Flexible Rectangular Plate Research Articles

Numerical investigation of smart material-based structures for vibration energy-harvesting applications

Abstract The present work deals with energy-harvesting devices, which are useful in scavenging power using piezoelectric materials. Utilizing classical beam theory and classical plate theory, finite element modelling has been carried out to optimize the performance of output power of a cantilever beam and a flexible rectangular plate. Harmonic oscillations and base excitation will be the two different forcing functions used to drive the system. Based on this, numerical investigations of the performance of output power for the piezoelectric cantilever beam and flexible rectangular plate with different cases are considered. The present work is also useful for designing the piezoelectric cantilever beams and plates to extract maximum output power within the frequency ranges from 0 to 200 Hz. Numerical investigations on the piezoelectric cantilever beam and flexible rectangular plate with different cases reveal that the performance of output power is influenced by factors like load resistance, applications of with and without host structures, and different design parameters like unimorph, bimorph, embedded, line- and cross-type piezoelectric patch arrangements.

Computing static behavior of flexible rectangular von Kármán plates in fast and reliable way

In this paper, several modifications of the Bubnov–Galerkin method for studying flexible rectangular plates have been constructed and validated. This is to overcome the challenges of applying the existing approaches to a specific class of problems, bordering on accuracy and machine time efficiency. Von Kármán’s (elliptic) equations consisting of a system of nonlinear PDEs of order eight, with respect to two functions, i.e., the deflection and the stress functions, are considered. The problem is reduced by employing several modifications of the Bubnov–Galerkin method combined with the Fourier approach (separation of variables) to a system of nonlinear ODEs. The paper deals with many of such techniques and a procedure that refines their solutions based on known discrepancies. A theorem on the convergence of the proposed iterative processes is formulated and proved. The developed iterative methods are used to solve rectangular plates problem, subjected to distributed and local loads for several kinds of boundary conditions. A comparison of the obtained solutions by various methods is carried out. The optimal calculation method in terms of accuracy and minimum machine time computation is revealed. In addition, we show that the proposed novel and efficient approaches can be used to estimate the static behavior of plates of arbitrary geometry.

A Novel Rigid Finite Element Formulation for Dynamic Analysis of Flexible Plates

This paper introduces a new rigid finite element method (RFEM) formulation for dynamic analysis of plates. In RFEM, a flexible body is divided into several rigid elements which are interconnected by spring and damping elements. Mainly, RFEM has been used to model systems with beam-like slender components, such as ropes and cables, and few RFEM formulations were developed to model plates and shells. In this study however, by means of the Timoshenko beam theory and cuboid rigid elements, a novel RFEM formulation is developed to model flexible flat plates with generic geometries of the surface, considering large deformation. For this purpose, an RFEM formulation is presented for straight beams with rectangular cross-section by means of the Timoshenko beam theory. Next, this formulation is expanded to model flexible rectangular plates with uniform thickness and then, it is elaborated on how to model non-rectangular plates with uniform thickness using only cuboid elements and consequently, the development of the proposed RFEM formulation is completed. After investigating various case studies, the competence of the proposed formulation is evaluated thoroughly. The results of RFEM came on a proper agreement with the FEM results in a way that the maximum difference between these results in dynamic analysis was about 3%.

The controlled impact of elastic plates on a quiescent water surface

The impact of flexible rectangular aluminum plates on a quiescent water surface is studied experimentally. The plates are mounted via pinned supports at the leading and trailing edges to an instrument carriage that drives the plates at constant velocity and various angles relative to horizontal into the water surface. Time-resolved measurements of the hydrodynamic normal force ($F_n$) and transverse moment ($M_{to}$), the spray root position ($\xi _r$) and the plate deflection ($\delta$) are collected during plate impacts at 25 experimental conditions for each plate. These conditions comprise a matrix of impact Froude numbers${Fr} = V_n(gL)^{-0.5}$, plate stiffness ratios$R_D= \rho _w V_n^2 L^3D^{-1}$and submergence time ratios$R_T= T_sT_{1w}^{-1}$. It is found that$R_D$is the primary dimensionless ratio controlling the role of flexibility during the impact. At conditions with low$R_D$, maximum plate deflections on the order of$1$ mm occur and the records of the dimensionless form of$F_n$,$M_{to}$,$\xi _r$and$\delta _c$are nearly identical when plotted vs$tT_s^{-1}$. In these cases, the impact occurs over time scales substantially greater than the plate's natural period, and a quasi-static response ensues with the maximum deflection occurring approximately midway through the impact. For conditions with higher$R_D$($\gtrsim 1.0$), the above-mentioned dimensionless quantities depend strongly on$R_D$. These response features indicate a dynamic plate response and a two-way fluid–structure interaction in which the deformation of the plate causes significant changes in the hydrodynamic force and moment.

