Deformable Rectangular Plates Research Articles

Serum Concentration and mRNA Expression of Serum Amyloid A and Serum Amyloid A1 in Pregnant Mares and Their Fetuses after Experimental Induction of Placentitis

In this paper, we analyze the dynamic stability of deformable rectangular plates (single and two parallel) that interact simultaneously with the external supersonic gas flow and the internal flow of an ideal fluid. The mathematical formulation of the dynamics of elastic structures is made using the variational principle of virtual displacements, which includes the expression for the work done by aero- and hydro-dynamic forces. The motion of the liquid in the case of small perturbations is described by the equations of potential theory, which are converted to a weak form using the Bubnov–Galerkin method. Numerical solution of the problem is based on the application of the procedures of the finite element method to the coupled system of equations. The estimation of stability involves computation and analysis of complex eigenvalues, obtained at gradually growing velocity of the fluid or gas flow. The diagrams showing the mutual influence of fluid and gas flow velocities on the boundary of aero-hydroelastic stability, are constructed and the effects of the kinematic boundary conditions specified at the edges of the structure and the height of the fluid layer are evaluated. It has been established that the violation of smoothness of the obtained dependences and stability diagrams can be attributed either to a change in the vibration mode, or to a change in the type of stability loss.

Stability analysis of rectangular parallel plates interacting with internal fluid flow and external supersonic gas flow

In this paper, we analyze the dynamic stability of deformable rectangular plates (single and two parallel) that interact simultaneously with the external supersonic gas flow and the internal flow of an ideal fluid. The mathematical formulation of the dynamics of elastic structures is made using the variational principle of virtual displacements, which includes the expression for the work done by aero- and hydro-dynamic forces. The motion of the liquid in the case of small perturbations is described by the equations of potential theory, which are converted to a weak form using the Bubnov–Galerkin method. Numerical solution of the problem is based on the application of the procedures of the finite element method to the coupled system of equations. The estimation of stability involves computation and analysis of complex eigenvalues, obtained at gradually growing velocity of the fluid or gas flow. The diagrams showing the mutual influence of fluid and gas flow velocities on the boundary of aero-hydroelastic stability, are constructed and the effects of the kinematic boundary conditions specified at the edges of the structure and the height of the fluid layer are evaluated. It has been established that the violation of smoothness of the obtained dependences and stability diagrams can be attributed either to a change in the vibration mode, or to a change in the type of stability loss.

Geometrically nonlinear rectangular simply supported plates subjected to a moving mass

The dynamic deformation of a geometrically nonlinear rectangular simply supported plate under a moving lumped mass is evaluated using mode expansion method. The governing differential equations of motion for a largely deformable rectangular plate are derived using Lagrange method based on appropriate in and out-of-plane spatial functions which satisfy the proposed boundary conditions. Although the proposed procedure is applicable for any arbitrary edge boundary conditions, only the simply supported plates are addressed in the present work. On the other hand, all inertial components of the moving mass are included in the derivation of the equations of motion. A numerical example is used to study the dynamic behavior of the plate, considering large deformations. The obtained results indicate that ignoring the geometric nonlinearity in determining the vertical deformations of plates under the effect of moving masses, especially those of considerable weight and velocity, results in excessively large linear amplitudes leading to an unfavorable conservative structural design.

Analytical calculation of in-plane response of plates with concentrated masses to impact and application to pyroshock simulation

