Flexible Plate Research Articles

Mathematical modeling of a submerged piezoelectric wave energy converter device installed over an undulated seabed

A piezoelectric wave energy converter (PWEC) device integrated with an impermeable breakwater placed over an undulated seabed is considered. The PWEC device is composed of a single submerged flexible plate with piezoelectric layers attached to both faces of the flexible plate. Due to the piezoelectric effect, this piezoelectric plate generates electricity when excited by the incident waves. A detailed analysis is done to investigate the effect of PWEC plate submergence depth, plate length, plate edge conditions, incident wave period, bottom ripple amplitude, and ripple number on the power generation by the PWEC device. It is seen that the PWEC device edge conditions, submergence depth, and plate length play a significant role in the resonating pattern associated with the wave power generation curve. The results demonstrate that the PWEC device having moderate plate length and with free and moored type front edges generates a higher amount of wave power for a wider range of incident wave frequencies.

Combined effect of bending stiffness and streamwise length of the attached flexible splitter plate on the vortex-induced vibration of a circular cylinder

A flexible splitter plate attached to the rear stagnation point of a circular cylinder is applied to control the vortex-induced vibration. The influence of its bending stiffness on the control effect is experimentally investigated for two streamwise lengths of 1D and 2D (D is the cylinder diameter). Five dimensionless bending stiffnesses are selected: 0.979 (P1), 3.123 (P2), 6.797 (P3), 23.087 (P4), and 47.484 (P5). The characteristics of cylinder vibration, flow field, lift, and plate dynamics are synchronously studied. The results show that the effect of the bending stiffness depends on the streamwise length of the plate. For the flexible splitter plate of 1D, the best suppression effect of cylinder vibration is achieved for the P2 case, where maximum vibration amplitude is reduced by 93%, while the galloping-like vibration is excited for P4 and P5 cases. According to the analysis of the plate dynamics and the wake development, a large plate vibration is expected to suppress the galloping-like vibration. The bending stiffness affects the vibration state of the flexible splitter plate, and consequently, the wake vortex development and dynamic response of the cylinder are changed. For the flexible splitter plate of 2D, the vibration amplitude response of the cylinder can be divided into three categories based on the variation of amplitude versus the reduced velocity. For P2-P5 cases, a beneficial phenomenon that a frequency branch is larger than the natural frequency occurs.

Investigation on the hydrodynamic load fluctuations through passive flexible leading edge

The study presents a numerical investigation of two-dimensional partly flexible plate dynamics. The structure is immersed in a turbulent fluid flow with a Reynolds number based on its chord of 104. The plate is animated by a forced pitching movement. The flexibility effects of the plate's leading edge are analyzed, as it deforms under the hydrodynamic loads. The fluid–structure interaction effects are considered by solving a coupled problem using a strong implicit procedure. Both fluid and solid dynamics are solved. The numerical results of the present study are validated with experimental ones with a good agreement between both approaches for the lower reduced frequencies. Differences are observable for high frequency that could be imputable to the three-dimensional aspects of the experiment. It has been shown that with an appropriate choice of the rigidity of the structure, it is possible to mitigate the unsteady load fluctuations without affecting the load mean values too much. Indeed, at low pitching frequency (drag mode), the leading-edge vortex generation is impacted by the flexible leading edge. As a result, it tends to decrease the hydrodynamic force fluctuation amplitude without really impacting the mean force value. Conversely, at high pitching frequency (propulsive mode), it was found that a flexible leading edge tends to increase both the magnitudes of the hydrodynamic forces and their mean values. Finally, it is shown that the load fluctuation mitigation, or amplification, is maximum for a specific flexibility value depending on the pitching frequency.

Microstructural effects on dynamic response of rigid and flexible pavements to moving load under plane strain

Granular soils and asphalt concrete materials are characterized by microstructural effects due to their micro-stiffness and micro-inertia. In this work, an attempt is made to investigate if the material microstructure in rigid and flexible pavements affects their dynamic response to loads moving with constant speed under conditions of plane strain and linearity. The rigid pavement is modeled by a homogeneous isotropic and linearly elastic with viscous damping flexural plate resting on a homogeneous isotropic microstructured linear elastic with hysteretic damping half-pane. The flexible pavement is modeled by a layered half-plane with linear isotropic microstructured layers. The top (asphalt concrete) layer is viscoelastic, while the other layers are elastic with hysteretic damping. The dynamic response of both of the above rigid and flexible pavement models to a distributed over a finite length moving load is obtained by using the complex Fourier series approach in conjunction with compatibility and equilibrium at the plate/soil (for rigid pavements) and soil layers (for flexible pavements) interfaces. The so obtained analytical/numerical response solutions for rigid and flexible pavement problems, after verification, are employed for conducting parametric studies in order to assess the microstructural effects on the response of those pavements to moving loads. It was found, at least for the pavement cases considered here, that the presence of the stiff plate in rigid pavements and the viscoelastic top layer in flexible pavements reduce considerably microstructural effects on the pavement response, rendering them not so important for usual vehicle speeds. However, for vehicle speeds very close to the critical speed, microstructural effects become very important.

