Applications of Holography to Dynamics: High-Frequency Vibrations of Plates

1970 ◽  
Vol 37 (4) ◽  
pp. 1083-1090 ◽  
Author(s):  
R. Aprahamian ◽  
D. A. Evensen
Keyword(s):  
High Frequency ◽  
Plate Theory ◽  
Mode Shapes ◽  
Vibration Modes ◽  
Classical Plate Theory ◽  
Simply Supported ◽  
Transverse Vibrations ◽  
Shear Effects ◽  
High Frequency Vibrations ◽  
Very High

The techniques of holographic interferometry are applied to study the high-frequency transverse vibrations of a simply supported rectangular plate. Over 110 vibration modes were identified using stored beam holography, at frequencies ranging from 162 cps to 20,000 cps. In addition, three very high modes were identified at frequencies up to 76.8 kcps. The corresponding modal numbers were m = 17, n = 31, for the highest mode identified. Time average holograms were made of these and other modes, and photographs made from the holograms are included herein. The experimental mode shapes and frequencies were generally in agreement with classical plate theory, except for the three highest modes. The latter agreed with Mindlin’s plate theory, which includes rotary inertia and shear effects. Other applications of holography to dynamics are briefly discussed.

scholarly journals Buckling and Vibration of Non-Homogeneous Rectangular Plates Subjected to Linearly Varying In-Plane Force

2013 ◽  
Vol 20 (5) ◽  
pp. 879-894 ◽  
Author(s):  
Roshan Lal ◽  
Renu Saini
Keyword(s):  
Differential Equation ◽  
Plate Theory ◽  
Differential Quadrature Method ◽  
Three Dimensional ◽  
Axial Direction ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Plate Material ◽  
Classical Plate Theory ◽  
Simply Supported

The present work analyses the buckling and vibration behaviour of non-homogeneous rectangular plates of uniform thickness on the basis of classical plate theory when the two opposite edges are simply supported and are subjected to linearly varying in-plane force. For non-homogeneity of the plate material it is assumed that young's modulus and density of the plate material vary exponentially along axial direction. The governing partial differential equation of motion of such plates has been reduced to an ordinary differential equation using the sine function for mode shapes between the simply supported edges. This resulting equation has been solved numerically employing differential quadrature method for three different combinations of clamped, simply supported and free boundary conditions at the other two edges. The effect of various parameters has been studied on the natural frequencies for the first three modes of vibration. Critical buckling loads have been computed. Three dimensional mode shapes have been presented. Comparison has been made with the known results.

Buckling and Vibration of Orthotropic Nonhomogeneous Rectangular Plates With Bilinear Thickness Variation

2011 ◽  
Vol 78 (6) ◽  
Author(s):  
Yajuvindra Kumar ◽  
R. Lal
Keyword(s):  
Orthogonal Polynomials ◽  
Plate Theory ◽  
Ritz Method ◽  
Cartesian Product ◽  
Mode Shapes ◽  
Thickness Variation ◽  
Rectangular Plates ◽  
Two Dimensional ◽  
Classical Plate Theory ◽  
Simply Supported

An analysis and numerical results are presented for buckling and transverse vibration of orthotropic nonhomogeneous rectangular plates of variable thickness using two dimensional boundary characteristic orthogonal polynomials in the Rayleigh–Ritz method on the basis of classical plate theory when uniformly distributed in-plane loading is acting at two opposite edges clamped/simply supported. The Gram–Schmidt process has been used to generate orthogonal polynomials. The nonhomogeneity of the plate is assumed to arise due to linear variations in elastic properties and density of the plate material with the in-plane coordinates. The two dimensional thickness variation is taken as the Cartesian product of linear variations along the two concurrent edges of the plate. Effect of various plate parameters such as nonhomogeneity parameters, aspect ratio together with thickness variation, and in-plane load on the natural frequencies has been illustrated for the first three modes of vibration for four different combinations of clamped, simply supported, and free edges correct to four decimal places. Three dimensional mode shapes for a specified plate for all the four boundary conditions have been plotted. By allowing the frequency to approach zero, the critical buckling loads in compression for various values of plate parameters have been computed correct to six significant digits. A comparison of results with those available in the literature has been presented.

