Finite Rectangular Plate Research Articles

Radiation efficiency of damped plates.

The radiation efficiency of damped plates is discussed in this letter. Below the critical frequency of a plate, numerical results show that the radiation efficiency is much influenced by damping. Some modifications of the classical formulas given by Cremer for an infinite plate and Leppington for a finite rectangular plate are proposed to include the influence of the damping on the radiation efficiency.

Magnetohydrodynamic (MHD) squeeze film characteristics between finite porous parallel rectangular plates with surface roughness

Purpose – The purpose of this paper is to consider magnetohydrodynamic (MHD) squeeze film characteristics between finite porous parallel rectangular plates with surface roughness. Design/methodology/approach – Based upon the MHD theory, this paper analyzes the surface roughness effect squeeze film characteristics between finite porous parallel rectangular plates lubricated with an electrically conducting fluid in the presence of a transverse magnetic field. Findings – It is found that the magnetic field effects characterized by the Hartmann number produce an increased value of the load carrying capacity and the response time as compared to the classical Newtonian lubricant case. The modified averaged stochastic Reynolds equation governing the squeeze film pressure is derived. Research limitations/implications – The present study has considered both Newtonian fluids and non-Newtonian liquids. Practical implications – The work represents a very useful source of information for researchers on the subject of MHD squeeze film with finite porous parallel rectangular plates lubricated with an electrically conducting fluid. Originality/value – This paper is relatively original and illustrates the squeeze film characteristics between finite porous parallel rectangular plates with MHD effects.

A numerical study of elastic bodies that are described by constitutive equations that exhibit limited strains

Recently, a very general and novel class of implicit bodies has been developed to describe the elastic response of solids. It contains as a special subclass the classical Cauchy and Green elastic bodies. Within the class of such bodies, one can obtain through a rigorous approximation, constitutive relations for the linearized strain as a nonlinear function of the stress. Such an approximation is not possible within classical theories of Cauchy and Green elasticity, where the process of linearization will only lead to the classical linearized elastic body.In this paper, we study numerically the states of stress and strain in a finite rectangular plate with an elliptic hole and a stepped flat tension bar with shoulder fillets, within the context of the new class of models for elastic bodies that guarantees that the linearized strain would stay bounded and limited below a value that can be fixed a priori, thereby guaranteeing the validity of the use of the model. This is in contrast to the classical linearized elastic model, wherein the strains can become large enough in the body leading to an obvious inconsistency.

Scalar Differential Equation for Slowly-Varying Thickness-Shear Modes in AT-Cut Quartz Resonators With Surface Impedance for Acoustic Wave Sensor Application

For time-harmonic motions, we generalize a 2-D scalar differential equation derived previously by Tiersten for slowly-varying thickness-shear vibrations of AT-cut quartz resonators. The purpose of the generalization is to include the effects of surface acoustic impedance from, e.g., mass layers or fluids for sensor applications. In addition to the variation of fields along the plate thickness, which is considered in the usual 1-D acoustic wave sensor models, the equation obtained also describes in-plane variations of the fields, and therefore can be used to study the vibrations of finite plate sensors with edge effects. The equation is compared with the theory of piezoelectricity in the special cases of acoustic waves and pure thickness vibrations in unbounded plates. An example of a finite rectangular plate is also given.

To solve the inverse Cauchy problem in linear elasticity by a novel Lie-group integrator

In this paper, we propose a simple, iteration free and easy-to-implement numerical algorithm for the solution of inverse Cauchy problem in linear or nonlinear elasticity. The bottom of a finite rectangular plate is imposed by overspecified boundary data, and we seek unknown data on the top side. A spring-damping transform method (SDTM) is introduced to the Navier equations, such that after a discretization by the differential quadrature method, we can apply a novel Lie-group integrator, namely the mixed group-preserving scheme (MGPS), to solve them as an initial value problem. Several numerical examples including nonlinear ones are examined to show that the MGPS can overcome the ill-posed behaviour of the inverse Cauchy problem in elasticity, which has good efficiency and stability against the noisy disturbance, even with an intensity large up to and .