Design and application of a new type of long snake moxibustion box

Long snake moxibustion, a kind of indirect moxibustion therapy for stimulating the midline part of the back of the patient's body after playing a layer of ginger or garlic mud, is frequently used to treat spine disorders and deficiency-cold type syndromes. In the present paper, we introduced our newly made moxibustion box for applying long snake moxibustion which is safe and easy to operate and can be freely moved at any time in the treatment of vertebral diseases, and conveniently makes the ignited-moxa close to the locus. This newly-designed moxibustion box is made of two lines of paralleled flexible connection side plates hinged ends to ends and two U-like connection plates at the opposite two ends to construct a rectangular frame device. The hinged design makes the long snake moxibustion box conform to the physiological curvature of human body and its length can be adjusted according to the height of patients. When used in clinical practice, it can enhance the patients' safety and reduce the operator's working intensity, and may be helpful to the popularization and development of long snake moxibustion therapy.

Nonlinear vortex dynamic analysis of flow-induced vibration of a flexible splitter plate attached to a square cylinder

Flow-induced vibration (FIV) of a flexible splitter plate attached to a square cylinder in laminar flow is investigated using a two-way fluid-structure interaction (FSI) simulation. The Reynolds number based on the edge length of the square cylinder is fixed at Re = 332.6. Numerical results show that the vortex structure falling off on both sides of the square cylinder will excite the vibration of the flexible plate. There is a noticeable phase difference between the flexible plate's hydrodynamic load (lift) and the vertical vibration displacement of the end of the plate. It is also found that there is a pronounced hysteresis effect on the vortex shedding on both sides of the square cylinder due to the action of the flexible plate. The viscous dissipation term dominates the vorticity transport in low Reynolds number flow regimes. The viscous dissipation term reflects the flow trend of the flow field to a certain extent. The results obtained are insightful to the design of FSI-based ocean engineering structures using flexible plates.

The dynamics of a rigid inverted flag

An ‘inverted flag’ – a flexible plate clamped at its trailing edge – undergoes large-amplitude flow-induced flapping when immersed in a uniform and steady flow. Here, we report direct numerical simulations of a related single degree-of-freedom mechanical system: a rigid plate attached at its trailing edge to a torsional spring. This system is termed a ‘rigid inverted flag’ and exhibits the dynamical states reported for the (flexible) inverted flag, with additional behaviour. This finding shows that the flapping dynamics of inverted flags is not reliant on their continuous flexibility, i.e. many degrees of freedom. The rigid inverted flag exhibits additional, novel states including a heteroclinic-type orbit that results in small-amplitude flapping, and a number of chaotic large-amplitude flapping regimes. We show that the various routes to chaos are driven by a series of periodic states, including at least two which are subharmonic. The instability and competition between these periodic states lead to chaos via type-I intermittency, mode competition and mode locking. The rigid inverted flag allows these periodic states and their subsequent interaction to be explained simply: they arise from an interaction between a preferred vortex shedding frequency and a single natural frequency of the structure. The dynamics of rigid inverted flags is yet to be studied experimentally, and this numerical study provides impetus for such future work.

Correction of pediatric angular deformities in lower limbs through guided growth using a novel flexible plate system

IntroductionTension band plates (TBP) for guided growth (GG) are the gold standard treatment for angular deformities around the knee. EPIFLEX® is a novel flexible TBP that adjusts to the patient's bone anatomy. HypothesisGG using a flexible TBP produces satisfactory correction rates with minor complications in the pediatric population with angular deformities around the knee. Materials and methodsA retrospective evaluation of 33 patients (60 knees) treated for genu varum and valgum with hemiepiphysiodesis using a flexible TBP between 2017 and 2020 was performed. The study aimed to assess correction and complication rates; patients who completed treatment were included regardless of the follow-up times after implant removal. ResultsThirteen females and 20 males with a median age of 10 years were included. The median treatment duration and follow-up were 10 and 22 months. The median monthly rate of change of mLDFA and mMPTA was 0.67° and 0.57°, respectively. A successful correction was achieved in 90% of the cases. There were no cases of infection or implant failure. Four cases presented overcorrection and two undercorrection; no significant relation with deformity or obesity was found. DiscussionGG using this flexible TBP showed satisfactory correction rates with a low incidence of complications and no implant failure. It provides flexibility through good adaptability to the bone anatomy and mobility of the screws avoiding implant protrusion or breakage. Level of evidenceIV; observational descriptive case series.

