Finite Rectangular Plate Research Articles

An Improved MSA Model for Evaluating the Sound Transmission Loss of a Rectangular Plate for Diffuse Field Incidence

This paper presents an approximate analytical model for estimating the transmission loss (TL) of a finite rectangular plate in the low frequency range, which is based on the modal summation approach (MSA) taking into account the modal radiation impedance and fluid loading. The mode-dependent radiation resistance is calculated using the Rayleigh integral. The fluid loading is taken into account through the natural frequency modified by the added mass. The results are compared with the ones of Statistical Energy Analysis (SEA) coupled with FEM and FEM coupled with BEM. In addition, the effects of the various vibration modes and the fluid loading on TL, and a way for reducing the calculation time are discussed.

Exact analytical results for the electrostatic potential due to a uniformly charged finite rectangular plate

We explain a general mathematical method that allows one to calculate the electrostatic potential created by a uniformly charged rectangular plate with arbitrary length and width at an arbitrary point in space. Exact analytical results for the electrostatic potential due to a uniformly charged finite rectangular plate are shown for special cases in order to illustrate the implementation of the formalism. Results of this nature are very important to various problems in physical sciences, applied mathematics, and potential theory.

Exact frequency-domain spectral element model for the transverse vibration of a reangular Kirchhoff plate

As an extension of the previous work on membranes by Kim and Lee (2020), an exact frequency-domain spectral element model is proposed for the transverse vibrations of a rectangular Kirchhoff plate, based on the following procedure. First, in the frequency-domain, the general solution of a finite rectangular plate element is derived in the spectral form in the spatial domain, after removing all the trigonometric terms that vanish at the four boundary edges. Second, as an important theoretical improvement, the boundary functions that represent the boundary conditions at the four boundary edges are also expressed in the spectral forms in the spatial domain. Next, by using the projection method based on the orthogonality of trigonometric functions, the spatial-domain spectral coefficients of the general solution are related to those of the boundary functions. Finally, from the force-displacement relationships, the spectral element matrix (or dynamic stiffness matrix) is formulated for a finite rectangular plate element. As both the general solution and boundary functions are expressed in the spectral forms in both the temporal and spatial domains, a fast Fourier transform (FFT) algorithm can be applied to efficiently simulate the vibration responses and waves in a plate. The accuracy and computational efficiency of the proposed SEM are evaluated by comparison with the exact theory, modal analysis method, and the commercial finite-element-analysis software ANSYS.

Bistatic scattering at soft–hard strips and plates: method of moments (MoM), theory of edge diffraction (TED), and physical theory of diffraction (PTD) analysis

Diffraction at a strip with one face soft (electric) and the other hard (magnetic) is studied. New results obtained by the method of moments (MoM) are compared with the asymptotic theory of edge diffraction (TED) for the totally soft and hard strips. Attention is given to diffraction of oblique incident waves including the grazing diffraction. Novel analytic estimations for the forward and backward grazing scattering are established via the advanced physical theory of diffraction (PTD) free of grazing singularities. These estimations are demonstrated for the infinitely long strips and finite size rectangular plates.

Sound transmission loss of double-wall piezoelectric plate made of functionally graded materials via third-order shear deformation theory

According to the third-order shear deformation assumption, this paper analytically investigates the sound transmission loss (STL) through a rigidly baffled finite rectangular double wall piezoelectric plate structure made of functionally graded materials (FGMs) with enclosed acoustic cavity, under harmonic plane sound excitation and initial external electric voltage. The effective piezoelectric material properties of each plate are supposed to be continuously variable across the thickness direction using a power law model in terms of the volume fractions of the material phases. The coupled vibroacoustic governing equations are achieved utilizing Hamilton’s principle and then solved analytically by applying the sound velocity potential method in conjunction with double Fourier series expansions to determine STL equation. Numerical studies are performed to illustrate the effects of the acoustic cavity depth, initial external electric voltage, incident elevation angle, gas type used in acoustic cavity, and material gradient on the changes of STL curves of simply supported double wall FGM piezoelectric plate. The profound effect of these factors on sound isolation performance is clearly shown.

