转子动力学的求解辛体系及其数值计算方法
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摘要
本文继承了应用力学对偶体系的辛数学方法,将它应用于陀螺转子动力学中。结合陀螺转子动力学自身的特点,构建了陀螺转子动力学的求解辛体系,并提出了相应的数值计算方法。这一套方法与传统的陀螺转子动力学方法不同,从一个全新的角度研究了陀螺转子动力学问题,为陀螺转子动力学的研究开辟了一条新路。
本学位论文的思想首先是将陀螺转子动力系统导入到哈密顿体系,然后在哈密顿体系下对陀螺转子动力学问题展开研究。主要研究了陀螺转子动力学的四个常规问题:陀螺系统的本征值问题、状态空间内的模态综合方法、陀螺系统的时间有限元方法以及一般线性哈密顿系统的本征摄动问题。从本论文的工作中可以看到,这种方法较好地解决了陀螺转子动力学的一些问题,大量的数值算例表明,该方法拥有其独特的优越性。主要研究工作如下:
1)对陀螺系统的本征值问题进行了研究
陀螺转子本征值问题一直是陀螺转子动力学的重要问题,在对应刚度阵正定的情况下已经提出了很多实用的方法。但对于不对称转子的情况,特别是在高转速情况下,采用相对坐标系,经常会出现系统刚度矩阵不正定的情况,特别是在自由度较多的情况下,其本征值问题是很难进行求解的。
本文第三章针对上述情况,首先将辛子空间迭代法的思想应用于陀螺系统,发展出了适合于不正定陀螺系统的辛子空间迭代法,这种方法继承了子空间迭代法的特点,具有很好的稳定性。之后,为了使已成熟的正定算法能够应用于不正定陀螺系统的本征值问题,本文提出了一种能够有效计算不正定陀螺系统本征值问题的方案。以上两种方法比较好地解决了不正定陀螺系统的本征值问题。
2)讨论了陀螺效应对转子系统的影响并建立了哈密顿框架下的模态综合方法
本文举例说明了在动力转子系统中陀螺效应对实际模型的影响。分析了转子陀螺效应对进动角速度、振型以及临界角速度的影响。数值结果表明,在一些工程问题中特别是高转速情况下,陀螺力对于转子系统振动是很重要的一项。
基于陀螺系统辛子空间迭代法,在哈密顿框架下提出了陀螺系统的模态综合方法(MSMGS)。可以看到此方法的转换矩阵为辛矩阵,保持了系统的哈密顿框架。算例证明了缩减后的系统能够比较好地近似原来的整体陀螺系统。
This dissertation inherits the symplectic method of duality system in applied mechanics, and it can be applied to gyroscopic rotor dynamics. Based on the characteristics of gyroscopic rotor dynamics, a new systematic methodology and the corresponding numerical computational methods are presented. This methodology is different from the traditional methodology and it studies some problems of gyroscopic rotor dynamics from a new viewpoint and a new way is developed for gyroscopic rotor dynamics.In this paper, gyroscopic systems can firstly be guided to Hamiltonian systems which constructs a perfect theoretical frame. In Hamiltonian systems, the theoretical analysis and computation of gyroscopic rotor systems can be studied. Four traditional problems of gyroscopic rotor dynamics are mainly studied in this dissertation: the eigenvalue problem of gyroscopic systems, modal synthesis method in the state space, time domain finite element method of gyroscopic systems, and a perturbation method for reanalysis of linear Hamiltonian systems. It is seen that some problems of gyroscopic rotor dynamics can conveniently be solved. The examples prove that this method have their own advantages. The main research work covers the following aspects:1) Study the eigenvalue problem of gyroscopic systemsThe eigenvalue problem of gyroscopic systems is always a typical mathematical problem of gyroscopic rotor dynamics. Many methods have been introduced when systemic stiffness matrix is positive definite, but when systemic stiffness matrix is not positive definite, i.e. the corresponding Hamiltonian function is not positive definite, the solving of the eigenvalue problem is very difficult.Firstly, using the idea of subspace iteration method of symmetric matrix and many parallel points in the characters of Hamiltonian matrix and symmetric matrix, an adjoint symplectic subspace iteration method of indefinite gyroscopic systems is proposed to solve the eigenvalue problem of indefinite gyroscopic systems. This method inherits the property of subspace iteration method and it has good stability. Secondly, the algorithms to solve eigenvalue problem of positive definite gyroscopic systems are wholesome. To use these algorithms to solve the eigenvalue problem of indefinite gyroscopic systems, a project is proposed to solve the eigenvalue problem of indefinite gyroscopic systems. This dissertation demonstrates by examples this algorithm is right. The above two methods can solve the eigenvalue problem of indefinite gyroscopic systems very well.2) Discuss influence of gyroscopic term to the vibration of rotor systems and propose modal synthesis method in Hamiltonian frame
