Thin Flexible Plate Research Articles

Semi-analytical approach to study wave interaction with a pair of obliquely submerged heterogeneous flexible structures

ABSTRACT Analyzing inclined flexible plates of varying thickness is crucial for optimising and attaining precise control over wave reflection and transmission. This is particularly significant in the context of breakwater construction. In contrast to horizontal breakwaters, inclined barriers can penetrate multiple layers of fluid with varying particle velocities, facilitating interactions and causing wave breaking, which results in the dissipation of wave energy. Moreover, compared to vertical structures, inclined ones demonstrate more effective wave attenuation. In the context of the present study, we investigate the interaction of water waves with a pair of obliquely submerged flexible thin plates characterised by non-uniform thickness. In order to model this complex physical scenario, we employ the principles of linear water wave theory in conjunction with Kirchhoff's thin plate theory. By employing repeated integration and Green's integral theorem, we reduce the boundary value problem to a system of coupled integral equations. Through appropriate approximations, we solve this system and obtain numerical values for various hydrodynamic quantities. This model provides comprehensive results for both horizontal and vertical plates, making it highly versatile. We examine the impact of varying thickness and the inclination of the two flexible plates to understand their roles in the wave scattering process and the associated physical parameters. The results obtained from this study shed light on the intricate dynamics involved in wave-structure interactions and can inform the design and optimisation of breakwater systems.

Flutter characteristics of a sheet with an elastic support in three-dimensional uniform flow

In the manufacturing processes of thin flexible plates (sheets) such as polarizing films, the flutter can be caused to the sheets due to the interaction between the motion of the sheets and fluid flow. Then, the flutter can cause serious damage to the sheets, leading to the wrinkles and scratches. Thus, it is crucial to investigate a detailed characteristics and excitation mechanism of the flutter. In the present study, a detailed flutter characteristics and excitation mechanism of a rectangular sheet with an elastic support is investigated. The elastic support is implemented using a fine flexible wire. The influence of bending stiffness of the support section on the flutter velocity and frequency is clarified through wind-tunnel experiments and numerical analysis. Moreover, the work done by the fluid force on the sheet surface was determined.The flutter velocity and frequency drastically decrease owing to decrease in bending stiffness of the support section regardless of the aspect and mass ratio. Then, the positive work region around the leading edge of the sheet expands owing to the large-amplitude oscillation around the leading edge. The expansion of positive work region owing to a large-amplitude oscillation around the leading edge of the sheet destabilizes the system.

New Approach of Normal and Shear Stress Components for Multiple Curvilinear Holes Which Weakened a Flexible Plate

In this article, a thin infinite flexible plate weakened by multiple curvilinear holes is considered. The strength shapes are mapped outside a unit circle with the assistance of particular conformal mapping under certain conditions. The mathematical model that governs the rounded forces of the current physical problem is the boundary value problem of elastic media. This study is applicable to many phenomena throughout nature, like tunnels, caves, and excavations in soil or rock. The Cauchy method for complex variables is used to get the closed forms of Gaursat functions and change the problem to a second-type integrodifferential equation with a Cauchy kernel, which is used for a large area of the contact problems. Then, the normal and shear stress components that act on the model are derived. Afterward, some of the physical applications are studied, and different stress components at specific values in each application are calculated and plotted using Maple 2023.

Hydrodynamic response of dual obliquely submerged non-uniform flexible thin plates

Examining inclined flexible plates with variable thickness becomes crucial when it comes to optimizing or achieving controlled reflection and transmission of waves, especially for the construction of breakwaters. Unlike horizontal breakwaters, inclined barriers have the ability to penetrate through numerous layers of the fluid, having different particle velocities and foster their interactions. This causes wave breaking which leads to loss of wave energy. Also, despite exhibiting a similar behavior, vertical structures do not attenuate waves as effectively as inclined ones do. Additionally, the resonant motion of the fluid trapped between two structures proves to enhance the attenuation, thus it is recommended to include dual structures in the model rather than just one. Therefore in the present study, we examine the water wave scattering phenomenon by a pair of symmetric flexible thin plates with non-uniform thickness obliquely submerged in deep water. We use the linear water wave theory and Kirchhoff’s thin plate theory to model the physical problem. The boundary value problem is converted into a system of coupled integral equations using repeated integration and Green’s integral theorem. Using appropriate approximations, this system is solved and its solutions are used to determine numerical values of different hydrodynamic quantities. Results of two horizontal as well as two vertical plates could be obtained from the present model, thus it is a very general model. Also results are illustrated to analyze the contribution of the thickness variation and the inclinations of the two flexible plates towards the wave scattering process and some related physical quantities.

