Thin Circular Plate Research Articles

Hygrothermoelastic analysis of the nano-circular plate with memory effect

In hygrothermal environments, the coupling effects of temperature and moisture substantially impact deflection and stresses play a significant role. This study presents a coupled hygrothermoelastic model with non-Fourier and non-Fick effects established by introducing relaxation times or phase lags of heat and moisture flux accompanied by memory-dependent derivatives. The boundary value problem is formulated by considering a thin circular plate as an exemplary example, where the perimetric edge is clamped. The upper and lower edges of the plate is subjected to zero temperature, whereas the curved surface is exposed to hygrothermal shock. The closed-form solution of temperature and moisture distribution is obtained via the integral transform approach. The Fourier series expansion approach is used to calculate the numerical Laplace inversion. The effects of both heat and moisture flux relaxation times on the thermal deflection/stresses of the plate are analyzed and illustrated graphically.

Deep subwavelength sound absorption by a buckling plate resonator

Conventional sound-absorbing materials face great difficulties in achieving deep subwavelength broadband sound absorption due to the causal constraint on the minimum thickness. A buckling plate resonator is proposed to alleviate the causal constraint on the thickness of sound-absorbing materials. The resonator consists of a back cavity sealed by an elastic circular thin plate. When the plate is subjected to a uniform in-plane compressive force up to its critical load, it buckles and behaves as a negative stiffness spring. The positive stiffness of the back cavity is counteracted by the negative stiffness of the buckling plate, resulting in resonance absorption at the deep subwavelength scale. The sound absorption performance of the buckling plate resonator under different in-plane loads was measured in an impedance tube. Quasi-perfect sound absorption was observed in a buckling plate resonator with a thickness less than 1% of the wavelength corresponding to the resonance frequency. This work provides a solution for the design of ultra-thin broadband sound absorbers.

THERMAL BEHAVIOUR OF A CIRCULAR PLATE UNDER CAPUTO-FABRIZIO FRACTIONAL IMPACT WITH SECTIONAL HEATING

Recent advances in the understanding of the precise physical thermal behaviour of various solids under the effect of fractional-order derivatives have boosted the study of thermoelasticity, which is primarily important in various industrial designs of usable structural materials. We investigated a thin circular plate that was subjected to additional sectional heating on its top and lower surfaces while creating thermal insulation around its outside border. In this work, we maintained the heat transfer equation while accounting for the impact of Caputo-Fabrizio fraction-order derivatives. According to specified boundary constraints, the integral transformation approach is used to assess the analytical solution of the displacement, temperature change, and thermal stresses. Furthermore, various functions and fractional parameters are computed using the material properties of aluminium metal plates for numerical purposes.

Study on the effect of the excited area on the vibration characteristics of a thin circular plate

Abstract In this paper, the vibration modes and radiated directivity of a thin circular plate excited by a sandwich longitudinal vibration transducer are studied. The finite element model is set up to study the effect of the contact area between the transducer and the circular radiation plate on the vibration modes and the radiation characteristics. The first six vibration modes of the thin vibration plate are analyzed, and the results show that the vibration modes of the area-excited thin circular plate are different from those of the free vibration thin circular plate and the point-excited thin circular plate. Simulation results show that under different contact areas, the thin circular plate vibrates in different modes. With the increase of the excitation area, the frequencies of each mode of the circular plate gradually increase. Moreover, the excited area also affects the radiative directivity. With the increase of the excited area, the main lobe width of the radiative directivity becomes narrower.