Fluid/Structure Interaction of Cantilevered Plate in Supersonic Separated Flow

The fluid-structure interaction of a cantilevered plate geometry in Mach 2 flow was studied experimentally to assess the effects of structural compliance on the surrounding flowfield. The test geometry, representative of a compliant control surface, consists of an overhanging plate that extends past the edge of a backward-facing step to create a separated region in the flow. This allowed for the study of recirculation effects and unsteady pressure forcing on the cantilevered plate without shock/boundary-layer interactions that would be present if the plate were inclined to the flow. Rigid and compliant test articles were studied to capture the fluid response with and without structural deformation. Schlieren photography and particle image velocimetry showed that under the unsteady conditions during startup of the wind tunnel the flexible plate exhibited a highly dynamic oscillatory response with frequencies similar to its natural vibration response. Under steady, started supersonic flow conditions, the flexible cantilever exhibited smaller oscillations around a mean deflection of two plate thicknesses. Oil flow visualization revealed nontrivial three-dimensionality of the test section flowfield. Modal decomposition of stereo digital image correlation measurements demonstrated that the distinct frequencies present in the flexible plate’s response consistently correspond to the same mode shapes.

A new, flexible tool concept for rotary die stretch bending

Mitigating manufacturing risks for profile-type parts formed using stretch bending operations is a considerable challenge in the industry, due to the investments required to manufacture the tooling for these processes. Traditionally such tooling has been made by solid steel blocks machined to the desired shape. This paper introduces a new tool concept for stretch bending processes, utilizing an adjustable multipoint supported flexible spring steel plate as a bend die, applied on a double rotary turntable stretch bender. The capability of the tool is investigated through a proof-of-concept experiment, assessing whether the tool design is capable of creating bends with the same quality as bends made using traditional solid steel dies. Overall, the experiment shows promising results as it allows for creating a large variation of bending shapes with good dimensional quality with low tooling investment, although the interface between the spring steel plate end and the aluminum profile creates some minor scuffing marks. The initial investigation found a pooled standard deviation of 0.25mm over 3 bend shapes with 3 samples for each, which is promising for further development. In industrial applications, the proposed tool would enable cost-efficient experimenting using production-intent tooling in the early product development phase, which could also be utilized for pre-production qualification processes, low volume manufacturing and multi-variant products.

Streamline penetration, velocity error, and consequences of the feedback immersed boundary method

This paper presents a study on streamline penetration, velocity error, and consequences of a fluid–structure interaction (FSI) solver based on the feedback immersed boundary method (IBM). In the FSI solver, the fluid dynamics is solved by the lattice Boltzmann method; the solid structure deformation is solved by the finite difference method and the finite element method for two- and three-dimensional cases, respectively; and the feedback IBM is used to realize the interaction between the fluid and the structure. The IBM is implemented in non-iterative and iterative ways. For the non-iterative version, two types of integration are discussed: without and with velocity prediction step. Five benchmark cases are simulated to study the performance of the three implementations: a uniform flow over a cylinder, flow-induced vibration of a flexible plate attached behind a stationary cylinder in a channel, flow through a two-dimensional asymmetric stenosis, a one-sided collapsible channel, and a three-dimensional collapsible tube. Results show that both the IBM with prediction step, the iterative IBM, and one iteration IBM with proper feedback coefficients can suppress the spurious flow penetration on the solid wall. While the velocity error does not significantly affect the force production and structure deformation for external flows, reducing it significantly improves the prediction of the force distribution and structure deformation for internal flows. In addition, the iterative IBM with smaller feedback coefficient has better numerical stability. This work will provide an important guideline for the correct use of the feedback IBMs.

Convergence Analysis of Havelock-Type Eigenfunction Expansions for Hydroelastic Problems in Water having Infinite Depth

The present paper demonstrates the point-wise convergence of the Havelock-type eigenfunction expansion to the velocity potentials associated with the water waves interaction with flexible plate and membranes in water having infinite depth. To consider the higher-order boundary condition at the mean free surface of the water domain, flexible plate and membranes are assumed to float in the mean water level. In the convergence analysis procedure, firstly, the havelock-type eigenfunction expansion for the unknown velocity potentials associated with the physical problems are obtained. Hereafter, a suitable Green's function is developed for the associated physical problem. Using the developed Green's function and the associated properties, the vertical components of the Havelock-type eigenfunction expansion is expressed in terms of the Dirac delta function. Finally, using appropriate properties of the Dirac delta function, the point-wise convergence of the Havelock-type eigenfunction expansion is demonstrated.