Fluid-structure interaction of a thin flexible heater inside a lid-driven cavity

The purpose of this study is to explore the characteristics of fluid-structure interaction and heat transfer enhancement from a hot flexible thin plate during mixed convection. The mixed convection condition of the plate heater is assumed by placing it inside a lid driven cavity with top and bottom cavity walls being fixed and insulated while left and right cavity walls maintained at constant low temperature move in downward and upward direction respectively. The Galerkin Finite Element Approach has been used to numerically solve equations describing unsteady flow, thermal and stress fields in the Arbitrary Lagrangian-Eularian (ALE) framework. Key parameters include Reynolds number ( Re) and Richardson number ( Ri), and Cauchy number ( Ca) which are varied in the range of 100 ≤ Re ≤ 500, 0.1 ≤ Ri ≤10, and 10−4 ≤ Ca ≤ 10−7 respectively. To investigate the effect of the heater’s geometry, the plate length ( l) has been varied within 0.1 ≤ l ≤ 0.7. The results have been presented in terms of the distribution of streamline and isotherm, heat transfer rate from the flexible heater as well as its deformation. The outcomes reveal that heat transfer enhances with the increase in Reynolds number and Richardson number while it degrades with the extension of the heater. Moreover, thermal frequency obtained by Fast Fourier Transform (FFT) indicates dominant peaks at higher Reynolds number and Richardson number although these become insignificant as the plate becomes rigid.

Complicated nonlinear oscillations caused by maneuvering of a flexible spacecraft equipped with hinged solar arrays

We present a general methodology for the nonlinear dynamical modeling of a three-axis stabilized spacecraft equipped with flexible solar arrays. The large-span multi-panel solar arrays are modeled as flexible thin plates that are connected to the rigid central body of the spacecraft by means of nonlinear flexible hinges. We construct a low-dimensional yet accurate dynamical model by using a Galerkin expansion in terms of global modes of the system that we compute first by using the Rayleigh–Ritz method. The hinged connections between the rigid and flexible parts of the system are imposed employing Lagrange multipliers. The model is used to study the spacecraft response triggered by various maneuvering scenarios. We in particular focus on the coupling between vibrations of the flexible components and the rigid motion of the spacecraft induced by hinge nonlinearities during these orbital and attitude maneuvering operations. In all cases considered, it is found that four global modes are sufficient to accurately compute the system’s response. We also observe other complicated nonlinear dynamical phenomena, such as hysteresis and superharmonic resonance that may be of concern in spacecraft design. Our modeling approach can be applied to other multibody systems directly.

Acoustic Analysis of Partially Flexible Cavity with Opening

In this study, the SPL at the boundary and within the domain of a rectangular-shaped container with a flexible laminated composite panel with opening have been investigated. Finite Element Analysis (FEM) for the flexible panel has been done and coupled with the acoustic domain using the Boundary Element Method (BEM) through the mobility relation. A MATLAB program has been developed to find flexible panel behavior by FEM and BEM to calculate the sound pressure level for a cavity. Eight-noded isoparametric serendipity elements have been used to model the boundary. A pressure-velocity formulation has been adopted to model the acoustic domain with radiation impedance for window boundary. It has been shown that the presence of a thin flexible plate and opening, drastically changes the SPL pattern inside and at the boundary compared to a rigid cavity due to the relative movement of flexible panel and energy dissipation through the opening by radiation.

Experimental Consideration on Suppression Effect of Elastic Vibration in Electromagnetic Levitation System for Flexible Thin Steel Plate with Curvature

Recently, research on non-contact conveyance systems using electromagnetic levitation technology has accelerated. We have constructed an electromagnetic levitation control system that keeps the relative distance between the electromagnet and steel plate constant. To investigate the levitation stability of thin steel plates, we performed magnetic levitation experiments on a thin steel plate with curvature. A physical disturbance was applied to the electromagnet units by vibrators. The electromagnet units were vibrated up and down by a vibrator. We investigated whether the bending magnetic levitation improved the levitation performance even if the magnetic levitation system was in a vibrating environment. We determined that it was possible to realize stable levitation for a steel plate under external disturbances during levitation at the optimal bending angle.

