Vibration of triangular plates of variable thickness

The problem of the natural frequencies and mode shapes of cantilevered triangular plates with variable thickness and arbitrary planform is solved using the finite element technique. This is done for various combinations of four non-dimensional geometric parameters, namely, the aspect ratio, the two thickness ratios along the two coordinate directions and the sweepback angle. The frequencies for the various cases are tabulated and a few typical mode shapes have been presented graphically.     

Three-dimensional vibration analysis of thick, circular and annular plates with nonlinear thickness variation

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, circular and annular plates with nonlinear thickness variation along the radial direction. Unlike conventional plate theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. Displacement components u_s, u_z, and u_θ in the radial, thickness, and circumferential directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the s and z directions. Potential (strain) and kinetic energies of the plates are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the plates. Numerical results are presented for completely free, annular and circular plates with uniform, linear, and quadratic variations in thickness. Comparisons are also made between results obtained from the present 3-D and previously published thin plate (2-D) data.     

Frequency response of shell-plate combinations

Natural frequencies of cylindrical shells with a circular plate attached at arbitrary locations are determined for various boundary conditions and L/D ratios. The semi-analytical finite element method is used for the analysis. A conical shell element with four degrees of freedom per node and two nodes per element is used. For clamped-clamped and simply-supported boundary conditions the plate is attached at the center of the shell. For a clamped-free boundary condition the plate is at the free end of the shell. The effects of plate thickness and L/D ratio of the shell on the frequencies of the shell-plate combination are investigated.     

Post-buckling behaviour of cylindrically orthotropic annular plates with uniform internal radial load

A finite element formulation for the study of the post-buckling behaviour of cylindrically orthotropic annular plates with uniform internal radial load is presented in this paper. The results for radial load ratios are presented in the form of empirical formulae in terms of the central deflection of the annulus to the thickness ratios of the plates for the first time in the literature for various values of the orthotropic parameters and radii ratios.     

Natural frequencies of vibration of cantilever sandwich beams

Theoretical and experimental studies were made in obtaining the natural frequencies of cantilever sandwich beams subjected to only gravity forces. The method of minimizing the total energy of the system was used for determining the frequencies. A vibration system made by Unholtz-Dickie was utilized to set the beam in vibration. Resonance occurred when the frequency of the shaker coincided with the natural frequency of the beam. The resonance frequencies were measured by transducers mounted at various locations on the beam. A total of sixteen beams of various lengths, thickness and core density were tested.

It was found that the natural frequency of a cantilever sandwich beam depends largely upon the thickness, length, core density and stiffness of the beam. In addition, the natural frequency has a nonlinear variation with the mode and for any particular mode, the value of the frequency increases as the length of the beam decreases.

Design factors were developed based upon the ratios of the theoretical frequencies of homogeneous beams having the same thicknesses and stiffnesses of that of sandwich beams and of the frequencies experimentally determined for similar sandwich beams.     

Post-buckling behaviour of cylindrically orthotropic annular plates

A finite element formulation for the study of the post-buckling behaviour of cylindrically orthotropic annular plates is presented in detail. The results for radial load ratios are presented in the form of empirical formulae in terms of the deflection at the inner edge to the thickness ratios of the plates for the first time in the literature for various values of the orthotropic parameters and radii ratios.     

Free vibration of laminated plates using a finite strip method based on a higher-order plate theory

The finite strip method based on the higher-order plate theory is developed for determining the natural frequencies of laminated plates. This method can accurately predict the through thickness effect of transverse shear deformation. Furthermore, only a few degress of freedom are required in the finite strip method. Some numerical results for various span-to-thickness ratios, material properties and stack sequences are presented for illustrative purposes. The present model provides a better way to obtain more accurate natural frequency results.     

Free vibration analysis of complete paraboloidal shells of revolution with variable thickness and solid paraboloids from a three-dimensional theory

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of solid paraboloids and complete (that is, without a top opening) paraboloidal shells of revolution with variable wall thickness. Unlike conventional shell theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. The ends of the shell may be free or may be subjected to any degree of constraint. Displacement components u_r, u_θ and u_z in the radial, circumferential, and axial directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the paraboloidal shells of revolution are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the complete, shallow and deep paraboloidal shells of revolution with variable thickness. Numerical results are presented for a variety of paraboloidal shells having uniform or variable thickness, and being either shallow or deep. Frequencies for five solid paraboloids of different depth are also given. Comparisons are made between the frequencies from the present 3-D Ritz method and a 2-D thin shell theory.     

