Isosceles Triangular Plates Research Articles

Numerical Evaluation of Frequencies and the Modes of Free Vibrations of Isosceles Triangular Plates with Free Edges

Free vibrations are analyzed for isotropic plates in the form of isosceles triangles. We deduce the formula for the evaluation of the frequencies of free vibrations for plates of regular triangular shape with free edges and compute the coefficients of vibration mode and boundary conditions. The frequencies and modes of free vibrations of the isotropic thin isosceles triangular plates with free edges and different apex angles are computed by the finite-element method.

Numerical evaluation of frequencies and modes of free vibrations of isosceles triangular plates with free edges

Free vibrations are analyzed for isotropic plates in the form of isosceles triangles. A formula for the evaluation of the frequencies of free vibrations of plates of regular triangular shape with free edges is derived, and the coefficients of the vibration mode and boundary conditions are computed. The frequencies and modes of free vibrations of isotropic thin isosceles triangular plates with free edges and different apex angles are calculated by the finite element method. Cite as: O. Ya. Grigorenko, M. Yu. Borysenko, E. V. Boychuk, “Numerical evaluation of frequencies and modes of free vibrations of isosceles triangular plates with free edges,” Mat. Met. Fiz. Mekh. Polya , 63 , No. 3, 28–39 (2020), https://doi.org/10.15407/mmpmf2020.63.3.28-39

Vibrational Behavior of Tapered Triangular Plate with Clamped Ends under Thermal Condition

In the present study, the effect of thermal gradient on the vibration of different types of triangular plates is discussed. Tapering in triangular plate is assumed bilinear. Variation in temperature of plate material is considered as linear in x-direction. The Rayleigh–Ritz technique is employed to evaluate the first two modes of frequency parameter for different values of thermal gradient, taper parameters and aspect ratios. The author reported frequency for both the modes of vibration for different types of clamped triangular plates, i.e., an isosceles triangular plate, right-angled isosceles triangular plate and scalene triangular plate. All numerical results are presented in tabular form.

Experimental and empirical study of large transverse deformation of triangular plates subjected to water hammer shock loading

In this paper, the dynamic plastic response of isosceles triangular plates under hydrodynamic loads was investigated experimentally using a drop hammer machine. To do this, a series of experimental tests were carried out on mild steel triangular plates with different thicknesses to bring insight into the effect of geometry and mechanical properties of the plate on the deformation of specimens, which were impacted by a piston-induced pressure wave inside a water tube. The effects of various impact loads originated from different drop hammer standoff distances, and hammer weights concerning variations of deflection of the center of mass were described. The experimental results were presented in terms of deflection of the center of mass of the plates and deflection profiles. The experimental results showed that the plate with lower thickness experienced higher deflection-to-thickness ratio. An empirical model was also proposed based on new dimensionless numbers for triangular plates in order to predict the ratio of deflection of the center of mass to thickness. The dimensionless numbers considered the effects of plate geometry, hydrodynamic applied load, and mechanical properties of materials. Comparison between the experimental results and empirical predictions demonstrated that the suggested model is accurate enough to predict of the response of isosceles triangular plates under hydrodynamic loads.

A unified formulation for free in-plane vibrations of arbitrarily-shaped straight-sided quadrilateral and triangular thin plates

The aim of this paper is to propose a unified formulation for the free in-plane vibration of arbitrarily-shaped straight-sided quadrilateral and triangular plates with arbitrary boundary conditions by means of improved Fourier series method (IFSM). The arbitrarily-shaped straight-sided quadrilateral and triangular plates are all mapped into a unit square plate uniformly, and accordingly, the problem can be solved directly by the existing vibration modeling method of rectangular plates. The displacement admissible function is generally expressed as the combinations of Fourier cosine and sine series. All the series expansion coefficients can be determined by the Rayleigh-Ritz procedure. A number of numerical examples of the quadrilateral and triangular plate with various shapes and boundary conditions are presented to assess the convergence and accuracy of the current solution method by comparing with those of literature or the finite element method (FEM). Furthermore, comprehensive studies on the influence of parameters of triangular plates on its in-plane vibration frequency characteristics are given, based on the angle of the isosceles triangular plate, the ratio of the length of the side of right-angled triangular plates and boundary restraint parameters.