Axisymmetric large deformation problems of thin shallow shells with different moduli in tension and compression

When materials are in tension or in compression, their mechanical properties are essentially different, however, in existing studies it is generally ignored because of the complexity of the analysis. In this paper, a theoretical study of the problem of axisymmetric large deformation of thin shallow shells with different moduli in tension and compression (bimodular thin shallow shells) under uniformly vertical loading is presented. Based on the Föppl–von Kármán equations of flexible thin plates, the governing equation of the axisymmetric large deformation of bimodular thin shallow shells are established by introducing an appropriate curvature modification term from thin shallow shells and a bending stiffness term from bimodular plates. By selecting the central deflection as a perturbation parameter, the perturbation solution of the axisymmetric large deformation problem is derived using the perturbation method. The applications of variation method as well as the implementation of numerical simulation verify, from the theoretical and numerical perspectives, the correctness of the perturbation solution obtained. This study shows that the introduction of different moduli in tension and compression not only affects the relationship of loads vs. central deflection, but also affects the critical values of the rise-thickness ratio of bimodular thin shallow shells when the snapping phenomenon occurs. In particular, the bimodular effect may accelerate or slow down the occurrence of the snapping phenomenon of thin shallow spherical shells, according to the relative magnitude of tensile and compressive moduli. The work presented here will be helpful to analyze the mechanical response of heavily loaded thin shallow shells with obvious bimodular effect.

Fluid structural interaction of a flexible plate submerged in the wake of a circular cylinder

The wake-induced vibration of a flexible thin plate is investigated in order to provide insight into the dynamic response of the plate due to wake vortices. The flexible plate is influenced by vortices with varying longitudinal spacing ratios in the wake of the cylinder, which is defined as the distance between the centre of the upstream cylinder and the tip edge of the plate (x* = x/D = 3.0–5.5). The flow is laminar and the Reynolds number is set at 200. The upstream circular cylinder is stationary and the downstream thin flexible plate is set as a cantilever beam in the wake of circular cylinder. The plate, therefore, is under the effects of upstream vortices and can be deflected due to the dynamic pressure of the flow. According to the numerical results, while both longitudinal (x*) and lateral (y*) distances can influence the dynamic reaction of the plate owing to approaching vortices, it is indicated that the effect of the lateral distance is greater than that of the longitudinal distance. Furthermore, it is demonstrated that the highest negative magnitude of pressure coefficient occurs with a 7% variation between y* = 0.0 and y* = 0.75.

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.

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.

Nonlinear large deformation problem of rectangular thin plates and its perturbation solution under cylindrical bending: Transform from plate/membrane to beam/cable

AbstractThe Föppl‐von Kármán equations describing deformation of flexible thin plates are established on the basis of moderately large‐deflection and small rotation angle. For many years, the equations have been regarded as a classical and effective model used for the analysis of flexible thin plate problems, and at the same time, this model has also been constantly evolving and improving. The improvements for the model, however, come mainly from properties of materials perspective, but seldom from the deformation of plate perspective. In this study, we revisit Föppl‐von Kármán equations from the viewpoint of deformation concerning rotation angle. A new form for the classical equations without a small‐rotation‐angle assumption is derived, for the first time, by giving up the basic assumption that the sine function of the rotation angle equals to the first‐order derivative of the corresponding displacement, thus improving the governing equations while enhancing the nonlinearity. The abandonment of the small‐rotation‐angle assumption reveals such a fact that the second‐order derivative of displacement in classical equations originates from the curvature and twist of plates. Due to the complexity of the equations derived, its perturbation solution is obtained under cylindrical bending with two opposites fully fixed. Results indicate that via cylindrical bending, a two‐dimensional plate or membrane problem is easily associated with a one‐dimensional beam or cable problem. Results also show that the abandonment of the small‐rotation‐angle assumption will contribute to the free development of the deflection curve rotation of the plate, while at the same time, the deflection value will tend to decrease to agree with the deformation of the plate.

Dynamics of flags over wide ranges of mass and bending stiffness

The flutter of a thin flexible plate or flag is a benchmark problem in fluid-structure interactions with applications to energy harvesting and flow sensing. Here we study flutter dynamics across a wide range of bending stiffnesses, and down to very small values of flag mass, e.g. very flexible fabric flags. We find that the flapping amplitude, frequency, and wavenumber are power laws of the flag mass and stiffness.