Comparison of elasticity theory and shell theory solutions

The axisymmetric bending of cylinder-cone shell junction has been analysed by the finite element method using three dimensional elasticity theory. Numerical results have been worked out for three shells having cylinder diameter to thickness ratios of 41, 21 and 13. These results have been compared with the results obtained by the classical thin shell theory of Sanders. The validity of the assumptions made in the classical thin shell theory is discussed. The nature of singularity at the re-entrant corner has been studied.     

Nonlinear axisymmetric static and transient analysis of orthotropic thin tapered circular plates

This study deals with the geometrically nonlinear axisymmetric static and transient analysis of cylindrically orthotropic elastic thin tapered circular plates subjected to uniformly distributed and discrete central loads. Differential equations in terms of transverse displacement w and stress function ψ have been employed. The displacement w and stress function ψ are expanded in finite power series. The orthogonal point collocation method in space domain and Newmark-β scheme in time domain have been used. Step function dynamic loads are considered. Static and dynamic results have been presented for isotropic and orthotropic immovable clamped and simply supported plates with linearly varying thickness for three values of taper ratios and the effect of varying thickness has been investigated. A simple approximate method is used to predict the maximum dynamic response to step load from the results for static loads and is found to yield sufficiently accurate results.     

Finite element free flexural vibration analysis of plates having various shapes and varying rigidities

Natural frequencies of skew, curved and tapered plates have been determined using the isoparametric quadratic plate bending element. Plates having linearly varying thickness in one direction and also those having parabolically varying thickness in orthogonal directions have been analysed. A simple stepped plate approach for a linearly tapered plate has also been considered to compare the results. Two approaches have been adopted for the solution of skew plates. In one case, the boundary conditions have been exactly satisfied by transforming the rotational displacements at the boundary nodes along and normal to the edge. In the other case, only the vertical deflection was locked. A curved plate has also been analysed by the first method. Results thus obtained from both cases have been compared.     

Application of the collocation method to vibration analysis of rectangular mindlin plates

The flexural vibration analysis of rectangular Mindlin plates using the collocation method is described. The results obtained by the present method are compared with published results for plates with uniform thickness and two opposite edges simply supported. The comparison shows that the method yields very good results with a relatively small number of collocation points, and that estimates for the higher modes can be obtained without any difficulties. Furthermore, the method is applied to plates with linearly varying thickness, and new findings are presented for the frequencies of plates.     

Post-buckling of linearly tapered,moderately thick circular plates by finite element method

The post-buckling behaviour of linearly tapered, moderately thick circular plates subjected to an external compressive load is studied in this paper through a finite element formulation. The results in the form of an empirical formula for the radial load ratios are presented for various values of the taper parameter and central thickness to radius ratio. Both simply supported and clamped boundary conditions are considered in the present study.     

Vibrations of skew plates immersed in water

Vibrations of skew plates immersed in water have been studied in the present work. Two skew prism fluid elements have been developed for the above study. As there is no information available on them, a few experiments have been conducted to measure the frequencies of vibration of these plates immersed in water. The results of both the theoretical and experimental investigations have been compared and found to be agreeing reasonably well. The general findings of these plates regarding their aspect ratios, thickness ratios and boundary conditions are discussed. In addition, the influence of skew angles on the vibration of skew plates have been investigated. It has been noticed that the effect of added mass is found to decrease with the increase of skew angle.     

Free vibration analysis of symmetric laminated composite plates by FSDT and radial basis functions

In this paper, we use the first-order shear deformation theory in the multiquadric radial basis function (MQRBF) procedure for predicting the free vibration behavior of moderately thick symmetrically laminated composite plates. The transverse deflection and two rotations of the laminate are independently approximated with the MQRBF approximation. The natural frequencies of vibration are computed for various laminated plates and compared with some available published results. Through numerical experiments, the capability and efficiency of the MQRBF method for eigenvalue problems are demonstrated, and the numerical accuracy and convergence are thoughtfully examined.     

