional vibration, whichcan intensify the bending vibration. Therefore, for the actual system,bending-torsion coupling vibration should be considered.
In subsequence, motion equations of low pressure cylinder-engine system underprimary resonance and combined resonance (the frequency of exciting force is thesum of natural frequency of bending vibration and the one of torsinal vibration) intosional direction are constructed, where anisotropic stiffness of shaft is considered.Similarly, the analytical solutions and bifurcation equations are obtained by usingaverage method. And the different bifurcation modals in persistent regions areobtained by singularity theory. The analytical solutions are tested by numericalcalculation. It is found that under the primary resonance and combined resonancethe amplitudes of bending vibration and torsional vibration both increase largely. Inthis case, primary resonance can be considered as the input energy of the torsionalvibration, and combined resonance can be considered as energy switch path. Whenthe primary resonance causes the torsional vibration of shaft, the energy istransmitted to bending vibration through switch path. Then the amplitude of bending vibration increases. For this case, the system can be destroyed both in bendingdirection and torsional direction. Therefore during the dynamical analysis of systemthe bending-torsion coupling vibration should be considered, and in the design phaseof the system this case is best to avoid.
The bending-torsion coupling vibration of shaft caused by parallel misalignmentof flange coupling is analyzed, where the nonlinear factor is caused by unbalancedmass. The bending-torsion coupling motion equations are constructed for the case ofcombined resonance, i.e. the ratio of the frequency of exciting force, naturalfrequency of bending vibration and natural frequency of torsinal vibration is1:1:2.By using average method and singularity theory, the analytical solutions andbifurcation equations are obtained. And the numerical solutions are simulated byRunge-Kutta method. For the combined resonance (the ratio of the frequency ofexciting force, natural frequency of bending vibration and natural frequency oftorsinal vibration is1:1:2), there is not only input energy (the ratio of the frequencyof exciting force and natural frequency of bending vibration is1:1), but also energyswitch path (the ratio of natural frequency of bending vibration and naturalfrequency of torsinal vibration is1:2), which can cause the damage of the system.Therefore, in this case the bending-torsion vibration should be considered. Duringthe design of the system, the combined vibration is best to avoid.
For the actual system, the state variables are usually constrained. There aremany studies on the one state variable singularity theory with constraints, whilelittle study on the two state variables singularity theory with constraints. In thispaper, the two state variables system with constraints is studied, where sevendifferent constraints are considered. The transition sets are obtained. In theunconstrained region, the bifurcation set and double limit set are confirmed with theones without constraints, but hysteresis set is different. The constraints arouse somenew transition sets. These new transition sets are associated with boundary.Therefore, they are called boundary induced transition sets. As an example, thebifurcation of a two dimensional system with constraints is analyzed. It is found thatthere are some new transition sets are induced by the constraints, which is a test ofthe bifurcation theory of two state variables system with constraints.
引文
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