Clamped Plate Research Articles

On dynamic response of rectangular sandwich plates with fibre-metal laminate face-sheets under blast loading

This paper focuses on the analytical and numerical analyses to predict the dynamic response of fully clamped rectangular sandwich plates with fibre-metal laminate face-sheets and metal foam cores under blast loading. Based on the rigid-plastic material approximation with modifications, simple models are developed for the dynamic response of sandwich plates with fibre-metal laminate face-sheets. The so-called ‘bounds’ and the membrane mode solution for dynamic response of rectangular sandwich plates are obtained. The effects of the composite volume fraction, the ratio of metal layer strength to composite layer strength, relative density and relative strength on the dynamic response are discussed. The finite element calculation is carried out to study the dynamic response of the sandwich plates under blast loading. Good agreement is achieved between the analytical solutions and numerical results. This work provides the efficient method that reasonably captures dynamic response of the rectangular sandwich plates subjected to the blast loading.

Evolution of the first eigenvalue of the clamped plate on manifold along the Ricci flow

ABSTRACTIn this article, we study the evolution, monotonicity for the first eigenvalue of the clamped plate on closed Riemannian manifold along the Ricci flow. We prove that the first nonzero eigenvalue is nondecreasing under the Ricci flow under certain geometric conditions and find some applications in 2-dimensional manifolds.

Effect of strain rate on dynamic responses of laterally impacted steel plates

Quasi-static punching tests and low-velocity impact tests have been conducted to examine the plastic response and failure of clamped rectangular steel plates struck by hemispherical indenters. The length, width and thickness of plate specimens are 127, 76 and 1.4 mm, respectively. Experimentally obtained quasi-static and dynamic force-displacement responses are compared to reveal the influence of dynamic material characteristics on the impact behaviour of steel plates. The failure mode of laterally impacted plates is exposed to further illustrate the dynamic material properties. The experimental results are compared with numerical simulations performed by the LS-DYNA finite element solver. The finite element model includes defining the material strain hardening, strain rate sensitivity and critical fracture strain. A simplified method is proposed to characterise the material strain rate effect at varied plastic strains in the simulations. A good agreement is observed between experimental and numerical results, demonstrating the rationality of the material definitions in low-velocity impact. The paper contributes to enhance the industry practice in impact simulations of steel structures when only material quasi-static tensile tests are available.

On the frequency response computation of geometrically nonlinear flat structures using reduced-order finite element models

This paper presents a general methodology to compute nonlinear frequency responses of flat structures subjected to large amplitude transverse vibrations, within a finite element context. A reduced-order model (ROM)is obtained by an expansion onto the eigenmode basis of the associated linearized problem, including transverse and in-plane modes. The coefficients of the nonlinear terms of the ROM are computed thanks to a non-intrusive method, using any existing nonlinear finite element code. The direct comparison to analytical models of beams and plates proves that a lot of coefficients can be neglected and that the in-plane motion can be condensed to the transverse motion, thus giving generic rules to simplify theROM. Then, a continuation technique, based on an asymptotic numerical method and the harmonic balance method, is used to compute the frequency response in free (nonlinear mode computation) or harmonically forced vibrations. The whole procedure is tested on a straight beam, a clamped circular plate and a free perforated plate for which some nonlinear modes are computed, including internal resonances. The convergence with harmonic numbers and oscillators is investigated. It shows that keeping a few of them is sufficient in a range of displacements corresponding to the order of the structure’s thickness, with a complexity of the simulated nonlinear phenomena that increase very fast with the number of harmonics and oscillators.

Magnetoelastic axisymmetric multi-modal resonance and Hopf bifurcation of a rotating circular plate under aerodynamic load

In this article, an investigation of magnetoelastic axisymmetric multi-mode interaction and Hopf bifurcations of a circular plate rotating in air and uniform transverse magnetic fields is presented. The expressions of electromagnetic forces and an empirical aerodynamic model are applied in the derivation of the dynamical equations, through which a set of nonlinear differential equations for axisymmetric forced oscillation of the clamped circular plate are deduced. The method of multiple scales combined with the polar coordinate transformation is employed to solve the differential equations and achieve the phase–amplitude modulation equations for the interaction among the first three modes under primary resonance. Then, the frequency response equation for the single-mode vibration, the steady-state response equations for three-mode resonance and the corresponding Jacobian matrix are obtained by means of the modulation equations. Numerical examples are presented to show the dependence of amplitude solutions as a function of different parameters in the cases of single mode and three-mode response. Furthermore, a Hopf bifurcation can be found in three-mode equilibrium by choosing appropriate parameters, where a limit cycle occurs and then evolves into chaos after undergoing a series of period-doubling bifurcations.

