Problems which require a study of the static and dynamic response of plates can be approached by first considering the plate to be a portion of an infinite plate, the prescribed boundary conditions being temporarily ignored. Once the plate's boundary has been defined in the infinite plate, a numerical solution is initiated by dividing this boundary into N segments of arbitrary length. For the static case the desired loading can then be applied to the infinite plate, and its effect on the deflection and stresses at the midpoint of the N boundary segments computed. To satisfy the boundary conditions of elementary plate theory, a concentrated force and moment are applied at the midpoint of each boundary segment. The magnitudes of these N equivalent forces and moments are determined by specifying that their combined effects, together with the applied loading, satisfy the boundary conditions at the N boundary points. This yields a set of 2N simultaneous equations whose solution constitutes the solution to the problem. A similar approach can be utilized for the vibrating plate. For the dynamic case the applied loading is assumed as zero, and a harmonically varying force and moment placed at the midpoint of each of the N boundary segments. The magnitudes of the N harmonically varying forces and moments are determined by specifying that their combined effects satisfy the boundary conditions at the N boundary points. This, coupled with the assumption of homogeneous boundary conditions, yields a set of 2N homogeneous equations. The frequency equation follows by setting the determinant of the coefficients equal to zero. The above approach to the solution of boundary value problems is formally known as the Reflection Method. Application of the Reflection Method to the static plate was previously accomplished by placing the equivalent forces and moments in the infinite plate at a finite distance from the midpoint of each boundary segment. This finite distance was called a retracted distance, and the curve along which the equivalent forces and moments were applied, a retracted boundary. In this investigation, the magnitude of the retracted distance was found to influence the condition of the coefficient matrix, while the solution remained relatively independent. The static response of plates by the Reflection Method as presented here applies the equivalent forces and moments directly to the boundary of the plate. This was found to impressively improve the condition of the coefficient matrix and reduce the number of significant figures necessary to obtain a numerical solution. With no increase in the number of boundary points, results were obtained comparable to those utilizing a retracted distance. The equations enabling the forces and moments to be applied directly to the boundary are developed and several examples presented. Application of the Reflection Method to the problem of determining natural frequencies is first illustrated for beams and then extended to plates. In each case the necessary equations are developed and sample problems presented. / Ph. D.

The in-plane vibration of a plate with an elliptic hole is studied. It is shown that standing waves whose wavelength is the same order of magnitude as the size of the hole are theoretically capable of causing microcracks which have been observed experimentally. Several approaches were used including reduction of the mixed boundary value problem to a Fredholm equation, and to a matrix eigenvalue problem. Contour curves of various stresses and displacements were obtained numerically. A new technique was developed for the solution of the wave equation appropriate for boundary conditions on an elliptical surface. / Ph. D.

A first-order shear deformation plate theory is used to predict free vibration frequencies in rotating laminated composite plates. The theory accounts for geometric non-linearity in the form of von Karman strains. The plate is permitted to have arbitrary orientation and offset from the axis of rotation. A finite element model is developed to obtain a solution to the problem. The model is validated by comparing the results for free vibration of non-rotating plates for various boundary conditions and material properties with the exact results based on the classical plate theory and the first-order shear deformable plate theory. Results are presented for free vibration of isotropic and laminated composite plates rotating at different angular velocities. A study has also been made on the change in the free vibration frequencies of the plate with angular velocity for different plate thicknesses and for different modulus ratios. / Master of Science

Natural frequencies of the double and triple-walled carbon nanotubes are determined exactly and approximately for both types. Approximate solutions are found by using Bubnov-Galerkin and Petrov-Galerkin methods. For the first time explicit expressions are obtained for the natural frequencies of double and triple-walled carbon nanotubes for different combinations of boundary conditions. Comparison of the results with recent studies shows that the above methods constitute quick and effective alternative techniques to exact solution for studying the vibration properties of carbon nanotubes. The natural frequencies of the clamped-clamped double-walled carbon nanotubes are obtained; exact solution is provided and compared with the solution reported in the literature. In contrast to earlier investigation, an analytical criterion is derived to establish the behavior of the roots of the characteristic equation. Approximate Bubnov-Galerkin solution is also obtained to compare natural frequencies at the lower end of the spectrum. Simplified version of the Bresse-Timoshenko theory that incorporates the shear deformation and the rotary inertia is proposed for free vibration study of double-walled carbon nanotubes. It is demonstrated that the suggested set yields extremely accurate results for the lower spectrum of double-walled carbon nanotube. The natural frequencies of double-walled carbon nanotubes based on simplified versions of Donnell shell theory are also obtained. The buckling behavior of the double-walled carbon nanotubes under various boundary conditions is studied. First, the case of the simply supported double-walled carbon nanotubes at both ends is considered which is amenable to exact solution. / Then, approximate methods of Bubnov-Galerkin and Petrov-Galerkin are utilized to check the efficacy of these approximations for the simply supported double-walled carbon nanotubes. Once the extreme accuracy is demonstrated for simply supported conditions, the approximate techniques are applied to two other cases of the boundary conditions, namely to clamped-clamped and simply supported-clamped double-walled carbon nanotubes. For the first time in the literature approximate expression for the buckling loads are reported for these boundary conditions. The dynamic deflection of a single-walled carbon nanotube under impact loading is analyzed by following a recently study reported on the energy absorption capacity of carbon nanotubes under ballistic impact. / by Demetris Pentaras. / Thesis (Ph.D.)--Florida Atlantic University, 2009. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2009. Mode of access: World Wide Web.

