# Application of transfer matrix method in alloy centrifugal casting

The basic idea of transfer matrix method is to divide the whole structure into several mechanical problems of docking and transfer of sub elements, establish the transfer matrix between the two ends of the element, and use matrix multiplication to analyze the static and dynamic characteristics of the structure (modal analysis and stability analysis). Its main characteristics are: the chain structure is divided into a series of elements, and the connection between each pair of elements is stable The order of the transfer matrix is equal to the order of the differential equation of motion of the element, so the calculation of the whole system by the transfer matrix method is decomposed into the calculation of each element with very low order, and then integrated, so that the calculation workload is greatly reduced.

The transfer matrix method was first proposed by Prohl and myklestad, which is called Prohl transfer matrix method. It is used to calculate the critical speed of rotor. Lundl uses the complex vibration quantity in the transfer matrix method, and introduces the damping and some unstable factors, so that the transfer matrix method can solve the complex eigenvalues and realize the stability analysis. Homer and Pilkey improved the transfer matrix method and proposed Riccati transfer matrix method to solve the problems of poor numerical stability and low accuracy when the system is complex. Later, Lee et al. Studied the steady-state response of nonlinear bearing rotor system by transfer matrix method, Liew et al. Analyzed the transient response of nonlinear bearing rotor system by transfer matrix method, and Gu et al. Studied the transient response as well as the stability analysis. Tsai et al. 1122] established the transfer matrix method of misalignment shaft with elastic support. In recent years, some scholars have also studied the transfer matrix method of continuous mass. It can be seen that the transfer matrix method is widely used in the study of rotating machinery dynamics.

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