Natural Frequencies Of Annular Plates Research Articles

Study on fundamental damped frequency parameters of plates made of FG porous material: a close-form solution

In this article, the natural frequencies of viscoelastic annular plates made of functionally graded porous (FGP) materials with assorted boundary conditions are investigated in a closed-form solution for the first time. The porosity variation is a function of the plate thickness according to symmetric and asymmetric distributions. The governing motion equations are derived by considering the first-order shear deformation theory (FSDT), energy method, and the calculus of variations. The viscoelastic properties of the structure are also captured by means of the standard linear solid (SLS) model. The equations of motion, which are a system of partial differential equations with variable coefficients, are then solved using the perturbations methodology. To do so, the equations with variable coefficients are converted to a system with constant coefficients, and the natural frequencies are found analytically in a closed-form solution without any approximation. In addition, a user-defined field (USDFLD) code is developed, which introduces solution-dependent material properties in Abaqus/Standard analyzes, for evaluating the accuracy of the analytical results. Finally, the impact of various parameters including inner-to-outer radius ratios, thickness-to-radius ratios, porosity coefficient, damping exponent, different combinations of boundary conditions, and porosity patterns on the natural frequencies of annular plates are examined.

Modal Analysis of Annular Plate with Crack and its Effect on Natural Frequency

The study of the dynamic behavior of annular plates with circumferential cracks can find many applications in several machine components such as flywheels, clutch plates, compact discs etc. A crack on a structural member introduces a local increase of flexibility in that region then; this affects the static and dynamic characteristics. The effects of cracks on the dynamic characteristics of structures, especially on the natural frequencies and modes, were extensively studied. In this study, the natural frequencies of annular plates with circumferential cracks are investigated by using finite element method. The cracks were non-propagating and open. The annular plate with different cracks was subjected to different boundary conditions and final variation in the natural frequency was obtained, which was compared with the theoretical frequency and the change in the natural frequency was studied. The results of this study with improved elements are compared with the theoretical results in the literature. It is observed that the location and the number of cracks have various effects on the natural frequencies related to the vibration modes.

Benchmark solution for transverse vibration of annular Reddy plates

The exact closed-form solution for freely vibrating annular thick plates is presented on the basis of the Reddy's higher-order shear deformation plate theory. Several combinations of classical boundary conditions including, free, soft simply supported, hard simply supported and clamped, are applied at the inner and outer edges of annular plates. Hamiltonian and minimum potential energy principles are employed to derive the equations of dynamic equilibrium and natural boundary conditions of the plate. To validate the accuracy and effectiveness of the present formulation, direct comparisons are made between our results and those from the literature. In parametric studies, the first eight natural frequencies of annular plates with different combinations of boundary conditions are tabulated for various values of the inner–outer radius ratios and thickness–radius ratios. A set of three-dimensional vibration mode shapes along with their corresponding contour plots are presented to illustrate transverse displacements of annular thick plates. Since the results of the present exact solution are the closest to those of exact three-dimensional elasticity solution, they can serve as a reliable benchmark for checking the accuracy of future numerical and analytical solutions.

Natural frequencies of annular plates with circumferential cracks by means of sector element

In this study, the natural frequencies of annular plates with circumferential cracks are investigated by using finite element method. The cracks were non-propagating and open. In the event of the selection of appropriate half sector angle, taking into account the closeness of the geometries of the sector and trapezoidal element, the flexibility matrix of the sector element with crack was obtained by the flexibility matrix of the trapezoidal element. Also, because of small half sector angles, the circumferential cracks in the elements were assumed to be straight cracks. The results of this study with improved elements are compared with the theoretical results in the literature. It is observed that the location and the number of cracks have various effects on the natural frequencies related to the vibration modes.

