Differential Equations Of Equilibrium Research Articles

Multimodal vortex-induced vibration mitigation and design approach of bistable nonlinear energy sink inerter on bridge structure

Large-scale structures, e.g., long-span bridge structures, are prone to induce multi-modal vibrations due to their densely spaced low modal frequencies. Due to the limited frequency bandwidth of linear dynamic absorbers, they are incapable of effectively mitigating vibrations across multiple modes. To this end, the bistable nonlinear energy sink inerter (BNESI) is used to mitigate the multimodal vortex-induced vibration (VIV) of the beam structure. The highly nonlinear equilibrium differential equations of the beam-BNESI system are numerically solved, and the simulated annealing (SA) algorithm is employed to determine the optimal VIV reduction ratio and BNESI parameters. In comparison to the cubic-type nonlinear energy sink inerter (CNESI), BNESI is found to possess more stable equilibrium positions, smaller stiffness coefficients, and higher VIV mitigation efficiency. The selection of design modes has been found to influence the efficiency of multimodal VIV mitigation, with the use of the intermediate modal order as the design mode resulting in the highest efficiency for multimodal VIV mitigation. The performance-based multimodal VIV mitigation design can be realized with three parameters, i.e., inertance ratio, damping coefficient, and stiffness coefficient. Moreover, the performance-based multimodal VIV mitigation approach and models proposed in this study demonstrate a high level of precision.

A modified quasi-3D theory and mixed beam element method for static behaviour analysis of functionally graded beams

This paper develops a modified quasi-3D theory and the mixed beam element model for static analysis of functionally graded beams. The modified quasi-3D theory encompasses two innovations. Firstly, the accurate distribution of transverse shear stress is derived from the differential equilibrium equation that describes the relationship between stresses, rather than the traditional one derived from geometric relations and constitutive equations. Secondly, the effect of distributed loads associated with transverse stretching deformation, a factor typically overlooked in the existing quasi-3D theory-based beam models, is involved. In the development of beam finite element model, the mixed variational principle is firstly utilized to formulate the model of a quasi-3D theory-based beam element, where generalized displacements and internal forces are treated as two types of independent field quantities. Two numerical examples are conducted to investigate the validity and accuracy of the proposed theory and beam element. Numerical results indicate that the proposed mixed beam element can accurately predict the stress distributions over the cross-section and produce accurate displacement solutions. Meanwhile, it is noted that the effect of distributed loads related to the transverse stretching deformation should be correctly considered in the analysis of functionally graded beams based on quasi-3D theory.

Bandgap Calculation and Experimental Analysis of Piezoelectric Phononic Crystals Based on Partial Differential Equations.

Focusing on the bending wave characteristic of plate-shell structures, this paper derives the complex band curve of piezoelectric phononic crystal based on the equilibrium differential equation in the plane stress state using COMSOL PDE 6.2. To ascertain the computational model's accuracy, the computed complex band curve is then cross-validated against real band curves obtained through coupling simulations. Utilizing this model, this paper investigates the impact of structural and electrical parameters on the bandgap range and the attenuation coefficient in the bandgap. Results indicate that the larger surface areas of the piezoelectric sheet correspond to lower center bands in the bandgap, while increased thickness widens the attenuation coefficient range with increased peak values. Furthermore, the influence of inductance on the bandgap conforms to the variation law of the electrical LC resonance frequency, and increased resistance widens the attenuation coefficient range albeit with decreased peak values. The incorporation of negative capacitance significantly expands the low-frequency bandgap range. Visualized through vibration transfer simulations, the vibration-damping ability of the piezoelectric phononic crystal is demonstrated. Experimentally, this paper finds that two propagation modes of bending waves (symmetric and anti-symmetric) result in variable voltage amplitudes, and the average vibration of the system decreases by 4-5 dB within the range of 1710-1990 Hz. The comparison between experimental and model-generated data confirms the accuracy of the attenuation coefficient calculation model. This convergence between experimental and computational results emphasizes the validity and usefulness of the proposed model, and this paper provides theoretical support for the application of piezoelectric phononic crystals in the field of plate-shell vibration reduction.

