Generalized hypergeometric function solutions for transverse vibration of a class of non-uniform annular plates

Duan, Wenhui; Quek, Ser Tong; Wang, Q.

doi:10.1016/j.jsv.2004.11.027

Cited by 21 publications

(5 citation statements)

References 23 publications

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“…The hypergeometric function has earlier been adopted to solve the transverse vibration problem of non-uniform annular plates (Duan et al, 2005), the lateral-torsional buckling problem of linearly tapered cantilever (Challamel et al, 2007), the axisymmetric bending problem of micro/nanoscale circular plates , the buckling problem of columns with allowance for self-weight (Duan and Wang, 2008), and axisymmetric transverse vibration problem of circular cylindrical shells with variable thickness (Duan and Koh, 2008), the flexural-torsional buckling problem of cantilever (Challamel et al, 2010), and the lateral-torsional stability boundaries for polygonally depthtapered strip cantilevers (Andrade et al, 2012). The Frobenius companion matrix of a second kind Chebyshev polynomial U nÀ1 ðx=2Þ is given by (Horn and Johnson, 1990) …”

Section: B Exact Solutions Via Matrix Decompositionmentioning

confidence: 99%

Development of analytical vibration solutions for microstructured beam model to calibrate length scale coefficient in nonlocal Timoshenko beams

Duan

Challamel

Wang

et al. 2013

Journal of Applied Physics

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The present study takes an analytical approach for solving the free vibration problem of a microstructured beam model, in which transverse displacement springs are added to allow for the transverse shear deformation effect in addition to the rotational springs. The exact vibration frequencies for the discrete microstructured beam model with simply supported ends are obtained via matrix decomposition. In addition, a general solution technique involving the use of Pad e approximants for the continualization procedure is proposed in order to obtain the continuous equivalent system for the discrete microstructured beam model. The analytical vibration solutions of the equivalent continuous system are obtained and their accuracy is assessed by using the exact solutions. It is found that the solutions of the equivalent continuous system have a first order accuracy when compared with the exact solutions of their discrete counterpart. The length scale coefficient in the nonlocal Timoshenko beam model is calibrated by using the analytical solutions. Two nonlocal Timoshenko beam models, i.e., the Wang model (without the length scale effect in the shear stress strain relation) and the Reddy model, are evaluated based on their ability to capture the nonlocal effect. V C 2013 AIP Publishing LLC. [http://dx

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Section: B Exact Solutions Via Matrix Decompositionmentioning

confidence: 99%

Development of analytical vibration solutions for microstructured beam model to calibrate length scale coefficient in nonlocal Timoshenko beams

Duan

Challamel

Wang

et al. 2013

Journal of Applied Physics

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“…Wang [19] derived the power series solutions for the axisymmetric vibration of circular Kirchhoff plates with power or shifted power variations in the thickness. Duan et al [20] showed the generalized hypergeometric function solutions for the flexural free vibrations of the annular Kirchhoff plate with its thickness varying monotonically in arbitrary power. Caruntu [21] found a class of non-uniform circular Kirchhoff plates with the Jacobi polynomials as their mode shapes of transverse free vibrations.…”

Section: Introductionmentioning

confidence: 99%

Exact solutions for axisymmetric flexural free vibrations of inhomogeneous circular Mindlin plates with variable thickness

Yuan

Chen

2017

Appl. Math. Mech.-Engl. Ed.

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Circular plates with radially varying thickness, stiffness, and density are widely used for the structural optimization in engineering. The axisymmetric flexural free vibration of such plates, governed by coupled differential equations with variable coefficients by use of the Mindlin plate theory, is very difficult to be studied analytically. In this paper, a novel analytical method is proposed to reduce such governing equations for circular plates to a pair of uncoupled and easily solvable differential equations of the Sturm-Liouville type. There are two important parameters in the reduced equations. One describes the radial variations of the translational inertia and flexural rigidity with the consideration of the effect of Poisson's ratio. The other reflects the comprehensive effect of the rotatory inertia and shear deformation. The Heun-type equations, recently well-known in physics, are introduced here to solve the flexural free vibration of circular plates analytically, and two basic differential formulae for the local Heun-type functions are discovered for the first time, which will be of great value in enriching the theory of Heun-type differential equations.

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“…However, this method can only determine a single frequency. Duan et al (2005) expressed the vibration solutions for an axisymmetric power law rigidity in terms of generalised hypergeometric functions. These functions are not readily available and their evaluation requires infinite series summations which are more tedious than direct infinite series solutions (e.g.…”

Section: Introductionmentioning

confidence: 99%

Exact closed form solutions for free vibration of non-uniform annular plates

Wang

Chen

2012

The IES Journal Part A: Civil & Structural Engineering

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Generalized hypergeometric function solutions for transverse vibration of a class of non-uniform annular plates

Cited by 21 publications

References 23 publications

Development of analytical vibration solutions for microstructured beam model to calibrate length scale coefficient in nonlocal Timoshenko beams

Development of analytical vibration solutions for microstructured beam model to calibrate length scale coefficient in nonlocal Timoshenko beams

Exact solutions for axisymmetric flexural free vibrations of inhomogeneous circular Mindlin plates with variable thickness

Exact closed form solutions for free vibration of non-uniform annular plates

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