Search results

In structural mechanics, systems with damping factor get converted to nonlinear eigenvalue problems (NEPs), namely, quadratic eigenvalue problems. Generally, the parameters of NEPs are considered as crisp values but because of errors in measurement, observation or maintenance-induced errors, the parameters may have uncertain bounds of values, and such uncertain bounds may be considered in terms of closed intervals. As such, this paper aims to deal with solving nonlinear interval eigenvalue problems (NIEPs) with respect to damped spring-mass systems having interval parameters.

Two methods, namely, linear sufficient regularity perturbation (LSRP) and direct sufficient regularity perturbation (DSRP), have been proposed for solving NIEPs based on sufficient regularity perturbation method for intervals. LSRP may be used for solving NIEPs by linearizing the eigenvalue problems into generalized interval eigenvalue problems, and DSRP may be considered as a direct solution procedure for solving NIEPs.

LSRP and DSRP methods help in computing the lower and upper eigenvalue and eigenvector bounds for NIEPs which contain the crisp eigenvalues. Further, the DSRP method is computationally efficient compared to LSRP.

The efficiency of the proposed methods has been validated by example problems of NIEPs. Moreover, the procedures may be extended for other nonlinear interval eigenvalue application problems.

The solution of dynamic problems of structures using finite element method leads to generalised eigenvalue problem. In general, if the material properties are crisp (exact) then we get crisp eigenvalue problem. But in actual practice, instead of crisp material properties we may have only bounds of values as a result of errors in measurements, observations and calculations or it may be due to maintenance induced error etc. Such bounds of values may be considered in terms of interval or fuzzy numbers. The purpose of this paper is to develop a fuzzy filtering procedure for finding real eigenvalue bounds of different structural problems.

The proposed fuzzy filtering algorithm has been developed in terms of fuzzy number to solve the fuzzy eigenvalue problem. The initial bounds of fuzzy eigenvalues are filtered to obtain precise eigenvalue bounds which are depicted by fuzzy (Triangular Fuzzy Number) plots using α-cut.

Previously, bounds of eigenvalues of interval matrices have been investigated by few authors. But when the structural problem consists of fuzzy material properties, then the interval eigenvalue bounds may be obtained for each interval of the fuzzy number. The proposed algorithm has been applied for standard fuzzy eigenvalue problems which may be extended to generalised fuzzy eigenvalue problems for obtaining filtered fuzzy bounds.

The developed fuzzy filtering method is found to be efficient for different structural dynamics problems with fuzzy material properties.