Nonlinear vibration characteristics of thermo-viscoelastic coupling of moving PET films with temperature-dependent elastic modulus

In order to improve the transport stability of roll-to-roll produced polyethylene terephthalate films (PET films) in high temperature environments, this paper investigates the nonlinear vibrational properties of moving PET films under a uniform temperature field by considering the temperature dependence of the elastic modulus. Firstly, constant tensile tests were conducted to test the elastic modulus values of PET films at different hot air temperatures, and the Wachtman model was improved to establish a temperature-dependent elastic modulus model for PET films. Secondly, based on the Kelvin-Voigt model, the nonlinear vibrational equilibrium differential equations of the system were established by applying the Bubnov-Galerkin method of discretization according to D’Alembert’s principle. Finally, numerical simulations based on the 4th-order Runge-Kutta method were carried out to analyze the effects of films width, external excitation frequency, external excitation amplitude, transfer velocity, ambient temperature, damping coefficient and viscoelasticity coefficient on the nonlinear vibration of the system. This provides a theoretical basis for the commissioning of stabilized transfer parameters for roll-to-roll production of PET films.

Stability analysis and design methods of steel pipe piles under non-uniform corrosion

Steel pipe piles (SPPs) are commonly applied for marine platform foundations due to high bearing capacity. However, for SPPs in corrosive marine environments, the corrosion on the steel tube is severe and shows a non-uniform distribution along the height direction, resulting in the significantdecrease of the stability bearing capacity. To investigate the stability performance of SPPs in marine corrosive environments, the corroded SPPs are simplified into five-segment stepped columns, and the equilibrium differential equations are established to obtain the rotational stiffness relationship. The analytical solutions of the critical buckling loads of five-segment stepped columns under arbitrary boundary conditions are obtained by iterative methods. Besides, considering the material and geometrical nonlinearity, the finite element (FE) model of corroded SPPs is established and then verified based on existing test results. The effects of the number of segments and stiffness ratios among different segments on the bearing capacity are analysed. Furthermore, considering the interactions among different segments, the calculation methods for the stability bearing capacity of stepped columns under axial compression and combined compression-bending are proposed respectively. The comparison between calculated and FE results indicates that the proposed calculation methods have good prediction accuracy and can be available for practical engineering reference.

Advanced Thick FGM Plates-Cylindrical Shells in Linear Unsteady Supersonic Flow

Introduction:: Thick functionally graded (FG) plates-cylindrical shells with third-order shear deformation theory (TSDT) of displacement models under thermal vibration in linear unsteady supersonic flow are investigated by using the generalized differential quadrature (GDQ) method. Method:: The nonlinear coefficient term of TSDT models and the advanced nonlinear shear correction coefficient expression are used to study the thermal vibration of thick FG plates and cylindrical shells in linear unsteady supersonic flow. At the intersection of FG plates-cylindrical shells, the displacements, stress resultants, and moment resultants could be reasonably assumed in the continuity condition for the differential volume in linear unsteady supersonic flow, respectively. Result:: The dynamic equilibrium differential equations of thick FG plates and cylindrical shells in linear unsteady supersonic flow, respectively in terms of partial derivatives of displacements and shear rotations subjected to partial derivatives of thermal loads, mechanical loads consist of the linear supersonic aerodynamic pressure and inertia terms can be presented. Two parametric effects including environment temperature and functionally graded material (FGM) power law index on the thermal stress and center displacement of thick FG plates-cylindrical shells in linear unsteady supersonic flow are investigated. Conclusion:: The linear unsteady supersonic flow over the outer surface of thick FG platescylindrical shells provides the additional aerodynamic effect on the values of displacements and stresses.

