# Application types of transfer matrix method in alloy centrifugal casting

In the 1980s, Huang Taiping, Gu jialiu and others successively proposed some improved transfer matrix methods for relatively complex rotor systems. The more common methods are transfer matrix impedance coupling method, transfer matrix mode synthesis method and transfer matrix direct integration method.

The transfer matrix impedance coupling method is mainly used to analyze and calculate the natural frequency and critical speed of complex multi rotor multi support system. The basic idea is to divide the complex rotor support system into several simple subsystems according to certain principles, and then establish the equations for the subsystems by using the transfer matrix method. The coupling relationship between the subsystems is based on the mechanical impedance at the dividing point A group of linear homogeneous equations is formed, and then the natural frequency and critical speed of the whole system are obtained.

The transfer matrix mode synthesis method is a combination of the transfer matrix method and the modal synthesis method. It makes use of the advantages of the transfer matrix method to analyze the chain structure simply and accurately and the modal synthesis method to reduce the order easily. It is suitable for analyzing the natural frequency and critical speed of the complex multi rotor multi support system. It can also calculate the unbalance response and optimize the rotor dynamics design. It is different from the transfer matrix impedance coupling method in that it constrains the subsystems at the dividing point, analyzes the constrained vibration mode and the constrained static displacement mode of each subsystem with the transfer matrix method, and then synthesizes the modes to obtain the dynamic characteristics of the system.

The transfer matrix direct integration method is aimed at the complex rotor support brake system. In the analysis, several characteristic disks are selected, and then the motion equations of the characteristic disks of each chain are established. The motion equations are established by the transfer matrix method, and the coupling equations are established according to the connection relationship between chains. Then the motion equations of the backup characteristic disks under the coupling force, that is, the motion equations of the system, are derived It’s the same. The natural frequency and mode shape of the system can be obtained by solving the homogeneous equation; the critical speed can be determined according to the unbalanced response; the stability can be studied according to the trajectory of the characteristic disk center; the transient response can be obtained by using the transient transfer matrix in the integration process. The transfer matrix direct integration method is a general calculation method of rotor dynamics.

In addition, in view of the complicated problem of rotor segmentation in the pre-processing work of transfer matrix method, Li Haifeng combined with the characteristics of modern CAD software to compile the pre-processing software, which can automatically extract the rotor graphics drawn by CAD software, and automatically segment the rotor according to the requirements of transfer matrix method, so as to prepare data for the calculation software of transfer matrix method, Later, Jiang Lingyun developed a complete set of rotor dynamics calculation software on the basis of his research, which is convenient for engineering application. Wan Haochuan et al. Proposed a method to select the state vector according to the vibration equation instead of the boundary condition, which greatly simplified the calculation process of the first derivative of the state vector. The incidence matrix was introduced between the state vector and the boundary condition, which made the solving process of the transfer matrix easier . Their method is particularly suitable for solving vibration equations of multi degree of freedom systems.

Li synthetically uses Riccati transfer matrix method and newmark-p method to solve the transient response of the shaft system of / 240mw hydro generator unit. Yinhailong 11431 and Yuanding 11441 compile transfer matrix calculation programs with MATLAB to calculate the critical speed and speed of multi rotor system with axial constraints and disc separator rotor system respectively Their research results have certain guiding significance for practical engineering application.

Meng and Wang Shaobo used the transfer matrix method to study the centrifugal unbalance response of gas turbine. The gas turbine rotor is a disk type pull rod rotor. The discontinuity of the pull rod rotor structure and the contact effect of the end face teeth between the disks will have a certain impact on the dynamic characteristics of the rotor. In view of this, they improved the transfer matrix method. The correctness of the proposed method is verified by comparing the theoretical results with the test results of the test bench The dynamic analysis and calculation of similar rotors can be used for reference.

At the beginning of the development of the computer age, the transfer matrix method did play an important role in the analysis of rotor dynamics. But for the complex rotor system, it needs a lot of professional experience and time-consuming when modeling with the beam model. In some cases, it also needs to write some special programs. Later, the finite element method gradually overcomes this shortcoming.

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