19. Comparison of Computational Generalized and Standard Eigenvalue Solutions of Rotating Systems

Modal analysis is regularly used to compute natural frequencies and mode shapes of structures via eigenvalue solutions in vibration engineering. In this paper, the eigenvalue problem of a 6 degrees of freedom rotating system with gyroscopic effects, including axial, torsional and lateral motion, is investigated using Timoshenko beam theory. The main focus thereby is the investigation of the computational time and the numerical errors in generalized and standard eigenvalue solutions of rotating systems. The finite element method is employed to compute the global stiffness, mass and gyroscopic matrices of the rotating system. The equations of motion is expressed in the state space form to convert the quadratic eigenvalue problem into the generalized and standard forms. The number of elements in the finite element model was varied to investigate the convergence of the natural frequencies and the computational performance of the two eigenvalue solutions. The numerical analyses show that the standard eigenvalue solution is significantly faster than the generalized one with increasing number of elements and the generalized eigenvalue solution can yield wrong solutions when using higher numbers of elements due to the ill-conditioning phenomenon. In this regard, the standard eigenvalue solution gives more reliable results and uses less computational time than the generalized one.