ДОСЛІДЖЕННЯ СИЛОВИХ ХАРАКТЕРИСТИК ПОЛІГРАДІЄНТНОГО ЕЛЕКТРОМАГНІТНОГО СЕПАРАТОРА ПРИ ЗМІНІ ФОРМИ ПЛАСТИН МАТРИЦІ

Research of influence of elements shape of a multicomponent lamellar matrix of polygradient electromagnetic separator on its strength characteristics. For analysis of magnetic field distribution in working area of electromagnetic separator the finite element method has been used, implemented in ELCUT 5.6 software package. A numerical-field analysis of magnetic field distribution in working interpolar zones of matrix with a polygradient medium on the basis of inserts in form of an isosceles trapezium and triangular plates with a rounded apex has been carried out. According to analysis results, the maximum values of power characteristics of electromagnetic polygradient separator have been established. For a multicomponent lamellar matrix of polygradient electromagnetic separator certain geometrical dimensions of matrix elements of various shapes, at which power characteristics of device acquire the maximum value have been defined. The results of research can be used for improvement of power characteristics of basic design of electromagnetic polygradient separator

Free vibration of arbitrary-shaped laminated triangular thin plates with elastic boundary conditions

Abstract This paper studies the free vibration of arbitrary shaped laminated triangular thin plates based on a modified Fourier series method. An arbitrary shaped triangular plate is mapped into a right-angled isosceles triangular plate with unit length by the coordinate transformation. By padding another right–angled isosceles triangular plate which is near zero thickness on the plate after transformation. The displacement functions are then generally expressed as the combinations of Fourier cosine series and supplementary functions introduced to eliminate the discontinuous or jumping phenomenon in the boundaries. The classical thin plate elasticity theory is employed to construct the energy expressions of the plate. The Rayleigh-Ritz method is used in the present method to obtain all the unknown series expansion coefficients. A number of results are presented to verify the convergence and accuracy of current solution method to laminated triangular thin plates with arbitrary shapes, different material parameters and different boundary conditions. Additionally, some new results are given as the benchmark for future research, which are based on the various boundary conditions and geometric dimensions.

Linear and non-linear vibration analysis of moderately thick isosceles triangular FGPs using a triangular finite p-element

BackgroundThe geometrically non-linear formulation based on Von-Karman’s hypothesis is used to study the free vibration isosceles triangular plates by using four types of mixtures of functionally graded materials (FGMs - AL/AL2O3, SUS304/Si3N4, Ti- AL-4V/Aluminum oxide, AL/ZrO2). Material properties are assumed to be temperature dependent and graded in the thickness direction according to power law distribution.MethodsA hierarchical finite element based on triangular p-element is employed to define the model, taking into account the hypotheses of first-order shear deformation theory. The equations of non-linear free motion are derived from Lagrange's equation in combination with the harmonic balance method and solved iteratively using the linearized updated mode method.ResultsResults for the linear and nonlinear frequencies parameters of clamped isosceles triangular plates are obtained. The accuracy of the present results are established through convergence studies and comparison with results of literature for metallic plates. The results of the linear vibration of clamped FGMs isosceles triangular plates are also presented in this study.ConclusionThe effects of apex angle, thickness ratio, volume fraction exponent and mixtures of FGMs on the backbone curves and mode shape of clamped isosceles triangular plates are studied. The results obtained in this work reveal that the physical and geometrical parameters have a important effect on the non-linear vibration of FGMs triangular plates.

Vibration of arbitrarily-shaped triangular plates with elastically restrained edges

A solution method is proposed for the vibration analysis of arbitrarily-shaped triangular plates with elastically restrained edges. An arbitrarily-shaped triangular plate is first mapped into a right-angled isosceles triangular plate and subsequently “stretched” to form a square plate by padding to it another (right-angled isosceles triangular) plate of near-zero thickness. The displacement function is then expressed as a two-dimensional Fourier cosine series supplemented by several one-dimensional series introduced to improve the convergence and accuracy of the displacement solution. Regardless of the shapes and boundary conditions of the triangular plate, the final stiffness and mass matrices can always be calculated analytically which makes the current solution method highly efficient in implementations. Numerical examples are presented to verify the applicability of the proposed method to triangular plates of different shapes, and, perhaps more importantly, different boundary conditions. The current method can be easily extended to plates with heterogeneous material properties, variable thickness, non-uniform elastic restraints, etc.

