On the Numerical Calculation of the Flexural Critical Speeds of a Rotating Multispan Shaft

摘要

Previous publications were concerned with numerical calculation of the critical speeds induced by the lateral vibration of a shaft running in two bearings. In these calculations, the known fourth-order homogeneous differential equation with homogeneous boundary conditions, which are dependent upon the structure of the bearings, was replaced by two simultaneous second-order differential equations. Following this, the problem is presented in the form of two simultaneous integral equations, of which the solution is replaced by calculation of the eigen-value of a matrix product. The method outlined above is now applied to a shaft running in a number of bearings, as follows: first of all, each span is treated as in the above case of two bearings. The final problem is accordingly that of solving a system of certain integral equations. The boundary conditions at each of the intermediate bearings, which are common to both of the consecutive spans, render it possible to present the whole problem as an eigen-value of a matrix product, as above in the case of only two bearings. The details of the method indicated above are elucidated below in the discussion of some typical examples. (Author)