On Lumping of Gyroscopic Matrix in Finite Element Analysis of Rotors

Abstract

Dynamic analysis of structures using Finite elements largely needs to handle Mass, Stiffness matrices and Excitation vector. Lumping of Mass matrix is well established and its computational advantage recognized.Dynamic analysis of rotors additionally requires handling of skew symmetric Gyroscopic matrix which makes the solution difficult and solvers complicated. The present work attempts to lump Gyroscopic matrix.In real co-ordinates, the Gyroscopic matrix is skew symmetric. If Gyroscopic effect of disc is only considered, off diagonal skew symmetric terms appear at limited degrees of freedom. However, if gyroscopic effect of shaft is also considered, the off diagonal skew symmetric terms are present throughout. Use of complex co-ordinates diagonalizes the skew symmetric terms due to discs only. However, if distributed gyroscopic effect of shaft is considered, the Gyroscopic matrix becomes symmetric but non-diagonal. Since gyroscopic effect like mass appears due to inertia, the present work attempts to lump Gyroscopic matrix in complex co-ordinates.