Experimental eigenvalues and mode shapes for flat clamped plates

Experimental eigenvalues of both square and rectangular clamped flat plates were measured using digital spectrum analysis. Individual mode shapes were recorded experimentally using holographic interferometry. Plate spectra showing the first 35 modes of vibration for each of the square and rectangular plates were recorded, allowing the experimentally determined eigenvalues to be compared with published theoretical predictions. Over 25 modes for a square plate and 16 modes for a rectangular plate with aspect ratio of 2/3 were recorded holographically. Selected recorded mode shapes are compared with beam mode shapes as well as with modified Bolotin mode shapes, both of which are popular assumed mode shapes in current numerical techniques. It was found that both of these assumed mode shapes agree favorably with the experimental results. The beam mode shapes agree better in some modes; the modified Bolotin mode shapes agree more favorably in others.     

Plastic buckling analysis of thick plates using p-Ritz method

This paper is concerned with the application of the p-Ritz method for the plastic buckling analysis of thick plates. In order to allow for the effect of transverse shear deformation in thick plates, the Mindlin plate theory is adopted. The plastic buckling behaviour of the plate is captured by using the incremental and deformation theories of plasticity. The material property of the plate is assumed to obey the Ramberg–Osgood stress–strain relation. The p-Ritz method will be applied to obtain the governing eigenvalue equation for the plastic buckling analysis of uniformly stressed plates with edges defined by polynomial functions. In the p-Ritz method, the displacement functions of the plate are approximately represented by the product of mathematically complete two-dimensional polynomial functions and boundary equations raised to appropriate powers that ensure the satisfaction of the geometric boundary conditions. The validity, convergence and accuracy of the method were demonstrated for various plate shapes such as rectangular, triangular and elliptical shapes. A parametric study was also undertaken to study the plastic buckling behaviour and the effect of transverse shear deformation.     

Experimental eigenvalues and mode shapes for flat clamped plates

Experimental eigenvalues of both square and rectangular clamped flat plates were measured using digital spectrum analysis. Individual mode shapes were recorded experimentally using holographic interferometry. Plate spectra showing the first 35 modes of vibration for each of the square and rectangular piates were recorded, allowing the experimentally determined eigenvalues to be compared with published theoretical predictions. Over 25 modes for a square plate and 16 modes for a rectangular plate with aspect ratio of 2/3 were recorded holographically. Selected recorded mode shapes are compared with beam mode shapes as well as with modified Bolotin mode shapes, both of which are popular assumed mode shapes in current numerical techniques. It was found that both of these assumed mode shapes agree favorably with the experimental results. The beam mode shapes agree better in some modes; the modified Bolotin mode shapes agree more favorably in others.     

Erratum to: Vibration Analysis of Perforated Plates Using Time-Average Digital Speckle Pattern Interferometry

Digital Speckle Interferometry is a non invasive full-field coherent optical technique used in mechanical vibration measurement. In this research, it is used for tuning resonant frequencies of vibrating plates in order to investigate the dynamical behavior of perforated plates. The plate was excited to resonant vibration by a sinusoidal acoustic source. Fringe pattern produced during the time-average recording of the vibrating plate, for several resonant frequencies were registered. Results of plates fixed at one edge having internal holes and attached masses are presented. Experimental natural frequencies and modal shapes are compared to those obtained by an analytical approximate solution based on the Rayleigh–Ritz method with the use of orthogonal polynomials as coordinate function. A high degree of correlation between computational analysis and experimental results was observed, proving the potentiality of the optical technique as experimental validation of the numerical simulations.     

Vibration Analysis of Perforated Plates Using Time-Average Digital Speckle Pattern Interferometry

Digital Speckle Interferometry is a non invasive full-field coherent optical technique used in mechanical vibration measurement. In this research, it is used for tuning resonant frequencies of vibrating plates in order to investigate the dynamical behavior of perforated plates. The plate was excited to resonant vibration by a sinusoidal acoustic source. Fringe pattern produced during the time-average recording of the vibrating plate, for several resonant frequencies were registered. Results of plates fixed at one edge having internal holes and attached masses are presented. Experimental natural frequencies and modal shapes are compared to those obtained by an analytical approximate solution based on the Rayleigh–Ritz method with the use of orthogonal polynomials as coordinate function. A high degree of correlation between computational analysis and experimental results was observed, proving the potentiality of the optical technique as experimental validation of the numerical simulations.     

