Parallelogram Plate Research Articles

Three-dimensional hygro-thermo-mechanical bending analysis of anisotropic skewed parallelogram plates on a tri-parameter foundation using a wavelet method

In the paper, a brand-new three-dimensional hygro-thermo-mechanical bending model of anisotropic skewed parallelogram plates with extreme skew angle in large deformation supported by a tri-parameter nonlinear elastic foundation is proposed. By investigating the mutual coupled interaction of hygrothermal effects between in-plane distribution and thickness variation, nine kinds of spatially analytical hygrothermal fields in parallelogram domain are established by analyzing the Dirichlet, Neumann and Robin thermal boundary conditions of sinusoidally induced heat conduction at top and bottom surfaces. With coordinate transformation in oblique system considering the Von Kármán membrane strains, the highly coupled and strongly nonlinear governing partial differential equations have been derived. A wavelet solving procedure has been developed with the convergence process validated and precision of obtained solutions verified in good accordance with published results. Parametric studies have been carried out to investigate the large-deflection behavior of simply supported or clamped skew plates on Winkler–Pasternak foundation under combined mechanical and hygrothermal loads. The influence of skew angle, material orthotropy and foundation contact nonlinearity is examined, which reveals the geometrically nonlinear responses of plates are highly affected by large oblique angle, principle orientation of material orthotropy and foundation parameters.

Analyzing vibration modes in non-homogeneous parallelogram plates

This study analyzes vibration modes in non-homogeneous orthotropic parallelogram plate with a one-dimensional circular thickness, focusing on SCCC edge condition, where C and S represent the clamped and simply supported edges of the plate, respectively. Circular Poisson’s ratio variation is considered, along with linear temperature changes. The study demonstrates the advantages of variable Poisson’s ratio over density parameter variation in obtaining shorter vibration time periods. Orthotropic parallelogram plate, thermal gradient, circular tapering, nonhomogeneity.

Time Period Analysis of Orthotropic Skew Plate with 2-D Circular Thickness and 1-D Circular Density

In the present study, the time period of vibration of an orthotropic parallelogram plate with 2 − D (two-dimensional) circular thickness under the effect of 2 − D parabolic temperature is investigated for the first time. The different edge conditions are C C C C , C C C F , C F C F , C S C F , and S F S F boundary conditions, where C, S, and F stands for clamped, simply supported, and free edges of the plate, respectively. The variation in density of plate material is considered to be 1 − D (one-dimensional) circular. The Rayleigh–Ritz technique is used to solve the differential equation and evaluate the time period for the first two modes of vibration. A convergence study of an orthotropic parallelogram, rectangle plate, and square plate for modes of frequency at various edge conditions is also carried out. The authors performed a comparative analysis of the time period and modes of frequency of orthotropic parallelogram, rectangle plate, and isotropic square plate with the available published results at various edge conditions. The main conclusion which we made from this study is that by choosing the above-mentioned plate parameters, we obtained fewer modes of frequency in comparison to other variations mentioned in the literature. Also, the study suggests that the variation in modes of frequency is less in comparison to other variations.

ГЕОМЕТРИЧЕСКОЕ МОДЕЛИРОВАНИЕ ФОРМЫ ПАРАЛЛЕЛОГРАММНЫХ ПЛАСТИН В ЗАДАЧЕ СВОБОДНЫХ КОЛЕБАНИЙ С ИСПОЛЬЗОВАНИЕМ КОНФОРМНЫХ РАДИУСОВ

The paper considers a method of geometric modeling applied when solving basic twodimensional problems of the theory of elasticity and structural mechanics, in particular the applied problems of engineering. The subject of this study is vibrations of thin elastic parallelogram plates of constant thickness. To determine a basic frequency of vibrations, the interpolation method based on the geometric characteristic of the shape of plates (membrane, cross sections of a rod) is proposed. This characteristic represents a ratio of interior and exterior conformal radii of the plate. As is known from the theory of conformal mappings, conformal radii are those obtained by mapping of a plate onto the interior and exterior of a unit disk. The paper presents basic terms, tables, and formulas related to the considered geometric method with a comparative analysis of the curve diagrams obtained using various interpolation formulas. The original computer program is also developed. The main advantage of the proposed method of determining the basic frequency of plate vibrations is a graphic representation of results that allows one to accurately determine the required solution on the graph among the other solutions corresponding to the considered case of parallelogram plates. Although there are many known approximate approaches, which are used to solve the considered problems, only geometric modeling technique based on the conformal radii ratio gives such an opportunity.