Sensitivity Analysis of the Factors Affecting the Ground Heave Caused by Jet Grouting

Jet grouted piles are widely used to reduce post-construction settlement of soft clay roadbeds. Nevertheless, it is easy to cause ground heave due to the jet grouted pile. According to the analytical method and numerical method, a sensitivity analysis of the factors affecting ground heave caused by a single jet grouted pile was performed. It is found that the influence of each parameter on ground heave is in the following order: grout pump pressure > embankment load > soil type (including the cohesion, friction angle, and Young’s modulus) > pile diameter > pile length. Considering the effect of the pump pressure on the ground heave is more significant, based on the analytical method of ground heave caused by a single jet grouted pile combined with the solution of small-deflection bending of a circular thin plate, the calculation method for the suggested limit grout pressure for construction under different embankment heights was established. Suggested values of theoretical grout pump pressure were given to prevent ground heave from harming the pavement of operating highways. This study provides some theoretical basis for the subsequent research on the jet grouted pile.

Study on the radiation directivity of a ring-excited thin circular plate with a fixed boundary

Air-coupled transducer with a flat plate structure have many applications in the fields of ground weather observation, ultrasonic defoaming, and directional strong acoustic radiation. In this paper, an analytical equation of the far-field radiation directivity of a fixed boundary ring-excited thin circular plate (RTCP) is deduced using Rayleigh integration method. A finite element model of the RTCP is established, and the relationship between the far-field radiation directivity and the excitation position, excitation area and working frequency is studied by considering the third-order axisymmetric flexural vibration of the RTCP. Computation results show that, for a RTCP, the excitation position has more effect on its radiation directivity. When the plate is excited at the positions between first two nodes, the directivity can be enhanced. When the excitation position is in the trough of the normal displacement curve along radius direction, the side lobes of the radiation directivity of the RTCP are minimized. The area of excitation region has smaller influence on the frequency and radiation directivity of the RTCP. However, working frequency has a great influence on the radiation directivity of the RTCP. When the working frequency is close to the vibration frequency of the circular plate, the sound radiation directivity is the best. A prototype fixed boundary circular plate excited by a longitudinal sandwich transducer was designed and manufactured. For comparison, its finite element model was also setup to simulate its acoustic radiation directivity. Experimental results were found to be in agreement with the theoretical calculations and finite element simulation results.

Integrated analytical, experimental, and numerical study of vibrational response in clamped circular plates with pressurised air cavities

This paper presents a comprehensive investigation, comprised of analytical, experimental and numerical approaches, into the interaction between the sound field produced within a pressurised air-filled back cavity and the vibration of a thin clamped circular plate subjected to its pressure. Previous investigations into the interaction of sound fields with vibrating boundaries have primarily been restricted to non-pressurised cavities. We extend the application of the weak modal coupling method to derive a solution for this unexplored physical problem by utilising classic plate theory, Fourier-Bessel series formulation and the uncoupled solutions for a pressurised cylindrical sound field and a thin clamped circular plate with uniform radial tension. We describe the vibration response of the coupled system in terms of the uncoupled cavity and plate modes. Using a set of geometric and physical properties for the system, the resonance frequencies and response functions are calculated and the coupling between the sound field and the plate is investigated as a function of cavity pressure, providing novel fundamental insights into the physical system. We construct an experimental rig to embody the conditions of the analytical investigation for the purpose of validation. Finite element analysis (FEA), employing modal analysis and transient dynamic analysis, is used to further validate and extend the analytical insights while more accurately mirroring the experimental conditions. The response functions, mode shapes, resonance frequencies and their dependence on the cavity pressure were experimentally measured and numerically simulated, with direct comparisons made with the analytical model. A high degree of accuracy for the analytical model is validated along with its ability to describe the underlying physical phenomena. The validated analytical model is then leveraged to perform fundamental explorations into the modal contributions as a function of cavity pressure and a parametric sensitivity analysis on the effects of plate radii and plate thickness on the coupled system resonance frequencies.