On the structure of the self-sustaining cycle in separating and reattaching flows

The separating and reattaching flows and the wake of a finite rectangular plate are studied by means of direct numerical simulation data. The large amount of information provided by the numerical approach is exploited here to address the multi-scale features of the flow and to assess the self-sustaining mechanisms that form the basis of the main unsteadinesses of the flows. We first analyse the statistically dominant flow structures by means of three-dimensional spatial correlation functions. The developed flow is found to be statistically dominated by quasi-streamwise vortices and streamwise velocity streaks as a result of flow motions induced by hairpin-like structures. On the other hand, the reverse flow within the separated region is found to be characterized by spanwise vortices. We then study the spectral properties of the flow. Given the strongly inhomogeneous nature of the flow, the spectral analysis has been conducted along two selected streamtraces of the mean velocity field. This approach allows us to study the spectral evolution of the flow along its paths. Two well-separated characteristic scales are identified in the near-wall reverse flow and in the leading-edge shear layer. The first is recognized to represent trains of small-scale structures triggering the leading-edge shear layer, whereas the second is found to be related to a very large-scale phenomenon that embraces the entire flow field. A picture of the self-sustaining mechanisms of the flow is then derived. It is shown that very-large-scale fluctuations of the pressure field alternate between promoting and suppressing the reverse flow within the separation region. Driven by these large-scale dynamics, packages of small-scale motions trigger the leading-edge shear layers, which in turn created them, alternating in the top and bottom sides of the rectangular plate with a relatively long period of inversion, thus closing the self-sustaining cycle.

Finite Plate with Circular and Square Hole under Partial Loading

In this paper a general analytical solution is obtained to find stress distribution in a finite elastic plate with a circular or square hole subjected to arbitrary biaxial partial loading using modified boundary condition by assuming plane stress conditions. The method employed is based on solution of circular hole in finite rectangular plate. This plate is mapped to circular ones and the partial loading is transformed to new boundaries as form of triangular functions. The Airy stress functions are selected according to these triangular functions and the unknown factors of Airy stress functions are derived by applying boundary conditions. The stresses in this plate with circular hole are mapped to plate with square hole using Muskhelishvili’s complex variable method. The results of this method are compared with theoretical solution of infinite plate and finite elemnt method solution of finite plate. The results showed the dimensions of plate and square hole and length of biaxial partial loading affected on Von Mises stress around the square hole. Von Mises stress increases around the square hole by decreasing length of the plate or increasing hole’s area.

Stress Analysis of Finite Steel Plate with a Rectangular Hole Subjected to Uniaxial Stress Using Finite Element Method

The orientation of a rectangular opening in a plate subject to a uniaxial stress has an important effect on stress distribution. In this paper, the effect of von mises stress has been investigated for various corner radius of rectangular opening to opening width ratio of a finite rectangular plate, from this analysis the suitable model has been selected and on the basis of that model the effect of von mises stress also has been investigated for various opening width to length ratio and from this analysis most desirable model has been selected when the value of opening width to length ratio is 0.5. Openings are reinforced to restore partially the effective cross section area and reduce stress concentration. In this paper, the effect of von mises stress on various types of reinforcements have been investigated and from this analysis it has been observed that single doubler plate gives the most suitable reinforced model. Finally, the convergence test has been performed for the most suitable reinforced model. The aim of this paper is to analyze a finite steel plate with a rectangular hole subjected to uniaxial stress and observe the effect of variation in reinforcement.

Audible Noise Characteristics of Filter Capacitors Used in HVDC Converter Stations

This paper presents a detailed investigation of audible noise characterization of filter capacitors by investigating the noise radiation characteristics and analyzing the effect of voltage frequency, phase angle, and magnitude on a single capacitor unit. First, the noise radiation for capacitor surfaces was characterized theoretically using finite rectangular plate model and experimentally measured in a semianechoic room. Second, voltage frequency impact on the noise distribution was investigated through simulation. The simulation results show that noise directivity increases with increasing frequency. The excitation frequency impact on the radiation ratio and the noise level was characterized through a sweep-frequency experiment. Our investigations show that the noise frequency response below 500 Hz is quite low, while from 500 to 2000 Hz, it is significant and increases slowly. Third, the impact of the voltage phase angle on the noise level was investigated. The results show a noise-level difference of 3.9 dB for different phase angles. Finally, we present the voltage magnitude impact on the noise level. Our theoretical analysis and experimental study show a linear relationship between the sound power level and the logarithm of the voltage magnitude with a slope of 40.

A New PSO-based Algorithm for Two-Dimensional Non-Guillotine Non-Oriented Cutting Stock Problem

ABSTRACTIn this paper, a new algorithm is proposed for the two-dimensional non-guillotine non-oriented cutting stock problem. The considered problem consists of cutting small rectangular pieces of predetermined sizes from large but finite rectangular plates. The objective is to generate cutting patterns that minimize the unused area and fulfill customer orders. The proposed algorithm is a combination of a new particle swarm optimization approach with a heuristic criterion inspired from the literature. The algorithm is tested on twenty-two instances divided into two sets. Corresponding results show the algorithm efficiency in optimizing the trim loss that is comprised between 2.6% and 7.8% for all considered instances.

