Numerical solution for frequencies and mode shapes of elliptical plate half of whose boundary is simply supported and the rest free

Free transverse vibrations of elliptical and circular plates of variable thickness with half of the boundary is simply supported and the rest is completely free have been computed till the first four frequencies converge to at least four significant digits. The thickness of the plate is taken varying linearly with space coordinates. Computations have been carried out by using the famous Rayleigh–Ritz method. Results are reported for various values of the taper parameters. Almost all the results reported here are entirely new. A table has been prepared to indicate the trend of convergence of the results with increasing the order of approximation. Comparison has been made with known results in special cases. For the sake of completeness, results for the cases when the entire boundary is simply supported or completely free are also reported. Three-dimensional mode shapes and the associated contour lines have been plotted in some selected cases using tools of Computer Graphics and Turbo C++.     

Free vibration analysis of point-supported plates by vibration testing technique

Sweep sine-wave testing is applied for the identification of dynamic characteristics of a four-point-supported square plate with free edges. The idea behind the method is to make use of free vibration time-response data such as acceleration to determine the natural frequencies and associated mode shapes. To compare with the theoretical results, detailed experimental results have been obtained for the various support locations lying at specified positions along the plate diagonals. The natural frequencies and associated mode shapes for the first five modes have been predicted, and the variation of natural frequencies with various support positions have been analyzed for the first symmetric and antisymmetric modes. It is found that the experimental results are generally in reasonable agreement with the theoretical ones.     

Transverse vibration of completely-free elliptic and circular plates using orthogonal polynomials in the Rayleigh-Ritz method

The Rayleigh-Ritz method using characteristic orthogonal polynomials has been applied to study the problem of transverse vibration of completely-free elliptic and circular plates of uniform thickness. A sequence of successive approximations has been generated to work out the frequencies and mode shapes. The process is continued until at least the first four frequencies converge to five significant digits. It has been found that the results agree completely with the known results for a circular plate. A comparison has also been made with some known experimental and theoretical results for an elliptic plate.     

Transverse vibrations of elliptical plate of linearly varying thickness with half of the boundary clamped and the rest simply supported

The Rayleigh–Ritz method has been applied to study the problem of transverse vibrations of elliptical plate with half of the boundary clamped and the rest simply supported. The thickness of the plate is varying linearly in the space coordinates. For the same thickness variation, the tabulated results in case of mixed boundary conditions lies between those reported for the two extreme cases when the entire boundary is either clamped or simply supported. Convergence of the results is indicated by increasing the order of approximation. Some results for a circular plate of linearly varying thickness with half of the boundary clamped and the rest simply supported have been obtained as special case. Comparisons have been made with the known results. Three-dimensional mode shapes have been drawn for some selected cases.     

Transverse vibration of a circular plate with arbitrary thickness variation

Rayleigh-Ritz method has been employed to obtain approximations to frequencies and mode shapes of circular plates with variable thickness. The boundary is either clamped, simply supported or completely free. The main distinguishing feature of the present investigations is that the thickness approximation is done by measuring thickness at a suitable set of sample points and then using interpolation to get the approximating polynomial. Thus, unlike other methods already available in literature where either linear or quadratic variation of thickness has been examined, here one can have a polynomial of arbitrary degree depending upon the number and locations of the sample points. The results have been tabulated in a large number of cases and three-dimensional mode shapes have been plotted for some selected cases. Comparison has been made with available results. A short tables are also given to depict the rate of convergence with the order of approximation.     

Transverse vibration of a circular plate with unidirectional quadratic thickness variation

Free transverse vibration of a circular plate with thickness varying as (1 + α× + βx²) has been studied when the edge of the plate is clamped or simply-supported. The Rayleigh—Ritz method with suitable choice of basis functions satisfying the essential boundry conditions, has been employed to find the fundamental frequency and the associated mode shapes for various values of α and β with different boundary conditions. The convergence of results is ensured by working out several approximations till the results converge to the desired accuracy. The results for uniform thickness, linear and parabolic variation of thickness have been obtained as special cases. Tables and graphs are given for frequency, mode shapes and for depicting the effect of parameters α and β on the frequency.     

