Dynamic modelling and chaos control for a thin plate oscillator using Bubnov–Galerkin integral method

The utilization of thin plate systems based on acoustic vibration holds significant importance in micro-nano manipulation and the exploration of nonlinear science. This paper focuses on the analysis of an actual thin plate system driven by acoustic wave signals. By combining the mechanical analysis of thin plate microelements with the Bubnov–Galerkin integral method, the governing equation for the forced vibration of a square thin plate is derived. Notably,the reaction force of the thin plate vibration system is defined as f=α|w|, resembling Hooke’s law. The energy function and energy level curve of the system are also analyzed. Subsequently, the amplitude–frequency response function of the thin plate oscillator is solved using the harmonic balance method. Through numerical simulations, the amplitude–frequency curves are analyzed for different vibration modes under the influence of various parameters. Furthermore, the paper demonstrates the occurrence of conservative chaotic motions in the thin plate oscillator using theoretical and numerical methods. Dynamics maps illustrating the system’s states are presented to reveal the evolution laws of the system. By exploring the effects of force fields and system energy, the underlying mechanism of chaos is interpreted. Additionally, the phenomenon of chaos in the oscillator can be controlled through the method of velocity and displacement states feedback, which holds significance for engineering applications.

Multiresolution method for bending of plates with complex shapes

A high-accuracy multiresolution method is proposed to solve mechanics problems subject to complex shapes or irregular domains.To realize this method,we design a new wavelet basis function,by which we construct a fifth-order numerical scheme for the approximation of multi-dimensional functions and their multiple integrals defined in complex domains.In the solution of differential equations,various derivatives of the unknown function are denoted as new functions.Then,the integral relations between these functions are applied in terms of wavelet approximation of multiple integrals.Therefore,the original equation with derivatives of various orders can be converted to a system of algebraic equations with discrete nodal values of the highest-order derivative.During the application of the proposed method,boundary conditions can be automatically included in the integration operations,and relevant matrices can be assured to exhibit perfect sparse patterns.As examples,we consider several second-order mathematics problems defined on regular and irregular domains and the fourth-order bending problems of plates with various shapes.By comparing the solutions obtained by the proposed method with the exact solutions,the new multiresolution method is found to have a convergence rate of fifth order.The solution accuracy of this method with only a few hundreds of nodes can be much higher than that of the finite element method(FEM)with tens of thousands of elements.In addition,because the accuracy order for direct approximation of a function using the proposed basis function is also fifth order,we may conclude that the accuracy of the proposed method is almost independent of the equation order and domain complexity.

Free Vibration Analysis of Rectangular Plate with Cutouts under Elastic Boundary Conditions in Independent Coordinate Coupling Method

Based on Kirchhoff plate theory and the Rayleigh-Ritz method,the model for free vibration of rectangular plate with rectangular cutouts under arbitrary elastic boundary conditions is established by using the improved Fourier series in combination with the independent coordinate coupling method(ICCM).The effect of the cutout is taken into account by subtracting the energies of the cutouts from the total energies of the whole plate.The vibration displacement function of the hole domain is based on the coordinate system of the hole domain in this method.From the continuity condition of the vibration displacement function at the cutout,the transition matrix between the two coordinate systems is constructed,and the mass and stiffness matrices are completely obtained.As a result,the calculation is simplified and the computational efficiency of the solution is improved.In this paper,numerical examples and modal experiments are presented to validate the effectiveness of the modeling methods,and parameters related to influencing factors of the rectangular plate are analyzed to study the vibration characteristics.

Displacement Field Variable Modeling Method for Heterogeneous Materials in Wind Power Blade Core Plates

In order to study the mechanical properties of the heterogeneous core plate of the wind turbine blade,a modeling method of the core plate based on displacement field variables is proposed.Firstly,the wind turbine blade core plate was modeled according to the theory of modeling heterogeneous material characteristics.Secondly,the three-point bending finite element model of the wind turbine blade core plate was solved by the display dynamic equation to obtain the deformation pattern and force-deformation relationship of the core plate.Finally,the three-point bending static test was conducted to compare with the finite element analysis.The test results show that:the damage form of the wind turbine blade core plate includes elasticity,yield,and failure stages.The main failure modes are plastic deformation,core material collapse,and panel-core delamination.The failure load measured by the test is 1.59 kN,which is basically consistent with the load-displacement result obtained by the simulation,with a difference of only 1.9%,which verifies the validity and reliability of the model.It provides data references for wind turbine blade structure design.

