Cantilever Rectangular Plate Research Articles

Nonlinear Vibrations of the Piezoelectric-Laminated Composite Cantilever Rectangular Plate Subjected to the Transverse and Parametric Excitation

The intelligent structure will be the key to the industrial manufacturing field in the future, piezoelectric materials are the most commonly used active materials in many mechatronic and vibration control systems. In this paper, we illustrated the nonlinear vibration behaviors of the piezoelectric-laminated composite cantilever rectangular plate subjected to transverse and parametric excitation. Two piezoelectric layers are attached to the upper and lower surfaces of graphite/epoxy composite laminated plate. The nonlinear dynamic equations are derived via the classical laminated plate theory, von Karman large deformation theory, the constitutive equation of the piezoelectric materials, and the Hamilton principle. Considering the Galerkin method, the nonlinear ordinary differential equations for the two-order vibration mode of the piezoelectric laminated composite cantilever rectangular plate are achieved. The multiple scale method is applied to obtain the average equations to study the nonlinear vibration characteristic of the stable motion. The primary resonance and the 1:1 internal resonance of the piezoelectric laminated composite cantilever rectangular plate are investigated. The amplitude-frequency response curves, force-amplitude response curves, time histories, phase curves, and bifurcation diagrams are given to investigate the nonlinear dynamic characteristic of the piezoelectric laminated composite cantilever rectangular plate. The detailed parametric analyses show that bubble response curves are separated from the amplitude-frequency response curves with the continuous increase of the external excitation due to the internal resonance. In addition, the results reveal the energy transform between the various order vibration modes of the system. This work is expected to provide theoretical guidance for the nonlinear vibration of the smart piezoelectric laminated composite structure.

Nonlinear vibrations and control of PFG-GR laminated composite cantilever rectangular variable cross-section plate with NPPF controller

This paper studies the nonlinear vibrations and control of the piezoelectric functionally graded graphene-reinforced laminated composite cantilever rectangular (PFG-GRLCCR) variable cross-section plate with the NPPF controller under the simultaneous resonant cases. The nonlinear dynamic governing equations of motion for the PFG-GRLCCR variable cross-section plate are obtained by using the extended Halpin-Tsai mode, linear constitutive equations of the piezoelectric materials, classical laminated plate theory, von Karman large deformation theory, and Hamilton principle. Considering Galerkin method, the nonlinear ordinary differential governing equations are obtained. The nonlinear positive position feedback (NPPF) controllers are designed according to the similarity principle. The multiple scale method is used to investigate the nonlinear vibration characteristics of the stable motion for the PFG-GRLCCR variable cross-section plate with the NPPF controllers. The amplitude-frequency response curves, force-amplitude response curves, time histories, phase portraits, and bifurcation diagrams are given to investigate the nonlinear dynamic characteristics of the PFG-GRLCCR variable cross-section plate with the NPPF controller. The results demonstrate that the NPPF controller can suppress the nonlinear vibrations of the structure well, and the effectiveness of the controllers is best when the NPPF controller has a similar nonlinear characteristic with the structure. The complex nonlinear vibration characteristics appear for the PFG-GRLCCR variable cross-section plate with the NPPF controller.

Adaptation of the Facchinetti, de Langre and Biolley model for the hydroelastic vortex-induced vibrations of a cantilevered flat plate

This paper examines the ability of coupled system models, derived from the Facchinetti, de Langre and Biolley model, to fit the vortex-induced vibrations observed on a cantilevered rectangular flat plate, with zero angle of attack, in a hydrodynamic tunnel. Two VIV areas have been scrutinized. The first one for 0.6×105≤Re≤105 and the second one for 4.5×105≤Re≤7×105 where two distinct hydrodynamic excitation mechanisms, characterized by two Strouhal numbers, successively lock the first bending and first torsional mode of the plate respectively. For the bending VIV, exhibiting weak fluid–structure coupling, good results were obtained using a hybrid model for which a structure oscillator forced at ωv1=2πSt1U/D is coupled with a wake oscillator associated to the second Strouhal number. Regarding the torsional VIV, amplitude response and successive frequency lock-in have been satisfactorily reproduced using a structure oscillator (in rotation) coupled with two distinct wake oscillators.

