Rapid Algorithm For Sensitivity Analysis Of Eigenvectors In Damped Linear Defective Vibration Systems

In the presence of general non-proportional viscous damping, it can be coupled under the non-diagonal terms of the mode or damped matrix with respect to motion equations in the modal coordinates. Of course the mode and structural have natural frequencies which are complex. In this process, it is represented that two methods make two routes: one is state-space, the other is approximation. As the size of the matrix is increasing, the more difficulty need more effort in mathematic, moreover, the deficient of form also represents in simplicity. Most of this reason result that more research should be developed in the effect of non-proportional viscous damping system.The sensitivity analysis of damped linear vibration systems are widely applied in the area of the structural control and design optimization and fault detection and system discrimination. The eigenproblem of matrix in sensitivity analysis consist of mode expansion and direct perturbation. By using the modal expansion in modification for structural and modal correction, the derivation of the eigenvector is supposed with linear combination of all modal, and defined the combination coefficient by calculation. In general, the property of complete system plays a important role in sensitivity analysis. The most discussion of them are assumed that the system is complete, however, in the practice, such as general damping system, flutter analysis of aero elasticity, the relative system don't own a set of eigenvector, the theories of the modal analysis can not adapt this condition.In this paper, it presents a new method to transform non-classical damped system into classical system with respect to the multi-degree-of-freedom equations and single-degree-of-freedom system. The main idea of the method proposed here lies in introducing a transformation. The transformed problem does get rid of the damping term. Then it analysis the vibration behaviors based on the matrix transfer function in the non-classical system. The property of similar matrix define that the characteristic value after the transformation is equal to the characteristic value before the transformation. As it can be seen that the transformed problem has changed the time-varying system into the quasi time-invariant system, it provides a analytical result about characteristic vector shows a better accuracy than before. A matrix function transformation is introduced to reduce a non-viscous damping system which have a complete eigenvector, so that the original non-proportional damped in non-proportionally damped linear vibration complete systems is transformed into a easier set of 2nd-order differential equations which does not possess the damping term. The new set of equations leads to a computational advantage and get rid of the effect of the damping part in solution of eigenproblem.It summarizes the basic theories and the conclusion of the sensitivity analysis about eigenprolem. Expressly, in damped linear defective vibration systems which does not possess complete eigenvectors in N-space, using the theory of generalized modes, authors have proposed methods to obtain a feature space about generalized eigenvectors, and determine a set of methods of sensitivity analysis towards eigenvector of defective systems, simultaneously introduce a brief and rapid analytical approach to calculate generalized modes by recursive solution.The end, a numerical test indicates the economy and availability of the algorithmic that developed in this paper.