Deflection of sandwich plates in terms of corresponding Kirchhoff plate solutions

Summary This study presents exact relationships between the deflections of isotropic sandwich plates and their corresponding Kirchhoff plates. The governing equilibrium equations for the sandwich plates are derived on the basis of the Reissner-Mindlin shear deformation plate theory. The considered plates are either (i) simply supported, of general polygonal shape and under any transverse loading condition or (ii) simply supported and clamped circular plates under axisymmetric loading. As the relationships are exact under the assumptions used in the plate theories, one may obtain exact deflection solutions of sandwich plates if the Kirchhoff plate solutions are exact. The relationships should also be useful for the development of approximate formulas for plates with other shapes, boundary and loading conditions, and may serve to check numerical deflection values computed from sandwich plate analysis software.     

Experimental determination of transverse vibration modes of thin I-shaped plates

The natural frequencies and the corresponding mode shapes of clamped thin I-shaped plates were experimentally determined by time-averaged holographic interferometry. The test plate was shaped as shown in Fig. 1, and eight kinds of plates were tested by taking a dimensionless length parameter ξ as a parameter. The natural-vibration modes exceeding 200 were identified using the real-time method. The corresponding natural frequencies ranged from 172 to 5606 Hz. In addition, in the case of a rectangular plate, the experimental results were shown to be good agreement with the theoretical ones.     

Experimental investigation on a turbulence generation system with high-blockage plates

An experimental study was conducted to develop and characterize systematically a new turbulence generator system to yield large turbulent Reynolds numbers in a compact configuration. The effect of the geometric parameters of two families of high-blockage plates on the resulting turbulent flow field was systematically studied: one series of plates was characterized by the number and distribution of circular openings; a second series had non-circular opening(s) with different shapes, distribution and position of the opening(s). The plates were placed upstream of a contoured contraction and the near field at the centerline of the resulting turbulent free jet was characterized by hot-wire anemometry in terms of mean axial velocity, turbulence intensity, turbulence length scales and corresponding Reynolds numbers. The plate with a central, non-circular opening produced the best compromise of highest turbulence levels along with excellent uniformity in average velocity and turbulence intensity, as evidenced by scan in the transverse direction. It appears to be the most promising one. By comparison with more traditional approaches to turbulence generation, we increased the turbulent Reynolds numbers based on the integral length scale to values on the order of 1000, which was one of the design objectives. Other plate geometries also yielded intense turbulence, but, in some cases, exhibited spurious frequency peaks in their power spectrum. The turbulent generation approach is to be adapted to combustion studies to reproduce conditions typical of practical system in relatively small experimental set-ups that are well-suited for bench-top experiments.     

Elasticity solutions for free vibrations of annular plates from three-dimensional analysis

This article, examines the vibrational characteristics of annular plates by using the three-dimensional elasticity theory. It aims to raise the quality of the investigation beyond that provided by the two-dimensional plate theories by resorting to a full three-dimensional analysis. A polynomials–Ritz model based on sets of orthogonally generated polynomial functions to approximate the spatial displacements of the plates in cylindrical polar coordinates is presented. The model is then used to extract the full vibration spectrum of natural frequencies and mode shapes. The vibration responses due to the variations of boundary conditions and thickness are investigated. Frequency parameters and three-dimensional deformed mode shapes are presented in vivid graphical forms. The accuracy of the method is validated through appropriate convergence and comparison studies.     