Nonlinear vibrations of FG-GRLCC rectangular variable cross-section plate subjected to transverse and parametric excitations in thermal environment

This paper investigates the linear and nonlinear vibration responses for the functionally graded graphene-reinforced laminated composite cantilever (FG-GRLCC) rectangular variable cross-section plate subjected to the transverse and parametric excitations under the thermal environment. Halpin-Tsai model is used to calculate the material properties of the graphene-reinforced structure. Using the classical laminated plate theory, von Karman large deformation theory, Hamilton principle and Galerkin method, the dynamic model is given for the FG-GRLCC rectangular variable cross-section plate. The natural frequencies and mode shapes are analyzed for the FG-GRLCC rectangular variable cross-section plate under the thermal environment through using Rayleigh-Ritz method. The averaged equations of the system are obtained based on the multiple scale perturbation (MSP) under the primary, 1/2 sub-harmonic and 1:1 internal resonances. The comparisons between the theoretical algorithm and finite element method are proposed to illustrate the accuracy of the present model. The results about the frequency veering phenomenon and mode shape interaction have been illustrated. The amplitude-frequency response curves and force-amplitude response curves are depicted for the FG-GRLCC rectangular variable cross-section plate. The nonlinear and chaotic vibrations for the FG-GRLCC rectangular variable cross-section plate is studied by using the bifurcation diagrams and max Lyapunov exponents.

FREE VIBRATIONS OF THIN ELASTIC ORTHOTROPIC CYLINDRICAL PANEL WITH RIGID – CLAMPED EDGE GENERATOR

Using the system of equations corresponding to the classical theory of orthotropic cylindrical shells, the free vibrations of thin elastic orthotropic cylindrical panel with rigid – clamped edge generator are investigated. In order to calculate the natural frequencies and to identify the respective natural modes, the generalized Kantorovich-Vlasov method of reduction to ordinary differential equations is used. Dispersion equations for finding the natural frequencies of possible types of vibrations are derived. An asymptotic relation between the dispersion equations of the problem at hand and the analogous problem for a cantilever rectangular plate is established. An algorithm for separating possible boundary vibrations is presented. As an example, the values of dimensionless characteristics of natural frequencies are derived for an orthotropic cylindrical panel.

An Analytical Solution on Vibration Reduction and Shear Force Mitigation of Cantilever Mindlin Plates Using Orthogonal Ribs

Vibration of aircraft wings and the dynamic stress concentration at the clamped edge are important research topics due to concerns on the safety of aircrafts. To have a better understanding of the problem, the free and forced vibration response of a ribbed rectangular cantilever plate representing a section of an aircraft wing is investigated in this study. A new analytical solution is developed for the vibration analysis of rib stiffened cantilever plates using Mindlin plate and Timoshenko beam theories alongside the finite integral transform technique. The one- and two-dimensional integral transforms are applied to the governing equations of beams and plates, respectively, where the coupling force components at the interface between the base plate and the beam(s) can be automatically defined during the integral transform. Eventually, the partial differential equations are transformed into a system of linear algebraic equations in which its derivation is rigorous and easily implemented. Good agreements are found between the results of analytical solution, finite element analysis (FEA) and related literature. The solution is then employed to study the vibration suppression of cantilever plates and the shear force at the clamped edge. It is found that the insertion of a pair of orthogonal ribs in the plate can effectively reduce its vibration. An optimum orthogonal ribbing pattern is obtained using multi-objective particle swarm optimization (MOPSO) algorithm, taking into consideration both the vibration suppression of the plate and the maximum induced shear force at the corners of the clamped edge.

An orthotropic continuum model with substructure evolution for describing bone remodeling: an interpretation of the primary mechanism behind Wolff's law.