The effect of discharge areas on the operational performance of a liquid-ring vacuum pump: Numerical simulation and experimental verification

To determine the effect of discharge area on the ability of a liquid-ring vacuum pump, the inspiratory capacity and shaft power are numerically investigated for different discharge areas adjusted by the number of discharge ports at different inlet pressures. Besides, the numerical results of these key operating parameters are verified by experiments in a developed testing system with a flexible valve plate on the discharge ports. The result shows that there exists an optimal discharge area of the liquid-ring vacuum pump at each inlet pressure, and good agreements are found between the experimental data and numerical results. For a 2BEC-40 vacuum pump, the maximum isothermal compression efficiencies can reach 39.99%, 46.80%, 49.16% and 40.82% for the inlet pressures of 20 kPa, 40 kPa, 55 kPa and 70 kPa with the optimum number of discharge ports is 1, 5, 12 and 12, respectively. The results suggest that the self-adaptive adjustment of discharge areas by the valve plates is of great importance to practical applications of a liquid-ring vacuum pump.

Local Mesh Refinement and Coarsening Based on Analysis-Suitable T-Splines Surface and Its Application in Contact Problem

AbstractIn contact analysis, reducing the computing time has been an issue under the premise of ensuring calculation accuracy around the region with violent stress changes. To improve computational efficiency for contact analysis in flexible multibody system, this paper proposes an adaptive local mesh refinement and coarsening approach based on analysis-suitable T-splines (ASTS). First, the kinematic model of thin plate is established based on analysis-suitable T-spline surface, and large deformation of flexible thin plate is described by the elastic model created by nonlinear Green–Lagrange strain. Second, to reduce computing time in contact analysis and ensure analysis accuracy, based on contact state and refinement distance, an effective adaptive local element mesh update method is proposed, which only refine locally on subject's refinement region and integrate redundant elements to reduce the degree-of-freedom (DOF) of system. Third, to analyze the system with varying mesh, a new solving algorithm with dynamic variables and geometry update routine is developed. Finally, performance of the proposed method in static and dynamic simulation is validated by four numerical examples. Results and consuming time of ASTS-based varying mesh prove the feasibility of the proposed method in contact problems.

Improvement of Calculation Technique for Flexible Orthotropic Plates on Elastic Base. Part 2. Calculation Results

The proposed paper is a continuation of the author's theoretical work presented earlier in the development of the theory and methodology for calculating flexible orthotropic reinforced concrete plates on an elastic foundation, taking into account the physical nonlinearity of the plate material. The paper presents numerical results of elastic and nonlinear calculations of an isolated rectangular orthotropic plate on an elastic foundation, modeled by an elastic homogeneous isotropic layer rigidly connected to a non-deformable foundation under the action of an external static load, taking into account the own weight of the reinforced concrete plate. The change in its stiffness at the time of cracking and further active opening of cracks has been taken into account in the calculation of the structure under study. The nonlinear calculation of the studied structure takes into account the change in its rigidity at the time of cracking and further active crack opening. The calculation of a flexible orthotropic plate on an elastic foundation in a nonlinear formulation is performed iteratively by the method of B. N. Zhemochkin. A mixed method of structural mechanics has been used to determine the coefficients of canonical equations and free terms. At the first iteration, the reinforced concrete plate is calculated as linearly elastic, homogeneous, and orthotropic; at the subsequent ones – as linearly elastic, inhomogeneous, and orthotropic at each Zhemochkin site. Camber plates with a clamped normal in the primary system of mixed method due to the action of a concentrated force are determined by the Ritz method when the deflections were represented as a power polynomial in a new original expression that has been proposed for the first time in the Part 1 of the paper. The solution algorithm has been implemented using the Wolfram Mathematica 11.3 computer program. Numerical and graphical results of elastic and non-linear calculations of sediment concrete road plate, contact stresses and bending moment diagrams on the plate are presented in the paper.