Free transverse vibration of elliptical plates of variable thickness with half of the boundary clamped and the rest free

First four frequencies have been computed for an elliptical plate half of whose boundary (y0) is clamped and the other half is free. The thickness of the plate is taken varying linearly with space coordinates. The Rayleigh–Ritz method has been used to obtain successive approximations utill convergence is achieved up to at least four significant figures. Results are tabulated for various values of the taper parameters. Three-dimensional mode shapes and the associated contour lines have been plotted in some selected cases. A table showing the rate of convergence with increasing order of approximation is given. Comparison has been made with known results in special cases. For the sake of completeness, results for the cases when the entire boundary is clamped or completely free are also reported.     

The free vibration analysis and optimal design of an adhesively bonded functionally graded single lap joint

In this study, three-dimensional free vibration and stress analyses of an adhesively bonded functionally graded single lap joint were carried. The effects of the adhesive material properties, such as modulus of elasticity, Poisson's ratio and density were found to be negligible on the first ten natural frequencies and mode shapes of the adhesive joint. Both the finite element method and the back-propagation artificial neural network (ANN) method were used to investigate the effects of the geometrical parameters, such as overlap length, plate thickness and adhesive thickness; and the material composition variation through the plate thickness on the natural frequencies, mode shapes and modal strain energy of the adhesive joint. The suitable ANN models were trained successfully using a series of free vibration and stress analyses for various random geometrical parameters and compositional gradient exponents. The ANN models showed that the support length, the plate thickness and the compositional gradient exponent played important role on the natural frequencies, mode shapes and modal strain energies of the adhesive joint whereas the adhesive thickness had a minor effect. In addition, the optimal joint dimensions and compositional gradient exponent were determined using genetic algorithm and ANN models so that the maximum natural frequency and the minimum modal strain energy conditions are satisfied for each natural frequency of the adhesively bonded functionally graded single lap joint.     

Free vibration analysis of corroded steel plates

Vibration analysis of unstiffened/stiffened plates has long been studied due to its importance in the design and condition assessments of ship and offshore structures. Corrosion is inevitable in steel structures and has been so far considered in strength analysis of structures. We studied the free vibration of pitted corroded plates with simply supported boundary conditions. Finite element analysis, with ABAQUS, was used to determine the natural frequencies and mode shapes of corroded plates. Influential parameters including plate aspect ratio, degree of pit, one-sided/both-sided corroded plate, and different corrosion patterns were investigated. By increasing the degree of corrosion, reduction of natural frequency increases. Plate aspect ratio and plate dimensions have no influence on reduction of natural frequency. Different corrosion patterns on the surface of one-sided corroded plates have little influence on reduction of natural frequency. Ratio of pit depth over plate thickness has no influence on the reduction of natural frequency. The reduction of natural frequency in both-sided corroded plates is higher than one-sided corroded plates with the same amount of total corrosion loss. Mode shapes of vibration would change due to corrosion, except square mode shapes.     

The influence of the slide-to-roll ratio on the friction coefficient and film thickness of EHD point contacts under steady state and transient conditions

The effect of different slide-to-roll ratios has been experimentally investigated under steady state and transient conditions using a steel ball in contact with the plane surface of a glass disc. Under transient conditions the entraining velocity has been varied with a sinusoidal law at two different frequencies. Measurements of the friction force and film thickness using optical interferometry have been made.For the same working conditions, different friction coefficient trends found for positive and negative slide-to-roll ratios can be related to different film thickness values and shapes. The combination of different thermal effects could be a possible explanation for the obtained results.     

