MODAL ANALYSIS PROCEDURE USING COMPLEX LEFT AND RIGHT EIGENVECTORS OF NON-PROPORTIONALLY DAMPED STRUCTURES

Abstract eng:
In the general case of non-proportionally damped structural model the associated quadratic eigenvalue problem leads to complex eigenvalues and eigenvectors. The modal decomposition of the equations of motion is usually to be performed in complex space. In this paper are presented possible variants of a general method [2] - [5] for modal transformation of damped multi-degree-of-freedom-systems (MDOFS) with non-modal symmetric damping matrix. The assembly of a modal transformation matrix in real space is based either on the conjugated complex left eigenvectors, or on the right eigenvectors, or on a combination of the left and right eigenvectors of the system. The eigenvector normalization can be performed with respect to the general mass or to the general stiffness matrix. The equations of motion are stated in state-space formulation. The developed real-space modal transformation matrix is always built by a combination of two complex transformations. Analytically expressions for all presented variants of the modal transformation basis are developed be the aid of computer algebra software. Those formulas operate with the real and the imaginary parts of the eigenvectors and the associated eigenvalues. All variants of the suggested modal procedure retain the common advantages of the classic modal decomposition of the equations of motion. The vibrations of a rotor blade of a wind turbine subjected to wind thrust loads have been calculated in two variants to demonstrate the performance of the presented modal analysis procedures. The initial computation of the complex eigenvalue solution of the FEM beam model and all subsequent computations are done by the aid of computer algebra software. The suggested procedures can be applied in structural systems containing different damping and energy-loss mechanism in various parts of the structure, described by non-proportional damping matrix.

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