Energy harvesting from unimorph piezoelectric circular plates under random acoustic and base acceleration excitations

Abstract Piezoelectric materials have been extensively employed in small scale energy harvesting devices. Commonly, the ambient excitation in these energy harvesting systems is considered to be harmonic. While in practical cases, these excitations are random in nature. The objective of the current paper is to use random vibration theory to investigate energy harvesting from a clamped circular plate interconnected to a circular piezoelectric layer. The excitations are assumed to be a random uniform pressure applied to the plate resulted from an acoustic wave and a random base acceleration of the system. Lagrange equations are utilized to derive the coupled equations describing the dynamic behavior of the plate and the circuit voltage. The random inputs are assumed to be band-limited uncorrelated white noises. Using the random vibration theory, the statistical parameters describing the plate’s dynamic and the harvested voltage are derived in terms of random characteristics of the inputs. A parametric study is carried out to investigate the effect of different design parameters on the harvested power. Also, an optimization problem is solved for maximizing the power density in the system. Using the results of the optimization problem, the power density can be significantly improved up to 3.2 mW/m2. The quantitative and qualitative results obtained in this paper can be effectively used for the optimal design of other piezoelectric energy harvesters under stochastic excitations.

Bending and free vibration of a circular magnetoelectroelastic plate with surface effects

Surface effects play a significant role in affecting the mechanical behavior of nanosized NEMS such as sensor, actuator, transducer. This paper studies the bending and free vibration of a magnetoelectroelastic plate with surface effects. By incorporating surface effects into the Kirchhoff theory of thin plates, the governing differential equation for bending and vibration of a magnetoelectroelastic plate is derived. Simply supported and clamped circular plates are analyzed for applied mechanical loading, electric voltage, and magnetic potential difference. An analytical method is presented to obtain closed form solutions for static deflection of bending and natural frequencies of free vibration. The well-known results of circular thin plates are recovered if ignoring the surface effects along with coupling coefficients. A comparison of the deflections and the natural frequencies with their counterparts in the absence of surface effects shows that surface residual tension plays a crucial role, which decreases the deflection and increases the natural frequencies. The influence of surface magnetoelectroelasticity is discussed and the small scale effect can be captured. The obtained results are helpful for design and application of NEMS.

Pore pressure and porosity effects on bending and thermal postbuckling behavior of FG saturated porous circular plates

Postbuckling and nonlinear bending of a geometrically perfect circular plate arisen by the temperature of one side of a axisymmetric circular plate made from a poroelastic solid whose matrix pores have been saturated by fluid have been numerically analyzed. The plate porosity varies continuously through the plate thickness according to some given specific functions. The postbuckling and nonlinear bending configurations of respectively clamped and simply-supported plates, as well as the critical postbuckling temperature, have been obtained under the influence of the plate one side surface temperature, thickness, type of pore distribution, porosity and compressibility of fluid confined by the pores. Equilibrium equations of the plate have been derived on the basis of the classical plate theory, Love–Kirchhoff hypothesis and the Sander’s nonlinear strain–displacement relationship. They have been discretized via differential quadrature method. The numerical results appropriately coincide with the relevant literature, namely the results of saturated porous and mathematically equivalent plates made from functionally grated materials.

Low-velocity impact of rectangular multilayer sandwich plates

The aim of the work is to investigate the low-velocity impact of fully clamped rectangular multilayer sandwich plates with metal foam cores struck transversely by a heavy mass. Analytical models considering interaction of bending and stretching induced by large deflections are developed for the dynamic response of multilayer sandwich plates. Numerical calculations of multilayer sandwich plates under low-velocity impact are conducted and analytical models can capture the numerical results reasonably. The cases of three typical impact locations are considered. Effects of loading location, the multilayer factor and shape of the striker on the structural response are discussed. It is demonstrated that the present analytical model can reasonably predict the low-velocity post-yield behavior of rectangular multilayer sandwich plates with metal foam core struck by a heavy mass.