DESIGN AND TECHNOLOGICAL PROPOSALS FOR IMPROVING A SINGLE-PHASE TRANSFORMER WITH LAMINATED MAGNETIC CORE

Goal. The purpose of the work is to analyze the features and substantiate the advantages of the spatial radial electromagnetic system of a single-phase transformer with rectangular sections of the rods of a charged magnetic circuit. Methodology. Improvement of single-phase transformers is possible by structural-geometric transformations of active elements. The justification of the advantages of such transformations is achieved by the method of invariant objective functions with dimensionless optimization components and universal relative controlled variables. Results. Replacing the rectangular contours of the variants of a planar magnetic circuit with a hexagonal circuit leads to a decrease in the volume of steel and the loss of three angular zones of changes in the direction of the magnetic flux relative to the texture compared to planar counterparts. The separation of the windings into sections with an arrangement on three rods leads to a decrease in the average length of the coil. Originality. A three-core magnetic core can consist of rod and jugular-angular parallelogram plates of anisotropic and isotropic electrical steel. It is advisable to combine a combination of steels by integral welding of billets with subsequent separation into chevron jugular-rod elements. The method called on has developed mathematical models of the mass, cost, and active power losses of variants of a single-phase electromagnetic system of a transformer with rod and armor planar and spatial radial three-core magnetic cores. Practical significance. It was found that the decrease in the extrema of the mass and cost indicators, as well as the calculated decrease, under the condition that the coefficients of additional losses are identical, the extrema of the losses of the single-phase radial three-rod system of the transformer relative to the planar analogue with a bar stocked magnetic wire from anisotropic steel, respectively 13,0-12,3 %, 15,3-10,1 % and 15,1-18,1 % with copper windings and 10,8-10,2 %, 12,8-13,7 % and 12,6-12,3 % with aluminum windings. The main indicators of traditional single-phase systems with rod and armored magnetic circuits differ insignificantly, which is consistent with early known optimization and calculation data of transformers.

Isogeometric three-dimensional vibration of variable thickness parallelogram plates with in-plane functionally graded porous materials

In this paper, the vibration behaviors of variable thickness parallelogram plates with in-plane functionally graded porous materials are investigated by a numerical approach for the first time. The unknown fields and geometry of the in-plane functionally graded porous variable thickness parallelogram plates are established numerically by non–uniform rational B–splines functions under the framework of isogeometric analysis with three-dimensional theory. The porosities of parallelogram plates are assumed to be distributed continually along the in-plane direction with different distribution modes. The weak form requirements with three-dimensional theory can be satisfied by the formulation easily. Fast convergence and good accuracy are demonstrated in several numerical examples. Additionally, some innovative results and parameter effects are carried out in the text. This work can shed some new light on the investigation of complex three-dimensional vibration behavior of in-plane functionally graded porous variable thickness parallelogram plates.

An exact GITT solution for static bending of clamped parallelogram plate resting on an elastic foundation

PurposeThe purpose of this study is to propose generalised integral transform technique (GITT) to obtain the exact solutions for bending of clamped parallelogram plate resting on elastic foundation.Design/methodology/approachThe GITT is used to solve the bending problem of the full clamped parallelogram plate under an elastic foundation. The auxiliary problem was developed and the corresponding eigenfunction and eigenvalue were calculated simultaneously. The original partial differential governed equation has been represented by the transformed ordinary differential equation system and solved by the subroutine DBVPFD from International Mathematics and Statistics Library.FindingsThe GITT has been proven to be an efficient approach to solve the bending problem of the plate with different loads, boundary conditions and elastic foundations. The parametric study indicates that the elastic foundation modulus has significant contribution in reducing the vertical deflections and moments for both rectangular and parallelogram plates. With the increasing of aspect ratio (a/b) and the elastic foundation modulus, the trends of the deflection and moment reduction decreased significantly.Originality/valueThe present hybrid analytical-numerical methodology was first used to solve the mechanics problem of the clamped parallelogram plate resting on elastic foundation. Excellent convergence and high accuracy was observed by comparing with the published results. It exhibits potential application to investigate the mechanics problem of the composite plate with different boundary conditions in the shipbuilding and civil engineering.