This paper demonstrates by example the effect of gyroscopic term to the true modal of rotor systems. The processional frequency, mode and critical speed are analyzed importantly. The state space method is used to solve the eigenvalue problem. This example demonstrates that gyroscopic effect can not be ignored for some vibration analysis.Based on an adjoint symplectic subspace iteration method of gyroscopic systems, modal synthefic method of large gyroscopic systems(MSMGS) is proposed. It shows that the whole transition matrix is composed of the eigenvector matrices of all the subsystems and a symplectic matrix which holds the Hamiltonian frame. The example proves that the whole gyroscopic system can be approximated by the reduced gyroscopic system.Finally, precise integration method is applied to the solving of the unbalance response of rotor systems that leads to high efficiency and good accuracy. The results demonstrate that the first few modes usually take main effects in rotor systems.3) Develop time domain finite element method of gyroscopic systemsBased on the variational principle, time domain finite element method of gyroscopic systems is presented. The corresponding trial function matrix, element stiffness matrix and inhomogeneous force are given. The interval combination method of time domain FEM is subsequently proposed which has higher efficiency. The method inherits the property of symplectic conservation and enhances computational accuracy. The examples comparing the numerical results obtained from three different methods: time domain FEM, 4th order Runge-Kutta method and Newmark method demonstrate the advantages of time domain FEM.Furthermore, time domain FEM is applied to nonlinear gyroscopic rotor systems. The computational results show that time domain FEM has good accuracy and stability.4) Propose a perturbation method for reanalysis of linear conservation systemsA perturbation method for reanalysis of linear Hamiltonian systems is studied via the self-adjoint simplectic orthonormality relation of Hamiltonian operator in this paper and a perturbation reanalysis method of Hamiltonian matrix(PRMHM) is proposed. The eigen-equation of Hamiltonian systems and the adjoint simplectic orthonormal relationship are presented. The second order eigensolutions of modified Hamiltonian systems are obtained. Based on the above method, the approximate method for computing 1st order eigenvector derivatives in general linear Hamiltonian systems is proposed, using the approximate method for computing eigenvector derivatives in free vibration. The examples prove that two algorithms are valid.
本学位论文的思想首先是将陀螺转子动力系统导入到哈密顿体系,然后在哈密顿体系下对陀螺转子动力学问题展开研究。主要研究了陀螺转子动力学的四个常规问题:陀螺系统的本征值问题、状态空间内的模态综合方法、陀螺系统的时间有限元方法以及一般线性哈密顿系统的本征摄动问题。从本论文的工作中可以看到,这种方法较好地解决了陀螺转子动力学的一些问题,大量的数值算例表明,该方法拥有其独特的优越性。主要研究工作如下:
1)对陀螺系统的本征值问题进行了研究
陀螺转子本征值问题一直是陀螺转子动力学的重要问题,在对应刚度阵正定的情况下已经提出了很多实用的方法。但对于不对称转子的情况,特别是在高转速情况下,采用相对坐标系,经常会出现系统刚度矩阵不正定的情况,特别是在自由度较多的情况下,其本征值问题是很难进行求解的。
本文第三章针对上述情况,首先将辛子空间迭代法的思想应用于陀螺系统,发展出了适合于不正定陀螺系统的辛子空间迭代法,这种方法继承了子空间迭代法的特点,具有很好的稳定性。之后,为了使已成熟的正定算法能够应用于不正定陀螺系统的本征值问题,本文提出了一种能够有效计算不正定陀螺系统本征值问题的方案。以上两种方法比较好地解决了不正定陀螺系统的本征值问题。
2)讨论了陀螺效应对转子系统的影响并建立了哈密顿框架下的模态综合方法
本文举例说明了在动力转子系统中陀螺效应对实际模型的影响。分析了转子陀螺效应对进动角速度、振型以及临界角速度的影响。数值结果表明,在一些工程问题中特别是高转速情况下,陀螺力对于转子系统振动是很重要的一项。
基于陀螺系统辛子空间迭代法,在哈密顿框架下提出了陀螺系统的模态综合方法(MSMGS)。可以看到此方法的转换矩阵为辛矩阵,保持了系统的哈密顿框架。算例证明了缩减后的系统能够比较好地近似原来的整体陀螺系统。
This dissertation inherits the symplectic method of duality system in applied mechanics, and it can be applied to gyroscopic rotor dynamics. Based on the characteristics of gyroscopic rotor dynamics, a new systematic methodology and the corresponding numerical computational methods are presented. This methodology is different from the traditional methodology and it studies some problems of gyroscopic rotor dynamics from a new viewpoint and a new way is developed for gyroscopic rotor dynamics.In this paper, gyroscopic systems can firstly be guided to Hamiltonian systems which constructs a perfect theoretical frame. In Hamiltonian systems, the theoretical analysis and computation of gyroscopic rotor systems can be studied. Four traditional problems of gyroscopic rotor dynamics are mainly studied in this dissertation: the eigenvalue problem of gyroscopic systems, modal synthesis method in the state space, time domain finite element method of gyroscopic systems, and a perturbation method for reanalysis of linear Hamiltonian systems. It is seen that some problems of gyroscopic rotor dynamics can conveniently be solved. The examples prove that this method have their own advantages. The main research work covers the