Howland’s isotropic Kts curve for plates with circular holes as a master curve for Kts in orthotropic plates with elliptical holes

The importance of the role played by the so-called stress concentration factors (or symbolically referred to as Kts) in analysis and design in both mechanical and structural engineering is a well-established fact, and accuracy and ease in their estimation result in significant aspects related to engineering costs, and additionally on both the reliability in the design of parts and/or in the analysis of failed members. In this work, rectangular finite width plates of both isotropic and orthotropic materials with circular and elliptical holes are considered. Based on two key observations reported herein, it is shown in a partially heuristic engineering sense, that Howland’s solution curve for the stress concentration factors for finite width plates with circular holes subjected to tension can be viewed as a master curve; accordingly, it can be used as a basis to rather accurately estimate stress concentration factors for isotropic finite width tension rectangular plates with centered elliptical holes and also rather accurately used to estimate stress concentration factors for orthotropic finite width rectangular plates under tension with centered elliptical holes. Two novel concepts are defined and presented to this effect: geometric scaling and material scaling. In all the examined and reported cases, the specific numerical results can be obtained accurately using a hand-held calculator making virtually unnecessary the need to program and/or use other complex programs based on the finite element method, just as an example. The maximum recorded average error for all the considered cases being 2.62% as shown herein.

Analytical approach to evaluate transmission loss of finite rectangular plate considering its eigenmodes.

Analytical approach to evaluate transmission loss of finite rectangular plate considering its eigenmodes.

STRESS INTENSITY FACTOR FOR DOUBLE EDGE CRACKED FINITE PLATE SUBJECTED TO TENSILE STRESS

A finite rectangular plate with double edge crack under uniaxial tension depends on the assumptions of Linear Elastic Fracture Mechanics LEFM and plane strain problem are studied in the present paper. The effect of crack position, crack oblique and the kinked crack orientation are investigated to predict if a crack starts to grow. These problems are solved by calculating the Stress Intensity Factor SIF for mode I (KI) and II (KII) near the crack tip theoretically using mathematical equations and numerically using finite element software ANSYS R15.A good agreement is observed between the theoretical and numerical solutions. The results show that the KI increases with increasing the relative crack length and tensile stress and these values are increased when the crack position draws near the plate edge while in case of parallel cracks the mutual shielding effect reduces KI in each crack.In mixed mode, it is shown that the maximum values of KI and KII occur at crack angle β=0o and 45o, respectively and the orientation of the kinked crack have significant effects on the KI and KII.

Frequency Domain Spectral Element Model for the Vibration Analysis of a Thin Plate with Arbitrary Boundary Conditions

We propose a new spectral element model for finite rectangular plate elements with arbitrary boundary conditions. The new spectral element model is developed by modifying the boundary splitting method used in our previous study so that the four corner nodes of a finite rectangular plate element become active. Thus, the new spectral element model can be applied to any finite rectangular plate element with arbitrary boundary conditions, while the spectral element model introduced in the our previous study is valid only for finite rectangular plate elements with four fixed corner nodes. The new spectral element model can be used as a generic finite element model because it can be assembled in any plate direction. The accuracy and computational efficiency of the new spectral element model are validated by a comparison with exact solutions, solutions obtained by the standard finite element method, and solutions from the commercial finite element analysis package ANSYS.

Analytical model for the temperature field around a nonuniform three-dimensional moving heat source: friction stir welding modelling

A transient semi-analytical solution is devised to find the temperature distribution around a nonuniform three-dimensional heat source in a finite rectangular plate with cooling surfaces and nonhomogenous boundary conditions. This solution can serve as a viable tool to find the temperature field and consequently provides useful insights into material modelling and response in a variety of industrial applications, including friction welding processes. As a numerical example, the established solution is applied to the friction stir welding as a solid-state welding method. The comparison between the obtained analytical results and the results found through FEM and experiment showed that the analytically obtained results and the numerical and experimental data are in good agreement.

STRESS INTENSITY FACTOR FOR DOUBLE EDGE CRACKED FINITE PLATE SUBJECTED TO TENSILE STRESS

A finite rectangular plate with double edge crack under uniaxial tension depends on the assumptions of Linear Elastic Fracture Mechanics LEFM and plane strain problem are studied in the present paper. The effect of crack position, crack oblique and the kinked crack orientation are investigated to predict if a crack starts to grow. These problems are solved by calculating the Stress Intensity Factor SIF for mode I (KI) and II (KII) near the crack tip theoretically using mathematical equations and numerically using finite element software ANSYS R15. A good agreement is observed between the theoretical and numerical solutions. The results show that the KI increases with increasing the relative crack length and tensile stress and these values are increased when the crack position draws near the plate edge while in case of parallel cracks the mutual shielding effect reduces KI in each crack.In mixed mode, it is shown that the maximum values of KI and KII occur at crack angle β=0o and 45o, respectively and the orientation of the kinked crack have significant effects on the KI and KII.