Transient heat transfer studies on a diesel engine valve

Numerical calculations based on finite difference approximations are carried out to assess the transient thermal response of the inlet and exhaust valves of a Tata-Mercedes, six cylinder, four stroke, water cooled diesel engine with a compression ratio of 19.5 and a rated power output of 110 hp at 3000 rpm for three different engine loadings. A detailed analysis has been given for estimating the boundary conditions of the inlet and exhaust valves of an internal combustion engine. The problem has been extended by applying a thin ceramic insulation coating of 2 mm thickness at the valve plate. The isothermal distribution in the valve bodies and the heat flow rates through the various cooling media for three different engine loadings have been depicted for each of the cases with and without insulation coating. The results indicate a reduction in heat loss through valves by use of an insulation coating on the valve plate.     

Dynamic Behaviors of Axially Moving Viscoelastic Plate with Varying Thicknessn

Structural components of varying thickness draw increasing attention these days due to economy and light-weight considerations. In view of the absence of research in vibration analysis of viscoelastic plate with varying thickness, this study devotes to investigate the dynamic behaviors of axially moving viscoelastic plate with varying thickness. Based on the thin plate theory and the two-dimensional viscoelastic differential constitutive relation, the differential equation of motion of the axially moving viscoelastic rectangular plate is derived, the plate constituted by Kelvin-Voigt model has linearly varying thickness in the y-direction. The dimensionless complex frequencies of axially moving viscoelastic plate with four edges simply supported are calculated by the differential quadrature method, curves of real parts and imaginary parts of the first three-order dimensionless complex frequencies versus dimensionless moving speed are obtained, the effects of the aspect ratio, thickness ratio, the dimensionless moving speed and delay time on the dynamic behaviors of the axially moving viscoelastic rectangular plate with varying thickness are analyzed. When other parameters keep constant, with the decrease of thickness ratio, the real parts of the first three-order natural frequencies decrease, and the critical divergence speeds of various modes decrease too, moreover, whether the delay time is large or small, the frequencies are all complex numbers.     

Natural frequencies of stepped thickness rectangular plates

This paper presents the natural frequencies of stepped thickness square and rectangular plates together with the mode shapes of vibration. The transverse deflection of a stepped thickness plate is written in a series of the products of the deflection functions of beams parallel to the edges satisfying the boundary conditions, and the frequency equation of the plate is derived by the energy method. By use of the frequency equation, the natural frequencies (the eigenvalues of vibration) and the mode shapes are calculated numerically in good accuracy for square and rectangular plates with edges simply supported or elastically restrained against rotation, having square, circular or elliptical stepped thickness, from which the effects of the stepped thickness on the vibration are studied.     

Free vibration analysis of isotropic and orthotropic triangular plates

A highly accurate and computationally efficient numerical method is developed for the flexural vibration of isotropic and orthotropic triangular plates. A set of two-dimensional orthogonal plate functions is used as an admissible displacement function in the Rayleigh-Ritz method to obtain the natural frequencies and mode shapes for the plates. From a prescribed starting function satisfying the boundary conditions, the higher terms in the orthogonal plate functions are constructed using the Gram-Schmidt orthogonalization process. Natural frequencies and mode shapes are obtained for three triangular plates with different support conditions. The obtained numerical results are presented, and the isotropic case is verified with other numerical methods in the literature.     

Closed-form vibration analysis of thick annular functionally graded plates with integrated piezoelectric layers

This paper employs an analytical method to analyze vibration of piezoelectric coupled thick annular functionally graded plates (FGPs) subjected to different combinations of soft simply supported, hard simply supported and clamped boundary conditions at the inner and outer edges of the annular plate on the basis of the Reddy's third-order shear deformation theory (TSDT). The properties of host plate are graded in the thickness direction according to a volume fraction power-law distribution. The distribution of electric potential along the thickness direction in the piezoelectric layer is assumed as a sinusoidal function so that the Maxwell static electricity equation is approximately satisfied. The differential equations of motion are solved analytically for various boundary conditions of the plate. In this study closed-form expressions for characteristic equations, displacement components of the plate and electric potential are derived for the first time in the literature. The present analysis is validated by comparing results with those in the literature and then natural frequencies of the piezoelectric coupled annular FG plate are presented in tabular and graphical forms for different thickness-radius ratios, inner-outer radius ratios, thickness of piezoelectric, material of piezoelectric, power index and boundary conditions.     