Variable angle tow-steered curvilinear fibres-based rotating arbitrarily layered composite beams- a coupled vibration of chordwise-flapwise-torsional motions by higher-order beam theory

In the present work, variable angle tow-steered curvilinear fibres-based straight composite rotating beam is studied for its coupled vibrational behaviours from different beam planes by a higher-order beam formulation satisfying plane stress situation and combining with finite element methodology. The structural theory is extended to include all the vibrational motions including chord- and flap-wise motions, axial and torsional vibrations of beam. The centrifugal force stiffening and gyroscopic effects stemming from rotational motions are taken into the formulation. The generic differential equations of equilibrium for the proposed structural theory are evolved adopting virtual dynamic work, Hamilton’s principle and a suitable C1 continuity based elemental equations are obtained. The newly constructed finite element formulation is tested for solved problems that are known in the literature. Based on in-depth analysis, vibration features of curvilinear fibre-based rotating generally layered composite beam including coupled dynamic motions in terms of vibrational modes and their frequencies pertaining to chord- and flap-wise, axial, torsional motions are detailed presuming variable angles along the curvilinear fibre path at the centre and layer edge, short and long beams, hub-radius, beam cross-section, and high rotational speed effects.

Stress state analysis of a cylindrical tank with water of an under water sports palace

The paper analyzes the stress state of a cylindrical tank, which is one of the main structural elements of a sports facility - an underwater sports palace. In the calculation of the cylindrical shell of the tank of constant thickness, the moment theory was used. The reservoir is, in fact, a swimming pool for conducting training sessions on underwater swimming in fins, diving, diving, is considered fixed in its base and filled with liquid (water of a special composition, homogeneous in depth, with a given specific gravity). It is known that integration of the calculation equations of the moment theory (of circular cylindrical shells), namely: a) differential equations of equilibrium; b) equations of deformations; c) equations of forces attributed to the unit length of the coordinate line (tangential forces, bending moments), is a complex mathematical problem associated with solving a system of partial differential equations with variable coefficients. Only the calculation of the curve of a cylindrical shell ultimately leads to a system of differential equations with constant coefficients. Under the assumption of significant depth of the tank compared to the thickness of its walls and the value of the radius of the cylinder base (which allows us to consider it as infinitely long), displacements and forces (tangential and transverse), bending moments are analytically found. The locations of the cylinder surface points, in which the indicated forces and moments acquire maximum values, have been determined; sketches of their diagrams, determining the character of change of these physical quantities (parameters) along the height of the tank, have been given. The results obtained in the work can be further used to clarify and improve engineering methods of calculation of structural elements of sports facilities of this type, both at the stages of their design and in the modes of real operation in order to prevent situations associated with the loss of their strength and stability.

A new set of explicit solutions for postbuckling behaviour of anisotropic shear deformable laminated cylindrical shells with general imperfection distribution and thermal loading

Thermal postbuckling analysis is presented for shear-deformable anisotropic laminated cylindrical shell with general imperfection distribution under several typical temperature fields are analyzed, including uniform temperature, tent-like linear distribution, axial and circumferential quadratic parabolic temperature fields over the shell surface and through the shell thickness. On this basis, the general temperature distribution can be obtained based on the three-dimensional steady-state heat transfer equation. At the same time, based on the high order shear deformation theory including anisotropic coupling effect, geometrically nonlinear large deflection displacement-strain relationship, the general initial geometric imperfections obtained by measurement and analytical means are introduced, and the thermal buckling equilibrium differential equation of anisotropic shear deformable laminated cylindrical shell with initial imperfections is established. A two-step perturbation-Galerkin integral method is used to obtain explicit solutions for the path development direction of post-thermal buckling equilibrium. The results of parameter analysis show that the material properties and temperature dependence, temperature distribution, lay-up and geometric parameters such as radius-to-thickness ratio have important effects on the thermal buckling load and postbuckling equilibrium path. The present method provides an effective theoretical prediction and analysis method for the design of the carrying capacity of submarine pipelines.