THREE-DIMENSIONAL ANALYSIS OF THICK PLATES BY MLS-RITZ METHOD

This paper presents a three-dimensional (3D) moving least-square Ritz (MLS-Ritz) formulation for the free vibration analysis of homogeneous elastic thick plates with mixed boundary constraints. The analysis is based on the linear elasticity theory. The Ritz trial functions are established through the moving least-square technique for the displacement fields of the plates. Vibration frequencies for thick square plates and right-angled isosceles triangular plates are obtained by the MLS-Ritz method. The reliability and accuracy of the presented method are examined by extensive convergence and comparison studies and it is established herein that the MLS-Ritz method is a powerful and effective numerical method for the 3D analysis of thick plates.

Three-dimensional free vibration analysis of plates using the h- p version of the finite element method

The h- p version of the finite element method based on a pentahedral p-element is applied to three-dimensional free vibrations of plates. The element's new hierarchical shape functions are expressed in terms of the shifted Legendre orthogonal polynomials. The accuracy of the solution can be improved by simultaneously refining the mesh and increasing the polynomial order. The method is capable of providing a full frequency spectrum including both the in-plane and the out-of-plane modes. Results of frequency calculations are first found for a skew plate on a soft edge support and a cantilevered right isosceles triangular plate and comparisons are made with other methods. Results are also found for a square plate on a hard edge support and comparisons are made with two-dimensional plate theories. Furthermore, highly accurate values for a free hexagonal plate are presented for the first time.

Moving least square Ritz method for vibration analysis of plates

This paper presents a novel numerical method, the moving least square Ritz method (MLS-Ritz), for free vibration analysis of classical thin plates. The proposed method utilizes the strength of the moving least square approach to define the Ritz trial function for the transverse displacement of the plates. A set of points is pre-selected on the calculation domain of a plate that forms the basis for the MLS-Ritz trial function. The edge support conditions of the plate are satisfied by forcing the boundary points to meet the geometric boundary conditions of the plate via a point substitution technique. Virtual points (points outside the plate domain) are introduced for clamped edges to improve the convergence and accuracy of the calculations. Square and right-angled isosceles triangular plates of various combinations of edge support conditions are selected to examine the validity and accuracy of the MLS-Ritz method. Extensive convergence studies are carried out to investigate the influence of the MLS mesh size, the MLS support radius, the number of Gaussian integration points and the shape of the MLS weight function on the proposed method. Comparing with the existing Ritz methods, the MLS-Ritz method is highly stable and accurate and is extremely flexible for dealing with plates of arbitrary shapes and boundary conditions.

Three-dimensional vibration analysis of cantilevered and completely free isosceles triangular plates

The free vibration of cantilevered and completely free isosceles triangular plates based on exact three-dimensional elasticity theory is investigated. The actual plate domain is first mapped onto a basic cubic domain. Then the Ritz method is applied to derive the eigenfrequency equation from the strain energy and the kinetic energy of the plate. A set of Chebyshev polynomial series multiplied by a boundary function chosen to satisfy the essential geometry boundary conditions of the plate is developed as the admissible functions of each displacement component. The convergence and comparison studies show that rather accurate results can be obtained by using this approach. The effects of thickness-to-width ratio and apex angle on eigenfrequencies of the plates are studied in detail. Sets of valuable results are presented, which may serve as the benchmark values for future numerical techniques in thick plate vibration analysis. Data for the completely free isosceles triangular plates are presented for the first time.

FREE VIBRATION ANALYSIS OF ISOSCELES TRIANGULAR MINDLIN PLATES BY THE TRIANGULAR DIFFERENTIAL QUADRATURE METHOD

The triangular differential quadrature method is applied to the free flexural vibration of isosceles triangular Mindlin plates. The first six frequencies are sought and the convergence of triangular differential quadrature method in vibrational analysis of triangular Mindlin plates is examined. In comparison with available results, good to excellent agreement is achieved for triangular plates with combinations of clamped and simply supported boundary conditions. The implementation of various other boundary conditions in triangular differential quadrature analysis is also discussed and the sensitivity of solution to the implementation of corner conditions in vibrational analysis is highlighted.