Analytical modelling and experimental validation of compliance-based low-frequency resonators for water circuits

Transmission losses of compact compliance-based resonators in water circuits are investigated. Experiments are performed to measure the anechoic transmission losses (TLan) of flexible-plate resonators and a gas resonator designed for frequencies between 10 and 100 Hz. The measurements are compared to theoretical results based on a lumped-element model and a finite-element model. The TLan is measured using a robust form of the multi-microphone method, which gave identical results for open and closed pipe acoustic terminations at the transmission side of the setup. When an estimate of the reflection coefficient at the termination is known, good results are obtained with only one transmission-side microphone. When TLan is high, a single microphone is sufficient on each side. For the flexible-plate resonators the TLan measurements are in agreement with theory except close to resonance, where the transmission signals are below the detection limit. Due to assumptions of a rigid cavity wall and a clamped top-plate, the theoretical resonance frequencies are too high except for the thinnest plate which displays static deformation stiffening. This deformation stiffening limits the possibility to lower the resonance frequency by using a thin flexible plate in a circuit with high static pressure. Low resonance frequencies are easier to reach with a gas resonator, in which a piston separates the water from a volume filled with air. For the gas-resonator, the measurements agree with the theoretical predictions when assuming a significant damping. The friction between the air-water-separation piston and cavity wall is suspected to cause this damping. Theory predicts that the TLan of both resonators designed for same resonance frequencies in absence of losses are equivalent. They therefore have quite similar performances except close to the resonance frequency. The flexible-plate resonator has a higher quality factor and higher (TLan) around the resonance frequency. The gas resonator is more complex and needs more maintenance but allows fine tuning of the resonance frequency by varying the gas volume.

DYNAMIC ANALYSIS OF VARIABLE MESH ISOGEOMETRIC THIN PLATE BASED ON T-SPLINE

The local strain of a thin plate with large displacement and large deformation will change dramatically under contact and collision working conditions. In order to ensure the accuracy and computational efficiency of flexible thin plate system’s dynamic analysis, this investigation integrates computer aided design (CAD) and computer aided engineering (CAE) systems, and proposes an isogeometric analysis (IGA) method for variable mesh flexible multibody system based on T-spline surface elements. Firstly, the kinematic description of a Kirchhoff thin plate based on T-spline surface elements is modeled, and the elastic model of a thin plate discretized by T-spline surface elements is established according to the nonlinear Green−Lagrange strain. Secondly, the goal of updating mesh of T-spline surface locally is achieved by inserting knots into the local region of the corresponding T-mesh. The transformation matrix, which is used to calculate the new generalized coordinates, generalized velocities and generalized accelerations of the refined system, is obtained by using the T-spline blending function refinement algorithm. The calculating solution algorithm for the dynamic equation of the system with variable degrees of freedom is created by combining the generalized α method with geometry update routine, and thus the local mesh refinement algorithm for the surface which is modeled by T-spline is formed. Finally, statics examples and flexible pendulum model verify the correctness for the elastic model of Kirchhoff thin plate based on T-spline surface, as well as computation precision and convergence for the proposed method in the dynamics analysis respectively. The dynamic analysis of the impacted flexible thin plate shows that the T-spline element and local refinement algorithm proposed in this paper can realize local mesh update only in the area where the strain changes violently, such as contact and collision, so as to control the degree of freedom of the system and improve the computational efficiency.

Theoretical study on the instability mechanism of flutter generated on a cantilevered flexible plate in three-dimensional uniform flow

The flutter of a thin flexible plate in uniform flow is an interesting theoretical problem in fluid–structure interaction phenomena, as well as an important engineering problem. Therefore, understanding of the instability mechanism is crucial to preventing fatal defects caused by the flutter. In this study, the instability mechanism of a flexible plate in three-dimensional uniform flow is investigated in terms of the energy transfer between the plate motion and fluid flow. A theoretical aerodynamic model of the plate is developed using the unsteady lifting surface theory and linear beam theory. The aerodynamic stability of the plate is investigated by eigenvalue analysis. The developed model is validated by comparison with previous experimental results and other theoretical models. Furthermore, to understand the instability mechanism, the work done by the fluid force on the plate surface is determined, and its influence on the stability of the system is examined.

Fish without Tail Fins-Exploring the Function of Tail Morphology of the First Vertebrates.