Investigation of a flutter of structures in gas flows with using energy method

Single-mode flutter is a type of panel flutter occurring at transonic and low supersonic speeds. Transition to instability in the form of single-mode flutter occurs without interaction between natural modes, in contrast to coupled-mode flutter, where coupling between the 1st and 2nd eigenmodes takes place. Blade flutter is one of the main issues that engine designers have to face. The danger of this phenomenon is a rapid increase of blades stresses, which can lead to their destruction. In this paper, a single-mode flutter of panels of rectangle and parallelogram panels and the influence of various design parameters on blade flutter boundary are investigated with using the energy method. It is shown that for parallelogram plates even at a small skew angle the aeroelastic stability increases significantly at transonic and low supersonic flight speeds. The study of blade flutter showed that effect of the inter-blade tension in the mid-span shroud on the flutter is significant in contrast to other investigated parameters.

Beam shaping design for fiber-coupled laser-diode system based on a building block trapezoid prism

For the fiber-coupled laser diodes, the high brightness and miniaturization have always been our goals to pursuit. An effective method of achieving a high power and high brightness fiber-coupled laser-diode system has been put forward by ZEMAX software simulation. We designed a simple beam shaping system to improve the beam symmetry and narrow the line width with a trapezoid prism and a set of parallelogram plates. Using this technique, a laser-diode stack consisted of eight bars was coupled into a fiber with a core diameter of 200 μm and a numerical aperture of 0.22. The simulation results demonstrated that the output power can reach 272 W with equalizing brightness of 7.48 MW/(cm2·sr), and the optical-optical efficiency was 85%.

Effect of density and Poisson’s ratio on thermal induced vibration of parallelogram plate

The present paper provides mathematical model for the study of natural (free) vibration of non homogeneous tapered parallelogram plate on clamped boundary condition. Here non homogeneity (in material) of the plate’s means that the density and Poisson’s ratio varies circularly and exponentially respectively. For tapered, we assumed that thickness of the plate varies linearly in one direction. Bi parabolic temperature (parabolic in ζ-direction and parabolic in ψ-direction) variation on the plate is being viewed. Rayleigh Ritz method is used to solve the model (governing differential equation of motion).

Vibration of Skew Plate with Circular Variation in Thickness and Poisson’s Ratio

Abstract The present study analyzes the natural vibration of non homogeneous visco elastic skew plate (parallelogram plate) with non uniform thickness under temperature field. Here non homogeneity in the plate’s material arises due to circular variation in Poisson’s ratio. Also the circular variation in thickness causes non uniformity in the shape of the plate. Bi linear temperature variation on the plate along both the axes is being viewed. The equation of motion related to frequency modes are solved by Rayleigh Ritz method. The findings of the present analysis are presented with the help of tables.

Structural analysis of the foldecture derived from racemic peptide foldamers

The molecular packing structure of an elongated parallelogram plate shaped foldecture composed of a 1:1 racemic mixture of 11-helical peptide foldamers was resolved by powder X-ray diffraction (PXRD) analysis. A comprehensive Rietveld refinement procedure compensated for powder texture and identified the principal face of the foldecture. Each foldamer makes head-to-tail intermolecular hydrogen bonds, creating extended chains of single enantiomers that form a network of hydrophobic close contacts with foldamers of both the opposite and the same chiralities. An isosurface for anisotropic microstrain was calculated and found to be smallest along the x-axis, which is parallel to the network of intermolecular hydrogen bonds. Comparison with the single crystal structure found molecular packing motifs to be almost identical—a result infrequently observed in enantiopure foldectures. This is the first powder X-ray diffraction structural analysis of a foldecture composed of multiple components.

Improving the Accuracy of Geometric Interpolation for Determining Fundamental Frequency of Parallelogram Plates Vibration

The paper considers a method of geometric interpolation of basic solutions for two-dimensional problems in the theory of elasticity and structural mechanics, particularly as applied to mechanical engineering. The scope of the study is the vibrations of thin elastic parallelogram plates of constant thickness. To determine the fundamental frequency of vibration it is suggested to use an interpolation method with a newly introduced geometric characteristic of plates which is a ratio of inner conformal radius to outer conformal radius. Taken from the theory of conformal mapping, the conformal radii of domains are obtained by mapping the plates onto the interior and exterior of a unit circle. The studies have shown that the use of the suggested ratio ​​gives up to twice more accurate results of the geometric interpolation. The paper presents the basic terms and formulas of the method with comparative analysis of the curve diagrams using a shape factor and a conformal radii ratio.