The plastic bending of clamped polygonal plates with an application to radial buckling in milk bottle tops

Solutions for the plastic bending of clamped polygonal plates are investigated. For an isosceles triangular plate, and for a truncated isosceles triangular plate, minima exist for plastic work of formation with respect to angles of the triangle, and location of the apex of the deformed pyramid shape. A characteristic number of radial buckles is formed when the foil circular top of British one-pint milk bottles is pushed in and material drawn over the rim of the bottle on opening. The plastic bending analysis provides an approximate means of predicting the number of buckles.

CORNER STRESS SINGULARITY EFFECTS ON THE VIBRATION OF RHOMBIC PLATES WITH COMBINATIONS OF CLAMPED AND SIMPLY SUPPORTED EDGES

An accurate method is presented for flexural vibrations of rhombic plates having all combinations of clamped and simply supported edge conditions. A specific feature here is that the analysis explicitly considers the bending stress singularities that occur in the two opposite, clamped–hingedand/or hinged–hinged, corners having obtuse angles in rhombic plates. The strength of these singularities increases significantly as the obtuse angles at the clamped–hinged and/or hinged–hinged corners increases. Stationary condition of single-field, classical thin-plate Lagrangian functional are derived using the Ritz method. Assumed transverse displacements are constructed from a hybrid set of (i) admissible and mathematically complete algebraic polynomials, and (ii) comparison functions (termed here “corner functions”) which account for both the kinematic boundary condition and the bending stress singularities at the obtuse clamped–hinged and/or hinged–hinged corners. Extensive convergence studies demonstrate that the corner functions accelerate the convergence of solutions, and that these function are required if accurate solutions are to be obtained for highly skewed plates. Accurate non-dimensional frequencies and normalized contours of the vibratory transverse displacement are presented for rhombic plates having a large enough skew angle of 75 ° (i.e., obtuse corner angles of 165 °), so that the significant influence of the corner stress singularities may be clearly understood. Accurate solutions for isosceles triangular plates with various combinations of clamped-hinged edges are also available from the frequency and mode shape data presented. The upper bound frequency solutions derived from the presented. The upper bound frequency solutions derived from the present analysis are shown to improve upon the existing upper bound results in the published literature.

Free vibration of isosceles triangular mindlin plates

The free flexural vibration of thick isosceles triangular plates based on the Mindlin shear deformation theory is investigated. The pb-2 Rayleigh-Ritz method, proposed earlier by the authors, is employed for solution. New sets of vibration frequency parameters for isosceles triangular Mindlin plates with various apex angles, thickness-to-width ratios and different combinations of free, simply supported and clamped edges are presented. These results are valuable to designers and may serve as the benchmark values for future numerical techniques and software packages in thick plate analyses. Since no results are available to date for such thick triangular plates, comparisons of results can only be made in the special case of small thickness-to-width ratios with existing thin plate results. In all cases, they are found to be in very close agreement. The influences of shear deformation and rotatory inertia on the frequency parameters are examined.

Stress distribution over a triangular tillage tool

To simplify the analysis of stress over a cultivator shovel moving at a certain depth, it was represented by an isosceles triangular plate of uniform thickness. To simulate the boundary conditions to which a cultivator shovel is subjected, the triangular plate was assumed as clamped along one of the edges while the other two edges were assumed as free. These assumptions made it possible to apply the theory of bending of laterally loaded plates for the computation of stresses set up in the tillage tool. This theory involves fundamental equations of elasticity, namely the equations of equilibrium, equations of continuity and surface conditions. The theoretically calculated values of stresses were fairly close to experimentally observed values.

Vibration and damping analysis of orthotropic triangular plates

The effect of fibre orientation and boundary conditions on the vibration and damping behaviour of thin orthotropic triangular plates is studied by using a three-noded triangular plate finite element. Isosceles triangular plates having three different included angles are analyzed.

Accurate analytical solution for free vibration of the simply supported triangular plate

Exploiting the superposition method developed earlier by the author, a highly accurate analytical-type solution is obtained for the free vibration of the general simply supported triangular plate. A new technique is introduced for the orderly storage of eigenvalues. It is shown how advantage can be taken of the more easily obtained isosceles triangular plate solutions. Accurate eigenvalues are tabulated for a wide range of plate geometries. This represents the first accurate and comprehensive treatment of this problem to appear in the literature. a b D E h K K* M W