Numerical Study for Vibration Analysis of Hole Flanged Rectangular Plate

Plates with flange are used in many engineering structures such as aircrafts, ships, and mechanical devices. In this paper, a numerical study for the vibration analysis of fully clamped rectangular Al-1100 alloy plates with different flange angles and different flange inner diameters was worked out. SolidWorks 2014 Simulation add-in was used to perform the analysis. It was found that increasing the flange angle and flange inner diameter increases the fundamental frequency and decreases the flange thickness. Adding the flange can increase the fundamental frequency up to (27%).

New analytic buckling solutions of moderately thick clamped rectangular plates by a straightforward finite integral transform method

A first endeavor is made in this paper to explore new analytic buckling solutions of moderately thick rectangular plates by a straightforward double finite integral transform method, with focus on typical non-Levy-type fully clamped plates that are not easy to solve in a rigorous way by the other analytic methods. Solving the governing higher-order partial differential equations with prescribed boundary conditions is elegantly reduced to processing four sets of simultaneous linear equations, the existence of nonzero solutions of which determines the buckling loads and associated mode shapes. Both numerical and graphical results confirm the validity and accuracy of the developed method and solutions by favorable comparison with the literature and finite element analysis. The succinct but effective technique presented in this study can provide an easy-to-implement theoretical tool to seek more analytic solutions of complex boundary value problems.

Analytical Modelling of Sound Transmission Through Finite Clamped Double-Wall Panels with Magnetic-Linked Stiffness

A theoretical modelling approach is proposed for the vibroacoustic problem of sound transmission across a rectangular double-wall panel clamp mounted on an infinite rigid baffle with magnetically connected stiffness. The magnetic stiffness is derived based on interaction energy between the two rectangular magnets attached to the clamped plates. The exact solution for the vibration of the clamped double plates is taken into account by the method of modal function, and the dynamic response of the structures is obtained by employing the weighted residual (Galerkin) method. This method enabled the coupling of the acoustic and the magnetic stiffness of the link to be done effectively. The accuracy of the theoretical predictions is checked against existing experimental data, with good agreement achieved. The model is then compared with the theoretical formulation in a duct based on a low-frequency range with and without magnetic stiffness. The sound transmission loss (STL) of the two models agrees perfectly in the first resonance, but the mass–air–mass resonance frequency for the current model tends to shift towards a higher value. Furthermore, the influence of magnetic stiffness on the elevation angle and azimuth angle of the STL is investigated. The present method is suitable for double-panel systems with connecting stiffness for practical plates and is applicable for both low- and high-frequency ranges.

Nonlinear tuning curve demonstration: Comparing a buried mine simulant with a clamped elastic plate—Cylindrical soil column oscillator

Experiments using soil-plate-oscillators (SPO) involve a cylindrical column of granular media (masonry sand) supported by a clamped circular elastic acrylic plate (12.7 cm diameter, 3.2 mm thick). The plate is clamped between two 20.3 cm O.D., 12.7 cm I.D., 6.4 cm thick flat toroidal brass “rings.” Two 15 cm diameter subwoofers (10 cm above the soil) are driven by an amplified swept sinusoidal chirp which drives the 2.5-cm soil column. A spectrum analyzer measures the laser Doppler vibrometer particle velocity versus frequency near the center of the column. The resonant frequency decreases with the increasing amplitude—representing softening in the nonlinear system. The back-bone curve (locus of the resonant frequency versus corresponding peak velocity coordinates) has a distinct arching shape where the slope of the velocity increases with the decreasing frequency. Here, the resonant frequency goes from 247 to 203 Hz. A lumped element bilinear hysteresis model describes the shape of the tuning curves and backbone curve. Next, a drum-like simulant (made by replacing the upper toroidal ring by a 0.64-cm thick ring) is buried 2.5-cm deep in an open square concrete tank (57 cm). Nonlinear tuning curve experiments “on” and “off” the mine are compared with the SPO results. Experiments using soil-plate-oscillators (SPO) involve a cylindrical column of granular media (masonry sand) supported by a clamped circular elastic acrylic plate (12.7 cm diameter, 3.2 mm thick). The plate is clamped between two 20.3 cm O.D., 12.7 cm I.D., 6.4 cm thick flat toroidal brass “rings.” Two 15 cm diameter subwoofers (10 cm above the soil) are driven by an amplified swept sinusoidal chirp which drives the 2.5-cm soil column. A spectrum analyzer measures the laser Doppler vibrometer particle velocity versus frequency near the center of the column. The resonant frequency decreases with the increasing amplitude—representing softening in the nonlinear system. The back-bone curve (locus of the resonant frequency versus corresponding peak velocity coordinates) has a distinct arching shape where the slope of the velocity increases with the decreasing frequency. Here, the resonant frequency goes from 247 to 203 Hz. A lumped element bilinear hysteresis model describes the shape of the tuning curves and ...