Wave propagation in linear viscoelastic plates of various thicknesses

Experiments were conducted to observe plate waves travelling in polymethylmethacrylate, polyethylene, and glass plates of various thicknesses at several distances of wave travel, when the plates have been subjected to a transverse surface impact by a steel ball. It has been shown that if Poisson's ratio is taken to be a real constant, the shapes of the plate waves can be calculated when the initial disturbance and the viscoelastic properties of the medium are known. It was found possible to predict quite well the shape of the plate waves observed in the experiments. The plate waves were shown, both theoretically and experimentally, to be of a different character than those observed on the free surfaces of large block materials. The effect of the height of ball drop and contacting ball radius on pulse shape was noted.     

Steady-state motion of a plate with a discontinuous mass

An “exact” solution employing the normal mode approach is presented for the steady-state motion of a plate of arbitrary shape and with arbitrary boundary conditions that is provided with a discontinuous mass. The plate is viscously damped and the excitation is furnished in the form of distributed and/or concentrated loads that vary sinusoidally. The rigid mass, which may act as a vibration damper, can be applied to any point in the structure. Experimental studies with a mechanical model corroborate the theoretical results. Results of the analysis are applied to cantilever and simply-supported square plates subjected to base excitation and discrete loads. The effects of all system parameters, including damper location and mass ratio, mode shapes, viscous damping, and excitation frequency are determined. It is found that the system under consideration is an efficient device for reducing the vibration of plates, particularly cantilevered ones.     

Simultaneous Measurement of Plate Natural Frequencies and Vibration Mode Shapes Using ESPI

The natural frequencies and vibration mode shapes of flat plates are simultaneously measured using ESPI. The method involves measuring the surface shape of a vibrating plate at high frame rate using a modified Michelson interferometer and high-speed camera. The vibration is excited here by impact; white (random) noise could alternatively be used. Fourier analysis of the acquired data gives the natural frequencies and associated mode shapes. The analytical procedure used has the advantage that it simultaneously identifies full-field quantitative images of all vibration modes with frequencies up to half the sampling frequency. In comparison, the ESPI time-averaging and the traditional Chladni methods both require that the plate be excited at each natural frequency to allow separate qualitative measurements of the associated mode shapes. The Instrumented Hammer method and Laser Doppler Vibrometry give quantitative measurements but require sequential sampling of individual points on the test surface to provide full-field results. Example ESPI measurements are presented to illustrate the use and capabilities of the proposed plate natural frequency and mode shape measurement method.     

Perturbation methods for the analysis of the dynamic behavior of damaged plates

This paper presents an analytical method for the analysis of the dynamic behavior of damaged plates. The proposed approach allows the derivation of mode shapes and corresponding curvature modes for plates with various kinds of defects. Damage is modeled as a localized reduction in the plate thickness. Both point and line defects are considered to model notches or line cracks and delaminations in the plate. Small thickness reductions are considered so that the dynamic behavior of the damage plate can be analyzed through perturbations with respect to the undamaged modes. Results are presented to demonstrate the sensitivity of the curvature modes with respect to the considered low damage levels. Also, the curvature modes are used for the estimation of the strain energy of the plate and for the formulation of a damage index which can be used to provide damage location and extent information.     

A comprehensive analytical solution for free vibration of rectangular plates with classical edge coditions: Experimental verification

The problem of obtaining free vibration frequencies and mode shapes of rectangular plates resting on combinations of classical (i.e., clamped, simply supported, or free) edge supports is one that has been investigated for more than one hundred years. More recently, the superposition method has been developed for obtaining accurate analytical-type solutions for this family of problems. The object of this paper is to report on the results of numerous experimental tests carefully performed in order to verify the superposition method and associated computer software. Experimental and computed results are compared for a wide range of plate configurations. Very good agreement between theory and experiment has been obtained with regard to both plate natural frequencies and mode shapes. It is concluded that this computational procedure constitutes a powerful new tool for analysis of rectangular plate vibration problems.     