Thesis. 1976. Sc.D.--Massachusetts Institute of Technology. Dept. of Mechanical Engineering. / Microfiche copy available in Archives and Engineering. / Vita. / Includes bibliographical references. / by Nathan Clay Martin II. / Sc.D.

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 1981. / Microfiche copy available in Archives and Barker. / Includes bibliographical references. / by Steven James Hollowell. / M.S.

Bibliography: leaves 159-169. / xii, 170 leaves : ill. (some col.) ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / This thesis is concerned with the transmission and distribution of vibratory power in built-up structures, and in particular, structures characteristic of naval ships. The study consists of a detailed investigation of the wave transmission properties of structural junctions, followed by an analysis of the wave propagation through plates with periodic stiffeners. The results are used to develop Statistical Energy Analysis (SEA) models for the prediction of vibration levels in structures characteristic of naval ships. / Thesis (Ph.D.)--University of Adelaide, Dept. of Mechanical Engineering, 1996

Whereas analytic and experimental investigations of plates subject to lateral vibrations have been rather thorough, the present study is an analytic determination of the various critical frequencies of vertically cantilevered thin elastic rectangular plates vibrating freely within their own planes. Within the restrictions imposed by excluding any motion perpendicular to the face of the plate, the upright edges are free to move in the other two directions, as is the top horizontal edge. Three different base conditions are imposed: • A clamped lower edge; • A lower edge which is freely vibrating transversely in the plane of the wall where the vertical fibers of the wall are fixed at their roots; and • A horizontally freely pulsating lower edge where the vertical fibers of the wall are fixed at their roots. The first two conditions are considered in relation to plate vibrations which are essentially vertical while the first and third conditions are each employed with essentially horizontal plate vibrations. In every case the effect of a uniform load placed along the upper edge is studied. Critical frequencies and associated amplitude coefficients are obtained for various ratios of base length to wall height. The solution, which is presented in tabular and graphic forms, is obtained by using the method of iteration on the Rayleigh-Ritz energy procedure. It is concluded that, for a wall with a clamped base vibrating in accordance with the given stipulations, the fundamental period is proportional to the square root of the face area of the wall. When the base of the wall is vibrating there is only one critical period, and it varies with the height of the wall. The factor of proportionality should take into account the material of which the wall is composed. For designing unframed walls, subjected to dynamic loads in their plane, where the applied shear is to be taken as some constant times the dead load at the base of the wall, the recommended lateral force requirements of the Seismology Committee of the Structural Engineers Association of California, as set forth in 1959, seem adequate as modified above. / Ph. D.

Sound pressure radiating from vibrating structures submerged in fluid, as in the case of a vibrating panel in a submarine hull, is usually undesirable. An optimal control methodology for the suppression of far-field acoustic radiation pressure from submerged structures has been developed by Meirovitch (ref. 1). The linear modal state feedback control law developed implies that the full state (displacements and velocities) is available, perhaps through measurements. However, in practice, it is not always feasible to measure the full modal state vector for feedback. To permit practical implementation of the feedback control law, an optimal stochastic state estimator, or Kalman-Bucy filter, has been developed here for incorporation into the control system design. The development has been specialized to a uniform simply supported rectangular plate. / Master of Science

The influence of tip clearance and angle of attack upon the mid-span unsteady pressure response of a vibrating flat plate was investigated experimentally. Unsteady pressure measurements were taken for a variety of incidence angles, vibration frequencies and tip clearances over a Mach number range of 0.2 to 0.6. It was found that changes in tip clearance had an effect on measured pressure fluctuations at higher angles of attack and larger Mach numbers. It was also observed that the amplitude of the unsteady pressure increased as the incidence angle was increased. The plate was mechanically induced to oscillate in translation, simulating the flISt bending mode. Averaged Fast Fourier Transforms were used to determine pressure oscillation amplitudes and phase lags with respect to the plate motion. / Master of Science