The CPCT based CBIE and HBIE for potential problems in three dimensions

In this paper, the authors present a more efficient and robust implementation of conventional and hypersingular BIEs for potential problems in three dimensions under the framework of boundary face method (BFM). The focus is laid on the accurate evaluation of singular curved surface integrals, and three aspects related are considered simultaneously: (a) the near singularity caused by distorted element shape; (b) the near singularity derived from the angular direction; (c) the singularity in the radial direction. A conformal polar coordinate transformation (CPCT) is employed to eliminate the shape effect of distorted integration cells, which can retain the shape characteristic. Besides, an improved sigmoidal transformation is introduced to alleviate the near singularity in the angular direction. By combination of the two strategies with previous singularity subtraction method, an efficient numerical integration scheme has been obtained for various orders of singularities. Some numerical examples including parallelogram plate, sphere and hollow cylinder examples with coarse meshes are presented to demonstrate the accuracy and flexibility of the proposed method.

Parallelogram plate shaped foldecture from the controlled self-assembly of α/β-peptide foldamer

Foldecture is a versatile material for the design of functional and well-defined 3D peptide architectures. In order to develop design principles for this emerging class of material, a systematic study of the relationship between foldecture morphology and molecular-level structure is necessary. Herein we report the formation of parallelogram plate shaped foldecture by the controlled self-assembly of heterogeneous α/β-peptide foldamer. Physical properties of foldecture were examined by IR, TGA, DSC, PXRD analysis. The foldecture packing motif was determined by high-resolution powder X-ray diffraction techniques and provides a link between the molecular-level structure and microscale morphologies.

Parametrisation and Application of Cube and Eggbox-Type Folded Geometries

There exist several kirigami, or non-developable folded plate patterns that share the most useful properties of the widely-used Miura-ori pattern, namely rigid-foldability, a tessellated unit geometry, and constituent elements composed of a single parallelogram plate. This paper first presents parametrisations of five such patterns, two cube-type and three eggbox-type, consistent with that previously developed for Miura-type patterns. For each pattern, the parametrisation establishes relationships between crease pattern, volumetric, and kinematic parameters. Along with the Miura-type patterns, these geometries form a set termed parallelogram-plate folded geometries and together they suggest a range of interesting structural engineering applications. This paper discusses two applications specifically enabled by the new parametrisations: space frame synthesis and hierarchical sandwich panels.

Vibration of Visco-Elastic Parallelogram Plate with Thickness Variation Linearly in one Direction and Parabolic in another Direction

The main objective of the present investigation is to study the vibration of visco-elastic parallelogram plate whose thickness varies bi-directionally. It is assumed that the plate is clamped on all the four edges and that the thickness varies linearly in one direction and parabolically in another direction. Using the separation of variables method and Rayleigh-Ritz technique with a two-term deflection function, the governing differential equation has been solved for vibration of visco-elastic parallelogram plate. For visco-elastic, the basic elastic and viscous elements are combined. We have taken Kelvin model for visco-elasticity that is the combination of the elastic and viscous elements in parallel. Here the elastic element means the spring and the viscous element means the dashpot. The assumption of small deflection is made. Visco-elastic of the plate is taken of the “Kelvin Type”. Time period and deflection function at different point for the first two modes of vibration are calculated for various values of taper constant, aspect ratio and skew angle and results are presented in tabular form. Alloy “Duralumin” is considered for all the material constants used in numerical calculations.

The Yield Line Method for Oblique Plate Moment Coefficient

This paper uses the yield line theory in plasticity, combined with the static equilibrium method, which derives bending coefficient calculation chart of parallelogram plate that exposes to uniform load in a variety of support conditions for designers .

Effect of Sinusoidal Thickness Variation on Vibrations of Non-homogeneous Parallelogram Plate with Bi-linearly Temperature Variations

The present study is the computational prediction about the effect of sinusoidal varying thickness on the vibrations of non-homogeneous parallelogram plate. Consideration of the plate's material is visco-elastic with clamped boundary. It is assumed that the temperature varies bi-linearly while the density of the plate's material varies linearly in one direction due to non-homogeneity. The general equation of motion and consecutive equations are solved by using the Rayleigh-Ritz method. To study the vibrational behavior of parallelogram plate, natural frequency for both the modes of vibration is calculated for different values of parameters and shown in the graphs. All the calculations are carried out for duralumin.

Determination of maximum deflection in transverse bending of parallelogram plates using the conformal radiuses ratio

A problem of transverse bending of elastic isotropic parallelogram plates under the action of uniformly distributed load is considered. The plate sides are either pivoted or rigidly restrained. To determine the value of maximum plate deflection, it is proposed that an interpolation technique with respect to form factor be used; and as a geometrical argument, an inner-outer conformal radius ratio can be applied instead of the form factor.