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# Solving Fully Fuzzy Nonlinear Eigenvalue Problems of Damped Spring-Mass Structural Systems Using Novel Fuzzy-Affine Approach

S. Rout1, S. Chakraverty1,*
1 Department of Mathematics, National Institute of Technology Rourkela, Odisha, India.
* Corresponding Author: S. Chakraverty. Email: .

Computer Modeling in Engineering & Sciences 2019, 121(3), 947-980. https://doi.org/10.32604/cmes.2019.08036

### Abstract

The dynamic analysis of damped structural system by using finite element method leads to nonlinear eigenvalue problem (NEP) (particularly, quadratic eigenvalue problem). In general, the parameters of NEP are considered as exact values. But in actual practice because of different errors and incomplete information, the parameters may have uncertain or vague values and such uncertain values may be considered in terms of fuzzy numbers. This article proposes an efficient fuzzy-affine approach to solve fully fuzzy nonlinear eigenvalue problems (FNEPs) where involved parameters are fuzzy numbers viz. triangular and trapezoidal. Based on the parametric form, fuzzy numbers have been transformed into family of standard intervals. Further due to the presence of interval overestimation problem in standard interval arithmetic, affine arithmetic based approach has been implemented. In the proposed method, the FNEP has been linearized into a generalized eigenvalue problem and further solved by using the fuzzy-affine approach. Several application problems of structures and also general NEPs with fuzzy parameters are investigated based on the proposed procedure. Lastly, fuzzy eigenvalue bounds are illustrated with fuzzy plots with respect to its membership function. Few comparisons are also demonstrated to show the reliability and efficacy of the present approach.

### Keywords

Fuzzy nonlinear eigenvalue problem, fuzzy set theory, affine arithmetic, interval overestimation problem, triangular fuzzy number, trapezoidal fuzzy number, fuzzy-affine approach.

Rout, S., Chakraverty, S. (2019). Solving Fully Fuzzy Nonlinear Eigenvalue Problems of Damped Spring-Mass Structural Systems Using Novel Fuzzy-Affine Approach. CMES-Computer Modeling in Engineering & Sciences, 121(3), 947–980.

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