复合板旋转约束刚度推导及箱型结构屈曲分析

基于结构稳定性理论,推导出正交各向异性纤维树脂增强复合材料箱型结构各板连接处旋转约束刚度的近似表达式．与以往的经验公式相比,考虑了材料的正交各向异性和截面尺寸的影响．通过对箱型结构的正交各向异性比、截面属性以及纵横比对旋转约束刚度的影响进行参数研究,验证了旋转约束刚度新公式的精确性和适用范围．通过比较采用不同近似表达式得到的结果与有限元计算结果的对比,表明本文的旋转约束刚度新公式可以更精确地用来计算箱型结构的临界屈曲载荷．     

Viscoelastic shear horizontal wave in graded and layered plates

In this paper, viscoelastic shear horizontal (SH) wave propagation in functionally graded material (FGM) plates and laminated plates are investigated. The controlling differential equation in terms of displacements is deduced based on the Kelvin–Voigt viscoelastic theory. The SH wave characteristics is controlled by two elastic constants and their corresponding viscous coefficients. By the Legendre polynomial series method, the asymptotic solutions are obtained. In order to verify the validity of the method, a homogeneous plate is calculated to make a comparison with available data. Through three different graded plates, the influences of gradient shapes on dispersion and attenuation are discussed. The viscous effects on the displacement and stress shapes are illustrated. The different boundary conditions are analyzed. The influential factors of the viscous effect are analyzed. Finally, two multilayered (two layer and five layer) viscoelastic plates that are composed of the same material volume fraction are calculated to show their differences from the graded plate.     

Nonlinear behavior of circular plates with work hardening

Tests were performed on two simply supported plates of aluminum alloy 2024-0, under a central concentrated load, with peak deflections up to 2.6 times the thickness. The load was provided by a small-diameter hard-steel rod. The plates had diameter-to-thickness ratios (D/h) of 20 and 41. Measurements were made of load, deflections and strains; membrane and bending strains were calculated from the test data. The test data are presented in comparison with theoretical predictions generated by the Grumman-developed finiteelement-computer code PLANS, which includes material and geometric nonlinearities. The theoretical prediction was excellent for deflections, and generally good for strains, when the central force was represented by a line load around the loading rod's contact circle. Using a uniform pressure as the central force caused the theory to slightly overpredict the peak deflections and greatly overpredict the peak strains at the plate center. The plates exhibited initial loss of stiffness under the plastic-bending behavior, followed by a restiffening when the large deflections caused a rapidly increasing membrane action that provided much additional resistance to the applied load.     

Three-dimensional vibration analysis of thick rectangular plates using Chebyshev polynomial and Ritz method

This paper describes a method for free vibration analysis of rectangular plates with any thicknesses, which range from thin, moderately thick to very thick plates. It utilises admissible functions comprising the Chebyshev polynomials multiplied by a boundary function. The analysis is based on a linear, small-strain, three-dimensional elasticity theory. The proposed technique yields very accurate natural frequencies and mode shapes of rectangular plates with arbitrary boundary conditions. A very simple and general programme has been compiled for the purpose. For a plate with geometric symmetry, the vibration modes can be classified into symmetric and antisymmetric ones in that direction. In such a case, the computational cost can be greatly reduced while maintaining the same level of accuracy. Convergence studies and comparison have been carried out taking square plates with four simply-supported edges as examples. It is shown that the present method enables rapid convergence, stable numerical operation and very high computational accuracy. Parametric investigations on the vibration behaviour of rectangular plates with four clamped edges have also been performed in detail, with respect to different thickness-side ratios, aspect ratios and Poisson’s ratios. These results may serve as benchmark solutions for validating approximate two-dimensional theories and new computational techniques in future.     

Tensile and compressive buckling of plates weakened by cracks

Plates are susceptible to buckling under compression when the thickness dimension becomes sufficiently small. Such mode of structure failure can prevail even if the plates were extended in tension. Wribkling of stretched thin sheets is a commonly observed phenomenon that leads to complex deflection patterns, particularly in regions close to crack-like imperfections. Predictions of the buckled displacement modes for plates weakened by cracks will be made on the basis of a theory formulated by application of variational calculus. Finite element method is used such that defects of other shapes can also be analyzed. Various buckled displacement modes of a center-cracked plate are determined and displayed graphically. The critical buckling loads are found to decrease with increasing crack size. Moreover, local wrinkling of the plate surface becomes less pronounced for the higher buckling modes. The method of solution applies equally well to plates that are initially curved.