Modeling and Analysis of Vibration Characteristics of Rectangular Plate with Circular Through Hole Under Complex Boundary Conditions

This paper presents a method to study the free vibration of a plate with circular holes.The circular hole is regarded as a virtual small plate in which the mass density and Young's modulus are zero.Therefore,the free vibration problem of the circular hole plate can be transformed into the free vibration problem of the equivalent rectangular plate with non-uniform thickness.The model is derived from the spectral geometry method(SGM),and the displacement of the plate with circular holes is expanded by the modified Fourier series.Virtual springs are added to the boundary of the plate to simulate the boundary conditions of simply supported and fixed supports.The accuracy of this method is verified by comparison with the finite element calculation results.The relationship between modal numerical solutions of plates with circular holes and boundary conditions and geometry of the plate is studied.

Study on damage mechanism and damage distribution of the rear plate under impact of debris cloud

The debris cloud generated by the hypervelocity impact(HVI)of orbiting space debris directly threatens the spacecraft.A full understanding of the damage mechanism of rear plate is useful for the optimal design of protective structures.In this study,the hypervelocity yaw impact of a cylindrical aluminum projectile on a double-layer aluminum plate is simulated by the FE-SPH adaptive method,and the damage process of the rear plate under the impact of the debris cloud is analyzed based on the debris cloud structure.The damage process can be divided into the main impact stage of the debris cloud and the structural response of the rear plate.The main impact stage lasts a short time and is the basis of the rear plate damage.In the stage of structure response,the continuous deformation and inertial motion of the rear plate dominate the perforation of the rear plate.We further analyze the damage mechanism and damage distribution characteristics of the rear plate in detail.Moreover,the connection between velocity space and position space of the debris cloud is established,which promotes the general analysis of the damage law of debris cloud.Based on the relationship,the features of typical damage areas are identified by the localized fine analysis.Both the cumulative effect and structural response cause the perforation of rear plate;in the non-perforated area,cratering by the impact of hazardous fragments is the main damage mode of the rear plate.

Exact solution for thermal vibration of multi-directional functionally graded porous plates submerged in fluid medium

An analytical method for analyzing the thermal vibration of multi-directional functionally graded porous rectangular plates in fluid media with novel porosity patterns is developed in this study.Mechanical properties of MFG porous plates change according to the length,width,and thickness directions for various materials and the porosity distribution which can be widely applied in many fields of engineering and defence technology.Especially,new porous rules that depend on spatial coordinates and grading indexes are proposed in the present work.Applying Hamilton's principle and the refined higher-order shear deformation plate theory,the governing equation of motion of an MFG porous rectangular plate in a fluid medium(the fluid-plate system)is obtained.The fluid velocity potential is derived from the boundary conditions of the fluid-plate system and is used to compute the extra mass.The GalerkinVlasov solution is used to solve and give natural frequencies of MFG porous plates with various boundary conditions in a fluid medium.The validity and reliability of the suggested method are confirmed by comparing numerical results of the present work with those from available works in the literature.The effects of different parameters on the thermal vibration response of MFG porous rectangular plates are studied in detail.These findings demonstrate that the behavior of the structure within a liquid medium differs significantly from that within a vacuum medium.Thereby,they offer appropriate operational approaches for the structure when employed in various mediums.

Steady Natural Convection from a Vertical Hot Plate with Variable Radiation

The natural convection from a vertical hot plate with radiation and constant flux is studied numerically to know the velocity and temperature distribution characteristics over a vertical hot plate.The governing equations of the natural convection in two-dimension are solved with the implicit finite difference method,whereas the discretized equations are solved with the iterative relaxation method.The results show that the velocity and the temperature increase along the vertical wall.The influence of the radiation parameter in the boundary layer is significant in increasing the velocity and temperature profiles.The velocity profiles increase with the increase of the radiation parameter.The temperature profiles near the wall plate parallel each other due to the constant heat flux applied to the wall.The influence of the radiation parameter is significant either in velocity or temperature characteristics.At the same time,the effect of the Prandtl number greater than 0.71 is not sensitive to the velocity and temperature variations elsewhere.

Enhancing Stress Intensity Factor Reduction in Cracks Originating from a Circular Hole in a Rectangular Plate under Uniaxial Stress through Piezoelectric Actuation

Circular holes are commonly employed in engineering designs;however, they often serve as locations where cracks initiate and propagate. This paper explores a novel approach to structural repair by utilizing piezoelectric actuators. The primary focus of this study is to investigate the influence of an adhesively bonded piezoelectric actuator patch placed above a circular hole on the stress intensity factor (SIF) in an aluminium plate. The plate is subjected to uniaxial tensile stress, while the piezoelectric actuator is excited with varying voltage levels. The analysis is conducted using the finite element method (FEM), a powerful numerical technique for simulating complex structures. The study assesses the stress distribution and employs the SIF as an adequate criterion for evaluating the impact of different patch configurations. The results indicate a strong correlation between the applied voltage and the SIF. Whether the SIF increases or decreases depends on the polarization of the piezoelectric actuator. Particularly noteworthy is the finding that rectangular patches in a horizontal orientation significantly reduce the SIF compared to other patch geometries. Moreover, double-sided patches exhibit a pronounced decrease in the SIF compared to single-sided patches. In summary, this research underscores the potential of piezoelectric actuators in mitigating stress intensity in structures with circular hole with crack initiation. It offers valuable insights into the influence of applied voltage, patch geometry, and patch placement on the SIF, thereby contributing to developing effective strategies for enhancing structural integrity.