A bulk silicon dissolved wafer process for microelectromechanicaldevices

A single-sided bulk silicon dissolved wafer process that has been used to fabricate several different micromechanical structures is described. It involves the simultaneous processing of a glass wafer and a silicon wafer, which are eventually bonded together electrostatically. The silicon wafer is then dissolved to leave heavily boron doped devices attached to the glass substrate. Overhanging features can be fabricated without additional masking steps. It is also possible to fabricate elements with thickness-to-width aspect ratios in excess of 10:1. Measurements of various kinds of laterally driven comb structures processed in this manner, some of which are intended for application in a scanning thermal profilometer, are described. They comprise shuttle masses supported by beams that are 160-360 μm long, 1-3 μm wide, and 3-10 μm thick. Some of the shuttles are mounted with probes that overhang the edge of the die by 250 μm. Resonant frequencies from 18 to 100 kHz and peak-to-peak displacements up to 18 μm have been measured     

Automatic design of tapered member gabled frames

A program for calculating member sizes to yield a minimum with grabled frame with tapered members has been developed. The design produced satisfies in all respects the requirements of Appendix D to the 1969 AISC specification which governs the design of tapered members. The frame design is symmetrical about the vertical centerline, although the loading need not by symmetrical. The rafter can have no, one or two changes of taper within its length, at the designer's option. The matrix force method is used to do the analyses necessary in the design process. The problem is solved by finding a design which minimizes frame weight subject to constraints imposed by the design specification,such as maximum stress and maximum width/thickness ratios, to minimum or interior penalty function approach using the variable metric method of Davidon, Fletcher and Powell. A number of example penalty function design are given to show the versatility of the technique, as well as demonstrating the effect parmeters such as purlin spacing, member length, rafter slope and member depth to width ratio have on resulting designs.     

Axisymmetric vibration of layered orthotropic spherical shells of variable thickness

Axisymmetric free vibration of thick orthotropic spherical shells with linearly varying thickness along the meridian is analysed. Both deep and shallow shells are considered for the analysis. The effect of thickness variation and lay-up are considered. The results are presented for clamped and hinged boundary conditions. A thick shell finite element is used for the analysis. It is observed that the thickness variation and lay-up have a pronounced effect on the natural frequencies and a considerable increase of the natural frequencies can be achieved by selecting a proper combination of lay-up and thickness variation.     

Multi-class fault diagnosis of induction motor using Hilbert and Wavelet Transform

The information extraction capability of two widely used signal processing tools, Hilbert Transform (HT) and Wavelet Transform (WT), is investigated to develop a multi-class fault diagnosis scheme for induction motor using radial vibration signals. The vibration signals are associated with unique predominant frequency components and instantaneous amplitudes depending on the motor condition. Using good systematic and analytical approach this fault frequencies can be identified. However, some faults either electrical or mechanical in nature are associated with same or similar vibration frequencies leading to erroneous conclusions. Genetic Algorithm (GA) is proposed and used successfully to find the most relevant fault frequencies in radial (vertical) frame vibration signal which can be used to diagnose the induction motor faults very effectively even in the presence of noise. The information obtained by Continuous Wavelet Transform (CWT) was found to be highly redundant compared to HT and thus by selecting the most relevant features using GA, the fault classification accuracy has considerably improved especially for CWT. Almost similar fault frequencies were found using CWT + GA and HT + GA for radial vibration signal.     

模糊化神经网络在摄像机标定中的应用

摄像机标定是精确的视觉系统的前提。该文提出了基于双目视觉的一种新型模糊化神经网络摄像机标定法。神经网络已经成功的运用于无参摄像机标定技术当中。由于镜头的畸变主要由径向畸变引起,且图像中点总的径向畸变正比于该点到图像中心距离的平方,所以通过对神经网络的输入层进行径向模糊化,可以进一步精确标定结果。通过与普通神经网络摄像机标定法、分割区间双神经网络摄像机标定法的仿真比较,证明了模糊化神经网络摄像机标定法的有效性。