A study on annular plates with radial through cracks by means of sector type element

In this study, the natural frequencies of annular plates with periodic radial through cracks are investigated by means of finite element method. The position of the crack in the plate surface was selected as parallel to radial direction. Both the cracks emanating from the inner and outer boundary of the plate is non-propagating and open. The crack in the element was modeled by an additional flexibility matrix, the terms of which were calculated using fracture mechanics. For this purpose, an isoparametric sector type element with radial through crack of four nodes and three degrees of freedom at each node is considered. In the event of the selection of appropriate half sector angle, taking into account the closeness of the geometries of the sector and trapezoidal type element, the flexibility matrix of the sector type element with crack are derived by means of the flexibility matrix of the trapezoidal type element. The natural frequencies of annular plates are found for different length and number of cracks. The theoretical results are obtained with different boundary conditions. The obtained results of improved elements are compared with experimental results in the literature. It is observed that the increment of the length and number of cracks has various effects on the natural frequencies and mode shapes of the annular plate.

Effects of Thickness Variation for the Dynamic Characteristics of Annular Plates Including the Shear Deformation and Rotary Inertia.

The effects of thickness variation, shear deformation and rotary inertia of annular plates are studied by deriving the equations of motion. To solve the differential equation governing the motion of annular plates the Chebyshev collocation method is employed. The dimensionless frequencies are evaluated for different values of taper constant, thickness ratio, radii ratio and power. The results of an experimental investigation are also presented, and the agreement between the experimental values and the predicted values in theory is remarkably good. As a result of this study, it is found that the effects of rotatory inertia and shear deformation reduce the natural frequencies of annular plates for three boundary conditions. This study also showed that the natural frequencies of annular plates with thickness expressed by the nth power function are higher than those by the (n-1)th power function for positive values of taper constant, and vice versa for negative values of taper constant for three boundary conditions.

Axisymmetric vibration analysis of rotating annular plates by a 3D finite element

The axisymmetric vibration of spinning isotropic annular plates is herein analyzed by a modified axisymmetric finite element. Due to the three-dimensional-elasticity nature of the present approach, all the initial stresses induced by the centrifugal force are calculated and taken into account in the augmented strain energy expression, while only two are considered in the conventional analyses. Also, the whole three velocity components are considered in the inertial term instead of only one by conventional approaches. Natural frequencies of annular plates with different ratios of inner-to-outer radius, ratios of thickness-to-radius, rotational speeds, and boundary conditions are derived by the present approach and are compared with those by a plate-theory approach.

Vibrational characteristics of annular plates and rings of radially varying thinkness

In this paper, annular plates having thickness variation are studied by deriving the equations of motion on the basis of the Mindlin plate theory. The Chebyshev collocation method is employed to solve the differential equation governing the transverse motion of such plates. The dimensionless frequencies are evaluated for different values of taper constant (α), thickness ratio (hu). radii ratio (e) and power (n). The results of an experimental investigation are also presented, and the agreement between these findings and the predicted values in theory is remarkably good. As a result of this study, it is found that the effects of rotatory inertia and transverse shear deformation reduce the natural frequencies for all boundary conditions and for all values ofn. ho, ∈, a ands (mode number). This study also showed that the natural frequencies of annular plates with thickness expressed by the nth power function are higher than those by the (n−1)th power function for positive values of α, and vice versa for negative values ofe for all three boundary conditions. Moreover, there is a proof that the natural frequencies of annular plates tend to be higher as the taper constant decrease and/or as the radii ratio increase for all three boundary conditions and for all values ofn, s andho.

FREE VIBRATIONS OF ANNULAR PLATES COUPLED WITH FLUIDS

The natural frequencies of annular plates on an aperture of an infinite rigid wall and in contact with a fluid on one side are theoretically obtained by using the added mass approach. The fluid is assumed to be incompressible and inviscid and the velocity potential describes its irrotational motion. The Hankel transform is applied to solve the fluid–plate coupled system; boundary conditions are expressed by integral equations. Mode shapes are first assumed not to be modified by the fluid. Accurate numerical results are given for different plate boundary conditions; they are suitable for engineering applications. The accuracy of the assumed-modes approach is theoretically studied by using the Rayleigh–Ritz method that removes the simplifying hypothesis that dry and wet mode shapes are the same. Eigenfunctions of the plate vibrating in vacuum are assumed as admissible functions and the Rayleigh quotient for coupled vibration is used to obtain a Galerkin equation. It was found that the fundamental mode and frequency, for all the plate boundary conditions considered, is well estimated by the assumed-modes approach; higher modes are computed with less accuracy by this formula and for some enhanced applications the Rayleigh–Ritz approach is necessary.