We use a series of hydrodynamic experiments on abstracted models to explore whether primitive vertebrates may have swum under various conditions without a clearly-differentiated tail fin. Cambrian vertebrates had post-anal stubby tails, some had single dorsal and ventral fins, but none had yet evolved a clearly differentiated caudal fin typical of post-Cambrian fishes, and must have relied on their long and flexible laterally-compressed bodies for locomotion, i.e., by bending their bodies side-to-side in order to propagate waves from head to tail. We approach this problem experimentally based on an abstracted model of Metaspriggina walcotti from the 506-million-year old Burgess Shale by using oscillating thin flexible plates while varying the tail fin geometry from rectangular to uniform, and finally to a no tail-fin condition. Despite a missing tail fin, this study supports the observation that the abstracted Metaspriggina model can generate a strong propulsive force in cruise conditions, both away from, and near the sea bed (in ground effect). However, when the abstracted Metaspriggina model moves in ground effect, a weaker performance is observed, indicating that Metaspriggina may not necessarily have been optimized for swimming near the sea bed. When considering acceleration from rest, we find that the Metaspriggina model's performance is not significantly different from other morphological models (abstracted truncate tail and abstracted heterocercal tail). Statistical analysis shows that morphological parameters, swimming modes, and ground effect all play significant roles in thrust performance. While the exact relationships of Cambrian vertebrates are still debated, as agnathans, they share some general characteristics with modern cyclostomes, in particular an elongate body akin to lampreys. Lampreys, as anguilliform swimmers, are considered to be some of the most efficient swimmers using a particular type of suction thrust induced by the traveling body wave as it travels from head to tail. Our current experiments suggest that Metaspriggina's ability in acceleration from rest, through possibly a similar type of suction thrust, which is defined as the ability to generate low pressure on upstream facing sections of the body, might have evolved early in response to increasing predator pressure during the Cambrian Explosion.

Dynamics of inverted flags: Experiments and comparison with theory

The study of the dynamics of cantilevered thin flexible plates in reverse axial flow – also known as inverted flags – has become of significant interest, partly due to their energy harvesting potential. This paper presents fresh experimental results, aiming to enhance our understanding of the dynamics of inverted flags, particularly concerning stability and global dynamics sensitivity to various system parameters, such as the aspect ratio (i.e. height-to-length ratio) and flag material. This is achieved through testing flags of various dimensions, made from different materials. The dynamics of the system is presented in the form of bifurcation diagrams in which the flag tip angle is shown as a function of the dimensionless flow velocity. The frequency content of oscillatory motions is presented in the form of spectrograms. Also, time-histories, phase-plane portraits, as well as power spectral density and probability density function plots are presented for different flow regimes. Interesting dynamical features, such as small- and large-amplitude flapping, static divergence, large-amplitude buckling, and chaotic motions, have been observed experimentally, some for the first time. It is shown that for flags of very low aspect ratio the undeflected static equilibrium is stable prior to a subcritical pitchfork bifurcation. For flags of large aspect ratio, on the other hand, the undeflected static equilibrium becomes unstable via a supercritical pitchfork bifurcation leading to static divergence (buckling) at a sufficiently high flow velocity; at higher flow velocities, past the pitchfork bifurcation, a supercritical Hopf bifurcation materializes leading to flapping motion around the deflected static equilibrium; at even higher flow velocities, flapping motion becomes symmetric around the undeflected static equilibrium. Interestingly, it is found that heavy flags may exhibit large-amplitude flapping right after the initial static equilibrium, provided that they are subjected to a sufficiently large disturbance. In addition to experiments, nonlinear analytical models for the two asymptotic cases of zero and infinite aspect ratios are outlined, the results of which are used for comparison with the experimental results. Theory and experiments are found to be in reasonably good agreement.

Free convective heat transfer of a non-Newtonian fluid in a cavity containing a thin flexible heater plate: an Eulerian–Lagrangian approach

In the present study, the free convection heat transfer of a power-law non-Newtonian fluid is considered in a cavity containing a flexible hot thin heater. The sidewalls of the square cavity are maintained at cold temperatures, while the hot heater is placed inside the cavity. The top and bottom walls of the cavity are kept insulated. The thin heater plate can undergo large deformations due to the interaction between the fluid flow and the heater. The arbitrary Lagrangian–Eulerian moving mesh method is employed to track the displacement of the heater in the fluid domain. Appropriate non-dimensional parameters are utilized to transform the governing equations into a general non-dimensional form. The equations governing fluid flow and heat transfer are solved using the finite element method with an automatic time-stepping scheme. The effect of control parameters such as the non-Newtonian power index (0.6 1) behavior reduces the fluid circulation and heat transfer rate in the cavity, but it increases the magnitude of the exerted tensions on the element. Moreover, raising Ra from 104 to 106 enhances the average heat transfer the value of Nuav by up to 3.5 times in pseudoplastic fluids and by 1.5 times in dilatant ones. In addition, it is found that shifting the heater upward deteriorates the heat transfer rate by suppressing the convection flow intensity. A 35% rise in the average heater can be obtained when the height of the plate was divided by 4 in the cse of dilatant fluid, and an increase by up to 100% is found for pseudoplastic fluids. Increasing the length of the element is also found to reduce the average Nusselt number and to increase the tensions in the heater. The average Nusselt number can be doubled when the length of the plate is reduced seven times.