Eigenvalue and eigenvector derivatives of fractional vibration systems

Abstract

Dynamic characterizations of fractional vibration systems have recently attracted significant research interest. Increasingly, successful applications of fractional derivatives have been found to the modeling of mechanical damping, vibration transmissions, improved fractional vibration controls and nonlinear vibration analyses. To facilitate further development, the eigenvalue problem including its derivatives, which are the central issues of vibration analysis, have to be fully established. This paper examines how eigenvalue and eigenvector derivatives of fractional systems can be derived when system matrices become functions of physical design parameters. First, new important orthonormal constraints are proposed since the modes are no longer orthonormal to the mass matrix, in this case due to its complex and frequency dependent nature. Next, new methods of eigenvector derivatives are developed for distinct eigenvalues for the cases of complete, incomplete and single mode modal data. Realistic and practical FE models incorporating fractional derivatives in the form of viscoelastic supports are employed to demonstrate the numerical accuracy and computational efficiency of the proposed methods. However, when repeated eigenvalues are considered due to structural spatial symmetries, the eigenvector space degenerates and further differentiation of system matrices are required in order to uniquely determine the eigenvector derivatives. Consequently, a new and effective general method is developed which can be applied to compute eigenvector derivatives of repeated eigenvalues with any multiplicity m. A simplified turbine bladed disk vibration model which is known to have repeated eigenvalues due to its cyclic symmetry, is then used to demonstrate the accuracy and salient features of the proposed method.