Fully analytic approach to study bifurcation phenomenon for a shape memory alloy beam under temperature variation and lateral white-noise excitation

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Abstract
The main aim of this article is to obtain the probability density function of the response of an SMA beam under lateral white-noise excitation and study the bifurcation phenomenon through a fully analytic approach. Firstly, an efficient constitutive stress–strain relation for a shape memory alloy material supporting both martensite and austenite phases is considered. Secondly, the governing equation of motion for a typical SMA simply supported beam is derived using some dimensionless parameters. Thirdly, the corresponding probability density function of the response is computed analytically using the powerful Fokker–Planck–Kolmogorov equation, and finally, the bifurcation phenomenon for parameter variations of the beam is investigated. The results show how geometric (beam aspect ratio), force (the mean value of the white-noise excitation), and environmental (working temperature) factors can affect the nonlinear behavior and response bifurcation of the beam. A numerical validation is also done that guarantees the correctness of the method and results.

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