Clamped Plate Research Articles

Comparison of the dynamic response of clamped plates under fully confined and unconfined air-blast loading

Comparison of the dynamic response of clamped plates under fully confined and unconfined air-blast loading

Double blast loadings of clamped square steel plates

This paper presents a combining experimental, numerical, and analytical efforts to elucidate the resultant effect of synchronised double blast loadings on square steel plates. Blast tests were performed to investigate the effect of the distance between the two identical charges upon the large inelastic deformation of fully-clamped steel plates at various standoff distances. Detailed three-dimensional finite element simulations using LS-Dyna were performed to provide insights into the equal blast wave collision and transient response of the ductile plates under the double blast loading. An analytical model was developed to predict the large inelastic deflection of the fully-clamped thin plate under either pulse pressure loading or impulsive loading (an instantaneous zero-period impulse). Experimental results found that splitting a 1kg TNT into 2 × 0.5 kg TNT (with a certain separating distance) would lead to greater permanent inelastic deformation of the ductile thin plates (by up to 19.6%). Both numerical and analytical results were validated against experimental data. Results from the parametric study showed that an optimum charging distance exists for maximised plate deflection under a given TNT mass and standoff distance. The results presented in this work shall be useful for evaluating the hazards of multiple blast loadings or for demolition applications.

Non-homogeneity effect on the vibration of the rectangular visco-elastic plate subjected to the linear temperature effect with quadratic thickness variation in both directions

The present work investigates the effect of non-homogeneity on the vibration of a rectangular visco-elastic plate. The plate is subjected to linear temperature variation in the x-direction with quadratic thickness variation in both directions. The quadratic variation has been considered in the material density of the plate only along the x-axis, and it is assumed that non-homogeneity transpires because of this variation. The governing differential equation is solved using the Rayleigh-Ritz technique. All four edges of the plate should be clamped to drive the frequency equation. Deflection and time period have been evaluated for several combinations of values of the thermal constant, constant of nonhomogeneity, taper constants, and length-to-width ratio (aspect ratio) for the first two vibrationmodes of the clamped plate. The results presented are compared with those found in the literature.

Improving the energy harvesting performance of clamped flexible plates using vortex shedding behind bluff bodies: A computational study

This study assesses flow-induced vibrations of clamped flexible plates with the objective of improving their energy harvesting performance. Toward this, a rectangular bluff body is placed between the two clamped flexible plates to harness the vortices generated behind the bluff body. The strain energy of the plate is used as a measure of energy harvesting performance. Computational studies are performed for different parameters of interest, such as dimensions and material properties of the plate and dimensions and locations of the bluff body. The effects of these stated parameters on flow-induced vibration response and vortical structures are investigated, and the optimal values for the location and geometry of the bluff body, as well as the aspect ratio and Young's modulus of the plate for energy harvesting performance, are determined. The results show that vortex shedding from the bluff body strongly influences the dynamic behavior and energy output of the flexible structures inside the wake region of the bluff body at different locations. Additionally, the aspect ratio and its effect on vorticity and energy harvesting are discussed in detail, along with the average displacement and average lift force experienced by the plates. The outcomes of this work offer significant insights into optimizing the design of clamped flexible plates for optimal energy generation by cleverly exploiting the vortex shedding behind fixed bluff bodies.

Quintic and biquintic B-spline finite element solutions of clamped structures

Recent studies on the control of smart material partial differential equation systems point toward a need for considering higher-ordered polynomial splines in the formulation of bases for approximate solutions obtained via the Galerkin finite element method. However, basis modifications are not well documented in the literature, nor are they well understood by researchers who are often utilizing software packages in their basis and control designs. In this study, solutions are generated using modified bases with quintic and biquintic B-splines for three clamped structures: an ordinary differential equation cantilevered beam, a partial differential equation cantilevered beam, and an isotropic clamped plate. Convergent solutions are obtained and verified for all three models.

Experimental evaluation of the use of cruciform specimens for biaxial stability analysis

Novel research is proposed about the suitability of cruciform specimens for estimating the stress-strain response and buckling modes of flat laminates under equibiaxial compression. The experimental challenge lies in determining whether the specimen design and test procedure can be adequate to observe geometric instability at the coupon scale with small dimensions of the tested region. Although the literature on composite buckling is extensive, the number of studies that include experimental biaxial testing of flat laminates is limited, particularly when considering strong flexural-torsional coupling. In this work the region of interest of the cruciform specimen consists of a carbon-fibre reinforced [∓45]S laminate with shape and boundary conditions close to a clamped square plate. For the first time, the buckling mode of an anisotropic laminate is observed in the central region of a cross-shaped specimen using 3D deflection surface data recorded by Digital Image Correlation. The effect of flexural-torsional coupling at the bifurcation onset is experimentally corroborated through the orientation of the buckling mode, which is analytically utilized to quantify the proportion between bending and twist moments. Non-linearities exhibited in the stress-strain evolution of the biaxially loaded region are verified to be independent of the response of the sample's arms. Furthermore, no previous experimental investigation has shown the differences in local strain gauge measurements induced by the stress state at the ply level. Additionally, this work adapts the method for strength reduction to estimate the real critical stress of a bi-compressed laminate.