Mode shapes and frequencies of thin rectangular plates with arbitrarily varying non-homogeneity along two concurrent edges

2016 ◽  
Vol 23 (17) ◽  
pp. 2841-2865 ◽  
Author(s):  
Roshan Lal ◽  
Renu Saini
Keyword(s):  
Eigenvalue Problems ◽  
Plate Theory ◽  
Differential Quadrature Method ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Simply Supported ◽  
Transverse Vibrations ◽  
Modes Of Vibration ◽  
Generalized Differential

Analysis and numerical results are presented for free transverse vibrations of isotropic rectangular plates having arbitrarily varying non-homogeneity with the in-plane coordinates along the two concurrent edges on the basis of Kirchhoff plate theory. For the non-homogeneity, a general type of variation for Young’s modulus and density of the plate material has been assumed. Generalized differential quadrature method has been used to obtain the eigenvalue problem for such model of plates for four different combinations of boundary conditions at the edges namely, (i) fully clamped, (ii) two opposite edges are clamped and other two are simply supported, (iii) two opposite edges are clamped and other two are free, and (iv) two opposite edges are simply supported and other two are free. By solving these eigenvalue problems using software MATLAB, the lowest three eigenvalues have been reported as the first three natural frequencies for the first three modes of vibration. The effect of various plate parameters on the vibration characteristics has been analysed. Three dimensional mode shapes have been plotted. A comparison of results with those available in literature has been presented.

Applications of Holography to Dynamics: High-Frequency Vibrations of Beams

1970 ◽  
Vol 37 (2) ◽  
pp. 287-291 ◽  
Author(s):  
R. Aprahamian ◽  
D. A. Evensen
Keyword(s):  
Cantilever Beam ◽  
Timoshenko Beam ◽  
Holographic Interferometry ◽  
High Frequency ◽  
Beam Theory ◽  
Mode Shapes ◽  
Resonant Frequencies ◽  
Transverse Vibrations ◽  
High Frequency Vibrations ◽  
Time Average

The relatively new experimental technique of holographic interferometry is described, and time-average holography is discussed. Time-average holography has been applied to study high-frequency transverse vibrations of a uniform cantilever beam. Modes from the 5th through the 34th were identified and recorded on holograms, the corresponding resonant frequencies ranged from 500–25,665 cps. In addition, the 77th mode was recorded at 99,395 cps, which demonstrates that the technique is workable at frequencies on the order of 100 kc. The experimental mode shapes and frequencies show good correlation with the Timoshenko beam theory. Other applications of holography to dynamics are briefly discussed.

Determination of vibration of single-layered graphene sheets using nonlocal modified couple stress theory

2021 ◽  
pp. 2250011
Author(s):  
F. Attar ◽  
R. Khordad ◽  
A. Zarifi
Keyword(s):  
Couple Stress ◽  
Plate Theory ◽  
Couple Stress Theory ◽  
Nonlocal Parameter ◽  
Modified Couple Stress Theory ◽  
Length Scale ◽  
Vibration Modes ◽  
Classical Plate Theory ◽  
Stress Theory ◽  
Modified Couple Stress

The free vibration of single-layered graphene sheet (SLGS) has been studied by nonlocal modified couple stress theory (NMCS), analytically. Governing equation of motion for SLGS is obtained via thin plate theory in conjunction with Hamilton’s principle for two cases: (1) using nonlocal parameter only for stress tensor, (2) using nonlocal parameter for both stress and couple stress tensors. Navier’s approach has been used to solve the governing equations for simply supported boundary conditions. It is found that the frequency ratios decrease with increasing nonlocal parameter and also with enhancing vibration modes, but increase with raising length scale parameter. The nonlocal and length scale parameters are more prominent in higher vibration modes. The obtained results have been compared with the previous studies obtained by using classical plate theory, the modified couple stress theory and nonlocal elasticity theory, separately.

