Thin Annular Plates Research Articles

Harmonic response analysis of a circular plate subjected to moving point loads using Green’s function approach

Dynamics of circular plates under moving loads are an important class of problems in manufacturing industry, for example, cutting thin annular plates using point cutting tools and wood/bar/pipe cutting saws. These tools are subjected to excessive vibration during operation. Its dynamic behavior needs to be analyzed rigorously. In the present work, dynamics of a circular plate subjected to moving harmonic point load is analyzed using Kirchhoff’s plate hypothesis. In order to capture the geometric nonlinearity, the von Kármán strain displacement relation is used. Constitutive relations are developed using a complex modulus. Further, extended Hamilton’s principle is used to obtain the nonlinear coupled governing equations in radial, tangential, and transverse directions. Governing equations are discretized using Galerkin’s method. The novelty in the present work is to develop Green’s function incorporating viscous damping to obtain the plate response subjected to arbitrary moving load for general boundary conditions. The proposed method is capable to obtain the plate response due to single/multi point moving loads. In order to verify the accuracy of the proposed method, Green’s function solution is compared with the numerical solution obtained from the RK4 method and shows good agreement. Further, resonance instabilities are studied, which occur when a single point load rotates with critical angular velocity. Subsequently, the response of the circular plate under three-point loads, each having a magnitude of 1/3rd of the single moving point load moving circularly in a constant phase has been discussed. In the end, a parametric study of hysteresis damping on the dimensionless deflection of the plate is carried out.

Dynamic characteristics of a relief valve captured by a combination of fluid–solid coupling and feedforward neural network (FNN)

Considering multidisciplinary characteristics of thin plate vibration, fluid–solid coupling, and other aspects of a relief valve controlled by annular thin plates, a dynamic finite element (FE) model in view of fluid–solid coupling is firstly established for capturing relationships between dynamic characteristics of crucial indexes and partial working conditions. Secondly, the partial dataset of FE model under different conditions is statistically analyzed, and it will be utilized to train the feedforward neural network (FNN) model. The training process of FNN could be completed if results drawn from the FNN model are highly consistent with those of the FE model. Thirdly, dynamic characteristics under more conditions will be predicted through such a trained model, and dynamic behaviors from the FE model for same conditions of the FNN model are also obtained. Finally, comparing with results from the FE model, the maximum absolute error of steady-state displacement from the FNN is 0.0052 mm in an instance, thus verifying the rationality of this combined method. Consequently, such a combination of the FE model and the FNN model presents high accuracy and avoids repeated calculations of FE model with long times.

Free Vibration of Annular Plates Constrained by Translational and/or Rotational Springs on Outer and Inner Edges

This paper presents comprehensive lists of natural frequencies for vibration of thin isotropic annular plates with outer and inner edges elastically constrained by translational and rotational springs. A method is extended from a previously presented Ritz approach to include effects of the springs. In numerical examples, sixteen models are considered to cover all general cases of locating translational and/or rotational springs on outer and/or inner edges in addition to classical boundary conditions of free and simple supported edges. After convergence study and comparison test in specific cases are made to validate solution accuracy, frequency parameters of the sixteen models are summarized and frequency variations with increasing spring stiffness are illustrated.

Free vibration analysis of circular and annular thin plates based on crack characteristics

This study presents the effect of vertical and horizontal oriented cracks on free vibration response for circular and annular thin plates. To investigate the dynamic behavior of the damaged circular structures the cracks are modeled separately considering horizontal/vertical orientations, ten different locations, and four crack sizes. For annular thin plates, vertical and horizontal oriented cracks are placed in the middle between the outer and inner edges to investigate the effect of crack directions. The first five resonant frequencies, and the corresponding mode shapes of the cracked circular and annular plates, are obtained by employing the Finite Element Method. The free vibration analyses are conducted considering clamped boundary conditions for circular plates and clamped-clamped, clamped-free, and free-clamped boundary conditions for annular plates. The results are presented and interpreted considering the differences in the non-dimensional frequencies and the mode shapes of those structures. According to the findings, it is seen that depending on its location and size, a crack can change the mode shapes by accumulating the bending regions around it. Besides, a crack may also change the number of bending regions that occurred in the mode shapes of the circular or annular structures.