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.

Improving resonant ice protection systems with substrate optimization

Electro-mechanical de-icing systems are low-energy ice protection solutions based on ice fracture mechanisms. This article focuses on resonant electro-mechanical de-icing systems that actuate modes of flexion, which require low energy compared to extension modes. However, fracture propagation limits are encountered when using such flexural modes, preventing the ice from being completely detached from the substrate. This study demonstrates the feasibility of extending the ice detachment area by optimizing the thickness of the substrate. First, the interest and the limits of flexural resonant modes are discussed. Then the de-icing of a simple metallic beam in free conditions using a flexural mode is improved owing to the parametric optimization of the substrate thickness. The optimization is verified by tests on aluminum prototypes. The optimization results are then extended to a clamped composite plate and then to a NACA profile, showing interest in the approach to fully de-ice structures with modes of flexion, even in the case of complex geometries. With this last example, the study also demonstrates the feasibility of electro-mechanical ice protection systems for carbon fiber reinforced Polymer composite structures.

Efficient method for calculating sound radiation from a circular source in an infinite baffle

The numerically efficient solution describing the sound radiation from a circular source located in a flat, rigid baffle is obtained. For this purpose, a half-space was divided into two subregions coupled through continuity equations. The Helmholtz equation was solved in both subregions. The proposed method can be used for circular baffled sources in the case when fluid–structure interactions are included, as well as for an input from a waveguide or cavity. The validity and numerical efficiency of the presented solution were tested assuming that a clamped circular plate is a source. The numerical simulations show that the results given by the proposed method agree with those given by the known integral solution. The exceptions are the field points at which the integral formulas fail and provide incorrect values of the sound pressure. Hence, the presented formulas can be the only method to perform accurate calculations at troublesome field points. The numerical efficiencies of the obtained formulas and integral ones were compared by estimating the value of the appropriate time ratios. The numerical analysis shows that the proposed method can be used to significantly improve calculations of the sound power and the sound pressure.

Sufficient conditions yielding the Rayleigh Conjecture for the clamped plate

Sufficient conditions yielding the Rayleigh Conjecture for the clamped plate

Sharp spectral gap estimates for higher-order operators on Cartan–Hadamard manifolds

The goal of this paper is to provide sharp spectral gap estimates for problems involving higher-order operators (including both the clamped and buckling plate problems) on Cartan–Hadamard manifolds. The proofs are symmetrization-free — thus no sharp isoperimetric inequality is needed — based on two general, yet elementary functional inequalities. The spectral gap estimate for clamped plates solves a sharp asymptotic problem from [Q.-M. Cheng and H. Yang, Universal inequalities for eigenvalues of a clamped plate problem on a hyperbolic space, Proc. Amer. Math. Soc. 139(2) (2011) 461–471] concerning the behavior of higher-order eigenvalues on hyperbolic spaces, and answers a question raised in [A. Kristály, Fundamental tones of clamped plates in nonpositively curved spaces, Adv. Math. 367(39) (2020) 107113] on the validity of such sharp estimates in high-dimensional Cartan–Hadamard manifolds. As a byproduct of the general functional inequalities, various Rellich inequalities are established in the same geometric setting.

Characterization and modeling of supercritical CO2 pulse pressures: Effects of activator mass and discharge plate thickness

Utilizing a bespoke CO2 phase transition pulse pressure experimental system, we conducted pulse pressure characterization tests across various activator masses, CO2 filling pressures, and energy discharge plate thicknesses. This approach enabled us to ascertain the pulse pressure's response characteristics and variation patterns under diverse conditions. The formula for calculating the peak supercritical CO2 pulse pressure was deduced by modeling the ultimate load calculation of the clamped circular plate, and then the time-course expression of the supercritical CO2 phase transition pulse pressure and energy was carried out by introducing the time factor and taking into account the parameters of the activator mass and the thickness of the energy discharging plate. Our findings reveal a four-stage pressure evolution in the cracking tube during initiation: a gradual increase, a rapid spike, swift attenuation, and eventual negative pressure formation. The activator mass and discharge plate thickness critically influence the peak pressure's timing and magnitude. Specifically, increased activator mass hastens peak pressure onset, while a thicker discharge plate amplifies it. The errors between calculated and experimental values for peak supercritical CO2 phase transition pressure fall within −5%–5%. Furthermore, the pressure peak and arrival time model demonstrates less than 10% error compared to experimental data, affirming its strong applicability. These insights offer theoretical guidance for controlling phase transition pressure and optimizing energy in supercritical CO2 systems.