3-D transient analytical solution based on Green’s function to temperature field in friction stir welding

Friction stir welding (FSW) is a relatively modern welding process, which not only provides the advantages offered by fusion welding methods, but also improves mechanical properties as well as metallurgical transformations due to the pure solid-state joining of metals. The FSW process is composed of three main stages; penetrating or preheating stage, welding stage and cooling stage. The thermal history and cooling rate during and after the FSW process are decisive factors, which dictate the weld characteristics. In the current paper, a novel transient analytical solution based on the Green’s function method is established to obtain the three-dimensional temperature field in the welding stage by considering the FSW tool as a circular heat source moving in a finite rectangular plate with cooling surface and non-uniform and non-homogeneous boundary and initial conditions. The effect of penetrating/preheating stage is also taken into account by considering the temperature field induced by the preheating stage to be the non-uniform initial condition for the welding stage. Similarly, cooling rate can be calculated in the cooling stage. Furthermore, the simulation of the FSW process via FEM commercial software showed that the analytical and the numerical results are in good agreement, which validates the accuracy of the developed analytical solution.

Analysis and experiment on transient dynamic response in finite mindlin plate

The transient wave propagation in the finite rectangular Mindlin plate is investigated by the analytical and experimental methods. The generalized ray method (GRM) which has been successfully applied to study the transient responses of beams, planar trusses, space frames and infinite layered media is extended to investigate the transient wave propagation and early short time transient response in finite Mindlin plate. Combining the wave solution, the shock source and the boundary conditions, the ray groups transmitted in the finite rectangular plate can be determined. Numerical simulations and experiments are performed and compared with each other. The results show that the transient wave propagation and early short time transient responses in the finite plate can be studied using the GRM. The early short time transient accelerations are very large for the finite plate subjected to the unit impulse, while the early short time transient displacements are very small. The early short time transient accelerations under the unit impulse are much larger than those under the unit step impulse. The thickness and material characteristics have remarkable effects on the early short time transient responses.

Evaluation of stress intensity factor of multiple inclined cracks under biaxial loading

A finite rectangular plate of unit thickness with two inclined cracks (parallel and non parallel) under biaxial mixed mode condition are modelled using finite element method. The finite element method is used for determination of stress intensity factors by ANYSIS software. Effects of crack inclination angle on stress intensity factors for two parallel and non parallel cracks are investigated. The significant effects of different crack inclination parameters on stress intensity factors are seen for lower and upper crack in two inclined crack. The present method is validated by comparing the results from available experimental data obtained by photo elastic method in same condition.

Acoustic radiation from a point excited multi-layered finite plate

The acoustic radiation from a finite rectangular composite plate is evaluated using eigenfunctions obtained through the use of three-dimensional equations of elasticity. The composite plate is made of perfectly bonded finite plates of identical lateral dimensions and of different thicknesses. The plate is free of shear stresses and is pinned on the in-plane displacements on all its boundaries and is baffled by an infinite rigid plane. The multi-layered plate is in contact with a different fluid medium on each of its two surfaces. The solution for the vibration response due to normal and shear surface forces is found in terms of the composite plate eigenfunctions that include heavy acoustic loading. The displacement vector field throughout the thickness of the plate is computed as well as the resultant near- and far-field radiated acoustic pressures for various ratios of thickness to plate dimensions over a broad frequency range. Initial results focus on a bilaminar plate. [Work supported by the ASEE Summer Faculty Research Program.]

Acoustic radiation from finite bilaminar submerged plates: Three-dimensional elasticity solution

The vibration and radiation of a submerged baffled finite rectangular bilaminar plate is obtained analytically using the three-dimensional equations of elasticity. The plate is composed of two perfectly bonded finite layers of identical size. The boundary conditions on the perimeter of the plate are free of shear stresses and pinned on the in-plane displacements. The plate is coupled to acoustic media on both sides and is baffled by an infinite rigid plane. The solution for the vibration response due to normal surface forces is found in terms of the bilaminar plate eigenmodes. The vibratory response and the associated near- and far-field radiated acoustic pressures are computed for various ratios of thickness to plate dimensions over a broad frequency range. [Work supported by the ASEE Summer Faculty Research Program.]

Effect of surface roughness on magnetohydrodynamic squeeze film characteristics between finite rectangular plates

The effect of surface roughness on the magnetohydrodynamic squeeze film behavior between two rectangular plates is analyzed. A generalized form of surface roughness pattern is considered. A stochastic random variable with non-zero mean, variance and skewness is used to model mathematically the surface roughness of the squeeze film bearings. The modified averaged stochastic Reynolds equation governing the squeeze film pressure is derived. The expressions for the squeeze film pressure, load carrying capacity and squeeze film time are obtained. Numerical computations of the results show that the negatively skewed surface roughness pattern increases the load carrying capacity and squeeze film time. On the contrary the performance of squeeze film suffers due to positively skewed surface roughness pattern. Further the magnetic effect characterized by the Hartmann number improves the performance of the squeeze film lubrication as compared to the classical non-conducting lubricant case.