Magnetoelastic nonlinear free vibration of an annular plate under a nonuniform magnetic field of the long straight current‐carrying wire

AbstractIn this article, the magnetoelastic nonlinear free vibration of a thin conductive annular plate under the nonuniform toroidal magnetic field is investigated, where the magnetic field is generated by a long straight current‐carrying wire. Based on electromagnetic theory, expressions of the nonuniform magnetic field, electromagnetic forces are deduced. The nonlinear free vibration equation of plate is derived by using the Hamilton principle. Considering different boundary conditions, the axisymmetric vibration differential equation is achieved by the Galerkin method. The method of multiple scales is employed to obtain the second‐order approximate analytical solutions and natural frequency. In numerical calculation, the characteristics of nonlinear natural vibration and singular points varying with different parameters, for example, current intensity and plate size, are presented, respectively. The results indicate that, with the increase of current, natural frequency increases and then stabilizes in the inner‐clamped and out‐free (C‐F) boundary, but decreases and stabilizes in the cases of the inner‐clamped and out‐clamped (C‐C) boundary or inner‐simply and out‐simply (S‐S) boundary. Moreover, when wire current is off, natural frequency maintains a constant, where the equilibrium solution of the system is a center. In addition, for all boundary conditions, natural frequency increases significantly with increase of internal radius and thickness, but decreases with the increase of outer radius.

Free Vibration Analysis of Spinning Sandwich Annular Plates with Functionally Graded Graphene Nanoplatelet Reinforced Porous Core.

This paper conducted the free vibration analysis of a sandwich annular thin plate with whirl motion. The upper and lower faces of the annular plate are made of uniform solid metal, while its core is porous foamed metal reinforced by graphene nanoplatelets (GPLs). Both uniform and non-uniform distributions of GPLs and porosity along the direction of plate thickness which leads to a functionally graded (FG) core are taken into account. The effective material properties including Young’s modulus, Poisson’s ratio and mass density are calculated by employing the Halpin–Tsai model and the rule of mixture, respectively. Based on the Kirchhoff plate theory, the differential equations of motion are derived by applying the Lagrange’s equation. Then, the assumed mode method is utilized to obtain free vibration behaviors of the sandwich annular plate. The finite element method is adopted to verify the present model and vibration analysis. The effects of porosity coefficient, porosity distribution, graphene nanoplatelet (GPL) distribution, graphene nanoplatelet (GPL) weight fraction, graphene nanoplatelet length-to-thickness ratio (GPL-LTR), graphene nanoplatelet length-to-width ratio (GPL-LWR), spinning speed, outer radius-to-thickness ratio and inner radius-to-thickness ratio of the plate, are examined in detail.

Reduced integration‐based solid and solid‐shell finite elements for gradient‐extended damage

AbstractThe present contribution is concerned with the incorporation of gradient‐extended damage into reduced integration‐based continuum finite elements. To this end, the purely mechanical low‐order solid and solid‐shell elements based on the isoparametric concept are combined with a gradient extended two‐surface damage plasticity model. Due to a tailored combination of reduced integration with hourglass stabilization, the enhanced assumed strain (EAS) method and in case of the solid‐shell the assumed natural strain (ANS) method, the most dominant locking phenomena are eliminated. A polynomial approximation of the strain‐like as well as the stress‐like quantities within the weak forms enables the definition of a suitable hourglass stabilization. In this way, the element stiffness contributions coming from the hourglass stabilization can be determined analytically, since they represent polynomials with respect to Cartesian coordinates. Two representative numerical examples of an elasto‐plastic asymmetrically notched specimen as well as an elastic thin annular plate reveal the accuracy and efficiency of the proposed methodology. Besides the ability to deliver mesh independent results, the framework is especially suitable for constrained situations in which conventional low‐order finite elements suffer from well‐known locking phenomena.

