A Symplectic Systematic Methodology For Rotor Dynamics And The Corresponding Numerical Computational Methods

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 frameThis 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.