Seismic response of permafrost slopes supported by the thermal anchor pipe frame: Theoretical analysis and shaking table test verification

To study the seismic response of the thermal anchor pipe frame supporting permafrost slope system, the coupled dynamic interaction model of the permafrost slope with the thermal anchor pipe frame in thawing and freezing periods was established by taking into account the stratification characteristics of the permafrost slope in different typical seasons, respectively. Based on D'Alembert's principle and Dirac's function, the coupled vibration equilibrium differential equations are given and the seismic response of the system is analyzed for different periods. The reliability of the proposed model was further verified through shaking table tests. The results show that the acceleration response at different positions during both freezing and thawing periods is similar to the initial loading seismic wave shape. The peak acceleration and velocity during the thawing period is greater than that during the freezing period. The shear force and bending moment of the frame column are distributed alternately positive and negative along its height. The incremental dynamic shear force and bending moment are greater during the thawing period. However, the peak bending moments and shear forces are greater during the freezing period when the combined effects of frost and earthquakes. The peak bending moment occurs at the position of the first layer of the thermal anchor pipe, and the shear force suddenly changes near the thermal anchor pipe. In seismic design, attention should be paid to checking the internal force of the frame column during the freezing period to prevent plastic bending failure. The results contribute to further understanding the dynamic interaction mechanism of the thermal anchor pipe frame permafrost slope system, which has important practical significance for its seismic resistance.

GRILLAGE ANALYSIS OF CELLULAR PLATE STRUCTURESOF VARYING DEPTH WITH INCL

A static analysis of cellular plate structures of linearly varying depth is presented by using thesimplified grillage method. The cellular plate structure in this study consists of top and bottomflange plates (non-parallel) separated by longitudinal webs having varying depth. A variationalformulation of thin-walled beam element with linearly varying depth has been carried out for thederivation of the stiffness matrix of the grillage members for various warping restraint effects. Theeffect of warping restraint in stiffness is included by two methods, firstly; the implicit method(through the torsional constant) and secondly; by the explicit method (as independent warpingdegree of freedom). Two differer,t approaches are used to derive the torsional stiffness matrix,namely:by the assumed displacement field and by the differential equation of equilibrium. Theeffect of secondary shear stresses from the warping restraint is also considercd.The varying cross section properties of the eguivalent grillage members are derived from theproperties of the actual cellular plate structure.Several numerical examples and applications are presented . The results are compared with thoseobtained from the flat plate / shell elements of MSCA.{ASTRAN program. Acceptable agreement isobtained.

Axisymmetric Pressing Problem of Discrete Metal Materials

The paper presents an analytically closed solution to the problem of axisymmetric pressing of discrete metal materials by the method of jointly solving the differential equations of equilibrium of the metal and the plasticity conditions of the porous body, taking into account all pressing factors without exception: the type and properties of the charge, loading conditions, porosity, temperature, friction, etc. The purpose of this work is to develop the foundations of the engineering theory of pressure processing of discrete materials using the example of solving the problem of axisymmetric pressing of structurally inhomogeneous metal chips in a movable closed matrix. The basis for constructing a physical and mathema-tical model of the process is the idealized case of uniform compaction of a porous body with the subsequent determination of the lateral pressure coefficient corresponding to the actual degree of compaction at various stages of loading. The resulting equation for the relationship between the stress tensor components and the yield stress and relative compaction density represents the cylindrical Mises plasticity condition, which in the limit at zero porosity transforms into the plasticity condition for compact metals. The boundary value problem is solved for tangential stresses, taking into account the magnitude and direction of action of contact friction forces, which in their physical nature do not differ from the friction forces in the depth of the pressed material. The physico-mathematical model makes it possible to calculate the stress fields and density of the body according to the coordinates of the deformation zone, as well as energy-power parameters (pressure, force, work of deformation) provided that three structural and rheological characteristics are determined: the yield strength, relative compression and the degree of deformation compaction. Due to the fact that the problem is solved in relation to bodies of rotation in a general form and in a general formulation, the solution itself should be considered as methodological for any axisymmetric loading scheme.

