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Exact Solutions and Mode Transition for Out-of-Plane Vibrations of Non-uniform Beams with variable curvature

Computers, Materials & Continua 2016年第1期51卷 1-19页

作者： Sen-Yung Lee Shueei-Muh Lin Kai-Ping Chang Department of Mechanical Engineering National Cheng Kung UniversityTainanTaiwanRepublic of China Mechanical Engineering Department Kun Shan UniversityTainanTaiwan 710-03Republic of China

The two coupled governing differential equations for the out-of-plane vibrations of non-uniform beams with variable curvature are derived via the Hamilton’s principle.These equations are expressed in terms of flexural and torsional displacements simultaneously.In this study,the analytical method is proposed.Firstly,two physical parameters are introduced to simplify the analysis.One derives the explicit relations between the flexural and the torsional displacements which can also be used to reduce the difficulty in experimental measurements.Based on the relation,the two governing characteristic differential equations with variable coefficients can be uncoupled into a sixth-order ordinary differential equation in terms of the flexural displacement only.When the material and geometric properties of the beam are in arbitrary polynomial forms,the exact solutions with regard to the outof-plane vibrations of non-uniform beams with variable curvature can be obtained by the recurrence formula.In addition,the mode transition mechanism is revealed and the influence of several parameters on the vibration of the non-uniform beam with variable curvature is explored.

关键词： Out-of-plane vibration variable curvature Non-uniform beam Exact solution Mode transition

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Non-uniform rational B-spline based free vibration analysis of axially functionally graded tapered Timoshenko curved beams

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Applied Mathematics and Mechanics(English Edition) 2020年第4期41卷 567-586页

作者： Zhiwei ZHOU Meixia CHEN Kun XIE School of Naval Architecture and Ocean Engineering Huazhong University of Science and Technology College of Engineering Huazhong Agricultural University

The free vibration of axially functionally graded(FG) tapered Timoshenko curved beams is studied with the numerical approach. By using the non-uniform rational B-spline(NURBS) basis functions, the exact geometry and the generalized displacement?eld are formulated. variable geometric parameters and material properties, including the curvature, cross-sectional area, area moment of inertia, mass density, and Young’s modulus, are expanded as functions of the coordinate in a parametric domain. Based on Hamilton’s principle, the weak formulation is derived by applying a re?ned constitutive relation which considers the thickness e?ect. Natural frequencies and mode shapes are obtained from the eigenvalue equation. Circular, elliptic, and parabolic curved beams are considered in numerical examples. The obtained results are in good agreement with those in the existing studies and those calculated by the ?nite element software ANSYS.Moreover, the e?ects of the material gradient, taper ratio, slenderness ratio, and heightspan ratio on vibration behaviors are discussed.

关键词： free vibration functionally graded(FG) material non-uniform crosssection curved beam variable curvature thickness effect

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