In aerospace missions pyroshocks occur due to controlled explosions of ordnance devices enabling the functionality of space modules. These shocks result from deployment mechanisms or opening solar sails and can cause failures of electronic devices and structures. Thus, essential components for assuring the reliability of modules are pyroshock tests for the completion of which strict requirements by the aerospace administrations have to be met. One of them is the definition of a specific acceleration signal and, based on this, the Shock Response Spectrum (SRS) for each part.So far, there is rather empirical than analytical knowledge about producing desired SRS with mechanical impacts and its characteristics due to the variation of input parameters. In this paper a widespread testing procedure for far-field pyroshocks is discussed which is realized by the in-plane impact of a hammer pendulum on a plate including the test specimen. The mechanical model consists of the contact between a rigid sphere and a free deformable rectangular plate with attached masses including subsequent propagation and reflection of longitudinal waves. In order to allow for a prediction of the acceleration field and the corresponding SRS due to the impact the problem is solved semi-analytically by using Hertzian contact theory, the Galerkin-procedure and numerical integration in time domain. The in-plane problem has, to the best of the authors' knowledge, not yet been treated in the literature in the way presented.The results calculated are compared with experimental data showing very good coincidence and allowing for a fast prediction of far-field pyroshock tests due to the impact excitation by a hammer pendulum. Hence, the framework of this paper is an enrichment for the current state of the art considering analytical pyroshock simulation. By better understanding the effect of pyroshocks to one and two dimensional structures a reduction of costs as well as durations for testing procedures seems promising.

A Ritz’s method based solution for the contact problem of a deformable rectangular plate on an elastic quarter-space

In this article, Ritz’s method is used to calculate with unprecedented accuracy the displacements related to a deformable rectangular plate resting on the surface of an elastic quarter-space. To achieve this required three basic steps. The first step involved the study of Green’s function describing the vertical displacements of the surface of an elastic quarter-space due to vertical force applied on its surface. For this case, an explicit formula was obtained by analytically resolving a complicated integral that did not previously have an analytical solution. The second step involved the study of the coupled system of a plate and an elastic quarter-space. This portion focused on determining reactive forces in the contact zone based on Hetenyi’s solution. After determination of the reactive forces, certain features were attributed to the plate’s edges. The final step involved the application of Ritz’s method to determine the deflections of the plate resting on the surface of the quarter-space. Finally, an example calculation and validation of results are given. This is the first semi-analytical solution proposed for this type of contact problem.

Free vibration analysis of shear deformable rectangular plates with a line hinge

A discrete method is proposed for analysing the natural vibration problem of shear deformable rectangular plates with a line hinge. The fundamental differential equations and the solutions of these equations are derived for two parts of the plate, which are obtained by dividing the plate along the line hinge. By transforming these equations into integral equations, and using numerical integration and the continuous conditions along the line hinge, the solutions of the whole plate can be expressed by the unknown quantities on the boundary and the quantities of the rotation along the hinge. Green function which is the solution of deflection of the bending problem of plate is used to obtain the characteristic equation of the free vibration. The effects of the position of the line hinge, the aspect ratio, the thickness-to-length ratio and the boundary condition on the natural frequency parameters are considered. By comparing the numerical results obtained by the present method with those previously published, the efficiency and accuracy of the present method are investigated.

Characteristic of electromechanical interaction of MEMS shuffling actuator with nonlinear plate electrode

When a voltage is applied between two electrodes consisting of a rigid ground and a deformable rectangular plate that has two parallel free edges, and one movable edge and one fixed edge, the plate bends under the electrostatic force generated in it. Accompanying to the bending deformation, the movable edge of the plate results in some displacement along the movable direction, which is the main clue of the MEMS actuator of shuffling movement. As the control voltage is applied up to a critical value, the phenomenon of snapping occurs at the plate electrode under the interaction of electromechanical coupling such that the shuffling displacement is obtained as possibly as large. Based on the nonlinear theory of plates with the von Karman’s type deformation and the electrostatic theory, the snapping and contact behavior is quantitatively analyzed by associating the increment finite element methods (IFEM) for deformation with the moment method for electrostatic fields as well as the arc-length control approach in the quantitative calculations. The numerical results for some case studies show that the path of bending deformation from stability into instability passing through the snapping criterion can be tracked well by this numerical code, while the characteristic curves of the maximum deflection in the plate and the shuffling displacement versus the control voltage, which are mainly concerned in the design of the MEMS shuffling actuator, are obtained additionally.