Hybrid isogeometric-based analysis and experimental investigation on the dynamic response characteristics of a clamped circular plate partially in contact with fluid

The aim of this study was to analyze the effect of fluid presence on the dynamic response characteristics of the thin circular plate and the distortions associated with the wet mode shapes. To this end, series of numerical calculations and experimental measurements were performed to investigate the free vibration characteristics of clamped circular plate under in-vacuo conditions and when it is partially in contact with fluid. In the experimental studies, the clamped boundary conditions are imposed on the circular end plate of the horizontal rigid cylindrical tank by closely-spaced bolts, and measurements were performed based on roving hammer impact technique. The proposed numerical approach was divided into two parts based on linear hydroelasticity theory. In the first stage, the thin circular plate is considered to be under in-vacuo conditions, and the mathematical model for this problem is developed based on the isogeometric finite element method (IGAFEM). The fluid environment is introduced in the second stage of the study in which the generalized in-vacuo modal displacements constitute the boundary conditions of the potential flow problem. The influence of fluid medium is incorporated in the system of equations in the form of fluid added mass, and the corresponding fluid forces are calculated by the isogeometric boundary element method (IGABEM). It is observed that the fluid presence has significant effects on the dynamic response characteristics of the test structure, and the specifically, the distortions of wet mode shapes were noticeable due to presence of free surface of the water. Overall, it is found that the natural frequencies and corresponding mode shapes obtained by conducted experiments and adapted numerical framework are in favorable agreement.

Indentation of circular hyperelastic membrane with hole by cylindrical indenter

We consider the problem of indentation of a thin circular elastic plate with a hole in the centre by a cylindrical indenter under large strains. We use the nonlinear theory of isotropic incompressible hyperelastic membranes and consider a quasi-static process of indentation. We use Coulomb’s law to describe the friction at a contact. The goals of this study are to determine and to compare the effects of friction, material parameters and pre-stretch of the membrane on the indentation process.

Hygrothermoelastic large deflection behaviour in a thin circular plate with non-Fourier and non-Fick law

The coupled fractional dual phase-lag hygrothermoelasticity theory, developed using fractional calculus principles, extends classical Fourier’s and Fick’s laws to a time-fractional differential equation. The concept is applied to a thin circular plate that is exposed to hygrothermal loadings. The finite integral transform method and decoupled technique are utilized to create closed-form expressions for various factors such as temperature, moisture, large deflection, and stresses. The study compares the results of the dual phase-lag model with those of classical and hyperbolic models. The phase-lags parameters play a crucial role in regulating the heat and moisture transfer mechanism

On the boundary conditions for a thin circular plate conjugated to a massive body

The problem of deformation under the action of uniform pressure of a circular plate coupled with a massive base is considered, while the condition for the coupling of the plate with the base is modeled using boundary conditions of the generalized elastic embedding type, i.e. the relationship between the bending moment and forces at the edge of the plate with displacements and rotation angles through the compliance matrix. The main goal of the work is to study the influence of the elasticity of the embedding on the elastic response of the plate. The solution to the problem was obtained in the formulation of the linear theory of plates, the theory of membranes in the approximation of homogeneity of longitudinal forces, and the Foppl — von Karman theory, also in the approximation of the assumption of homogeneity of longitudinal forces. The values of the coefficients of the compliance matrix were obtained using the finite element method for the auxiliary problem and compared with the values of the coefficients obtained for related problems by analytical methods. Numerical results were obtained for an aluminum wafer on a silicon base. The obtained solution was compared with the solution obtained for the rigid embedment condition for all three models used. It is shown that in the case of large deflections (several plate thicknesses), taking into account the compliance of the embedment becomes essential.

