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Technical Brief

# Filtering Algorithm for Real Eigenvalue Bounds of Interval and Fuzzy Generalized Eigenvalue Problems

[+] Author and Article Information
Nisha Rani Mahato

Department of Mathematics,
National Institute of Technology Rourkela,
Rourkela, Odisha 769008, India

S. Chakraverty

Department of Mathematics,
National Institute of Technology Rourkela,
Rourkela, Odisha 769008, India
e-mail: sne_chak@yahoo.com

Manuscript received July 12, 2015; final manuscript received March 9, 2016; published online August 19, 2016. Assoc. Editor: Athanasios Pantelous.

ASME J. Risk Uncertainty Part B 2(4), 044502 (Aug 19, 2016) (8 pages) Paper No: RISK-15-1085; doi: 10.1115/1.4032958 History: Received July 12, 2015; Accepted March 09, 2016

## Abstract

This paper deals with an interval and fuzzy generalized eigenvalue problem involving uncertain parameters. Based on a sufficient regularity condition for intervals, an interval filtering eigenvalue procedure for generalized eigenvalue problems with interval parameters is proposed, which iteratively eliminates the parts that do not contain an eigenvalue and thus reduces the initial eigenvalue bound to a precise bound. The same iterative procedure has been proposed for generalized fuzzy eigenvalue problems. In general, the solution of dynamic problems of structures using the finite element method (FEM) leads to a generalized eigenvalue problem. Based on the proposed procedures, various structural examples with an interval and fuzzy parameter such as triangular fuzzy number (TFN) are investigated to show the efficiency of the algorithms stated. Finally, fuzzy filtered eigenvalue bounds are depicted by fuzzy plots using the $α$-cut.

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## Figures

Fig. 1

Triangular fuzzy number

Fig. 2

Multistory frame structure

Fig. 3

Plot of first eigenvalue bounds

Fig. 4

Plot of second eigenvalue bounds

Fig. 5

Plot of third eigenvalue bounds

Fig. 6

Plot of fourth eigenvalue bounds

Fig. 7

Plot of fifth eigenvalue bounds

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