Effectiveness of line type and cross type piezoelectric patches on active vibration control of a flexible rectangular plate

Abstract In the present work, vibration control of a simply supported plate with line type and cross type piezoelectric (PZT) patches are investigated with and without actuation voltage. The plate is modeled under the assumption of Kirchhoff’s Plate theory. The mass of PZT patches remain constant in all cases. In case of actuation, applied voltage considered are 1, 2 and 3 mV. The external excitation to the plate is in the form of harmonically varying point load of 1 mN. It is noticed that cross type PZT patch is more effective in deflection suppression of plate than that of line type PZT patch at 3 mV of actuation at patch thickness of 0.75 μm. Suppression of central deflection of plate for line type and cross type PZT patches are obtained in different frequency bands of (175–185 Hz) and (870–880 Hz) respectively.

Geometrically nonlinear models of static deformation of micropolar elastic thin plates and shallow shells

AbstractThe present paper considers micropolar plates and shallow shells, the elastic deflections of which are comparable with their thickness and are small in comparison with characteristic cross‐section size. At the same time, both rotation angles of the normal to the median surface before deformation and their free rotations are small. Also, in the tensors of deformation and flexures‐torsions, the nonlinear members in the gradients of the displacement are considered. Hypothesis method is developed, on the basis of which general applied models of static deformation of the micropolar elastic flexible plates and shallow shells are constructed. On the basis of these models, specific problems for the micropolar elastic flexible rectangular plates and shallow shells rectangular in cross‐section are solved in case, when bounds are hinge‐supported. On the basis of numerical analysis, certain effective features of the micropolar material are established in comparison with the corresponding classical material.

Analysis and modeling of a flexible rectangular cantilever plate

Flexible plate structures with large deflection and rotation are commonly used structures in engineering. How to analyze and solve the cantilever plate with large deflection and rotation is still an unsolved problem. In this paper, a general nonlinear flexible rectangular cantilever plate considering large deflection and rotation angle is modeled, solved and analyzed. Hamilton's principle is applied to obtain the nonlinear differential dynamic equations and boundary conditions by introducing a coordinate transformation between the Cartesian coordinate system and the deformed local coordinate system. Stress function relating to in-plane force resultants and shear forces is given for the first time for complex coupling equations caused by coordinate transformation. The nonlinear equations and the solving method are validated by experiments. Then, harmonic balance method is adopted to get the nonlinear frequency-response curves, which shows strong hardening spring characteristic of this system. Runge–Kutta methods are used to reveal complex nonlinear behaviors such as 5 super-harmonic resonance, bifurcations and chaos for general nonlinear flexible rectangular cantilever plate.

Active disturbance rejection and Lyapunov redesign approaches for robust boundary control of plate vibration

ABSTRACTIn this paper, two approaches, namely active disturbance rejection control (ADRC) and Lyapunov redesign, are utilised to stabilise the vibration of a boundary-controlled flexible rectangular plate in the presence of exogenous disturbances. Based on ADRC, an estimation/cancellation strategy is applied where disturbance is estimated online by an extended state observer (ESO) and cancelled by injecting the output of ESO into the feedback loop. By the Lyapunov redesign, on the other hand, the control law intended for a nominal system is redesigned by adding a (discontinuous) control component that makes the system robust to large uncertainties. Both control algorithms are designed directly based on partial differential equation model of the plate so that spillover instabilities that are a result of model truncation are avoided. The established control schemes are able to stabilise the plate vibration by actuating and sensing only along the plate boundary while accounting for the dynamical effects of Gaussian curvature integral, in-plane membrane force and actuator mass. The stability of each control approach is proven using Lyapunov analysis. The efficacy of each proposed control is illustrated by simulation results.

Nonlinear analysis of flexible plates lying on elastic foundation

This article describes numerical procedures for analysis of flexible rectangular plates lying on elastic foundation. Computing models are based on the theory of plates with account of transverse shear deformations. The finite difference energy method of discretization is used for reducing the initial continuum problem to finite dimensional problem. Solution procedures for nonlinear problem are based on Newton-Raphson method. This theory of plates and numerical methods have been used for investigation of nonlinear behavior of flexible plates on elastic foundation with different properties.