Demonstration of nonlinear tuning curve vibration response using a homemade analog sweep spectrum analyzer

A super-heterodyne sweep spectrum analyzer has been designed (using analog circuitry), built and tested to cover the 5 to 1000 Hz range. The analyzer employs a linear ramp voltage versus time, driving a voltage-controlled oscillator VCO that generates a sweep from 32.768 kHz to 33.768 kHz in 70 s. An analog AD734 multiplies the VCO signal by a sinusoidal signal (generated by a watch crystal 32.768 kHz oscillator). The “mixer” signal is low-pass filtered and amplified to generate a 5-1000 Hz swept tone using a 2-in. speaker. The accelerometer (mounted on a thin circular clamped acrylic plate) vibration response involves multiplying this signal by the VCO, then filtering this “mixer” signal using a 4-stage watch crystal ladder filter with a 1 Hz bandwidth. This signal is squared and low-pass filtered to generate the time (converted to frequency) vs. mean-squared voltage response on an oscilloscope. In the demonstration, 300 g of 6 mm diameter glass beads are supported by the 11.4 cm diam, 3.2 mm thick plate and upper rigid cylindrical wall column. The nonlinear tuning curve vibration response is recorded for various incremental drive amplitudes to demonstrate that the tuning curve resonant frequency significantly decreases with increasing drive amplitude.

AXISYMMETRIC DYNAMIC BEHAVIOUR OF THIN OBLATE SHELLS

This paper presents a theoretical investigation of the axisymmetric free vibrations of an isotropic thin oblate spheroid shell. The analysis depends on the Rayliegh – Ritz's method. The non – shallow shell theory is used for the analysis. The analysis based on considering the oblate spheroid as a continuous system constructed from two spherical shell elements matched at the continuous boundaries.Throughout the results, it is shown that when the eccentricity reaches zero, an exact thin sphere solution is emerged and when the eccentricity equals one an exact thin circular plate solution is emerged. Therefore, the eccentricity of an oblate shell at medium value lies between these two values.It was found that the Rayleigh – Ritz's method is suitable for all eccentricities, while the literature showed that the Rayleigh's method is suitable for eccentricities less than 0.6.

Mechanical behavior of nanocircular plates under coupled surface and nonlocal effects by using molecular dynamics simulations

The equations of control for a circular thin plate under simple support are obtained by coupling the surface effect and non-local effect using the equilibrium method. Molecular dynamics simulations were used to calculate the residual surface strain energies at different crystal orientations. Different crystal orientations of the same material would affect its surface energy and thus result in different mechanical behaviors. The effects of residual surface strain energy and non-local parameters on the mechanical behavior of circular thin plates were analyzed. The analysis reveals that the surface effect and non-local effect on the deflection of the nanoplates are significant. The surface effect parameter of tensile stress increases the deflection, while the opposite is true for the surface effect parameter of compressive stress, and the non-local parameter decreases the deflection. The influence of surface effect and non-local effect is significant for smaller thicknesses and weakens for larger thicknesses.

Gyroscopically coupled in-plane and transverse vibrations of an annular free-clamped microplate

In this work, we construct and study a model of coupled plane-transverse oscillations of a circular thin plate with a concentric hole under the action of Coriolis and centrifugal inertia forces caused by the rotation of the system along an axis located in the plane of the plate. Equations of vibrations in partial derivatives are obtained using the variational principle of Hamilton - Ostrogradsky. Assuming the smallness of the angular velocity of rotation with respect to the frequency of the working skew-symmetric flexural form of the plate oscillations, an approximate analytical solution is found for both the radial and circumferential, and transverse components of the displacement field in the free oscillation mode. Using the Galerkin projection method, the problem was reduced to a system of two second-order linear differential equations for modal coordinates of mutually orthogonal basic skew-symmetric modes of the plate vibrations. It is found that the regime of initially excited harmonic oscillations in the presence of rotation is transformed into the regime of amplitude-modulated beats. Analytical expressions are found both for the frequency of the slow beat envelope and for the relative depth of their amplitude modulation. The fundamental possibility of determining the modulus of the projection of the angular velocity vector onto the plane of the plate from the measured value of the envelope frequency is shown. The problem of choosing the optimal geometric shape of the resonator from the point of view of maximizing the sensitivity of the system to changes in the value of the angular velocity of rotation is studied. The question of determining the direction of the projection of the angular velocity vector onto the plane of the plate from the measured depth of the amplitude modulation of the beat regime is considered.

