Compliance Under a Small Torsional Couple of an Elastic Plate Pressed Between Two Identical Elastic Spheres

1966 ◽  
Vol 33 (2) ◽  
pp. 377-383 ◽  
Author(s):  
J. J. O’Connor
Keyword(s):  
Integral Equation ◽  
Elastic System ◽  
Plate Thickness ◽  
Small Region ◽  
Thin Plates ◽  
Flexible Plate ◽  
Surface Shear ◽  
Shear Moduli ◽  
Contact Circle ◽  
Contact Stress Distribution

The object of the analysis is to calculate the surface shear traction and the torsional compliance of an elastic system comprising a plate pressed between identical spheres. The problem is formulated in terms of an integral equation which is solved numerically. The parameters are the plate thickness and ratio of shear moduli. Solutions are obtained for all except extremely thin plates, for which a previous approximate solution is shown to be valid. The contact stress distribution is always close to the well-known distribution appropriate to the half-space, with a singularity at the edge of the contact circle, unless the plate is simultaneously thin and flexible. A thin flexible plate confines the singularity to a very small region at the edge of the contact circle, the stress elsewhere being essentially proportional to radius. The torsional compliance predicted by the analysis agrees well with experiment.

scholarly journals Tailoring Plate Thickness of a Helmholtz Resonator for Improved Sound Attenuation

2016 ◽  
Author(s):  
Mohammad Kurdi ◽  
Shahin Nudehi ◽  
Gregory Scott Duncan
Keyword(s):  
Transmission Loss ◽  
Plate Thickness ◽  
Helmholtz Resonator ◽  
Element Model ◽  
Flexible Plate ◽  
End Plate ◽  
Receptance Coupling ◽  
Coupling Techniques ◽  
Gradient Based ◽  
Optimization Search

A Helmholtz resonator with flexible plate attenuates noise in exhaust ducts, and the transmission loss function quantifies the amount of filtered noise at a desired frequency. In this work the transmission loss is maximized (optimized) by allowing the resonator end plate thickness to vary for two cases: 1) a non-optimized baseline resonator, and 2) a resonator with a uniform flexible endplate that was previously optimized for transmission loss and resonator size. To accomplish this, receptance coupling techniques were used to couple a finite element model of a varying thickness resonator end plate to a mass-spring-damper model of the vibrating air mass in the resonator. Sequential quadratic programming was employed to complete a gradient based optimization search. By allowing the end plate thickness to vary, the transmission loss of the non-optimized baseline resonator was improved significantly, 28 percent. However, the transmission loss of the previously optimized resonator for transmission loss and resonator size showed minimal improvement.

scholarly journals The features of the use of the waveguide radiators in smart antenna systems

2018 ◽  
Vol 26 (1) ◽  
pp. 89-92
Author(s):  
V. M. Morozov ◽  
V. I. Magro
Keyword(s):  
Integral Equation ◽  
Antenna Array ◽  
Antenna Arrays ◽  
Smart Antenna ◽  
Integral Equation Method ◽  
Thin Plates ◽  
Coupling Region ◽  
Five Elements ◽  
Antenna Systems ◽  
Selection Of

The features of the use of finite waveguide antenna arrays in the structure of modern smart antenna systems are considered. The paper deals with the problem of diffraction of an electromagnetic wave on a finite waveguide antenna array scanning in the E-plane. Antenna array consists of five radiating elements. The open ends of the waveguides are surrounded by a metal screen. The resonator coupling region was chosen as matching elements. The solution of the problem is carried out by the integral equation method on the basis of the selection of overlapping regions. The problem reduces to solving the Fredholm integral equation of the second kind. An array of infinitely thin plates and that of waveguides with a finite wall thickness are considered. The main regularities for choosing the optimal geometric dimensions of the antenna array are established. Studies were carried out for arrays with a number of elements from five to fifteen. The analysis of edge effects in the final antenna array is carried out. It is shown that the introduction of a resonator region into a five-element lattice makes it possible to expand the sector of the radiation angles and avoid the effect of blinding. It is shown that this statement is valid not only for five-element lattices, but also for arrays with a large number of radiating elements. The radiation patterns are calculated. The  coefficients of mutual coupling in an array with five elements are investigated. General recommendations for choosing optimal sizes of the resonator coupling region of radiators are considered.

Anti-Plane Deformation Uniformly Piecewise Homogeneous Space with a Periodic System of Semi-Infinite Internal Cracks

2020 ◽  
pp. 13-20
Author(s):  
V.N. Hakobyan ◽  
A.A. Grigoryan
Keyword(s):  
Integral Equation ◽  
Numerical Calculation ◽  
Homogeneous Space ◽  
Plane Deformation ◽  
Contact Stresses ◽  
Antiplane Deformation ◽  
Intensity Factors ◽  
Shear Moduli ◽  
Mechanical Quadrature ◽  
Equal Thickness

In this paper, we have constructed a solution for the problem of antiplane deformation of a uniformly piecewise homogeneous space of two alternately repeating heterogeneous layers of equal thickness from different materials, which are relaxed on their median planes by two semi-infinite, periodic parallel tunneling cracks. A system of defining equations of the problem is derived in the form of a system of two singular equations of the first kind, with respect to contact stresses acting in the contact zones on the median planes of heterogeneous layers, the solution of which, in the general case, is constructed by the method of mechanical quadrature. In the particular case when the cracks in the heterogeneous layers are the same, the solution of the problem is reduced to the solution of two independent equations and their closed solutions are constructed. The defining singular integral equation of the problem is also obtained in the case when there are no cracks in one of the heterogeneous layers. In the general case, a numerical calculation was carried out and patterns of changes in contact stresses and intensity factors of destructive stresses at the end points of cracks were determined depending on the physical and mechanical and geometric parameters of the problem, which are the ratios of the shear moduli of the layers and the ratio of the layer thickness and crack lengths.