Nonlinear vibrations of plates in axial pulsating flow

A theoretical approach is presented to study nonlinear vibrations of thin infinitely long and wide rectangular plates subjected to pulsatile axial inviscid flow. The flow is set in motion by a pulsating pressure gradient. The case of plates in axial uniform flow under the action of constant transmural pressure is also addressed for different flow velocities. The plate is assumed to be periodically simply supported in both in-plane directions with immovable edges and the flow channel is bounded by a rigid wall. In this way the system under study is a finite rectangular plate with conditions on the fluid and the plate boundaries coming from the periodicity of the infinite system. The equations of motion are obtained based on the von Kárman nonlinear plate theory retaining in-plane inertia via Lagrangian approach. The fluid model is based on the potential flow theory. The resulting Lagrange equations of motion of the coupled system contain quadratic and cubic nonlinear terms and are studied by using a code based on the pseudo-arc-length continuation and collocation scheme. The effect of different system parameters such as flow velocity, pulsation amplitude, pulsation frequency and channel pressurization on the stability of the plate and its geometrically nonlinear response to pulsating flow are fully discussed. It has been found that the presence of positive transmural uniform pressure and small pulsation frequency would destroy the pitchfork bifurcation (divergence) that flat plates exhibit when subjected to uniform flow. Moreover, in case of zero uniform transmural pressure numerical results show a hardening type behavior for the entire flow velocity range when the pulsation frequency is spanned in the neighborhood of the plate׳s fundamental frequency. On the contrary, a softening type behavior is presented when a uniform transmural pressure is introduced.

Ground cross-modal impedance as a tool for analyzing ground/plate interaction and ground wave propagation.

An analytical approach is investigated to model ground-plate interaction based on modal decomposition and the two-dimensional Fourier transform. A finite rectangular plate subjected to flexural vibration is coupled with the ground and modeled with the Kirchhoff hypothesis. A Navier equation represents the stratified ground, assumed infinite in the x- and y-directions and free at the top surface. To obtain an analytical solution, modal decomposition is applied to the structure and a Fourier Transform is applied to the ground. The result is a new tool for analyzing ground-plate interaction to resolve this problem: ground cross-modal impedance. It allows quantifying the added-stiffness, added-mass, and added-damping from the ground to the structure. Similarity with the parallel acoustic problem is highlighted. A comparison between the theory and the experiment shows good matching. Finally, specific cases are investigated, notably the influence of layer depth on plate vibration.

Influence of crack offset distance on the interaction of multiple cracks on the same side in a rectangular plate

In the present work finite element method has been employed to study the interaction of multiple cracks in a finite rectangular plate of unit thickness with cracks on the same side under uniaxial loading conditions. The variation of the stress intensity factor and stress distribution around the crack tip with crack offset distance has been studied. Due to the presence of a neighbouring crack, two types of interactions viz. intensification and shielding effect have been observed. The interaction between the cracks is seen to be dependent on the crack offset distance. It is seen that the presence of a neighbouring crack results in the appearance of mode II stress intensity factor which was otherwise absent for a single edge crack. It can be said that the proximity of cracks is non-desirable for structural integrity. The von-Mises stress for different crack orientations has been computed. Linear elastic analysis of state of stress around the crack tip has also been done.

Planar stress wave attenuation in plates with circular voids and inclusions

This paper addresses planar stress wave attenuation in finite rectangular aluminum plates with circular voids and two types of inclusions; namely, high-density polyethylene (HDPE) and steel. The voids and inclusions act as a source of impedance mismatch and scatter the waves within the plates. The plates studied here have multiple numbers of circular voids and inclusions with different dimensions and positions, and are subjected to short duration transient loadings. To find the most efficient layout and dimensions of the circular voids and inclusions within these plates, a heuristic optimization methodology is developed based on Genetic Algorithms (GA). This methodology is coupled with the Finite Element (FE) method to analyze the wave propagation behavior of the plates under elastic response. The developed optimization methodology is capable of updating the FE model of each candidate design during the optimization procedure, as each in general has different geometric characteristics. The results show that the attenuation capacity of the plates with circular voids and inclusions is a complex function of the dimensions of the plates and the wavelength associated with the transient loading, as well as, the type, positions and dimensions of the voids and inclusions. It is observed that significant amounts of planar stress wave attenuation can be achieved, if the dimensions and positions of the circular voids and inclusions are selected appropriately.