A Comparison of Two Macroinvertebrate Multi-Plate Sampling Methods to Inform Great Lakes Monitoring and Remediation Efforts

Hester-Dendy (HD) multi-plate samplers have been widely used by state and federal government agencies for bioassessment of water quality through use of macroinvertebrate community data. To help guide remediation and restoration efforts at the Niagara River Great Lakes Area of Concern site, a multi-agency study was conducted in 2014 to assess the contribution of seven major urban tributaries on the US side of the river toward the impairment of the Niagara River. As part of this study, macroinvertebrate communities were sampled using two co-located versions of HD samplers: one version used by the New York State Department of Environmental Conservation (NYSDEC) and another by the US Environmental Protection Agency Office of Research and Development. Samplers were deployed in tributaries in highly developed watersheds with high percent impervious surface. The two sampling methods varied in terms of number and size of plates, between-plate spacing, and deployment method. Comparison of the similarity/grouping of communities with multivariate ordination techniques, Nonmetric Multidimensional Scaling and Multi-Response Permutation Procedure, showed that both methods were able to detect differences in communities at stations, despite some grouping by month and method. The indices and metrics derived from the two HD methods were found to give comparable but not identical assessments of water quality. Despite their differences, the methods were robust with respect to water quality categories derived from indices used nationally (HBI) and by NY state (BAP). For the common richness metrics, total taxa and EPT richness, there was no statistical difference between means from 3 samplings. Some metrics, especially percent tolerant collector-gatherer individuals, did show significant differences at certain stations. Indicator Species Analysis showed some taxa associated with each method. The observed community differences were thought mostly due to the difference in sampler deployment position. .

Unified Solution Method of Rectangular Plate Elastic Bending

The bending of rectangular plate is divided into the generalized statically determinate bending and the generalized statically indeterminate bending based on the analysis of the completeness of calculating condition at the corner point. The former can be solved directly by the equilibrium differential equation and the boundary conditions of four edges of the plate. The latter can be solved by using the superposition principle. Making use of the recommended method, the bending of the plate with all kinds of...

A TIME DOMAIN METHOD FOR QUASI-STATIC ANALYSIS OF VISCOELASTIC THIN PLATES

Based on the Boltzmann’s superposition principles of linear viscoelastic materials and the von K*-rm*-n’s hypotheses of thin plates with large deflections, a mathematical model for quasi-static problems of viscoelastic thin plates was given. By the Galerkin method in spatial domain, the original integro-partial-differential system could be transformed into an integral system. The latter further was reduced to a differential system by using the new method for temporal domain presented in this paper. Numerical results show that compared with the ordinary finite difference method, the new method in this paper is simpler to operate and has some advantages, such as, no storage and quicker computational speed etc.

SIMPLIFIED ANALYSIS METHOD FOR ULTIMATE LOAD CAPACITY OF WEB PLATES OF BOX GIRDERS

The prosperous post buckling load capacity of web plates of box girders can be used.In this article,the post buckling behaviour of web plates of box girders under different loading conditions is theoretically analyzed and on the basis of domestic and overseas design codes of steel structures,the corresponding simplified analysis methods are put forward for the engineering design or code revision.It is proved that the simplified methods are safe,efficient and practicable through the comparison between several results.

New exact solutions for free vibrations of rectangular thin plates by symplectic dual method

The separation of variables is employed to solve Hamiltonian dual form of eigenvalue problem for transverse free vibrations of thin plates, and formulation of the natural mode in closed form is performed. The closed-form natural mode satisfies the governing equation of the eigenvalue problem of thin plate exactly and is applicable for any types of boundary conditions. With all combinations of simplysupported （S） and clamped （C） boundary conditions applied to the natural mode, the mode shapes are obtained uniquely and two eigenvalue equations are derived with respect to two spatial coordinates, with the aid of which the normal modes and frequencies are solved exactly. It was believed that the exact eigensolutions for cases SSCC, SCCC and CCCC were unable to be obtained, however, they are successfully found in this paper. Comparisons between the present results and the FEM results validate the present exact solutions, which can thus be taken as the benchmark for verifying different approximate approaches.

