On the free vibration analysis of spinning structures by using discrete or distributed mass models

Abstract

The analysis of the natural frequencies and modes of vibration of a spinning structure is in general complicated by the presence of gyroscopic, or Coriolis, forces, leading to a complex Hermitian dynamic stiffness (or impedance) matrix. It is shown that if the analysis is performed by using a discrete model with N degrees of freedom, the leading principal minors of the Nth order dynamic stiffness matrix exhibit a type of Sturm sequence property. This leads to a theorem which can be used for the systematic calculation of the natural frequencies of either a discrete system which is assembled from sub-structures, or an assembly of distributed mass members. In both of these cases the order n of the matrix is less than the number of degrees of freedom, and its determinant possesses poles as well as zeros. The theorem is identical with a corresponding one for non-spinning structures previously derived by the authors. The application of the theorem to a spinning two-dimensional frame, with distributed mass members, is discussed in some detail.