Exact mixed-classical solutions for the bending analysis of shear deformable rectangular plates

The thin plate model leads to a differential equation of fourth-order, while the mixed first-order shear deformation plate is modeled by a system of differential equations of second-order. The first model does not provide a very good analysis of shear deformity in plates in which the thickness-to-length ratio is relatively large. Nevertheless, the latter is more difficult and much more accurate. In this paper, the relationships between the findings of the two models allowing shear deformation results to be obtained from the results of classical thin theories are displayed without using any shear correction factors. The exact solutions are presented for bending of six types of rectangular plates having two opposite edges simply supported, and the other two edges may be quite general. The accuracy of the present model is demonstrated via problems for which the exact solutions and numerical results are available, and solutions are also presented as benchmark solutions for researchers to use in checking their numerical thick plate solutions.

Numerical analysis on snapping induced by electromechanical interaction of shuffling actuator with nonlinear plate

When a voltage is applied between two electrodes consisting of a rigid ground and a deformable rectangular plate that has two parallel free edges, and one movable edge and one fixed edge, the plate bends under the electrostatic force generated in it. Accompanying to the bending deformation, the movable edge of the plate results in some displacement along the movable direction, which is the main clue of the MEMS actuator of shuffling movement. As the control voltage is applied up to a critical value, the phenomenon of snapping occurs at the plate electrode under the interaction of electromechanical coupling such that the shuffling displacement is obtained as possibly as large. Based on the nonlinear theory of plates with the von Karman’s type deformation and the electrostatic theory, a theoretical analysis for the snapping behavior is quantitatively displayed in this paper. For this purpose, a numerical code is proposed by associating the increment finite element methods for deformation with the moment method for electrostatic fields as well as the arc-length control approach in the quantitative calculations. The numerical results for some case studies show that the path of bending deformation from stability into instability passing through the snapping criterion can be tracked well by this numerical code, while the characteristic curves of the maximum deflection in the plate and the shuffling displacement versus the control voltage, which are mainly concerned in the design of the MEMS shuffling actuator, are obtained additionally.

Differential cubature method for analysis of shear deformable rectangular plates on Pasternak foundations

In this paper, the differential cubature method (DCM) was applied to the bending analysis of shear deformable plates resting on Pasternak foundation. An attractive advantage of the DCM is that it can produce the acceptable accuracy of numerical results with very few grid points in the solution domain and therefore can be very useful for rapid evaluation in engineering design. The detailed procedures for discretizing the governing equations and boundary conditions of the title problems using the DCM are presented. Numerical solutions for rectangular thick plates on Pasternak foundation and subjected to different boundary conditions are obtained. The convergence studies are carried out to establish the minimal grid points needed for achieving accurate solutions. Next, the solutions for some selected cases are presented and verified by comparing them with the published values. It is observed that the DCM is able to furnish convergent solution with relatively fewer grid points than the more established differential quadrature method (DQM).

Stability of shear deformable rectangular plates using refined finite element model

Stability characteristics of moderately thick rectangular isotropic plates subjected to in-plane loads, such as uniaxial or biaxial compression, shear and biaxial compression-tension, are studied in this paper. For this purpose, a simple higher-order theory involving only four unknowns is employed. This theory allows parabolic variation of transverse shear strains and stresses through the thickness and satisfies the condition of vanishing of transverse shear stresses at the top and bottom surfaces of the plate. The components of the in-plane displacements are expressed as in classical plate theory. However, transverse displacement is assumed as a function of the square of the thickness coordinate. A four node rectangular C 1 continuous finite element having eight degrees of freedom per node is developed. The merits and disadvantages of the higher-order theory and corresponding finite element employed here are demonstrated through a number of numerical examples.

A comprehensive analysis of the state of stress of elastic anisotropic flat plates using refined theories

This paper establishes, on a quantitative basis, the correlation between two apparently different higher-order theories of anisotropic plates. In addition it provides a strong mathematical tool allowing one to determine, in an exact and unified manner, the state of stress of orthotropic shear deformable rectangular plates exhibiting a variety of edge conditions.