A Deep Collocation Method for the Bending Analysis of Kirchhoff Plate

In this paper,a deep collocation method(DCM)for thin plate bending problems is proposed.This method takes advantage of computational graphs and backpropagation algorithms involved in deep learning.Besides,the proposed DCM is based on a feedforward deep neural network(DNN)and differs from most previous applications of deep learning for mechanical problems.First,batches of randomly distributed collocation points are initially generated inside the domain and along the boundaries.A loss function is built with the aim that the governing partial differential equations(PDEs)of Kirchhoff plate bending problems,and the boundary/initial conditions are minimised at those collocation points.A combination of optimizers is adopted in the backpropagation process to minimize the loss function so as to obtain the optimal hyperparameters.In Kirchhoff plate bending problems,the C^1 continuity requirement poses significant difficulties in traditional mesh-based methods.This can be solved by the proposed DCM,which uses a deep neural network to approximate the continuous transversal deflection,and is proved to be suitable to the bending analysis of Kirchhoff plate of various geometries.

THE 3-D BOUNDARY ELEMENT METHOD OF ROLLER BEARING BY PLATE ELEMENT ANALOGUE

The mill roller bearing is made up of an internal ring, middlerolls and an external ring, the analysis of which is a multi-bodiescontact problem. In this paper, based on the three-dimensionalelastic contact BEM without friction, and using the structuralcharacteristics of roller bearings, middle rolls are de- scribed byelastic plate units of different shapes, which is placed on theinternal ring.

A 3D shell-like approach using element-free Galerkin method for analysis of thin and thick plate structures

A new efficient meshless method based on the element-free Galerkin method is proposed to analyze the static deformation of thin and thick plate structures in this paper. Using the new 3D shell-like kinematics in analogy to the solid-shell concept of the finite element method, discretization is carried out by the nodes located on the upper and lower surfaces of the structures. The approximation of all unknown field variables is carried out by using the moving least squares （MLS） approximation scheme in the in-plane directions, while the linear interpolation is applied through the thickness direction. Thus, different boundary conditions are defined only using displacements and penalty method is used to enforce the essential boundary conditions. The constrained Galerkin weak form, which incorporates only dis- placement degrees of freedom （d.o.f.s）, is derived. A modified 3D constitutive relationship is adopted in order to avoid or eliminate some self-locking effects. The numeric efficiency of the proposed meshless formulation is illustrated by the numeric examples.

Mixed finite element and differential quadrature method for free and forced vibration and buckling analysis of rectangular plates

This paper presents a combined application of the finite element method （FEM） and the differential quadrature method （DQM） to vibration and buckling problems of rectangular plates. The proposed scheme combines the geometry flexibility of the FEM and the high accuracy and efficiency of the DQM. The accuracy of the present method is demonstrated by comparing the obtained results with those available in the literature. It is shown that highly accurate results can be obtained by using a small number of finite elements and DQM sample points. The proposed method is suitable for the problems considered due to its simplicity and potential for further development.

MODIFIED LAYER REMOVAL METHOD FOR MEASUREMENT OF RESIDUAL STRESS DISTRIBUTION IN THICK PRE-STRETCHED ALUMINUM PLATE

The integrated structure parts are widely used in aircraft. The distortion caused by residual stresses in thick pre-stretched aluminum plates during machining integrated parts is a common and serious problem. To predict and control the machining distortion, the residual stress distribution in the thick plate must be measured firstly. The modified removal method for measuring residual stress in thick pre-stretched aluminum plates is proposed and the stress-strain relation matrix is deduced by elasticity theory. The residual stress distribution in specimen of 7050T7451 plate is measured by using the method, and measurement results are analyzed and compared with data obtained by other methods. The method is effective to measure the residual stress.

Nonlinear vibration analysis of a circular micro-plate in two-sided NEMS/MEMS capacitive system by using harmonic balance method

In this study, forced nonlinear vibration of a circular micro-plate under two-sided electrostatic, two-sided Casimir and external harmonic forces is investigated analytically. For this purpose, at first, von Karman plate theory including geometrical nonlinearity is used to obtain the deflection of the micro-plate. Galerkin decomposition method is then employed, and nonlinear ordinary differential equations (ODEs) of motion are determined. A harmonic balance method (HBM) is applied to equations and analytical relation for nonlineaT frequency response (F-R) curves are derived for two categories (including and neglecting Casimir force) separately. The analytical results for three cases:(1) semi-linear vibration;(2) weakly nonlinear vibration;(3) highly non linear vibration, are validated by comparing with the numerical solutio ns. After validation, the effects of the voltage and Casimir force on the natural frequency of two-sided capacitor system are investigated. It is shown that by assuming Casimir force in small gap distances, reduction of the natural frequency is considerable. The influences of the applied voltage, damping, micro-plate thickness and Casimir force on the frequency response curves have been presented too. The results of this study can be useful for modeling circular parallel-plates in nano /microelectromechanical transducers such as microphones and pressure sensors.