Green's function of the clamped punctured disk

Experiments using a soil-plate-oscillator (SPO) involve a vertical cylindrical column of granular medium (masonry sand, glass spheres, uncooked brown rice, un-popped popcorn kernels, or even “Toasty Oats” ™ cereal) that is supported by an air-backed thin circular elastic acrylic plate (20.3 cm diameter and 3.2 mm thick) that is rigidly clamped to the bottom of a thick-walled aluminum tube. The soil column is driven from below using an electrodynamic system. Here, an AC coil placed on axis and below the plate, drives a 1 cm diameter 1.5 cm long rare earth magnet that is fastened to the underside center of the plate. The coil is electrically driven by an amplified swept sinusoidal slowly varying chirp. A small accelerometer attached to the magnet is used to measure the vibration. In nonlinear tuning curve experiments the resonant frequency decreases significantly with increased amplitude—representing a softening in the nonlinear system. For fixed amplitude the resonant frequency vs. the granular medium mass loading (over the plate) reaches a minimum and then increases with increased loading due to the granular medium’s flexural stiffness—which overcomes the mass loading effects. For water loading, the frequency always decreases since there is no bending stiffness.

The soil plate oscillator apparatus (SPO) consists of two circular flanges (5 inch ID, 8 inch OD, 2.5 inch thick) sandwiching and clamping a thin (1/8 inch) circular elastic acrylic plate. Granular media (soil, masonry sand, glass beads, or even edible granular material) is supported at the bottom by the plate and side walls of the upper flange. A magnetic disk (1 cm) fastened below the plate (on center) is driven by a swept sinusoidal current in a coil placed below the magnet. This electrodynamic SPO, placed in a Wheatstone bridge, was used to measure the on-axis motional electrical impedance Zmot with and without granular loading. Nonlinear tuning curves near resonance exhibit softening with increasing drive amplitude. Measurements of Real{Zmot} and Imag{Zmot} were used to predict the nonlinear mechanical point impedance (Zmech = constant / Zmot) behavior of the SPO system. A lumped element model (granular media and elastic plate system) help determine the mesoscopic nonlinear properties of the granular media alone. Nonlinear tuning curves of the magnet’s acceleration were also measured. A bilinear hysteresis model fits the tuning curve results. See I. T. Perez-Miravete et al., JASA 125, 1302 (2009) for a nonlinear mesoscopic analysis of a flexural bar.

Nonlinear deflections and radial surface stresses in thin elastic circular plates laterally deformed into symmetrical concave shapes are analyzed. The deformations are induced by loading each plate with a small center ring while the plate is resting on a ring of nearly the same diameter as the plate. The rings and plate are coaxial, i.e. concentric. Center deflections up to 3 and 4.5 times the plate thickness were predicted for ring‐diameter ratios of 0.5 and 0.2, respectively. The predicted deflection profiles at various loads agreed quite well with those which were determined experimentally on a chemically strengthened glass plate. This analysis provides a new criterion for using the coaxial‐ring loading method for flexural strength testing of brittle materials.

Views Icon Views Article contents Figures & tables Video Audio Supplementary Data Peer Review Share Icon Share Twitter Facebook Reddit LinkedIn Tools Icon Tools Reprints and Permissions Cite Icon Cite Search Site Citation M. S. Korman, D. V. Duong, A. E. Kalsbeck; Electrodynamic soil plate oscillator: Modeling nonlinear mesoscopic elastic behavior and hysteresis in nonlinear acoustic landmine detection. AIP Conf. Proc. 28 October 2015; 1685 (1): 080003. https://doi.org/10.1063/1.4934456 Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentAIP Publishing PortfolioAIP Conference Proceedings Search Advanced Search |Citation Search

By means of Green's functions, an exact solution has been found for the thin elastic rectangular plate, built-in along one edge and simply supported along the other edges, for various static loading conditions. The results for deflections and moments have been compared with an approximate method by Timoshenko and the finite element method for different loading cases. It has been shown that results given by Timoshenko and finite element methods agree well with the exact solution, presented in this work.

The introduction of effective elastic moduli allows us to apply the classical nonlinear von Karman theory for modeling the behavior of nanoplates with surface stresses taken into account according to the Gurtin–Murdoch model. In particular, this allows us to apply the known solutions and methods for macroplates to nanoplates. So, earlier the authors solved the problem of buckling an thin elastic annular plate stretched by two point loads. In this paper, we address buckling this annular plate having nanosize. In previous studies, the manifestation of the size effect under buckling of nanoplates was shown mainly in the case of a homogeneous stress field. Here, the size effect is considered when a nanoplate is buckling under an inhomogeneous stress field.The introduction of effective elastic moduli allows us to apply the classical nonlinear von Karman theory for modeling the behavior of nanoplates with surface stresses taken into account according to the Gurtin–Murdoch model. In particular, this allows us to apply the known solutions and methods for macroplates to nanoplates. So, earlier the authors solved the problem of buckling an thin elastic annular plate stretched by two point loads. In this paper, we address buckling this annular plate having nanosize. In previous studies, the manifestation of the size effect under buckling of nanoplates was shown mainly in the case of a homogeneous stress field. Here, the size effect is considered when a nanoplate is buckling under an inhomogeneous stress field.

