The flexibility function approach to vibration analysis of rectangular plates with arbitrary multiple point supports on the edges

Free vibrations of rectangular plates with interior point supports

A new type of admissible function is used to study free, flexural vibration problems for rectangular plates with arbitrary interior elastic point supports. The one-dimensional deformation function of a beam with point load is applied in forming the admissible deformation shape function of a plate with a corresponding boundary condition. This study provides an extension of the classical solution in the vibration analysis of a plate with interior elastic point supports. The extended solution allows one to solve this type of problem easily when compared to other solutions available in the literature. Numerical results are presented for a number of specific problems, including convergence and accuracy of the approach, which include natural frequencies and mode shapes of some typical plates with interior point supports.

This article presents a simple and efficient procedure for the natural vibration analysis of rectangular plates and stiffened panels in contact with fluid on one side. The assumed mode method is applied, where the natural frequencies and mode shapes are obtained by solving an eigenvalue problem of a multi-degree-of-freedom system matrix equation derived by using Lagrange’s equation of motion. The Mindlin thick plate theory is applied for a plate, and in the case of stiffened panels, the effect of framing is taken into account by adding its strain and kinetic energies to the corresponding plate energies. Potential flow theory assumptions are adopted for the fluid, and free surface waves are ignored. The fluid velocity potential is derived from the boundary conditions for the fluid and structure and is utilized for the calculation of added mass using the assumed modes. The applicability and accuracy of the developed procedure are illustrated with several numerical examples using a developed in-house code. A comparison of the results with those obtained by general purpose finite element analysis software is provided, where very good agreement is achieved.

Three-dimensional vibration solutions are presented for rectangular plates with mixed boundary conditions, based on the small strain linear elasticity theory. The analysis is focused on two kinds of rectangular plates, the boundaries of which are partially fixed while the others are free. One of those studied is a rectangular plate with partially fixed boundaries symmetrically arranged around four corners and the other one is a rectangular plate with partially fixed boundaries around one corner only. A global analysis approach is developed. The Ritz method is applied to derive the governing eigenvalue equation by minimizing the energy functional of the plate. The admissible functions for all displacement components are taken as a product of a characteristic boundary function and the triplicate Chebyshev polynomial series defined in the plate domain. The characteristic boundary functions are composed of a product of four components of which each corresponds to one edge of the plate. The R-function method is applied to construct the characteristic boundary function components for the edges with mixed boundary conditions. The convergence and comparison studies demonstrate the accuracy and correctness of the present method. The influence of the length of the fixed boundaries and the plate thickness on frequency parameters of square plates has been studied in detail. Some valuable results are given in the form of tables and figures, which can serve as the benchmark for the further research.

New analytic buckling solutions of rectangular thin plates with two free adjacent edges by the symplectic superposition method

Using the superposition and small deflection theories, thermal change perpendicular to a surface is considered. A CRTP (concrete rectangular thin plate) with two adjacent CEDs (clamped edges) and one SSED (simply supported edge) and one FED (free edge) under thermal change is seen as a RTP’s superposition with three SSEDs and one FED in the case of temperature change and a RTP with three SSEDs and one FED under the bending moment on two adjacent edges. Adopting the Levy method, by supposing the deflection function with undetermined parameter on the FED, the theoretical solution of a CRTP (concrete rectangular thin plate) with three SSEDs and one FED is obtained; the theoretical solution of a CRTP with three SSEDs and one FED is obtained under thermal change; taking the theoretical solution of a CRTP with four SSEDs under the bending moment on two adjacent edges, the theoretical solution of a CRTP with three SSEDs and one FED under the bending moment on two adjacent edges is obtained; the theoretical solution for the bending moment and deflection of a CRTP with two adjacent CEDs and one SSED and one FED under temperature change is obtained.

Natural vibration analysis of plates represents an important issue in engineering applications. In this paper, a new and simplify method for vibration analysis of circular and rectangular plates is presented. The design of an effective robust controller, which consistently attenuates transverse vibration of the plate caused by an external disturbance force, is given. The dynamics of the plate is modeled as a distributed parameter system. We have studied the control vibration of the plate using quantitative feedback theory method by determining the transfer functions between various factors of control system. In this method we have developed the general distributed parameter system method for uncertainty problem for simply supported rectangular plate and clamped circular plate. The quantitative feedback method is one of the robust control methods which is capable to solve problems despite structural and non-structural uncertainty. Quantitative Feedback Theory introduces the new technique to design one-point feedback controllers for distributed parameter systems. The results demonstrate that the control law provided a significant reduction in the plate vibration. The numerical simulation of the designed controller demonstrates that the QFT controller can consistently attenuate the vibration compared to a passive system.

A nine-node isoparametric plate element in conjunction with first-order shear deformation theory is used for free vibration analysis of rectangular plates with central cutouts. Both thick and thin plate problems are solved for various aspect ratios and boundary conditions. In this article, primary focus is given to the effect of rotary inertia on natural frequencies of perforated rectangular plates. It is found that rotary inertia has significant effect on thick plates, while for thin plates the rotary inertia term can be ignored. It is seen that the numerical convergence is very rapid and based on comparison with experimental and analytical data from literature, it is proposed that the present formulation is capable of yielding highly accurate results. Finally, some new numerical solutions are provided here, which may serve as benchmark for future research on similar problems.

Performance of similitude methods for structural vibration analyses of rectangular plates

Several sets of static beam functions have been established by the authors, which are the static solutions of beams under arbitrary static loads. The beam is a strip taken from the rectangular plate to be considered, and the static load acting on the beam is expanded into a series which may be a series of point loads or a series of sinusoid loads or a series of polynomial loads. These sets of static beam functions have been successfully applied to the vibration analysis of rectangular plates with varying complexity by the Rayleigh-Ritz method and finite strip method. Some numerical results are given. By comparing the results with those available from literature it can be shown that the present method is extremely effective for solving the aforementioned problems and usually a few static beam functions will be sufficient to give acceptable results.

Free vibration analysis of plate assemblies using the dynamic stiffness method based on the higher order shear deformation theory

Bending of rectangular orthotropic plates with rotationally restrained and free edges: Generalized integral transform solutions

Free vibration analysis of beams and rectangular plates with free edges by the discrete singular convolution

Abbreviated. Preface (W.E. Diewert). Editor's introduction to volume (W.A. Barnett, J. Binner). Part 1: Structural function system specification: consumer demand. Section 1.1: Editors' overview of part 1 (W.A. Barnett, J. Binner). Section 1.2: The differential approach. Section 1.3: The locally flexible functional form approach. 1.4: The globally flexible functional form approach. Section 1.5: Recursively nested, homothetic, and inverse function structures. Part 2: Structural function system specification: production. Section 2.1 Editors' overview of part 2. Section 2.2 Production by firms. Section 2.3: Household production. Part 3: Relevant econometric theory. Section 3.1: Editors' overview of part 3. Section 3.2: Asymptotic statistical theory. Part 4: Nonlinear structure in time series. Section 4.1: Editors' overview of part 4. Section 4.2: Chaos. Section 4.3: Frequency domain methods. Section 4.4: A competition.

Remarks of some problems for rectangular thin plates with free edges on elastic foundations