Clamped Thin Elastic Plate Research Articles

Bending analysis of simply supported and clamped thin elastic plates by using a modified version of the LMFS

The localized method of fundamental solutions is a recent domain-type meshless collocation method with the fundamental solutions of governing equations as the radial basis functions. This approach forms a sparse system matrix and has a higher efficiency than the traditional method of fundamental solutions. In this paper, a modified version of the localized method of fundamental solutions is developed for bending analysis of simply supported and clamped thin elastic plates. Some auxiliary nodes on the boundary are firstly introduced to provide additional weight coefficients, which contribute to the construction of the determined system of equations and avoid the over-determined equation system of the localized method of fundamental solutions for bending analysis of thin elastic plates. Several numerical experiments with simply supported or clamped boundary conditions are provided, and numerical results are in good agreement with the analytical or COMSOL Multiphysics solutions.

Modeling nonlinear acoustic landmine detection using a soil plate oscillator apparatus

In modeling nonlinear acoustic landmine detection, there will be a cylindrical drum-like buried target. This target, which has a clamped thin elastic plate, rigid sidewalls, and bottom plate, is buried in an open concrete “soil tank” containing dry sifted masonry sand. Subwoofers located 40 cm above the surface are driven with a swept tone from 50 to 450 Hz to generate airborne sound that couples into the sand exciting vibration in the buried target. A small geophone is used to measure the “ground” vibration profile across the buried target. Nonlinear tuning curves of vibration particle velocity vs. frequency are used to compare “over—off the target” results. The backbone curves (peak velocity vs. corresponding resonant frequency) are different in these cases—such that resonances due to soil alone or soil over a compliant buried target can be distinguished. Next, a soil plate oscillator SPO (consisting of two circular flanges sandwiching and clamping a thin circular elastic plate that supports a cylindrical level column of sand above the plate) is driven by airborne sound. Nonlinear tuning curve vibration at points across the sand are compared with the results in the “idealized” landmine detection experiment to develop a lumped element model of the system.

Free vibration analysis of an elliptical plate with eccentric elliptical cut-outs

The elaborated collocation multipole method is employed to obtain a semi-analytical solution, involving proper products of angular and radial Mathieu functions, for the free flexural vibrations of a fully clamped thin elastic plate of elliptical planform containing multiple elliptical cutouts of arbitrary size, location, and orientation. The problem boundary conditions are satisfied by uniformly collocating points on the boundaries, and exactly calculating the normal derivative of plate displacement at the collocation points through use of appropriate directional derivative in each coordinate system. The multipole expansion is truncated to yield a coupled algebraic linear system of equations that is then solved for the nontrivial eigensolutions. Extensive numerical simulations present the first three calculated natural frequencies and the associated deformed mode shapes of an elliptical plate with elliptical/circular cutouts, for a wide range of plate/cutout aspect ratios, and cutout location/orientation parameters. The accuracy of solutions is checked through appropriate convergence studies, and the validity of results is established with the aid of a commercial finite element package as well as by comparison with the data available in the existing literature.

Meshless LBIE formulations for simply supported and clamped plates under dynamic load

Simply supported and clamped thin elastic plates under dynamic loads are analyzed. Both harmonic and impact loads are considered. Viscous damping is taken into account. The governing partial differential equation (PDE) of fourth order is decomposed into two coupled PDEs of second order for the deflection and its Laplacian. Both equations contain time-dependent variables. The Laplace transform is used to eliminate the time dependence of the variables. Unknown Laplace transforms are computed from the local boundary integral equations. The meshless approximation based on the moving least square method is employed for the implementation. Time-dependent values are obtained by the Durbin inversion technique.

Meshless local boundary integral equation method for simply supported and clamped plates resting on elastic foundation

Simply supported and clamped thin elastic plates resting on a two-parameter foundation are analyzed in the paper. The governing partial differential equation of fourth order for a plate is decomposed into two coupled partial differential equations of second order. One of them is Poisson’s equation whereas the other one is Helmholtz’s equation. The local boundary integral equation method is used with meshless approximation for both the Poisson and the Helmholtz equation. The moving least square method is employed as the meshless approximation. Independent of the boundary conditions fictitious nodal unknowns used for the approximation of bending moments and deflections are always coupled in the resulting system of algebraic equations. The Winkler foundation model follows from the Pasternak model if the second parameter is equal to zero. Numerical results for a square plate with simply and/or clamped edges are presented to prove the efficiency of the proposed formulation.

Meshless formulations for simply supported and clamped plate problems

AbstractIn this paper, simply supported and clamped thin elastic plates are analysed. The biharmonic differential equation, representing the basis of the Kirchhoff theory, is decomposed into two Poisson equations. Local boundary integral equations are derived for this system of equations. The meshless approximation based on the moving least‐squares method is employed for the implementation. In the case of simply supported plates, it is sufficient to use the local boundary integral equations. For the case of clamped plates we propose to use a combination of the local boundary integral equations and the global ones. Then, two groups of nodal unknowns are computed separately. This leads to a reduced system of algebraic equations. Copyright © 2002 John Wiley & Sons, Ltd.

Nonlinear acoustic techniques for landmine detection: Experiments and theory, Part II

A nonlinear acoustic technique for detecting buried landmines has been suggested by Donskoy [SPIE Proc. 3392, 211 (1998); 3710, 239 (1999)] and verified in a number of experiments by Sabatier et al. [J. Acoust. Soc. Am. 110, 2757, 2758 (2001)]. Airborne sound involving two primary frequencies f1 and f2 undergo acoustic-to-seismic coupling which interacts with the compliant mine and soil to generate a difference frequency component that can effect the vibration velocity at the surface. An LDV profile across a buried mine at the difference frequency (chosen at a resonance) showed more relative sensitivity than a linear profile. A layer of soil in a large tube, terminated by an airbacked clamped thin elastic plate, is used to model nonlinear effects including Donskoy’s ‘‘bouncing’’ soil–mine interface. The frequency spectrum of modes of the plate’s response is investigated vs the amount of soil mass loading. In particular, the lowest mode frequency first decreases with loading and then increases. Linear response curves near resonance compare well with the difference frequency response as one primary is varied. [Work supported by the United States Army Communications-Electronics Command Night Vision and Electronic Sensors Directorate.]

Nonlinear acoustic techniques for landmine detection: Experiments and theory

A nonlinear acoustic technique for detecting buried landmines has been suggested by Donskoy [SPIE Proc. 3392, 211 (1998); 3710, 239 (1999)]. Airborne sound at two primary frequencies f1 and f2 undergo acoustic-to-seismic coupling and a superimposed ‘‘slow’’ compressional wave interacts with the compliant mine and soil. The nonlinear mechanism involves a simple model of the top surface of the mine–soil planar surface separating two elastic surfaces. During the compression phase of the wave, the surfaces stay together and then separate under the tensile phase due to a relatively high compliance of the mine. This ‘‘bouncing‘‘ soil–mine interface is thought to be a bimodular oscillator that is inherently nonlinear. Geophone measurements scanning the soil’s surface (at the difference frequency) profile the mine, but off the mine some nonlinearity exits. Amplitude-dependent frequency response curves for a harmonically driven mass-soil oscillator are used to find the nonlinearity of the soil acting as a ‘‘soft’’ spring. Finally, a layer of soil in a large tube, terminated by a thin elastic plate, is used to model the bimodular nonlinearity. [Work supported by the United States Army Communications-Electronics Command Night Vision and Electronic Sensors Directorate.]