On Some Generalization of the Eigenvectors Properties

Abstract

In this paper the generalization of the classical mode orthogonality and normalization relationships known for undamped systems to non – classical and non – viscously damped systems were established and investigated. Classical mode orthogonality relationships known for undamped systems were generalized to non – viscously damped systems. It was shown that there exists unique relationship which relates the system matrices to the natural frequencies and modes of non – viscously damped systems. These relationships, in return, enable us to reconstruct the system matrices from full set of modal analysis. The non – viscously damping model is such that the damping forces depends on the past history of motion via convolution integrals over some kernel functions. Classical modal analysis is extended to deal with general non – viscously damped of multiple degree- of- freedom (MDOF) linear dynamic systems. The concept (complex) of non – viscous mode was introduced and further shown that the system response can be obtained exactly in terms of these modes.

Keywords: Eigenvectors, eigenvalues, viscously undamped, orthogonality, normalization.