Elastodynamic Green's functions for sandwich panels with aluminum foam core and transversely isotropic face sheets using potential functions method

Green's functions of stress and displacement due to wave propagation in a sandwich panel consisting of transversely isotropic face-sheets and aluminum foam core are developed in the present paper. Dynamic partial differential equations of equilibrium for the considered face-sheets and core are expressed in cylindrical coordinates (r, θ, z), and converted into two separate equations using two unknown potential functions. Then, by writing the potential functions as Fourier series in the circumferential direction and using Hankel transform in the radial direction, an analytical solution to the potential functions in Hankel transformed space is developed. Using the boundary conditions and continuity of the problem as well as the place of application of harmonic forces, Green's functions for displacements and stresses in the frequency domain are derived using Hankel's inverse integral transform. These functions are expressed as complex integrals and calculated numerically. To verify the methodology and results, a comparison study is performed for particular cases showing a high level of the accuracy. Green's functions obtained from this paper will give information about the loading models or materials causing the maximum normal stress, shear stress, vertical and radial displacements. In addition, based on the results, with an increase in the core thickness compared to that of face-sheets in a Sandwich panel, an increment in Green's functions of stress and displacement is observed in both real and imaginary parts.

A quasi-3D hyperbolic formulation for the buckling study of metal foam microplates layered with graphene nanoplatelets-embedded nanocomposite patches with temperature fluctuations

The work analyzes the buckling behavior of size-dependent microplates with a metal foam core, covered by graphene nanoplatelets (GNPs)-embedded nanocomposite patches. Microplates rest on a bi-parameter elastic substrate, and they are immersed in a thermal environment to observe the effect of temperature fluctuations on their elastic buckling performance. All material properties for each layer of the microstructure are thickness-dependent. A novel quasi-3D shear and normal (Q-3D S-N) hyperbolic theory is here proposed to describe the kinematic relations, accounting for the transverse normal strain. At the same time, a modified couple stress theory (MCST) is employed to account for the size dependence of the mechanical behavior due to the presence of a material length-scale parameter. Using the energy method and virtual work principle, the differential equilibrium equations are derived and solved analytically, where solutions are verified against the existing literature. The study focuses on the impact of different parameters on the normalized critical buckling load (NCBL). Based on the results from the systematic investigation, it is found that the addition of GNPs to the microplate enhances its stiffness, leading to increased values of NCBL, which in turn reduce for an increased imperfection ratio.

A mesh-free Hermite-type approach for buckling analysis of functionally graded polygonal thin plates

This work aims to establish a developed Meshless process which uses the point interpolation and the Hermite-type point interpolation methods for instability analysis of functionally graded material (FGM) thin polygonal plates. The structure’s material characteristics should gradually change according to the thickness direction. The minimization principle is used to obtain the governing nonlinear static equilibrium differential equations and related boundary conditions. To obtain the discrete weak form, point interpolation method (PIM) is used to approximate membrane components, while Hermite-type point interpolation method (HtPIM) ones are used for flexural components. The critical buckling loads and the associated eigenmodes are obtained by transforming the resulting equations into an eigenvalues problem. To evaluate the validity and effectiveness of the presented methodology, several examples concerning specimens of FGM rectangular and polygonal plates of various geometric planar shapes (L, T and C) subjected to uniaxial and biaxial in-plan loads are studied. The outcomes are contrasted with those of the finite element approach and with some data from the literature. Based on the analysis of the comparative study, the proposed methodology presents a very good agreement with the validated and previously published numerical results.

Predicting monotonic lateral load-displacement behavior of offshore flexible piles in saturated MCC soils: An application of the cavity expansion theory

This paper proposes an application of the cavity expansion theory to predict the load-displacement response of monotonically laterally loaded piles and the effective stress distribution of saturated modified Cam-clay (MCC) soil surrounding the pile. This approach incorporates geometric relationship, equilibrium equation, and boundary conditions to derive governing equations. These equations are subsequently solved to obtain monotonic p-y curves, which encompass the elastic, elastoplastic, and critical processes. To determine the load-displacement behavior and effective stress distribution of the soil, the proposed approach combines the deflection equilibrium differential equation with the finite-difference method. The accuracy and effectiveness of the approach are validated through a comparison with a Finite Element Method (FEM) simulation. Parametric studies are also presented to investigate the effects of overconsolidation ratio, in-situ coefficient, and mean effective stress. The validity and capacity of the proposed model are further demonstrated using model pile tests. The results affirm that this approach can well predict the monotonic load-displacement response of the pile and effectively captures important phenomena observed in pile tests.