Data-driven approaches for damage-type classification in vibrating square plates

Acoustic radiation from a mechanical structure due to broadband forcing is inherently dependent on the structure’s material, geometry, boundary conditions, and mechanical status. Measurements of this acoustic or vibrational response can be used to detect mechanical changes (i.e., damage) when compared to known baseline measurements of the structure. Often, however, knowledge of the type of damage is useful for subsequent considerations. Even for relatively simple structures, like flat plates, the lack of simple solutions to problems involving localized damage makes analytical treatments challenging. This is further complicated since the location and severity of damage are typically unknown a priori. We present a data-driven approach for classification of various forms of damage—including cuts, localized mass changes, delamination, and boundary changes—in a vibrating clamped square plate. Frequency response curves are generated using FEA of a 30-by-30-by-0.3-cm aluminum plate with various types and severity of damage. Features (including changes in peak-locations, -amplitudes, and -widths compared to baseline) are extracted and used to train classifiers, with performance quantified using auxiliary test data. Comparisons are made between classifier types, including nearest neighbor methods, discriminant analysis, and support vector machines. [Work supported by NAVSEA through the NEEC and the US DoD through an NDSEG Fellowship].Acoustic radiation from a mechanical structure due to broadband forcing is inherently dependent on the structure’s material, geometry, boundary conditions, and mechanical status. Measurements of this acoustic or vibrational response can be used to detect mechanical changes (i.e., damage) when compared to known baseline measurements of the structure. Often, however, knowledge of the type of damage is useful for subsequent considerations. Even for relatively simple structures, like flat plates, the lack of simple solutions to problems involving localized damage makes analytical treatments challenging. This is further complicated since the location and severity of damage are typically unknown a priori. We present a data-driven approach for classification of various forms of damage—including cuts, localized mass changes, delamination, and boundary changes—in a vibrating clamped square plate. Frequency response curves are generated using FEA of a 30-by-30-by-0.3-cm aluminum plate with various types and severit...

Design of The Vibrostabilisation Stand For Reducing Residual Stresses in Discs Used In The Construction of Multi-Plate Clutches and Brakes

Abstract Heavy-duty, oil-cooled brake discs (MMOTs) are often used in heavy-duty brake systems manufactured by companies such as Caterpilar, Clark, Komatsu and Liebherr. These discs are usually made of special steels, and in most cases, the flatness of the working surfaces should not exceed 0.15–0.30 mm. Although the technological processes of friction disc production include several stages of heat treatment and grinding, the required accuracy is not achieved in some cases. In addition, the remaining residual stresses lead to the deformation of the discs during their lifetime. In production practice, three methods are used to reduce residual stresses: thermo-fixing, dynamic stabilisation and vibratory stabilisation consisting in bringing discs to transverse resonance vibrations and maintaining resonance until significant stress reduction. The article proposes a method of stabilising the discs using the resonance phenomenon at the first few frequencies. In this article, Cauchy’s function method and characteristic series method are used to develop solution value problem for clamped circular plates with discrete inclusions as concentrated masses and springs. Calculation methods for quick estimation of the own frequency of discs with additional ring mass enabling the use of low power vibration inductors are presented. The use of a special membrane and a pneumatic cushion in the construction of the stand allows to induce vibrations of higher frequencies.