A Relation Between Two Biharmonic Green's Functions on Riemannian Manifolds

In this work we study the von Karman system for a thin circular elastic plate fixed to the elastic base and subjected to the compressing force along its boundary. The system is composed of two fourth-order nonlinear partial differential equations that give a valid mathematical description of the buckling of the plate. We intend to demonstrate the applicability of nonlinear functional analysis in the study of this problem. We describe the solution set of the von Karman equations in a small neighbourhood of a simple bifurcation point.

In this paper, a new deformation measurement method based on fiber bragg grating strain monitoring is proposed. Aiming the four-sided fixed elastic thin plate structure, an algorithm for reconstructing deformation field based on fiber bragg grating strain monitoring is proposed. The mechanical model of elastic thin plate was established. According to the degree of deformation, it can be divided into small deflection deformation and large deflection deformation. The relationship between surface strain and curvature of a thin plate was deduced. On this basis, a curvature iteration method was proposed to reconstruct the deformation field of thin plates. Aiming at the non-linear problem of large deflection deformation, a subsidence correction algorithm based on curvature iteration method was proposed. The experimental platform for strain monitoring of thin plate and several different loading conditions was established. Based on the discrete point strain data obtained by fiber bragg grating strain monitoring, the strain field is reconstructed to obtain detailed strain information. Based on the curvature iteration method and strain information, the deformation curves of thin plates under different loads are obtained. The experimental results show that the average error of reconstruction of the elastic thin plate structure is 6%, and the reconstructed results can accurately reflect the deformation surface of elastic thin plate. It also provides a basis for the study of deformation field reconstruction under complex constraints.

We consider the motion of a thin elastic plate with non-uniform thickness. The plate is either submerged and has some inclination with the vertical or is floating on the upper surface of the water. Green's function arising from the fourth-order boundary condition for the non-uniform plate (which we refer to as plate Green's function) is determined using two different methods in terms of the vibrating modes of the plate. These, in turn, are derived from the modes of a plate with constant thickness. The problem is finally reduced to a boundary integral equation involving the plate Green's function and the fundamental Green's function. This equation is hypersingular in the case of a submerged plate. A numerical solution to the integral equation is used to find results for elastic plates with variable thicknesses. The results are validated by comparing them with those of an elastic plate with uniform thickness. We also present simulations of the time-domain motion when the plate–fluid system is subject to an incident wave pulse using Fourier transform.

An experimental apparatus, designed to study flexural vibrations of a soil loaded plate, consists of a thin circular elastic plate clamped at the boundary between thick round flanges. The soil column is supported by the plate and upper cylindrical wall. A small magnet attached to the center of the plate is driven by a rigid ac coil (located coaxially below the plate) to complete the electrodynamic soil plate oscillator (SPO) design. Measurements of the electrical motional impedance Zmot of the SPO versus frequency are made using the complex output to input response of a Wheatstone bridge that has an identical coil element in one of its legs. The mechanical impedance Zmech (force/particle velocity, at the plate’s center) is inversely proportional to Zmot. Experiments, with and without mass loading at the plate’s center, verify Snowden’s theory for a clamped plate [J. Acoust. Soc. Am. (1971)] and help establish mechanical parameters. Resonant oscillations (for various point mass loadings) provided effective mass, spring, damping and coupling constant parameters of the system. “Tuning curve” behaviors of real {Zmot} and imaginary {Zmot} at successive vibration amplitudes (for various soil column heights) are used to develop a general mesoscopic nonlinear elastic model for granular media.

摘要 Abstract 作者及机构信息 黄择言 1. 北京地质学院理论力学教研室 Authors and contacts Huang Ze-yan 参考文献 [1] 施引文献

Hybrid isogeometric-based analysis and experimental investigation on the dynamic response characteristics of a clamped circular plate partially in contact with fluid

We consider the motion of a thin elastic plate with non-uniform thickness. The plate is either submerged and has some inclination with the vertical or is floating on the upper surface of the water. Green's function arising from the fourth-order boundary condition for the non-uniform plate (which we refer to as plate Green's function) is determined using two different methods in terms of the vibrating modes of the plate. These, in turn, are derived from the modes of a plate with constant thickness. The problem is finally reduced to a boundary integral equation involving the plate Green's function and the fundamental Green's function. This equation is hypersingular in the case of a submerged plate. A numerical solution to the integral equation is used to find results for elastic plates with variable thicknesses. The results are validated by comparing them with those of an elastic plate with uniform thickness. We also present simulations of the time-domain motion when the plate–fluid system is subject to an incident wave pulse using Fourier transform.

We consider the motion of a thin elastic plate with non-uniform thickness. The plate is either submerged and has some inclination with the vertical or is floating on the upper surface of the water. Green's function arising from the fourth-order boundary condition for the non-uniform plate (which we refer to as plate Green's function) is determined using two different methods in terms of the vibrating modes of the plate. These, in turn, are derived from the modes of a plate with constant thickness. The problem is finally reduced to a boundary integral equation involving the plate Green's function and the fundamental Green's function. This equation is hypersingular in the case of a submerged plate. A numerical solution to the integral equation is used to find results for elastic plates with variable thicknesses. The results are validated by comparing them with those of an elastic plate with uniform thickness. We also present simulations of the time-domain motion when the plate–fluid system is subject to an incident wave pulse using Fourier transform.