Method of R-function for investigation of parametric vibrations of orthotropic plates of complex shape

Kurpa, L. V.; Mazur, Olga

doi:10.1007/s10958-011-0296-8

Cited by 3 publications

(9 citation statements)

References 4 publications

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“…According to the R-function theory. [23][24][25][26] v 0 can be taken 4 , y 4 , x 2 y 2 , and y 5 . When programming with MATLAB, the T-shaped thin plate is divided into three rectangles.…”

Section: Example 3 T-shaped Thin Platementioning

confidence: 99%

“…According to R-function theory, [23][24][25][26] the v 0 describing a complex region can be described by Boolean operations _ a and^a of each subdomain, which are corresponding to the union set and intersection set. The function v l describing each subdomain satisfies the first-order normalization equation, and v 0 also satisfies the first-order normalization equation.…”

Section: R-function Theorymentioning

confidence: 99%

“…When n = 9 and m = 10, the results of the paper and FEM are compared which shows a good agreement in Figure 9, and that proves the convergence and correctness of this method. According to the R-function theory, [23][24][25][26] v 0 can be taken…”

Section: Example 3 T-shaped Thin Platementioning

confidence: 99%

“…21 Chen and Xiao 22 studied the numerical characteristics of the preconditioned GMRES algorithm in solving the boundary element method equations of elastic mechanics problems. 22 Kurpa and Mazur 23 proposed a method of R-function theory for studying parametric vibration of complex shape plates with irregular shapes and different boundary conditions. Kurpa et al applied the R-function theory for the free vibration of shallow shells, combined the spline approximation method, and the Ritz method to analyze the stress-strain state of the laminated shallow shell and the vibration of the laminated shallow shell with complex notch.…”

Section: Introductionmentioning

confidence: 99%

“…The key is to express the boundary of the problem by standardized equation v = 0, and the region of the problem is expressed by inequality v.0. 23 The introduction of the R-function theory can easily build the deflection function w of the irregular thin plate to meet the boundary conditions, and then the specific deflection function of the concave-convex irregular thin plate can be obtained by combining the variational method.…”

Section: Introductionmentioning

confidence: 99%

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R-Function and variation method for bending problem of clamped thin plate with complex shape

Xia

2021

Advances in Mechanical Engineering

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Solving ordinary thin plate bending problem in engineering, only a few analytical solutions with simple boundary shapes have been proposed. When using numerical methods (e.g. the variational method) to solve the problem, the trial functions can be found only it exhibits a simple boundary shape. The R-functions can be applied to solve the problem with complex boundary shapes. In the paper, the R-function theory is combined with the variational method to study the thin plate bending problem with the complex boundary shape. The paper employs the R-function theory to express the complex area as the implicit function, so it is easily to build the trial function of the complex shape thin plate, which satisfies with the complex boundary conditions. The variational principle and the R-function theory are introduced, and the variational equation of thin plate bending problem is derived. The feasibility and correctness of this method are verified by five numerical examples of rectangular, I-shaped, T-shaped, U-shaped, and L-shaped thin plates, and the results of this method are compared with that of other literatures and ANSYS finite element method (FEM). The results of the method show a good agreement with the calculation results of literatures and FEM.

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“…According to the R-function theory. [23][24][25][26] v 0 can be taken 4 , y 4 , x 2 y 2 , and y 5 . When programming with MATLAB, the T-shaped thin plate is divided into three rectangles.…”

Section: Example 3 T-shaped Thin Platementioning

confidence: 99%

Section: R-function Theorymentioning

confidence: 99%

Section: Example 3 T-shaped Thin Platementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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R-Function and variation method for bending problem of clamped thin plate with complex shape

Xia

2021

Advances in Mechanical Engineering

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Buckling and free vibration analysis of functionally graded sandwich plates and shallow shells by the Ritz method and the R-functions theory

Kurpa

Shmatko

2020

Proceedings of the Institution of Mechanical Engineers, Part C:

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The purpose of the paper is to study stability and free vibrations of laminated plates and shallow shells composed of functionally graded materials. The approach proposed incorporates the Ritz method and the R-functions theory. It is assumed that the shell consists of three layers and is loaded in the middle plane. The both cases of uniform as well as non-uniform load are possible. The power-law distribution in terms of volume fractions is applied to get effective material properties for the layers. These properties are calculated for different arrangements and thicknesses of the layers by the analytical formulae obtained in the paper. The mathematical formulation is carried out in framework of the first-order shear deformation theory. The proposed approach consists of two steps. The first step is to define the pre-buckling state by solving the respective elasticity problem. The critical buckling load and frequencies of functionally graded material shallow shells are determined in the second step. The highlight of the method proposed is that it can be used for vibration and buckling analysis of plates and shallow shells of complex shape. The numerical results for frequencies and buckling load of plates and shallow shells of complex shape and different curvatures are presented to demonstrate the potential of the method developed. Different functionally graded material plates and shallow shells composed of a mixture of metal and ceramics are studied. The effects of the power law index, boundary conditions, thickness of the core, and face sheet layers on the fundamental frequencies and critical loads are discussed in this paper. The main advantage of the method is that it provides an analytical representation of the unknown solution, which is important when solving nonlinear problems.

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Dynamic Stability of Orthotropic Viscoelastic Rectangular Plate of an Arbitrarily Varying Thickness

et al. 2021

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The research object of this work is an orthotropic viscoelastic plate with an arbitrarily varying thickness. The plate was subjected to dynamic periodic load. Within the Kirchhoff–Love hypothesis framework, a mathematical model was built in a geometrically nonlinear formulation, taking into account the tangential forces of inertia. The Bubnov–Galerkin method, based on a polynomial approximation of the deflection and displacement, was used. The problem was reduced to solving systems of nonlinear integrodifferential equations. The solution of the system was obtained for an arbitrarily varying thickness of the plate. With a weakly singular Koltunov–Rzhanitsyn kernel with variable coefficients, the resulting system was solved by a numerical method based on quadrature formulas. The computational algorithm was developed and implemented in the Delphi algorithmic language. The plate’s dynamic stability was investigated depending on the plate’s geometric parameters and viscoelastic and inhomogeneous material properties. It was found that the results of the viscoelastic problem obtained using the exponential relaxation kernel almost coincide with the results of the elastic problem. Using the Koltunov–Rzhanitsyn kernel, the differences between elastic and viscoelastic problems are significant and amount to more than 40%. The proposed method can be used for various viscoelastic thin-walled structures such as plates, panels, and shells of variable thickness.

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Method of R-function for investigation of parametric vibrations of orthotropic plates of complex shape

Cited by 3 publications

References 4 publications

R-Function and variation method for bending problem of clamped thin plate with complex shape

R-Function and variation method for bending problem of clamped thin plate with complex shape

Buckling and free vibration analysis of functionally graded sandwich plates and shallow shells by the Ritz method and the R-functions theory

Dynamic Stability of Orthotropic Viscoelastic Rectangular Plate of an Arbitrarily Varying Thickness

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