We propose a variational approach that employs a generalized principle of virtual work to estimate both the mechanical response and the changes in living bone tissue during the remodeling process. This approach provides an explanation for the adaptive regulation of the bone substructure in the context of orthotropic material symmetry. We specifically focus upon the crucial gradual adjustment of bone tissue as a structural material that adapts its mechanical features, such as materials stiffnesses and microstructure, in response to the evolving loading conditions. We postulate that the evolution process relies on a feedback mechanism involving multiple stimulus signals. The mechanical and remodeling behavior of bone tissue is clearly a complex process that is difficult to describe within the framework of classical continuum theories. For this reason, a generalized continuum elastic theory is employed as a proper mathematical context for an adequate description of the examined phenomenon. To simplify the investigation, we considered a two-dimensional problem. Numerical simulations have been performed to illustrate bone evolution in a few significant cases: the bending of a rectangular cantilever plate and a three-point flexure test. The results are encouraging because they can replicate the optimization process observed in bone remodeling. The proposed model provides a likely distribution of stiffnesses and accurately represents the arrangement of trabeculae macroscopically described by the orthotropic symmetry directions, as supported by experimental evidence from the trajectorial theory.

Nonlinear vibrations of four-degrees of freedom for piezoelectric functionally graded graphene-reinforced laminated composite cantilever rectangular plate with PPF control strategy

The nonlinear vibration suppression of the piezoelectric functionally graded graphene-reinforced laminated composite cantilever (PFG-GRLCC) rectangular plate with positive position feedback (PPF) control strategy is investigated firstly. The material properties of the graphene-reinforced structure are calculated through the Halpin–Tsai micromechanical model. Considering the transverse external excitation and the converse effect of piezoelectricity, the governing equations of motion are formulated through von Karman large deformation theory, the classical laminated plate theory, and Hamilton principle. After adding the PPF controllers, a four-degrees of freedom model for the close-loop vibration control system is achieved via Galerkin truncation technique. The average equations in the case of the primary resonance and 1:1:3:3 internal resonance can be obtained using the multiple scale perturbation (MSP) method. The amplitude–frequency response curves are studied by the numerical continuation algorithm. The detailed parametric analyses show that the PPF controller can effectively reduce the nonlinear vibration response amplitudes of the PFG-GRLCC rectangular plate. In addition, the results reveal the energy transform between the host system and the PPF controller. This work is expected to provide theoretical guidance for nonlinear large amplitude vibration reduction of graphene-reinforced structure.

Free Vibration of Rectangular Composite Cantilever Plate and Its Application in Material Degradation Assessment

Many composite cantilever plate-like structures have found engineering applications in different industries. For attaining a meaningful assessment of the plate vibration characteristics, it is important to have efficient and effective methods for determining the natural frequencies/mode shapes of composite cantilever plates. In this paper, a method formulated on the basis of the Ritz method and a simple first-order shear deformation theory (SFSDT) is presented to analyze the free vibration of thin as well as thick rectangular composite cantilever plates for determining their natural frequencies. In the SFSDT, the total deflection is the sum of two deflection components, namely, bending and through-thickness shear-deformation-induced deflections. The successful application of the Ritz method together with the SFSDT for the free vibration analysis of thick composite plates relies on the selection of two independent sets of characteristic functions for the bending and through-thickness shear-deformation-induced deflections, respectively, to satisfy the requirements for the displacement and force conditions at the fixed edge of the plate. The novelty of the proposed method is that two independent sets of characteristic functions, namely, polynomials and trigonometric functions, which satisfy the displacement and force conditions at the fixed edge have been identified and used in the variational method to construct the eigenvalue problem for extracting the modal characteristics (natural frequencies and mode shapes) of the plate. It has been shown that the uses of the selected characteristic functions can produce excellent natural frequencies for both thin and thick composite cantilever plates. Some existing theoretical and experimental natural frequencies of thin as well as thick composite plates have been used to demonstrate the accuracy of the proposed method in predicting natural frequencies. The significant effects of through-thickness shear deformation on the natural frequencies of composite cantilever plates are studied to show the merit of the present method. Finally, for illustrating the application of the proposed method in free vibration analysis, a novel procedure established on the basis of the sensitivity analysis of natural frequencies is presented to assess the material degradation of composite cantilever plates. The numerical examples have shown that fewer than 10 iterations are required in the identification process to produce a good estimation of the current value for each material constant.