Dynamic behaviour of thick plates resting on Winkler foundation with fourth order element

This paper focuses on the study of dynamic analysis of thick plates resting on Winkler foundation. The governing equation is derived from Mindlin\'s theory. This study is a parametric analysis of the reflections of the thickness / span ratio, the aspect ratio and the boundary conditions on the earthquake excitations are studied. In the analysis, finite element method is used for spatial integration and the Newmark-B method is used for the time integration. While using finite element method, a new element is used. This element is 17-noded and it\'s formulation is derived from using higher order displacement shape functions. C++ program is used for the analyses. Graphs are presented to help engineers in the design of thick plates subjected to earthquake excitations. It is concluded that the 17-noded finite element is used in the earthquake analysis of thick plates. It is shown that the changes in the aspect ratio are more effective than the changes in the aspect ratio. The center displacements of the reinforced concrete thick clamped plates for b/a=1, and t/a=0.2, and for b/a=2, and t/a=0.2, reached their absolute maximum values of 0.00244 mm at 3.48 s, and of 0.00444 mm at 3.48 s, respectively.

Generalization of Philippin’s results for the first Robin eigenvalue and estimates for eigenvalues of the bi-drifting Laplacian

In the present paper, we first consider the weighted eigenvalue problem $$\Delta _f u=\lambda _{f}u$$ in M with the Robin boundary condition $$\frac{\partial u}{\partial \nu }+\beta u=0$$ on $$\partial M$$ , where $$(M^n,g,e^{-f})$$ is a compact n-dimensional weighted Riemannian manifold of nonnegative Bakry–Emery Ricci curvature. We derive under some convexity condition of the boundary $$\partial M$$ , an explicit lower bound of the first weighted Robin eigenvalue $$\lambda _{1,f}(\beta )$$ depending only on the geometry of M and the constant $$\beta $$ appearing in the boundary condition. Another new estimate for $$\lambda _{1,f}(\beta )$$ with respect to the first nonzero Neumann eigenvalue $$\mu _{2,f}$$ of the weighted Laplacian $$\Delta _f$$ is also obtained. Furthermore, we provide some lower bounds for the first buckling and clamped plate eigenvalues of the bi-drifting Laplacian on weighted manifolds.

Optimal fuzzy iterative learning control based on artificial bee colony for vibration control of piezoelectric smart structures

Combining P-type iterative learning (IL) control, fuzzy logic control and artificial bee colony (ABC) algorithm, a new optimal fuzzy IL controller is designed for active vibration control of piezoelectric smart structures. In order to accelerate the learning speed of feedback gain, the fuzzy logic controller is integrated into the ANSYS finite element (FE) models by using APDL (ANSYS Parameter Design Language) approach to adjust adaptively the learning gain of P-type IL control. For improving the performance and robustness of the fuzzy logic controller as well as diminishing human intervention in the operation process, ABC algorithm is used to automatically identify the optimal configurations for values in fuzzy query table, fuzzification parameters and defuzzification parameters, and the main program of ABC algorithm is operated in MATLAB. The active vibration equations are driven from the FE equations for the dynamic response of a linear elastic piezoelectric smart structure. Considering the vibrations generated by various external disturbances, the optimal fuzzy IL controller is numerically investigated for a clamped piezoelectric smart plate. Results demonstrate that the proposed control approach makes the feedback gain has a fast learning speed and performs excellent in vibration suppression. This is demonstrated in the results by comparing the new control approach with the P-type IL control.

NUMERICAL INVESTIGATION OF NATURAL FREQUENCIES FOR CLAMPED LONGITUDINAL COMPOSITE PLATES

In this paper, two finite element models were performed. The fiber and matrix represented as two different materials in the first plate, while the second showed as a composite plate. The boundary conditions included clamped the plates on four ends and the dimensions of the composite plates were changed in this study. The finite element was performed and ANSYS 16.1 was employed in modeling. By comparing the results between the frequency ratio and mode numbers for different plate thickness, the results showed that natural frequencies calculated by these two model thickness of the (length/width) ratio will be more uniform and the error will be small. The numerical equations that performed from this study used to investigate the natural frequencies for longitudinal clamped composite plate.

Additive Schwarz preconditioners for the obstacle problem of clamped Kirchhoff plates

When the obstacle problem of clamped Kirchhoff plates is discretized by a partition of unity method, the resulting discrete variational inequalities can be solved by a primal-dual active set algorithm. In this paper we develop and analyze additive Schwarz preconditioners for the systems that appear in each iteration of the primal-dual active set algorithm. Numerical results that corroborate the theoretical estimates are also presented.