Vibration Of Annular Sector Plates Research Articles

Nonlinear large-amplitude vibration analysis of annular sector plates made of FGMs subjected to cooling shock

A numerical approach is developed in the present article to study the geometrically nonlinear vibrations of annular sector plates made of functionally graded materials (FGMs) due to being exposed to cooling shock. The first-order shear deformation plate theory (FSDT) is used in the modeling of plate, and the governing equations of motion are derived based on Hamilton's principle considering von Kármán nonlinear kinematic relations. It is also considered that the properties of FGMs are dependent on temperature and distribution of materials. According to the uncoupled thermoelasticity theory, the temperature distribution is obtained via 1D Fourier-type transient heat conduction equation. To numerically solve the problem, the generalized differential quadrature (GDQ) method and Newmark-beta integration scheme are utilized. Considering two different types of thermal loading as cooling shock, the effects of various parameters including thermal load rapidity time, geometry, magnitude of thermal load and material properties on the large-amplitude vibrations of annular sector plates are investigated.

Rapid heating vibrations of FGM annular sector plates

In this paper, thermally induced vibration of annular sector plate made of functionally graded materials is analyzed. All of the thermomechanical properties of the FGM media are considered to be temperature dependent. Based on the uncoupled linear thermoelasticity theory, the one-dimensional transient Fourier type of heat conduction equation is established. The top and bottom surfaces of the plate are under various types of rapid heating boundary conditions. Due to the temperature dependency of the material properties, heat conduction equation becomes nonlinear. Therefore, a numerical method should be adopted. First, the generalized differential quadrature method (GDQM) is implemented to discretize the heat conduction equation across the plate thickness. Next, the governing system of time-dependent ordinary differential equations is solved using the successive Crank–Nicolson time marching technique. The obtained thermal force and thermal moment resultants at each time step from temperature profile are applied to the equations of motion. The equations of motion, based on the first-order shear deformation theory (FSDT), are derived with the aid of the Hamilton principle. Using the GDQM, two-dimensional domain of the sector plate and suitable boundary conditions are divided into a number of nodal points and differential equations are turned into a system of ordinary differential equations. To obtain the unknown displacement vector at any time, a direct integration method based on the Newmark time marching scheme is utilized. Comparison investigations are performed to validate the formulation and solution method of the present research. Various examples are demonstrated to discuss the influences of effective parameters such as power law index in the FGM formulation, thickness of the plate, temperature dependency, sector opening angle, values of the radius, in-plane boundary conditions, and type of rapid heating boundary conditions on thermally induced response of the FGM plate under thermal shock.

Effects of CNTs waviness and aspect ratio on vibrational response of FG-sector plate

This paper is motivated by the lack of studies in the technical literature concerning to the influence of carbonnanotubes (CNTs) waviness and aspect ratio on the vibrational behavior of functionally graded nanocomposite annular sector plates resting on two-parameter elastic foundations. The carbon nanotube-reinforced (CNTR) plate has smooth variation of CNT fraction based on the power-law distribution in the thickness direction, and the material properties are also estimated by the extended rule of mixture. In this study, the classical theory concerning the mechanical efficiency of a matrix embedding finite length fibers has been modified by introducing the tube-to-tube random contact, which explicitly accounts for the progressive reduction of the tubes\' effective aspect ratio as the filler content increases. Parametric studies are carried out to highlight the influence of CNTs volume fraction, waviness and aspect ratio, boundary conditions and elastic foundation on vibrational behavior of FG-CNT thick sectorial plates. The study is carried out based on three-dimensional theory of elasticity and in contrary to two-dimensional theories, such as classical, the first- and the higher-order shear deformation plate theories, this approach does not neglect transverse normal deformations. The annular sector plate is assumed to be simply supported in the radial edges while any arbitrary boundary conditions are applied to the other two circular edges including simply supported, clamped and free. For an overall comprehension on 3-D vibration of annular sector plates, some mode shape contour plots are reported in this research work.

Free in-plane vibrations of annular sector plates with elastic boundary supports

In this investigation, a generalized Fourier series method is proposed for the in-plane vibration analysis of annular sector plates with elastic restraints along each of its edges. The in-plane displacement fields are universally expressed as a new form of trigonometric series expansions with a drastically improved convergence as compared with the conventional Fourier series. The expansion coefficients are considered as the generalized coordinates, and determined using the Rayleigh-Ritz technique. Several examples are presented to demonstrate the effectiveness and reliability of the current method for predicting the modal characteristics of annular sector plates with various cutout ratios and sector angles under different boundary conditions. It is also shown that annular and circle plates can be readily included as the special cases of the annular sector plates when the sector angle is set equal to 360°.

