Vibration of deploying rectangular cross-sectional beam made of functionally graded materials

• Published in: Journal of low frequency noise, vibration, and active control Vol. 37; no. 4; pp. 748 - 761
• Main Authors: Zhao, Feng-Qun, Wang, Zhong-Min
• Format: Journal Article
• Language: English
• Published: London, England SAGE Publications 01.12.2018; Sage Publications Ltd; SAGE Publishing
• Subjects: Cantilever beams; Cross-sections; Deflection; Differential equations; Divergence; Dynamic stability; Eigenvalues; Eigenvectors; Equations of motion; Functionally gradient materials; Galerkin method; Ordinary differential equations; Time dependence; Transverse oscillation; complex frequency; divergence instability; Deploying functionally graded material beam; transverse vibration
• ISSN: 1461-3484; 2048-4046
• DOI: 10.1177/1461348418765957

Transverse vibration and stability of deploying rectangular cross-sectional cantilever beam made of functionally graded material are investigated. The functionally graded material beam is assumed to be constructed with ceramics and metal phases, and the corresponding equivalent parameters of functionally graded material are found to continuously vary across the cross-sectional height with a simple power law. The differential equations of motion of deploying functionally graded material cantilever beam are derived by Hamilton’s principle. Based on the assumed modal method, the beam deflection function is expanded into a series, in which each term is expressed to admissible function multiplied by generalized coordinate. The eigenfunctions of cantilever beam in which the length of the beam is time-dependent are chosen as admissible functions. Galerkin method is employed to discretize the differential equation to a set of time-coordinate-dependent ordinary differential equations, and then the eigenvalue problem depending on time is obtained. The deployment motion of beam is considered as a constant speed in this study, and some numerical results, which are variations of tip deflection response and complex frequencies, are obtained. Finally, the effects of gradient index of functionally graded material, deploying speed, initial length and protruded length, the cross-sectional height on dynamical characteristics, and divergence instability of the deploying functionally graded material beam are discussed.