Experimental Study of Flexible Plate Vibration Control by Using Two-Loop Sliding Mode Control Strategy

It is well known that intelligent control theory has been used in many research fields, novel modeling method (DROMM) is used for flexible rectangular active vibration control, and then the validity of new model is confirmed by comparing finite element model with new model. In this paper, taking advantage of the dynamics of flexible rectangular plate, a two-loop sliding mode (TSM) MIMO approach is introduced for designing multiple-input multiple-output continuous vibration control system, which can overcome uncertainties, disturbances or unstable dynamics. An illustrative example is given in order to show the feasibility of the method. Numerical simulations and experiment confirm the effectiveness of the proposed TSM MIMO controller.

Thrust generation by a heaving flexible foil: Resonance, nonlinearities, and optimality

Flexibility of marine animal fins has been thought to enhance swimming performance. However, despite numerous experimental and numerical studies on flapping flexible foils, there is still no clear understanding of the effect of flexibility and flapping amplitude on thrust generation and swimming efficiency. Here, to address this question, we combine experiments on a model system and a weakly nonlinear analysis. Experiments consist in immersing a flexible rectangular plate in a uniform flow and forcing this plate into a heaving motion at its leading edge. A complementary theoretical model is developed assuming a two-dimensional inviscid problem. In this model, nonlinear effects are taken into account by considering a transverse resistive drag. Under these hypotheses, a modal decomposition of the system motion allows us to predict the plate response amplitude and the generated thrust, as a function of the forcing amplitude and frequency. We show that this model can correctly predict the experimental data on plate kinematic response and thrust generation, as well as other data found in the literature. We also discuss the question of efficiency in the context of bio-inspired propulsion. Using the proposed model, we show that the optimal propeller for a given thrust and a given swimming speed is achieved when the actuating frequency is tuned to a resonance of the system, and when the optimal forcing amplitude scales as the square root of the required thrust.

Experimental Study of Adaptive Sliding Mode Control for Vibration of a Flexible Rectangular Plate

This paper aims to address the intelligent active vibration control problem of a flexible rectangular plate vibration involving parameter variation and external disturbance. An adaptive sliding mode (ASM) MIMO control strategy and smart piezoelectric materials are proposed as a solution, where the controller design can deal with problems of an external disturbance and parametric uncertainty in system. Compared with the current ‘classical’ control design, the proposed ASM MIMO control strategy design has two advantages. First, unlike existing classical control algorithms, where only low intelligence of the vibration control system is achieved, this paper shows that high intelligent of the vibration control system can be realized by the ASM MIMO control strategy and smart piezoelectric materials. Second, the system performance is improved due to two additional terms obtained in the active vibration control system. Detailed design principle and rigorous stability analysis are provided. Finally, experiments and simulations were used to verify the effectiveness of the proposed strategy using a hardware prototype based on NI instruments, a MATLAB/SIMULINK platform, and smart piezoelectric materials.

11006 可撓性平板の抗力特性

The body with flexibility and elasticity in flow undergo fluid force, then deforming its shape. Drag characteristic of such a flexible and elastic body in flow is expected to be different from that of solid body. In this study, we performed the drag measurements using flexible rectangular plates of three different densities, as test pieces. Mainflow velocity was set in the range from 5 m/s to 15 m/s. Drag measurement was performed by using wind tunnel balance. It seems that the drag act on the flexible rectangular plate was increased linearly.

Modelling of sound radiation from a beam-stiffened plate and a clamped rectangular plate based on a modal method

The farfield acoustic radiation efficiency and power of a flexible rectangular plate coupled to a relatively stiffer beam are investigated. A numerical model based on a modal method that consists of a plate with sliding edges surrounded by four stiff beams is studied. Assuming that each beam is a heavy mass, a plate with clamped edges is realised, and this model is verified. This model is then extended to a beam-stiffened plate. If the bending stiffness of the excited beam is large, the radiation efficiency increases in the corner- and edge-mode frequency regions and is higher than that of the clamped plate in terms of the averaged response for randomly selected excitations. The reason for this effect is that the corner and edge areas that radiate sound are broader because the behaviour of the plate is governed by the motion of the stiff beam. This is explained in terms of the wavenumber and the wavelength of a stiff beam and a flexible plate. It is shown that this is true only when the excitation is applied to the beam, and the radiation efficiency is similar if the plate is excited. In addition, it was found that the radiation power decreases with increasing beam stiffness because the vibration of the plate actually decreases. In addition, it was shown that the variation in the radiation efficiency of the beam-stiffened plate is smaller when the beam is excited than when the plate is excited.