3-D vibration analysis of annular sector plates using the Chebyshev–Ritz method

The three-dimensional free vibration of annular sector plates with various boundary conditions is studied by means of the Chebyshev–Ritz method. The analysis is based on the three-dimensional small strain linear elasticity theory. The product of Chebyshev polynomials satisfying the necessary boundary conditions is selected as admissible functions in such a way that the governing eigenvalue equation can be conveniently derived through an optimization process by the Ritz method. The boundary functions guarantee the satisfaction of the geometric boundary conditions of the plates and the Chebyshev polynomials provide the robustness for numerical calculation. The present study provides a full vibration spectrum for the thick annular sector plates, which cannot be given by the two-dimensional (2-D) theories such as the Mindlin theory. Comprehensive numerical results with high accuracy are systematically produced, which can be used as benchmark to evaluate other numerical methods. The effect of radius ratio, thickness ratio and sector angle on natural frequencies of the plates with a sector angle from 120° to 360° is discussed in detail. The three-dimensional vibration solutions for plates with a re-entrant sector angle (larger than 180°) and shallow helicoidal shells (sector angle larger than 360°) with a small helix angle are presented for the first time.

Vibration of annular sector plates from three-dimensional analysis

Accurate three-dimensional elasticity solutions are presented for the free-vibration analysis of annular sector plates with arbitrary (but continuous) boundary conditions. The Ritz procedure is used to minimize the integral energy functional, which is formulated in a cylindrical polar coordinate system, in the derivation of the governing eigenvalue equation. The vibratory characteristics of annular sector plates with variations in boundary condition, sector angle, and thickness are examined. The accuracy of the method is validated through appropriate convergence and comparison studies. The first known results presented here provide a basis for understanding the physics of vibrations of annular sector plates and can be used to assess numerical results obtained from various refined two-dimensional theories.

On the use of 2-D orthogonal polynomials in the Rayleigh-Ritz method for flexural vibration of annular sector plates of arbitrary shape

This paper presents a simple, computationally efficient and relatively accurate numerical model for the study of free vibration behaviour of sectorial plates of arbitrary shape. The approach is developed based on the Rayleigh-Ritz method with a set of 2-D orthogonal polynomials as the admissible displacement function where the governing eigenvalue equation is derived by minimizing the energy functional with respect to each of the assumed unknown coefficients. In this paper, several examples are solved and the results obtained are compared with the established solutions to demonstrate the accuracy and flexibility of the present method. New results are also determined for several sectorial plates with different geometrical shapes and combinations of boundary conditions.

Vibration of annular sector plates using spline strip method

AbstractThis paper presents the vibrations of annular sectorial plates with two opposite radial edges simply supported using the spline strip method. To demonstrate the convergence and accuracy of the present method, several examples are solved, and results are compared with those obtained by the analytical method and by other numerical methods. Stable convergence and excellent accuracy are obtained using the high‐order spline strip models. Accurate frequencies of annular sector plates with different sector angles, radii ratios and tapered thickness in the angular direction are shown in tabular form.

Application of the spline element method to analyze vibration of annular sector plates

This paper deals with the vibrations of isotropic annular sector plates with arbitrary boundary conditions using the spline element method. To demonstrate the accuracy of the methods, several examples are solved, and results are compared with those obtained by analytical methods and by other numerical methods. Good accuracy is obtained. Frequencies of annular sector plates with free edges and with point supports are presented.

Free vibration of annular sector plates by an integral equation technique

A numerical method is presented for the free vibration analysis of polar orthotropic clamped annular sector plates. The results are compared with analytical and experimental values of other investigators. A parametric study has been done by varying the sector angle and radii ratio. The frequencies for isotropic and orthotropic cases are presented in the form of graphs.

Free vibration of annular sector plates with edges possessing different degrees of rotational restraint

Results for the natural frequencies of annular sector plates possessing different degrees of elastic restraint along the edges are presented. The analysis is based on a numerical method developed by the author. The functions in the circumferential direction satisfying the boundary conditions along the radial edges, which are required in the analysis, are indicated. To the best of the author's knowledge, no previous results exist for such plates.

A semi-analytic solution for free vibration of annular sector plates

The paper describes a semi-analytical method in which the basic function in the circumferential direction satisfying the boundary conditions of the radial edges is substituted into the free vibration equation of the curved plate. By a suitable transformation, an ordinary differential equation is obtained. The resulting equation is solved by a finite difference technique. Tabulated results have been presented for annular sector plates possessing different boundary conditions. Excellent accuracy has been obtained wherever comparisons have been possible.