Analysis of sound radiation from a vibrating elastically supported annular plate using compatibility layer and radial polynomials

This study investigates the problem of sound radiation from a thin vibrating annular plate. The plate is supported elastically at its circumference and embedded into the bottom of a circular cavity in a rigid plane. The selected model provides results covering the occurrence of sound radiation from the thin annular plates with any arbitrary boundary configurations. The cavity and the half-space are filled with air, and the air in the cavity plays the role of the compatibility layer. Therefore, the vibrations of the plate are coupled with that of air. Application of the compatibility layer allows the decomposition of the coupled problem of sound radiation to two problems: one for the vibration of the plate coupled with the cavity and the other for the sound radiation from the cavity to the half-space. Consequently, the plate is not coupled to the half-space directly. Therefore, the modal impedance coefficient of the plate is not required to solve the problem. The coefficients of the cavity are sufficient for the calculations, and they can be calculated significantly faster than in the case of performing a numerical integration while simultaneously satisfying the assumed accuracy by using radial polynomials. This also makes it possible to simplify numerical calculations. Finally, the total acoustic power, the acoustic pressure, and the vibration velocity have been obtained using the proposed approach and analyzed numerically. In the specific case when the cavity depth is reduced to zero, the nondimensionalized added virtual mass incremental as well as the modal impedance coefficients of the plate are calculated. Using the obtained results, the resonant frequencies can be accurately determined within a broad frequency band. The described method can be used for calculating the sound radiation for any arbitrary boundary configuration of the thin annular plates. In addition, the proposed approximation reduces the time required for the numerical calculations by roughly 2000 times compared to the time required for the numerical integration.

Monitoring electron energies during FLASH irradiations

When relativistic electrons are used to irradiate tissues, such as during FLASH pre-clinical irradiations, the electron beam energy is one of the critical parameters that determine the dose distribution. Moreover, during such irradiations, linear accelerators (linacs) usually operate with significant beam loading, where a small change in the accelerator output current can lead to beam energy reduction. Optimisation of the tuning of the accelerator’s radio frequency system is often required. We describe here a robust, easy-to-use device for non-interceptive monitoring of potential variations in the electron beam energy during every linac macro-pulse of an irradiation run. Our approach monitors the accelerated electron fringe beam using two unbiased aluminium annular charge collection plates, positioned in the beam path and with apertures (5 cm in diameter) for the central beam. These plates are complemented by two thin annular screening plates to eliminate crosstalk and equalise the capacitances of the charge collection plates. The ratio of the charge picked up on the downstream collection plate to the sum of charges picked up on the both plates is sensitive to the beam energy and to changes in the energy spectrum shape. The energy sensitivity range is optimised to the investigated beam by the choice of thickness of the first plate. We present simulation and measurement data using electrons generated by a nominal 6 MeV energy linac as well as information on the design, the practical implementation and the use of this monitor.

Generalized Reissner analysis of large axisymmetric deflections of thin circular and annular plates

A generalized Reissner theory for axisymmetric problems of thin circular and annular plates is applied to various problems. The plates are assumed to be linearly elastic, and shear deformation is neglected. Large deflections, rotations, and strains are allowed. The plate edges may be clamped or simply supported, movable or immovable. The types of loading considered are uniform pressure, central concentrated load, uniform edge moment, full ponding, and central rigid hub pushed downward (with or without an elastic foundation). In some cases, the effects of Poisson's ratio and the radius-to-thickness ratio are investigated, and the maximum deflections are compared to those based on von Kármán theory and Reissner theory.

Free vibration of FGM annular sectorial plates with arbitrary boundary supports and large sector angles

An original work is presented for the free vibration analysis of thin functionally graded material (FGM) annular sectorial plates with arbitrary boundary supports using the weak form quadrature element method (QEM). Novel C 1 quadrature annular sectorial plate element is developed based on the physically neutral surface thin plate theory in cylindrical coordinate system. Three displacement functions are used to avoid the introduction of the mixed second order derivative degree of freedom at corner points. The derived explicit formulations with a varying node number can be applied for any type of node. A way to define the elastic spring constants is proposed to ensure the independent of the non-dimensional frequency parameter on the power-law exponent. The convergence and accuracy of the present model are validated by comparing the obtained results with those available in literatures. It is demonstrated that the proposed model is simple, straightforward and has good performance. A wide range of numerical results are reported to demonstrate the effect of the sector angle, ratio of inner/outer radii, and boundary conditions on the vibration behavior of the thin annular sectorial plate. Some tabulated results should be new which may serve as a reference for verifying numerical solutions obtained by other methods.