Multiple Crack Analysis in Finite Plates

In this study, interactions of multiple cracks in finite rectangular plates are analyzed. A finite plate is treated as a polygon-shaped plate which can be created by contiguous crack segments embedded in an infinite plate forming an enclosed region so there are no true crack tips remaining around the plate. The problem is formulated in terms of integral equations expressed by edge dislocation distributions representing opening displacement profiles for the cracks with both normal and tangential components. To solve this boundary value problem we employ the superposition approach based on the dislocation distributions requiring the determination of crack opening displacement profiles that satisfy the traction- free condition on interior crack faces and the given boundary tractions on four exterior cracks defining the rectangular plate. Applying the method with a new point allocation scheme reducing the number of evaluation points leads to a set of linear algebraic equations in terms of unknown weighting coefficients of opening displacement profiles. These opening displacement profile components are categorized in terms of polynomial terms which are integers and wedge terms which are rational numbers approximated from irrational wedge eigenvalues to evaluate integrals in closed forms. Once the coefficients of opening displacement profile components are obtained, stress and displacement fields on the entire body, as well as stress intensity factors at the crack tips and kinks are calculated. After the accuracy of the method is shown by comparing the results for different sample problems, the strong interactions of multiple cracks in a rectangular plate under various loading conditions are demonstrated with figures and tables.

T-stress Solutions for Cracks Emanating from a Circular Hole in a Finite Plate

The boundary element method is employed to obtain T-stress solutions for cracks emanating from a circular hole in finite rectangular plates. Numerical values of the T-stress are obtained using the M-contour integral approach. A range of crack lengths are analyzed for two hole sizes, and the cases of a single crack and double-cracks emanating from the hole in the plate under both uniform remote tension and simple bending are considered. For completeness, stress intensity factor solutions are also presented. These results will be useful for failure assessments using two-parameter linear elastic fracture mechanics.

Crack propagation analysis with Galerkin boundary element method

AbstractThis paper describes the application of symmetric Galerkin boundary element methods (SGBEM) for the analysis of a 2D crack propagation problem. The sub‐domain SGBEM for crack problem is derived. The coefficient matrix is completely symmetric. Cohesive crack model is used to simulate crack propagation. The increment control method for crack propagation and the method for unknown crack propagation path have been derived for high‐order element. Two‐stage interpolation method called the ‘quasi‐higher order element method’ (QHOEM) is then proposed to solve the double integrals. In the initial stage, it uses higher order elements to interpolate the field variables, and for the numerical integration involved, it further uses interpolation functions to decompose the higher order elements into lower order elements so that the existing analytical integration can be applied. A finite rectangular plate containing a centre crack growth and four‐points bending beam problem have been analysed to check the accuracy of the proposed method. For actual application, a dam buttress with an edge crack has been analysed and the results are found to be in agreement with the other numerical and experimental results. Copyright © 2004 John Wiley & Sons, Ltd.

Variability of the coupling loss factor between two coupled plates

Within statistical energy analysis (SEA), the coupling loss factor (CLF) is a statistical quantity defined in terms of the average behaviour of an ensemble of similar systems. Thus, the ‘effective’ CLF for a given realization may differ from the ensemble average. The effective CLF of two coupled plates with low modal density and low modal overlap fluctuates significantly compared with the ensemble average CLF. Accordingly, the CLF is the main parameter expected to determine the confidence intervals in the SEA prediction. The aim of the present paper is to quantify the variability of the effective CLF for the case of two finite rectangular plates. Extensive parameter investigations have been performed by using the dynamic stiffness method (DSM) and a ‘numerical’ power injection method (PIM) using the SEA power balance equations. The effects of frequency bandwidth and the number of modes in a band were separated by using frequency averages at a series of constant bandwidths rather than one-third octave averages, while the modal overlap is made independent of frequency by setting the damping loss factor to be inversely proportional to frequency. Finally, an improved empirical model for the variability of the CLF is derived, from these numerical results, in terms of the modal overlap factor and the number of modes in a frequency band. This model can subsequently be used to evaluate the uncertainty of the CLF of such a system.

Multiple-crack analysis with the quasi-higher-order symmetric Galerkin boundary element method

This paper describes the application of the symmetric Galerkin boundary element method (SGBEM) for the analysis of a two-dimensional linear elastic crack problem. The direct SGBEM for the multiple-crack problem is derived using the Betti reciprocal theorem, resulting in a completely symmetric matrix. The auxiliary fictitious state is used to obtain the equations. This approach can deal with any number of traction-free cracks within a single domain. Two-stage interpolation method called the ‘quasi-higher-order element method’ (QHOEM) is then proposed to solve the double integrals. In the initial stage, it uses higher-order elements to interpolate the field variables and, for the numerical integration involved, it further uses interpolation functions to decompose the higher-order elements into lower-order elements so that the existing analytical integration can be applied. Thus, it can be embedded into the present codes with ease and also reduces the computational cost. A finite rectangular plate containing a slanting crack and a kinked crack has been analysed to check the accuracy of the proposed method. A load-carrying fillet welded joint containing several cracks is then analysed, and the stress intensity factors at the crack tips are found to be in complete agreement with the dual-boundary-element method results.