Experimental Study of Adaptive Fuzzy Sliding Mode Control for Vibration of a Flexible Rectangular Plate

In this paper, an approach is developed for active vibration control of flexible rectangular plate structures using control theory. The flexible rectangular plate system is first modeled through a new type of modeling method—discretization and reduced-order modeling method (DROMM); and second, the validity of the obtained new model is investigated by comparing the plate natural frequencies, static analysis, and forced vibration response analysis between a finite-element model and the new model. After validating the model, an adaptive fuzzy sliding mode multiple-input multiple-output (MIMO) controller is applied to the plate dynamic equation through the MATLAB/Simulink platform. The simulation and experimental results clearly demonstrate an effective vibration suppression capability that can be achieved using an adaptive fuzzy sliding mode MIMO controller.

20512 物体後流中のエネルギー散逸計測の試み

It is important to understand the effect of flow-induced vibration on the wake behind bluff body. The purpose of this study is to investigate the energy dissipation in bluff body wake by PIV measurement. In order to evaluate the reliability of dissipation measurement, the energy dissipations in the circular cylinder wake obtained by both PIV and hot wire measurement were compared. In addition, energy dissipation in the wake behind the flexible rectangular plate with the shape deformation and the flow induced vibration was measured to make clear the effect of flow-induced vibration in energy dissipation mechanism.

Regular and Chaotic Dynamics of Flexible Plates

Nonlinear dynamics of flexible rectangular plates subjected to the action of longitudinal and time periodic load distributed on the plate perimeter is investigated. Applying both the classical Fourier and wavelet analysis we illustrate three different Feigenbaum type scenarios of transition from a regular to chaotic dynamics. We show that the system vibrations change with respect not only to the change of control parameters, but also to all fixed parameters (system dynamics changes when the independent variable, time, increases). In addition, we show that chaotic dynamics may appear also after the second Hopf bifurcation. Curves of equal deflections (isoclines) lose their previous symmetry while transiting into chaotic vibrations.

Robust Nominal Model-Based Sliding Mode Robust Control for Vibration of Flexible Rectangular Plate

In this paper, we developed an approach for active vibration control of flexible rectangular plate structures using control theory. The flexible rectangular plate system is firstly modelled and simulated via a finite element method; and secondly, a new type of modelling method, and the state-space model are involved in the development of the equation of motion in state-space, which is efficiently used for the analysis of the system and design of control laws with a modern control framework. Then, the validity of the obtained new model is investigated by comparing the plate natural frequencies and forced vibration response analysis predicted by the finite element model with the calculated values obtained from new model. After validating the model, nominal model-based sliding mode robust MIMO controller is applied to the plate dynamics via the MATLAB/Simulink platform. The simulation results clearly demonstrate an effective vibration suppression capability that can be achieved using nominal model-based sliding mode robust MIMO controller.

Thermal failure of flexible rectangular viscoelastic plates with distributed sensors and actuators

The problem of thermal failure of simply supported flexible rectangular plates with distributed sensors and actuators as a result of dissipative heating by forced resonant bending vibrations is considered. The plates are loaded by a uniformly distributed, harmonic-in-time pressure. The mechanical problem is solved by the Galerkin method and the method of harmonic balance. The influence of dissipative heating on the efficiency of active damping of the forced resonant vibrations of the flexible plates is investigated. By equating the temperature to the Curie point, a simple expression for the critical mechanical load is obtained. When the load is more this critical value the active materials lose the piezoelectric effect and become passive. This is a specific type of thermal failure, where the body keeps its integrity but the electromechanical system loses its functionality.

Fuzzy control for geometrically nonlinear vibration of piezoelectric flexible plates

This paper presents a LMI(linear matrix inequality)-based fuzzy approach of modeling and active vibration control of geometrically nonlinear flexible plates with piezoelectric materials as actuators and sensors. The large-amplitude vibration characteristics and dynamic partial differential equation of a piezoelectric flexible rectangular thin plate structure are obtained by using generalized Fourier series and numerical integral. Takagi-Sugeno (T-S) fuzzy model is employed to approximate the nonlinear structural system, which combines the fuzzy inference rule with the local linear state space model. A robust fuzzy dynamic output feedback control law based on the T-S fuzzy model is designed by the parallel distributed compensation (PDC) technique, and stability analysis and disturbance rejection problems are guaranteed by LMI method. The simulation result shows that the fuzzy dynamic output feedback controller based on a two-rule T-S fuzzy model performs well, and the vibration of plate structure with geometrical nonlinearity is suppressed, which is less complex in computation and can be practically implemented.