Analytical Calculation of Large Deflection of Annular Thin Plate of Damping Valve

Aiming at the phenomenon of bending deformation caused by the pressure difference of the throttle plate in the working process of the damping valve, the analytical calculation of the large deflection of the annular thin plate was carried out. On the basis of extracting the physical model of the thin plate, the boundary conditions and initial conditions of the annular thin plate were set by using the Qian’s perturbation method, and the nonlinear large deflection deformation analytical formula was derived; Through the finite element modeling, the accurate numerical solution of thin plate deformation was obtained, and the key parameters of the analytical formula were corrected by the least squares method. Through verification, the modified analytical formula had the characteristics of easy calculation and high calculation accuracy, and the influence law of the large deflection of the throttle valve was studied, which provided the necessary theoretical support for the accurate design of the damping valve performance.

Analysis of Thermal-Bending Stresses in a Simply Supported Annular Sector Plate

The present article deals with the analysis of thermal-bending stresses in a heated thin annular sector plate with simply supported boundary condition under transient temperature distribution using Berger’s approximate methods. The sectional heat supply is on the top face of the plate whereas the bottom face is kept at zero temperature. In this study, the solution of heat conduction is obtained by the classical method. The thermal moment is derived on the basis of temperature distribution, and its stresses are obtained using thermally induce resultant moment and resultant forces. The numerical calculations are obtained for the aluminium plate in the form of an infinite series involving Bessel functions, and the results for temperature, deflection, resultant bending moments and thermal stresses have been illustrated graphically with the help of MATHEMATICA software.

Free vibration of circular annular plate with different boundary conditions

This paper deals with the numerical simulation of free vibration analysis of a thin circular annular plate for various boundary conditions at the outer edge and inner edge. Classical plate theory is used to derive the governing differential equation for the transverse deflection of the thin isotropic plate. The finite element method is used to evaluate the first six natural frequencies and mode shapes of the thin uniform circular annular plate with radius ratios (r1/r2) for different boundary conditions. These natural frequencies results are compared with those available in the literature. The results are verified with classical plate theory with our Abaqus results and checked with the previous research literature on the topic.

Sound radiation by a vibrating annular plate using radial polynomials and spectral mapping.

This study deals with sound radiation from a thin clamped annular plate. The fluid loading effect is included. Although it is a classical problem, using direct numerical integration to solve such a problem is usually troublesome and time consuming since the coupling integrals have certain singularities. Therefore, this study applies radial polynomials to improve the time efficiency of numerical calculations with no loss of accuracy. As radial polynomials are usually used for circular geometry, spectral mapping has been proposed to perform the necessary conversion from circular to annular geometry. The conversion is performed in the wavenumber domain. Spectral mapping makes it possible to efficiently obtain a number of numerical results for the acoustic power and acoustic field generated.

Free vibration and buckling analyses of functionally graded annular thin sector plate in-plane loads using GDQM

In the present study, buckling and free vibration analyses of annular thin sector plate made of functionally graded materials (FGMs) resting on visco-elastic Pasternak foundation, subjected to external radial, circumferential and shear in-plane loads is investigated. Material properties are assumed to vary along the thickness according to an power law with Poisson\'s ratio held constant. First, based on the classical plate theory (CPT), the governing equation of motion is derived using Hamilton\'s principle and then is solved using the generalized differential quadrature method (GDQM). Numerical results are compared to those available in the literature to validate the convergence and accuracy of the present approach. Finally, the effects of power-law exponent, ratio of radii, thickness of the plate, sector angle, and coefficients of foundation on the fundamental and higher natural frequencies of transverse vibration and critical buckling loads are considered for various boundary conditions. Also, vibration and buckling mode shapes of functionally graded (FG) sector plate have been shown in this research. One of the important obtained results from this work show that ratio of the frequency of FG annular sector plate to the corresponding values of homogeneous plate are independent from boundary conditions and frequency number.