Dynamic behavior of a clamped circular plate and strain energy representation (part I)

The first part of the paper presents a study regarding the dynamics behavior of thin circular plate clamped all around. An analytical investigation is performed and the results in terms of mode shapes are used to highlight the plate’s dynamics. The modal shapes are obtained using Bessel functions and their graphic representation is presented in 3D by using MS Excel software.

The Radiation Efficiency Measurements of Real System of a Thin Circular Plate Embedded Into a Thick Square Baffle

Most of sound sources are complex vibroacoustic objects consist of numerous elements. Some coupled vibrating plates of different shapes and sizes can be easily found in urban environments. The main aim of this study is to determine the sound radiation of coupled plates system of practical importance. The investigated vibroacoustic system consist of a thin circular plate coupled with a thick flat baffle with a circular hole. The circular plate has been mounted to the baffle’s hole using screws and two steel rings. The measurement setup was located inside a semi-anechoic chamber to assure the free field conditions. It was necessary to take into account the whole system surface to obtain the radiation efficiency based on the Hashimoto’s method. Such an approach can be troublesome and time-consuming. Therefore, the criterion has been proposed which allows the vibration velocity measurements and calculations to be performed only for the thin plate’s area. An alternative approach has been proposed based on the classical Rayleigh integral formula. Its advantage is a simpler implementation in a computer code. The obtained results have been compared with the theoretical results obtained for the elastically supported circular plate. A good agreement has been obtained at low frequencies.

Variational Solution and Numerical Simulation of Bimodular Functionally Graded Thin Circular Plates under Large Deformation

In this study, the variational method and numerical simulation technique were used to solve the problem of bimodular functionally graded thin plates under large deformation. During the application of the variational method, the functional was established on the elastic strain energy of the plate while the variation in the functional was realized by changing undetermined coefficients in the functional. As a result, the classical Ritz method was adopted to obtain the important relationship between load and maximum deflection that is of great concern in engineering design. At the same time, the numerical simulation technique was also utilized by applying the software ABAQUS6.14.4, in which the bimodular effect and functionally graded properties of the materials were simulated by subareas in tension and compression, as well as the layering along the direction of plate thickness, respectively. This study indicates that the numerical simulation results agree with those from the variational solution, by comparing the maximum deflection of the plate, which verifies the validity of the variational solution obtained. The results presented in this study are helpful for the refined analysis and optimization design of flexible structures, which are composed of bimodular functionally graded materials, while the structure is under large deformation.

Higher-Order Finite Element Vibration Analysis of Circular Raft on Winkler Foundation

On a Winkler foundation, solid circular plate vibration is examined using a higher-order finite element in polar co-ordinate system. The present formulation has developed a Mat-lab code to handle any boundary conditions. Validation of the code is carried out after the convergence studies. The results are compared to other researchers and show excellent conformity. Furthermore, a parametric analysis gave the first 10 natural frequency characteristics in tabular and graphical form. The authors conclude that the present formulation is straightforward, behaves exceptionally well for thin solid circular plates on elastic foundations with reasonable convergence rate and accuracy, and requires less computational effort, resources, and time.

Thermally-Induced Stresses in a Pre-Buckling State of a Circular Plate within the Fractional-Order Framework

This paper considers a transient thermoelastic problem in an isotropic homogeneous elastic thin circular plate with clamped edges subjected to thermal load within the fractional-order theory framework. The prescribed ramp-type surface temperature is on the plate's top face, while the bottom face is kept at zero. The three-dimensional heat conduction equation is solved using a Laplace transformation and the classical solution method. The Gaver–Stehfest approach was used to invert Laplace domain outcomes. The thermal moment is derived based on temperature change, and its bending stresses are obtained using the resultant moment and resultant forces per unit length. The results are illustrated by numerical calculations considering the material to be an Aluminum-like medium, and corresponding graphs are plotted.