Effect of Shear Deformations on the Bending of Rectangular Plates

1960 ◽  
Vol 27 (1) ◽  
pp. 54-58 ◽  
Author(s):  
V. L. Salerno ◽  
M. A. Goldberg
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Plate Theory ◽  
Order Partial Differential Equation ◽  
Rectangular Plates ◽  
Partial Differential ◽  
Classical Plate Theory ◽  
Simply Supported ◽  
Wide Range

The three partial differential equations derived by Dr. E. Reissner2, 3 have been reduced to a fourth-order partial differential equation resembling that of the classical plate theory and to a second-order differential equation for determining a stress function. The general solution for the two partial differential equations has been applied to a simply supported plate with a constant load p and to a plate with two opposite edges simply supported and the other two edges free. Numerical calculations have been made for the simply supported plate and the results compared with those of classical theory. The calculations for a wide range of parameters indicate that the deviation is small.

High-frequency vibrations of circular and annular plates with the Mindlin plate theory

2020 ◽  
Vol 90 (5) ◽  
pp. 1025-1038
Author(s):  
Hui Chen ◽  
Rongxing Wu ◽  
Longtao Xie ◽  
Jianke Du ◽  
Lijun Yi ◽  
...  
Keyword(s):  
High Frequency ◽  
Plate Theory ◽  
Mindlin Plate ◽  
Annular Plates ◽  
Mindlin Plate Theory ◽  
High Frequency Vibrations

scholarly journals NEMS Resonators for Detection of Chemical Warfare Agents Based on Graphene Sheet

2019 ◽  
Vol 2019 ◽  
pp. 1-23
Author(s):  
Nikola Anđelić ◽  
Zlatan Car ◽  
Marko Čanađija
Keyword(s):  
Graphene Sheet ◽  
Plate Theory ◽  
Chemical Warfare Agents ◽  
Chemical Warfare ◽  
Single Layer Graphene ◽  
Mode Shapes ◽  
Graphene Sheets ◽  
Classical Plate Theory ◽  
Layer Graphene ◽  
Warfare Agents

Graphene sheets are the basis of nanoelectromechanical (NEMS) mass resonators that are in recent years experimentally used for detection of specific gas molecules in air. The aim of this paper is to theoretically investigate the possibility of application of graphene sheets in detection of Chemical Warfare Agents (CWA’s). By application of the nonlocal theory of elasticity and classical plate theory, equations that describe natural vibrations of the simply supported single-layer graphene sheet (SLGS) and double-layer graphene sheet (DLGS) were derived. In order to detect attached CWA molecule, the frequency shift method was applied. The results indicate that the proposed methodology could be successfully implemented in order to detect an attached CWA molecule. However, only specific mode shapes could be employed for such detection.

Effect of Pasternak Foundation on Axisymmetric Vibration of Polar Orthotropic Annular Plates of Varying Thickness

2010 ◽  
Vol 132 (4) ◽  
Author(s):  
Seema Sharma ◽  
U. S. Gupta ◽  
R. Lal
Keyword(s):  
Plate Theory ◽  
Mode Shapes ◽  
Thickness Variation ◽  
Axisymmetric Vibration ◽  
Classical Plate Theory ◽  
Energy Principle ◽  
Annular Plates ◽  
Edge Conditions ◽  
Modes Of Vibration ◽  
Polar Orthotropic

Free axisymmetric vibrations of polar orthotropic annular plates of variable thickness resting on a Pasternak-type elastic foundation have been studied based on the classical plate theory. Hamilton’s energy principle has been used to derive the governing differential equation of motion. Frequency equations for an annular plate for two different combinations of edge conditions have been obtained employing Chebyshev collocation technique. Numerical results thus obtained have been presented in the form of tables and graphs. The effect of foundation parameter and thickness variation together with various plate parameters such as rigidity ratio, radius ratio, and taper parameter on natural frequencies has been investigated for the first three modes of vibration. Mode shapes for specified plates have been presented. A close agreement of results with those available in the literature shows the versatility of the present technique.

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