following aspects:1) Study the eigenvalue problem of gyroscopic systemsThe eigenvalue problem of gyroscopic systems is always a typical mathematical problem of gyroscopic rotor dynamics. Many methods have been introduced when systemic stiffness matrix is positive definite, but when systemic stiffness matrix is not positive definite, i.e. the corresponding Hamiltonian function is not positive definite, the solving of the eigenvalue problem is very difficult.Firstly, using the idea of subspace iteration method of symmetric matrix and many parallel points in the characters of Hamiltonian matrix and symmetric matrix, an adjoint symplectic subspace iteration method of indefinite gyroscopic systems is proposed to solve the eigenvalue problem of indefinite gyroscopic systems. This method inherits the property of subspace iteration method and it has good stability. Secondly, the algorithms to solve eigenvalue problem of positive definite gyroscopic systems are wholesome. To use these algorithms to solve the eigenvalue problem of indefinite gyroscopic systems, a project is proposed to solve the eigenvalue problem of indefinite gyroscopic systems. This dissertation demonstrates by examples this algorithm is right. The above two methods can solve the eigenvalue problem of indefinite gyroscopic systems very well.2) Discuss influence of gyroscopic term to the vibration of rotor systems and propose modal synthesis method in Hamiltonian frame
This paper demonstrates by example the effect of gyroscopic term to the true modal of rotor systems. The processional frequency, mode and critical speed are analyzed importantly. The state space method is used to solve the eigenvalue problem. This example demonstrates that gyroscopic effect can not be ignored for some vibration analysis.Based on an adjoint symplectic subspace iteration method of gyroscopic systems, modal synthefic method of large gyroscopic systems(MSMGS) is proposed. It shows that the whole transition matrix is composed of the eigenvector matrices of all the subsystems and a symplectic matrix which holds the Hamiltonian frame. The example proves that the whole gyroscopic system can be approximated by the reduced gyroscopic system.Finally, precise integration method is applied to the solving of the unbalance response of rotor systems that leads to high efficiency and good accuracy. The results demonstrate that the first few modes usually take main effects in rotor systems.3) Develop time domain finite element method of gyroscopic systemsBased on the variational principle, time domain finite element method of gyroscopic systems is presented. The corresponding trial function matrix, element stiffness matrix and inhomogeneous force are given. The interval combination method of time domain FEM is subsequently proposed which has higher efficiency. The method inherits the property of symplectic conservation and enhances computational accuracy. The examples comparing the numerical results obtained from three different methods: time domain FEM, 4th order Runge-Kutta method and Newmark method demonstrate the advantages of time domain FEM.Furthermore, time domain FEM is applied to nonlinear gyroscopic rotor systems. The computational results show that time domain FEM has good accuracy and stability.4) Propose a perturbation method for reanalysis of linear conservation systemsA perturbation method for reanalysis of linear Hamiltonian systems is studied via the self-adjoint simplectic orthonormality relation of Hamiltonian operator in this paper and a perturbation reanalysis method of Hamiltonian matrix(PRMHM) is proposed. The eigen-equation of Hamiltonian systems and the adjoint simplectic orthonormal relationship are presented. The second order eigensolutions of modified Hamiltonian systems are obtained. Based on the above method, the approximate method for computing 1st order eigenvector derivatives in general linear Hamiltonian systems is proposed, using the approximate method for computing eigenvector derivatives in free vibration. The examples prove that two algorithms are valid.
引文
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