Measurement of fluid film thickness on the valve plate in oil hydraulic axial piston pumps (I)-bearing pad effects-

The tribological mechanism between the valve plate and the cylinder block in oil hydraulic axial piston pumps plays an important role on high power density. In this study, the fluid film thickness between the valve plate and the cylinder block was measured with discharge pressure and rotational speed by use of a gap sensor, and a slip ring system in the operating period. To investigate the effect of the valve plate shapes, we designed two valve plates with different shapes : the first valve plate was without a bearing pad, while the second valve plate had a bearing pad. It was found that both valve plates behaved differently with respect to the fluid film thickness characteristics. The leakage flow rates and the shaft torque were also experimented in order to clarify the performance difference between the valve plate without a bearing pad and the valve plate with a bearing pad. From the results of this study, we found out that in the oil hydraulic axial piston pumps, the valve plate with a bearing pad showed better film thickness contours than the valve plate without a bearing pad.     

弹性关节柔性操作臂的频率及振型特性分析

为了研究柔性操作臂关节弹性对频率及振型的作用机理,建立了柔性操作臂的弹性约束模型。根据弹性约束模型,得到了弹性关节柔性操作臂的频率方程及振型函数。采用数值方法,对弹性关节柔性操作臂的前三阶频率及振型特性进行了分析。结果表明：关节的弹性对柔性操作臂的频率及振型具有明显的影响,将关节视为理想刚性约束会产生明显的误差;由灵敏度分析可知,线性约束对频率的影响大于扭转约束,且高阶频率段表现较为明显;扭转约束对振型的影响比线性约束更为显著,随着扭转约束的增大,振型发生改变,且高阶振型表现较为明显。通过实验对建立的弹性约束模型进行验证,实验结果表明,所建立的弹性约束模型能够合理地表征关节的弹性约束作用,分析结果更接近实际情况,误差较小。     

Spline finite strip analysis of the buckling and vibration of composite prismatic plate structures

A spline finite strip capability is described for predicting the buckling stresses and natural frequencies of vibration of prismatic plate structures which may be of composite laminated construction with arbitrary lay-ups. The plate structures may have general boundary conditions. The capability embraces analyses based on the use of first-order shear deformation plate theory and of classical plate theory, and utilizes substructuring procedures which include the use of superstrips. The theoretical development is not detailed since the present paper reports a very direct extension of a theoretical study developed for the analysis of single plates in an earlier paper in this Journal. A considerable range of buckling and vibration applications is documented and comparison of spline finite strip numerical values of buckling stresses and frequencies is made with results generated using the semi-analytical finite strip method and, in some cases, the finite element method. Buckled and vibrational mode shapes are presented for some applications.     

Vibration of stepped thickness plates

The vibration characteristics of stepped thickness plates of rectangular geometry with simply supported edges are investigated through an exact analysis. To illustrate the effect of sudden variation of thickness in the step form on the dynamic characteristics of plates, numerical calculations have been made for various combinations of thickness step, step-length ratio and side ratio. Natural frequencies for the first eight to nine modes are presented in the form of tables. Nodal lines are also plotted for a few selected cases. Several interesting features such as that of the thickness step splitting the degenerate frequencies of uniform rectangular plates into distinct ones, and the “frequency crossing” of some of the modes are observed.     

Three-dimensional free vibration analysis of perforated super elliptical plates via the p-Ritz method

A three-dimensional free vibration analysis of a perforated plate with rounded corners is presented. A solution method based on the linear, small strain, three-dimensional elasticity theory and the p-Ritz algorithm is employed. This analysis method uses sets of uniquely defined one- and two-dimensional polynomial functions as the trial displacements in the thickness and surface directions to arrive the eigenvalue equation which yields the natural frequencies and mode shapes for the perforated super elliptical plates. The accuracy of these results, if possible, is validated through comparison with the available literature. Parametric investigations on the vibration behaviours of the perforated super-elliptical plates with respect to different thickness ratios, cutout sizes and boundary constraints are examined.