Application of the Spline Element Method to Analyze Vibration of Skew Mindlin Plates With Varying Thickness in One Direction

Mizusawa, T.; Kondo, Yae

doi:10.1006/jsvi.2000.3303

Cited by 21 publications

(9 citation statements)

References 16 publications

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“…In the recent decade, Mizusawa and Kondo [12] reported the natural frequencies of skew Mindlin plates with linearly varying thickness in the longitudinal direction by the spline element method. Woo et al [13] studied the vibration behaviour of skew Mindlin plates with and without cut-outs by the p-version finite element method using integrals of Legendre polynomials.…”

Section: Introductionmentioning

confidence: 99%

Application of the DSC-Element method to flexural vibration of skew plates with continuous and discontinuous boundaries

Lai

Zhou

Zhang

et al. 2011

Thin-Walled Structures

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Section: Introductionmentioning

confidence: 99%

Application of the DSC-Element method to flexural vibration of skew plates with continuous and discontinuous boundaries

Lai

Zhou

Zhang

et al. 2011

Thin-Walled Structures

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“…This system of ordinary differential equations can be normalized: We used the method of spline approximation to determine dimensionless frequencies for different number of collocation points (N = 8, 10,12,14,16,18,20,22). Tables 1 and 2 summarize these frequencies for shells with four clamped edges and a = -0.4 and for shells with four hinged edges and a = 0.4, respectively.…”

mentioning

confidence: 99%

“…The spline-approximation method was used in [6,[8][9][10][11][12][13]16] to analyze the stress-strain state of shells and to determine their resonant frequencies.…”

mentioning

confidence: 99%

Free vibrations of shallow orthotropic shells with variable thickness and rectangular planform

Grigorenko

Parkhomenko

2011

Int Appl Mech

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The free vibrations of shallow doubly curved orthotropic shells with rectangular planform and varying thickness is solved using a refined formulation and the spline-approximation method. Various boundary conditions are considered. The effect of the curvature of the mid-surface on the spectrum of natural frequencies is examined. The natural frequencies and modes of orthotropic shells of constant and varying thickness are compared and analyzed Introduction. Anisotropic shells of variable thickness are widely used, as structural members, in industry and construction. Modern design and engineering solutions require high strength and reliability of new structures and mechanisms. In this connection, it is necessary to develop efficient numerical methods for the calculation of the load-bearing capacity of structural members, including shells with different physical and geometrical parameters and for the determination of their resonant frequencies.The natural frequencies of shells are often calculated using the classical Kirchhoff-Love theory, which is characterized by a simple analytical tools and high accuracy. However, development and widespread use of advanced composite materials necessitate using refined theories [1-3, 5, etc.].The stress-strain state of shallow shells was examined in [8, 10, 12] using a nonclassical problem formulation. The free vibrations of plates were studied in [6,7,9,11,15] following a refined approach. Most nonclassical theories allow for transverse shear. Of such theories, we choose the nonclassical Timoshenko-Mindlin model.In designing shallow orthotropic shells of constant thickness with hinged edges, use may be made of the procedure of separation of variables by expanding the unknown functions into double trigonometric series. However, this procedure cannot be applied to other boundary conditions, including clamped ends. Because of this, the posed problem has to be solved by numerical methods.The spline-approximation method has recently become the most popular in determining the dynamic characteristics of plates and shells. It has the following advantages: stability of splines against local perturbations; good convergence of spline-interpolation; and simplicity and convenience of numerical implementation of spline algorithms. When used in various variational, projective, and other discrete-continuous methods, spline-functions yield optimal results, simplify the numerical implementation of these methods, and ensure high accuracy of the solution.The present paper analyzes the free vibrations of shallow doubly curved orthotropic shells with rectangular planform and varying thickness with various boundary conditions using a refined problem formulation and the nonclassical Timoshenko-Mindlin model.The problem will be solved by using spline-approximation with respect to one coordinate to reduce the original system of partial differential equations to an eigenvalue boundary-value problem for systems of ordinary differential equations of high

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“…The natural vibrations of rectangular plates with varying thickness were studied using Mindlin's theory with lesser activity than in similar investigations fulfilled within the framework of the classical theory of plates. Let us note some works, such as [Mindlin 1951;Mizusava 1993;Mizusava and Condo 2001;Roufacil and Dawe 1980], dedicated to this scientific trend. The collocation method based on orthogonal polynomials was used in [Mikami and Yoshimura 1984] to analyze vibrations of a plate with linearly varying thickness.…”

Section: Introductionmentioning

confidence: 99%

“…In [Mizusava 1993;Mizusava and Condo 2001], in order to solve one-dimensional boundary-value problems or those reduced to them, which describe bending, stability, and vibrations of plates and shells, the solution is approximated by splines of the third or fifth power and the problem is reduced to a system of algebraic equations. This is more advantageous than other methods from the viewpoint of calculation time and accuracy.…”

Section: Introductionmentioning

confidence: 99%

Spline-based investigation of natural vibrations of orthotropic rectangular plates of variable thickness within classical and refined theories

Grigorenko

Efimova

2008

JOMMS

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A spline-collocation approach is proposed for studying the natural vibrations of orthotropic rectangular plates of variable thickness. The approach is based on the spline-approximation method and the method of discrete orthogonalization coupled with the step-by-step search method. The study is carried out within the framework of the classical and refined theories of plates. The dynamic response of the plates is studied depending on the variation of the plate thickness, the mechanical parameters, and the type of boundary conditions.

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Application of the Spline Element Method to Analyze Vibration of Skew Mindlin Plates With Varying Thickness in One Direction

Cited by 21 publications

References 16 publications

Application of the DSC-Element method to flexural vibration of skew plates with continuous and discontinuous boundaries

Application of the DSC-Element method to flexural vibration of skew plates with continuous and discontinuous boundaries

Free vibrations of shallow orthotropic shells with variable thickness and rectangular planform

Spline-based investigation of natural vibrations of orthotropic rectangular plates of variable thickness within classical and refined theories

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