Influence of support properties on the sound radiated from the vibrations of rectangular plates

The control of the forced vibration response of structures through the optimal tuning of its supports is desirable in many applications. Tuning may enhance the dissipation of vibration energy within the supports, thereby reducing fatigue and structure-borne noise. Two different models were developed to calculate the optimal support stiffness that minimizes the velocity response of homogeneous plates. The first model, based on the wave propagation at the edge, yields a good first cut approximation of the optimal properties. The optimal viscous and viscoelastic support stiffness for minimal reflection at the edge was calculated. Maximum absorption of the incident waves occurs when the viscous support stiffness matches the characteristic mechanical impedances of the plate. The second model, based on the Rayleigh–Ritz method, yields more accurate estimates of the optimal support stiffness required to minimize the forced velocity response of the finite rectangular plate. The optimal support properties calculated from the two different methods were in good agreement. This suggested that the modal response of the plate is strongly influenced by the wave reflections at the edges. Finally, the effects of support properties on the sound radiated from the plate were investigated. The optimal support stiffness that minimizes the radiated sound power was found to be smaller than the value that minimizes the velocity response. The results show that both the velocity response and sound radiation are strongly influenced by dissipation of vibration energy at the edges, and demonstrate that support tuning can yield significant noise and vibration reduction.

Power flow model of flexural waves in finite orthotropic plates

In this paper, an approximate power flow model is developed for the analysis of the flexural waves in finite orthotropic plate transversely vibrating in the medium-to-high frequency ranges. The derived energy equation for the model is expressed with time- and locally space-averaged farfield wave energy density. It could be the more general form than that of the conventional power flow analysis model seen in the isotropic plate. With the derived model, dynamic characteristics varying with the direction can be expressed. To verify the validity and accuracy of the model, numerical analyses are performed for the case where a finite rectangular plate is excited by a harmonic point force, and the calculated results expressed with the energy levels are compared with classical modal solutions by changing the frequency and the damping loss factor of the plate. The dominant power transmission paths in the plate are also predicted from the distribution of the approximate intensity fields.

Magneto‐hydrodynamic squeeze film characteristics for finite rectangular plates

The squeeze‐film characteristics between two parallel rectangular plates with an electrically conducting fluid in the presence of a transverse magnetic field are analyzed. The squeeze‐film Reynolds equation applicable to the curved surfaces is derived using the continuity equation and the magneto‐hydrodynamic (MHD) motion equations. A closed‐form solution is obtained for the squeeze‐film pressure of parallel rectangular plates, and applied to predict the squeeze‐film behavior. According to the results, the presence of magnetic fields signifies an enhancement in the squeeze‐film pressure. On the whole, the magnetic‐field effect characterized by the Hartmann number provides an increase in value of the load‐carrying capacity and the response time as compared to the classical non‐conducting lubricant case, especially for larger values of the aspect ratio or smaller values of film height.

Fluid forces on kayak paddle blades of different design

Three kayak paddle blades of different design (Conventional, Norwegian, Turbo) were tested in a low-speed wind tunnel at a maximum chord Reynolds number of Re = 2.2–2.7 × 105 (corresponding to speed through water of ≈1 m/s). The mean drag force and side force acting on each blade were measured, as the yaw and pitch angles were varied. The results were compared with those recorded for a finite rectangular flat plate of similar area and aspect ratio. For zero pitch angle of the blades, the results indicate that the drag coefficient was mostly independent of the blade design as the yaw angle was varied between ± 20°, with only the Norwegian blade design displaying a marginally higher drag coefficient than either of the other two blades or the flat plate. Increasing the pitch angle to 30°, while maintaining the yaw angle at zero, resulted in a 23% reduction of the drag coefficient for the flat plate, but only a 15% reduction of the drag coefficients for the three blades. For all designs, the drag coefficient reduction followed a simple cosine relationship as the pitch angle or yaw angle was increased. The wind tunnel experiments revealed that the side force coefficients for all three paddle blade designs were entirely independent of the blade design and were indistinguishable from those recorded for a flat plate. In summary, the study showed that the nondimensional force coefficients are largely independent of the paddle blade design.