Spectral collocation method for free vibration of sandwich plates containing a viscoelastic core

Viscoelastic constrained layer damping is a simple and efficient method for reducing noise, vibration, and fatigue in metal structures. In this work, a spectral collocation method utilizing a layer-wise plate theory was established to inspect the vibration characteristics of a sandwich plate with a viscoelastic core. The displacements of each layer satisfied the Mindlin plate theory. The core's transverse shear stress remained constant. The three layers’ transverse displacements were assumed to be identical. The displacement fields were reduced to nine variables by utilizing the interfaces’ continuity of the displacements. The equations of motion were obtained by utilizing Hamilton’s principle. The viscoelastic core’s material properties were considered as frequency-dependent. To address the complex eigenvalue problem, an iterative algorithm was used. The theoretical results were compared with published theoretical and experimental data for fully clamped rectangular sandwich plates to validate the proposed method. The modified Oberst beam method was applied by fitting the impulse response function to obtain the adhesive’s frequency-dependent storage modulus and loss factor. The natural frequencies and corresponding loss factors of three sandwich plates with different adhesives were measured by conducting impact tests. The numerical results from the present theoretical method agreed well with the measured results.

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.

Comparison of free vibration behaviors for simply supported and clamped T-shaped thin plate resting on Winkler elastic foundation

The analysis of free vibration problems for thin plates is essential for the design of various structural systems. However, it is difficult to find analytical solutions due to the complexity of mathematical computing. Based on the symplectic superposition method, the T-shaped thin plate on the Winkler elastic foundation is divided into four sub-plates and are solved by using the symplectic eigen expansion method, and the modes and frequencies are studied. The method begins directly with the fundamental equations and undergoes a rigorous mathematical derivation without assuming the form of the solution beforehand. This approach helps circumvent the drawbacks associated with traditional semi-inverse solution methods. In addition, the theoretical calculation model and finite element analysis model of T-shaped thin plates on elastic foundation are established by using Mathematic software and ABAQUS software in present paper. It proves that the symplectic superposition method converges very fast and has a good consistency with the finite element simulation results. Results show that the boundary condition, foundation stiffness and aspect ratio have great influences on vibration frequency and mode shape for T-shaped structures.

Bending Behavior of Clamped Skew Plates

The deflection analyses performed using MSC/NASTRAN on skew plates under different loads such as (concentrated and uniformly distributed loads) for clamped boundary conditions are presented in the present study. The CQUAD4 and CQUAD8 components of MSC/NASTRAN are verified using values from the literature. The CQUAD8 element was employed in the present experiments because it had been found to produce more effective results than the CQUAD4 element. For isotropic skew plates, the variation of deflection with regards to aspect ratio and length to thickness ratio is provided. With an increase in the skew angle, it appears that the deflections decrease.

Nonlinear dynamics response of porous functionally graded annular plates using modified higher order shear deformation theory

In this investigation, the nonlinear dynamics of porous annular plates that have been functionally graded are examined under varying time-dependent loads. Both simple supported and fully clamped boundary conditions are taken into account. The mechanical properties of the functionally graded plate in its thickness are considered according to the distribution function of the modified law. In addition, porosity, as a stress-relieving property, is also applied throughout the plate thickness by different functions and pore volume fractions. By adopting Hamilton’s principle, the equations of motion are obtained based on the modified higher-order shear deformation plate theory. Then these partial differential equations are solved using the viscous dynamic relaxation method in conjunction with Newmark’s implicit integration method. The present findings are compared and successfully confirmed with those available in the literature. Finally, the effects of some key factors such as porosity distribution type, pore volume fraction, power-law index, loading conditions, and thickness-to-radius ratio on the dynamic behavior of both simple and fully clamped plates have been studied in detail. The study’s findings indicate that porosity pattern X, characterized by a greater prevalence of interconnected pores, exhibited enhanced resistance to deflection. Conversely, pattern O, featuring a more uniformly distributed pore size, effectively mitigated stress in FG porous materials. These observations offer valuable insights for optimizing plate design and elevating its performance.

Modelling of bidirectional functionally graded plates with geometric nonlinearity: A comparative dynamic study using whole domain and finite element method

Present work attempts to study the nonlinear dynamic behaviour of bidirectional functionally graded plates (BFG) along with unidirectional functionally graded plates (UFG). For BFG plates the material is graded along the thickness and length simultaneously and two types of UFG plates are considered namely thickness-wise (TFG) and length-wise (AFG) plates. The dynamic problem is formulated using two different methods namely, whole domain method and finite element method. The constitutive relation for the former is generated using classical plate theory whereas Mindlin plate theory is used in the later. Geometric nonlinearity is addressed here by using Von Karman type strain-displacement relation. Both clamped and simply supported boundary conditions are considered and the plate is assumed to be under action of uniformly distributed external load. The results of free and forced vibration analyses are presented as backbone curves and frequency response curves respectively. The results show that all three plates exhibit hardening type nonlinearity. It is also seen that the effect of material gradation parameters is more profound in simply supported plates compared to clamped plates. Specifically, these parameters do not seem to have much effect on BFG clamped plates in free vibration. The outcome of the analysis is compared with those in existing literature and fair agreement is observed between them.

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.