Aeroelastic analyses of functionally graded aluminium composite plates reinforced with graphene nanoplatelets under fluid-structural interaction

This study presents a comprehensive investigation on functionally graded (FG) graphene reinforced aluminium cantilever rectangular plates under aerodynamic loads. Two-way loosely coupled fluid structure interaction (FSI) is implemented by coupling finite element analysis (FEA) and computational fluid dynamics (CFD). FEA models are developed using FG graphene nanoplatelets (GPL) reinforced aluminium composite structures with different GPL distribution patterns. Fluid domain is modelled using finite volume method and FSI module is then used to connect FEA with the CFD code. The simulation results of FSI indicate that the maximum stress of the plate can be efficiently reduced with satisfactory aerodynamic performance after aeroelastic tailoring, depending on specific GPL distribution patterns. It is shown that in conjunction with aeroelastic tailoring technique, the composite plate can achieve optimized structural and aerodynamic performance. The present results of the FG graphene reinforced aluminium plates offer useful design guideline for their applications as the structural components in aeronautical fields.

Modelling and nonlinear dynamics of dielectric elastomer sandwich cantilever-plate actuators

Inspired from the bimorph actuators for flapping and propelling swimming vehicles, a soft actuator made of two dielectric elastomer layers and a hyperelastic material core is presented. This actuator is also termed as a sandwich cantilever rectangular plate, which is stimulated by alternating voltages with a phase difference. Based on the classical plate theory and the variational method, the governing equation describing the nonlinear vibration of the sandwich cantilever plate is formulated. Employing the harmonic balance method and the arc-length continuation method, periodic solutions are obtained, then the convergence analyses of the numbers of discrete modes and harmonics are conducted. In order to illustrate the frequency features more clearly, a modified mode numbering scheme is developed. The numerical results manifest that, the combined DC-AC voltage excitation and the AC voltage excitation can excite abundant nonlinear behaviors of the soft actuator, including the local peak and the isolated bubble aroused by the nonlinear coupling. Additionally, the fundamental mode response keeps dominating the dynamic behaviors of the cantilever-plate actuator. The nonlinear dynamics broaden the resonant domain significantly, which can be potentially applied to the design and manufacture of high-performance soft actuators through the vibration enhancement strategy.

Forward and inverse analysis of transient responses for a cantilevered rectangular plate under normal and oblique impact loadings

This study investigates the transient responses of a cantilevered rectangular aluminum plate under normal and oblique impact loadings. The transient displacement and strain histories of the aluminum plate are forward calculated as a superposition of the normal modes using first-order shear deformation plate theory and the spectral collocation method considering both in-plane and out-of-plane dominant vibrations. A laser displacement meter and strain gauges are used to measure the displacement and strain histories of the same aluminum plate when subjected to normal and oblique loadings. Subsequently, frequency-dependent damping ratios are obtained from the associated wavelet spectrograms. The measured histories correlate well with those estimated using the theoretical analysis considering the damping ratios. Additionally, the inverse problem of determining the impact force history from the transient response is investigated. First, the transfer function between the impact force and measured response is derived using a theoretical analysis developed in the time domain. Subsequently, a wavelet deconvolution method is applied to reconstruct the impact force using a discrete wavelet transform with different scale and shift components. Thus, the impact force is obtained by reconstructing the wavelet expansion coefficients using these components, thereby demonstrating the validity of the wavelet deconvolution technique utilizing the theoretical transfer function.

An approach to determining parameters for active vibration control of piezoelectric-based smart-systems

In this paper, based on the numerical solution to the problem of forced steady-state vibrations of piecewise homogeneous electro-viscoelastic bodies, an analysis of the mechanical response of a system to the action of a number of parameters characterizing the applied mechanical or/and electrical action was carried out. The object of research was a cantilever rectangular plate with a piezoelectric element located on its surface. Functional relations that allow determining optimal parameters of electrical impact providing minimal value of a mechanical response were obtained for cases of action of two loading factors. A condition was established that allows reaching such a result: the resulting responses in case of action of a single load factor should be equal and loads must be applied in opposite phase.