Refined Model of the Dynamic Behavior of Flexible Reinforced Shallow Shells Made from Nonlinear Elastic Materials

An initial-boundary-value problem is formulated for the dynamic deformation of flexible reinforced shallow shells made of composite materials whose components are nonlinearly elastic. The geometric nonlinearity is taken into account in the Karman approximation. The resulting equations allow one, with different degrees of accuracy, to determine the strain state of such thin-wall structures with account of their weakened resistance to transverse shears. The relatios of the traditional nonclassical Reddy theory follow from these equations as a special case. Numerical integration of the problem stated is carried out on the basis of the method of steps in time by using an explicit “cross”-type scheme. Thin shallow spherical shells and flexible annular plates are considered. They are reinforced in the radial and circumferential directions, contain an absolutely rigid insert, and are clamped along the outer edge. The dynamic behavior of such structures is studied in relation to the level of pressure caused by the blast of air waves and to the form (convex or concave of the face) to which this pressure is applied. It is revealed that, at time intervals exceeding 0.1 s, mechanical behavior of the reinforced shallow shells and plates calculated by the Reddy theory significantly differs from the dynamic response determined according to the refined theory proposed.

Differential Quadrature Free Vibration Analysis of Functionally Graded Thin Annular Sector Plates in Thermal Environments

In this paper, the free vibration behavior of functionally graded (FG) thin annular sector plates in thermal environment is studied using the differential quadrature method (DQM). The material properties of the FG plate are assumed to be temperature dependent and vary continuously through the thickness, according to the power-law distribution of the volume fraction of the constituents. The nonlinear temperature distribution along the thickness direction of the plate is considered. Based on the classical plate theory, the governing differential equations of motion of the plate are derived and solved numerically using DQM. The natural frequencies of thin FG annular sector plates in thermal environment under various combinations of clamped, free, and simply supported boundary conditions (BCs) are presented for the first time. To ensure the accuracy of the method, the natural frequencies of a pure metallic plate are calculated and compared with those existing in the literature for the homogeneous plate where the results are in good agreement. The effects of temperature field, BCs, volume fraction exponent, radius ratio, and the sector angle on the free vibrations of the FG-plate are examined.

Comparisons of methods for solving static deflections of a thin annular plate

A highly accurate method for obtaining static deflections of a thin annular plate is helpful to effectively design the complicated structures with these plates. There have been numerous methods to achieve such a target. However, there is no direct technical literature for comparing these methods comprehensively. Therefore, the current study aims at performing comparison of three methods, optimization method (OM), finite element method (FEM), and harmonic differential quadrature (HDQ) method. Combining an instance, the comparisons give us insight into high accuracy and consistency of each other, showing high accuracy of the methods in this field. Compared with the results of FEM, the maximum error, less than 1%, demonstrates that the accuracy of the OM is high enough. Combining the small errors, the excellent stability of those brings a reliable method in this field. The maximum error and fluctuation drawn from the HDQ are evidently larger than those of the OM, and it is difficult to obtain results with higher accuracy based on the HDQ. Finally, the work described here suggests that the OM can be utilized to deal with such a complex problem in view of engineering and theory, and the HDQ method is more suitable to study the method for solving very complex partial differential systems of high order.

A simplified approach for the thermoelastic large deflection in the thin clamped annular sector plate

ABSTRACTThe present article is concerned with analysis of large deflection of a heated thin annular sector plate with clamped edges under transient temperature distribution using Berger’s approximate methods. The prescribed surface temperature is at the top face of the plate whereas the bottom face is kept at zero temperature. In this study, the Laplace transform as well as the classical method have been used for the solution of heat conduction equation. The thermal moment is derived on the basis of temperature distribution, and its stresses are obtained using resultant bending moment and resultant forces per unit length. The calculations are obtained for the aluminium plate in the form of an infinite series involving Bessel functions, and the numerical results for temperature, deflection, resultant bending moments, and thermal stresses have been illustrated by graphs.