Proportional integral derivative controller based on ant colony optimization for vibration cancellation of horizontal flexible plate structure

<p>Flexible plate structure provides many benefits as compared to their rigid counterparts including lower energy consumption, effective, lightweight, and quick response. However, the vibration easily affects the flexible plate structure resulting in structural damage. This study introduces the modelling of a flexible plate structure based on a system identification technique known as ant colony optimization (ACO) algorithm for vibration control. Firstly, the input-output vibration data that represent the actual structure of flexible plate was achieved from the experiment. Next, the acquired vibration data was used to develop a dynamic model of the flexible plate structure. The performances of the ACO algorithm were assess based on mean squared error (MSE), pole-zero plot and correlation test in order to get a precise and reliable outcome. The results show that ACO algorithm achieved the minimum MSE which was 6.7613×10<sup>−6</sup>, high stability of pole-zero plot and excellent correlation test. Subsequently, the best model of ACO was chosen to create controller based on an active vibration control technique. It was noticed that the controller managed to obtain a 6.19 dB reduction at the first mode vibration in which the percentage of attenuation of the controller was 10.63% for sinusoidal disturbances and 9.64% for multiple sinusoidal disturbances.</p>

Enhanced absorption in structural acoustic silencers using piezo-shunting.

Structural acoustic silencers (sas) consist of a thin flexible plate that is introduced into a wall of an otherwise rigid-walled duct in which acoustic noise is propagating. Transmission loss (TL) in sas arises from reflection due to impedance mismatch, as well as partial absorption due to material damping in the plate. Reflection-based TL is less preferable than absorption-based TL, especially where back-pressure is undesirable. In this study, the TL arising out of absorption is increased by increasing the effective plate damping using the piezo-shunting method. Experiments are conducted on a custom-made sas with copper sheets of two different thicknesses. Piezo-shunting to increase the damping of the copper plate is done by connecting an electrical resistor to the external circuit of a thin piezoelectric bimorph attached to the copper plate. The effect of piezo-shunting on the sas TL is analyzed using a model. Piezo-shunt increases the fraction of power absorbed and decreases the reflected power fraction. The dependency of these fractions on the value of the electrical resistor and the impedance of the plate are discussed. This study demonstrates that piezo-shunting can be used to improve absorption-based TL in sas.

An application of cantilevered plates subjected to extremely large amplitude deformations: A self-starting mechanism for vertical axis wind turbines

The use of Darrieus straight-bladed vertical axis wind turbines (VAWTs) are becoming increasingly attractive for wind energy harvesting thanks to their distinctive advantages, even though these lift-force driven wind turbines suffer from anaemic self-starting capability, especially in areas with irregular wind regimes. In this paper, a novel self-starting mechanism is proposed for VAWTs. The proposed concept centres on the use of pliable plates in either of two distinct configurations: (i) with plates attached to the trailing edge of the rigid airfoil-shape blades of the VAWT; (ii) with plates installed inside the VAWT at a distance from the rotor. The feasibility of these designs for self-starting has been investigated theoretically. In particular, flexible blades are idealized as cantilevered beams placed at an angle to a uniform steady flow. Using a Hamiltonian framework, a geometrically-exact fluid-elastic continuum model is developed for the extremely large-amplitude deflections of the flexible blades. The fidelity of the developed framework is first validated against two sets of experimental data available in the literature. The developed framework is then used to estimate the static torque coefficients generated by the flexible plates in both aforementioned VAWT configurations. Finally, the self-starting efficacy of the proposed design is compared against a Savonius based mechanism used in a well-studied benchmark Darrieus–Savonius turbine. The numerical simulations reveal that, regardless of the initial static positions of the turbine, proper engineering of the pliable plates results in a torque generation capability which spins the turbine, even at low wind speeds. This finding proves the feasibility of the proposed concept as a potential self-starting mechanism for VAWTs. The advantages of this mechanism, such as easier design, manufacturing and implementation, enhanced reliability and economical maintenance, make this concept worthwhile for further exploration.

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.

Flexible Design of the Connecting Plate in Pneumatic Soft Manipulators

This paper presents a novel connecting plate for achieving a flexible connection in a soft manipulator. The main material used in the connecting plate is silicone, which functions similarly to the bionic oblique muscle (BOM) of an octopus arm and takes over the two actuator segments. Each segment of the actuator consists of three chambers, and the chambers are assembled with the connecting plate via bolts and loops to be a manipulator. By using a flexible plate instead of a rigid one, the manipulator achieves more dexterous movements. This paper establishes a nonlinear model for analyzing the shear and compression deformation of a silicone connecting plate. Additionally, finite element method was used to analyze the mechanical performance of the connecting plate. The deformation of a soft material can reduce the stress concentration during movement of the manipulator. Therefore, in the experiment, the BOM exhibited better motion performance compared with the rigid connection plate.