Comparative Study of Beam and Plate Theories for Moderate Aspect Ratio Wings

The main objective of the present study is to identify the best structural model for both aeroelastic and structural analysis of unswept rectangular wings with aspect ratios between 2 to 5. The paper presents a detailed study of the similarities and differences between the classical beam and plate theories, which provides new insight into this classical problem. To perform the aeroelastic analysis, the three-dimensional Peters aerodynamic model is coupled with classical plate theory. This combination leads to a new simpler three-dimensional aeroelastic model that can reduce the computational time. The results have been evaluated using both the unpublished and highly cited experimental data. Interestingly, by performing both structural and aeroelastic analyses, a comment can be stated that, even for relatively low aspect ratios () of thin rectangular cantilever plates, the model based on the beam theory is closer to the experimental model compared to the plate theory. This finding may be unexpected by many researchers. Also, using a novel approach, it has been shown that the most significant difference between the beam and plate theories results arises from a subtle difference in the displacement field, which has not been given attention so far.

Performance Evaluation of Novel Piezoelectric Cantilever Beam Structure for Energy Harvesting

This study proposes an elastic compact Piezoelectric energy harvester (PEH) in multiple configurations, including a single cantilever beam, a single cantilever beam with a hole, two cantilever beams with a hole contributing towards a wider operating bandwidth. The cantilever beams are designed in multiple structural forms and section shapes, including symmetrical uniform section rectangle frames. Natural frequencies, vibration modes, and output voltages of PEH are estimated using Finite element software, with the frame acting as a rigid body and the cantilever plate acting as a deformed structure. A PEH with a rectangular cantilever plate surrounded by a rigid frame is designed and attached to the vibration platform with a customized fixture in the simulation software. The design meets the requirements of natural frequency and high output voltage in the low-frequency band. The PEH structure is simulated using a variety of piezoelectric materials, primarily PZT4 and PZT5H, barium titanate, lithium niobate, zinc oxide, and polyvinylidene difluoride (PVDF) using the finite element method. In the structural configuration of two cantilever beams with holes, PEH exhibited the maximum output voltage of 32.832 V for Zinc oxide compared to other materials. So, it can be concluded that insertion of hole and variation in the geometry of cantilever structure along with frame plays a vital role in achieving a large voltage from the vibrations in PEH. HIGHLIGHTS The finite element approach was used to develop and simulate the cantilever-based piezoelectric energy harvesting structure Multiple elastic compact Piezoelectric energy harvester (PEH) designs, including a single cantilever plate, a single cantilever plate with a hole, and two cantilever plates with a hole, are simulated using different materials The frame, hole insertion, and geometry modification of the cantilever structure all play important roles in generating a substantial voltage from the vibrations in PEH This study presents an Eigen frequency analysis of different PEH structures GRAPHICAL ABSTRACT

Effect of ribs on vibration characteristics of cantilever plate

Analytical solution for the vibration response results of the ribbed rectangular cantilever plate using the Kirchhoff plate theory is presented in this paper. The validity of the analytical solution is verified by comparison with the results of finite element analysis. The analytical model is then used to study the effect of the single rib, periodic ribs and orthogonal ribs on the modal vibration of the cantilever plate. According to the minimum inertia axis and torsion center line of the cantilever plate, the modal vibration shapes generated by the cantilever plate are divided into flexural and torsional vibration mode. The effect of the insertion of single rib on the modal vibration of the cantilever plate and the curvature of the insertion position are analyzed. It can be found that the effect of single rib on modal vibration of cantilever plate depends on the vibration mode and the location of the rib insertion, where a rib can be used as a mass loading, a stiffness attachment, or a combination of both. And the effect of periodic ribs on the vibration characteristics of the cantilever plate is further investigated. When periodically ribbed, the effect of the periodic ribs on the modal vibration of the cantilever plate depends on the dominant of the total of the mass loading or the stiffness enhancement played by each of these ribs. However, when another rib is added to the torsion center line of the periodic ribbed cantilever plate, it shows stiffness enhancement in both flexural and torsional vibration.

Estimation of Virtual Exciting Source by Free Vibration Response and Application of Structural Design for Noise Reduction

Exciting force on structures is required for estimation of vibration and noise after structural modification. Conventionally, the vibration response after the modification is obtained by measuring or estimating the excitation force before the modification and assuming that the force is invariant after the modification. However, in general, the force changes before and after the structural modification. Virtual exciting source (VES) has been proposed to change the force on structure. VES consists of connecting vibration system and structural invariant force. In this paper, practical method of the identification of the position of VES, the characteristics of the system and the structural invariant force is proposed. First, for a target structure with known modal parameters, the ratio of measured forced vibration response at multiple response points and the calculated forced vibration response at assumed excitation point is calculated. The true excitation point is obtained as the point which make the variance of the ratio minimize. Second, characteristics of connecting vibration system are optimized as minimized error between natural frequency measured by free vibration response and calculated natural frequency. Finally, structural invariant force is calculated from the measured acceleration and the characteristics of the structure. For cantilever rectangular plate, the validity of this method is shown in numerical simulations and experiments.