A Numerical Solution for an Axially Symmetric Contact Problem

1967 ◽  
Vol 34 (2) ◽  
pp. 283-286 ◽  
Author(s):  
Yih-O Tu
Keyword(s):  
Stress Distribution ◽  
Contact Problem ◽  
Contact Stress ◽  
Plate Thickness ◽  
Contact Radius ◽  
Axially Symmetric ◽  
Pressure Distributions ◽  
Linear Algebraic Equations ◽  
Axially Symmetric Contact ◽  
Contact Stress Distribution

A numerical scheme for the axially symmetric contact problem of a plate pressed between two identical spheres is given. The axially symmetric contact stress distribution is represented by a finite set of pressure distributions linearly varying with the radius between values defined in a set of concentric circles. The normal displacements of the bodies in contact resulting from these pressure distributions are matched at every radius of the discrete set of radii of these circles. The integral equation for the unkown contact stress distribution is thus approximated by a set of linear algebraic equations whose solution yields the unknown pressure values of the approximate distribution. The contact radius, relative approach, and the maximum contact stress are then computed numerically from this solution and are presented in terms of the total load, the radius of the sphere, and the plate thickness.

Turbulent Flow Around a Bluff Rectangular Plate. Part I: Experimental Investigation

1991 ◽  
Vol 113 (1) ◽  
pp. 51-59 ◽  
Author(s):  
N. Djilali ◽  
I. S. Gartshore
Keyword(s):  
Shear Layer ◽  
Rectangular Plate ◽  
Separation Bubble ◽  
Plate Thickness ◽  
Mixing Layer ◽  
Low Frequency ◽  
Surface Shear Stress ◽  
Constant Growth Rate ◽  
Lower Growth ◽  
Surface Shear

Measurements are reported for the separted reattaching flow around a long rectangular plate placed at zero incidence in a low-turbulence stream. This laboratory configuration, chosen for its geometric simplicity, exhibits all of the important features of two-dimensional flow separation with reattachment. Conventional hot-wire anemometry, pulsed-wire anemometry and pulsed-wire surface shear stress probes were used to measure the mean and fluctuating flow field at a Reynolds number, based on plate thickness, of 5 × 104. The separated shear layer appears to behave like a conventional mixing layer over the first half of the separation bubble, where it exhibits an approximately constant growth rate and a linear variation of characteristic frequencies and integral timescales. The characteristics of the shear layer in the second half of the bubble are radically altered by the unsteady reattachment process. Much higher turbulent intensities and lower growth rates are encountered there, and, in agreement with other reattaching flow studies, a low frequency motion can be detected.

scholarly journals Formulation of a New Mixed Four-Node Quadrilateral Element for Static Bending Analysis of Variable Thickness Functionally Graded Material Plates

2021 ◽  
Vol 2021 ◽  
pp. 1-23
Author(s):  
Pham Van Vinh
Keyword(s):  
Functionally Graded Material ◽  
Variable Thickness ◽  
Mixed Finite Element ◽  
Plate Thickness ◽  
Functionally Graded ◽  
Thin Plates ◽  
Quadrilateral Element ◽  
Graded Material ◽  
Element Technique ◽  
Power Law Index

A new mixed four-node quadrilateral element (MiQ4) is established in this paper to investigate functionally graded material (FGM) plates with variable thickness. The proposed element is developed based on the first-order shear deformation and mixed finite element technique, so the new element does not need any selective or reduced numerical integration. Numerous basic tests have been carried out to demonstrate the accuracy and convergence of the proposed element. Besides, the numerical examples show that the present element is free of shear locking and is insensitive to the mesh distortion, especially for the case of very thin plates. The present element can be applied to analyze plates with arbitrary geometries; it leads to reducing the computation cost. Several parameter studies are performed to show the roles of some parameters such as the power-law index, side-to-thickness ratio, boundary conditions (BCs), and variation of the plate thickness on the static bending behavior of the FGM plates.

Electroelastic Finite Element Modeling and Experimental Validation of Structurally-Integrated Piezoelectric Energy Harvester

2013 ◽  
Author(s):  
Ugur Aridogan ◽  
Ipek Basdogan ◽  
Alper Erturk
Keyword(s):  
Finite Element ◽  
Energy Harvesting ◽  
Energy Harvester ◽  
Electrical Power ◽  
Element Model ◽  
Thin Plates ◽  
Flexible Plate ◽  
Numerical Predictions ◽  
Piezoelectric Energy ◽  
Host Structure

Vibration-based energy harvesting has attracted interest of researchers from various disciplines over the past decade. In the literature of piezoelectric energy harvesting, the typical configuration is a unimorph or a bimorph cantilevered piezoelectric beam located on a vibrating host structure subjected to base excitations. As an alternative to cantilevered piezoelectric beams, piezoelectric layers structurally integrated on thin plates can be used as vibration-based energy harvesters since plates and plate-type structures are commonly used in aerospace, automotive and marine applications. The aim of this paper is to present experiments and electroelastic finite element simulations of a piezoelectric energy harvester structurally integrated on a thin plate. The finite element model of the piezoceramic patch and the all-edges-clamped plate are built. In parallel, an experimental setup is constructed using a thin PZT-5A piezoceramic patch attached on the surface of all-edges-clamped rectangular aluminum plate. The electroelastic frequency response functions relating voltage output and vibration response to forcing input are validated using the experimentally obtained results. Finally, electrical power generation of the piezoceramic patch is investigated using the experimental set-up for a set of resistive loads. The numerical predictions and experimental results show that the use of all-edge-clamped flexible plate as host structure for piezoelectric energy harvester leads to multimodal vibration-to-electricity conversion.

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