Optimal sensor configuration for structural response prediction by a modified Nelder–Mead simplex method

The optimal configuration of the sensor system is extremely critical to effectively control the test cost and ensure the quality of the collected data for structural health monitoring (SHM) system. Different from the traditional sensor configuration criterion, the information entropy measure aims to minimize the uncertainty of structural model parameters to be identified, which can ensure that the collected data provides the maximum amount of information for the model identification. However, most of the current researches within the framework of entropy measure are aimed at the model identification, while studies that consider response prediction are still rare. In this paper, by constructing the posterior predictive distribution function of the frequency response functions (FRFs) at the unmeasured positions of structures and also the corresponding information entropy representation, the optimal sensor configuration problem for the purpose of structural response prediction at the unmeasured position is studied. A modified bound-constrained Nelder–Mead simplex method to handle the binary encoded optimization variables is also proposed, which can effectively solve the combinatorial optimization problem in the optimal sensor configuration. The methodology proposed is validated by a forced vibration study conducted for a laboratory rectangular cantilever plate. The obtained results show that the prediction error and uncertainty of the higher-order modes of the FRFs at the unmeasured positions is generally larger than that of the lower ones. Also, by comparing the performance of worst and optimal sensor layouts, the importance of the optimal configuration for response prediction at unmeasured locations is clearly revealed especially for a small number of sensors.

Stability of rectangular cantilever plates with high elasticity

The problem of cantilever plate stability has been little studied due to the difficulty of solving the corresponding boundary problem. The known approximate solutions mainly concern only the first critical load. In this paper, stability of an elastic rectangular cantilever plate under the action of uniform pressure applied to its edge opposite to the clamped edge is investigated. Under such conditions, thin canopies of buildings made of new materials can be found at sharp gusts of wind in longitudinal direction. At present, cantilever nanoplates are widely used as key components of sensors to create nanoscale transistors where they are exposed to magnetic fields in the plate plane. The aim of the study is to obtain the critical force spectrum and corresponding forms of supercritical equilibrium. The deflection function is selected as a sum of two hyperbolic trigonometric series with adding special compensating summands to the main symmetric solution for the free terms of the decomposition of the functions in the Fourier series by cosines. The fulfillment of all conditions of the boundary problem leads to an infinite homogeneous system of linear algebraic equations with regard to unknown series coefficients. The task of the study is to create a numerical algorithm that allows finding eigenvalues of the resolving system with high accuracy. The search for critical loads (eigenvalues) giving a nontrivial solution of this system is carried out by brute force search of compressive load value in combination with the method of sequential approximations. For the plates with different side ratios, the spectrum of the first three critical loads is obtained, at which new forms of equilibrium emerge. An antisymmetric solution is obtained and studied. 3D images of the corresponding forms are presented.

Damage Accumulation in a Novel High-Throughput Technique to Characterize High Cycle Fatigue

Abstract A multiple-insert carrier plate assembly has been developed to increase the throughput of vibration-based fatigue testing. Typically, in vibration-based fatigue testing, a cantilevered rectangular plate is excited in a high-frequency resonant mode until failure. In this manner, an S-N curve for bending fatigue can be constructed. The concept behind vibration-based fatigue testing has been improved by employing a U-shaped carrier plate and fastening multiple smaller specimens into the channel of the carrier plate, such that fatigue tests can be conducted in tandem. However, because of the presence of a variable stress field over the surface of the assembly, the strain in the specimens is unequal. To account for this unequal strain distribution, a framework is provided for estimating how damage accumulates within each specimen. Using this framework, the efficiency gained by adopting a multi-insert approach is evaluated. By optimizing the strategy with which specimens are replaced as they fail, a throughput 5.49 times greater than sequential testing is shown to be feasible.