Investigation of Passive Pile Groups’ Responses Induced by Combined Surcharge-Induced and Excavation-Induced Horizontal Soil Loading

Excessive lateral forces and the deformation of pile groups caused by adjacent surcharge loading and excavation can potentially lead to the failure or collapse of nearby pile foundations. This paper presents a simplified analytical method to investigate the lateral displacement of passive pile groups caused by combined surcharge-induced and excavation-induced horizontal soil loading and validates it by utilizing two published centrifuge tests and numerical modeling. Firstly, the passive load resulting from surcharge-induced horizontal soil loading is determined using the improved formula of Boussinesq and the theory of local plastic deformation. The additional horizontal force exerted on the pile axis subjected to excavation-induced horizontal soil loading is also taken into account using the Mindlin solution. Furthermore, the shielding effect between pile groups is also proposed to analyze the response of a laterally loaded pile group due to excavation unloading. Secondly, according to the Pasternak foundation model of soil–pile interaction, the equilibrium differential equations are solved by the finite difference method. Finally, parametric analyses are undertaken to evaluate the behavior of different constraints on the pile head and surcharge procedure. The results demonstrate that the capped-head pile groups can provide greater restraint compared to the free-head groups for the lateral displacement of pile groups. The front pile experiences greater forces than the rear one in passive pile groups as a result of the smaller soil pressure on the rear pile. Additionally, increasing the horizontal stiffness of the capped head with fully fixed ends can significantly decrease the horizontal deformation of the pile groups.

Thermoelastic creep evolution in a variable thickness functionally graded piezoelectric rotating annular plate considering convection and radiation heat transfer

The thermoelastic time-dependent creep progress of a rotating annular plate with nonuniform thickness made of functionally graded piezoelectric material considering convection heat transfer and solar radiation was investigated. The nonlinear differential equation of heat transfer considering conduction, convection, and solar radiation was solved using the differential transformation method. Then, the differential equation of equilibrium for the annular plate containing creep strains was derived. Firstly, the differential equation was solved analytically for the zero time . Then, the stress and strain rates were achieved using Norton’s law with Prandtl–Reuss equations. Finally, the time-dependent creep stresses and deformation redistributions were obtained employing an iterative method.

Face sheet defects in plastic forming of aluminum foam sandwich panel: theoretical models, numerical and experimental studies

In this paper, the theoretical models of face wrinkling and dimpling defects occurred in the plastic forming of aluminum foam sandwich panel (AFSP) were constructed and the influences of AFSP parameters on defect were investigated to predict and avoid the surface forming defects. The wrinkling and dimpling behaviors of shaped AFSPs were analyzed at different scales first, and the analytical models of critical wrinkling and dimpling stresses of face sheet for the forming of AFSP were developed using energy method and differential equilibrium equation method respectively. Finite element simulations of curved AFSPs forming were performed using macroscopic and mesoscopic AFSP models, and plastic forming experiments of spherical and saddle-shaped AFSPs with different core relative densities were conducted. The surface profiles of simulated parts and the face wrinkles and dimples in experimental parts indicated that the established theoretical models can predict the occurrence of face sheet defects in curved AFSPs forming accurately. Furthermore, the effects of geometric and material parameters of AFSP on face wrinkling suggested that the seriousness of face wrinkling can be reduced by decreasing the thicknesses of face sheet and core, and the wrinkling can be suppressed when the elastic Poisson's ratio of core is close to 0 and 0.5.

Improving shape formation under conditions of plane tensile stress

Thin-walled axisymmetric truncated parts made of sheet billets are actively used in rocket and aerospace engineering. Improvement to their shape formation, based on directed material thickness change will ensure the production of parts with minimum thickness variation. This will also enable aviation and space industry enterprises to attain leading positions, as well as reduce labor costs. This work studies the possibility of obtaining thin-walled axisymmetric parts of truncated tapered shape using one of the methods of sheet metal stamping under flat tensile stress conditions (flanging). The mechanism was identified and the analysis of the stress-strain state of the billet during deformation was carried out. This takes into account the minimizing of the difference between the specified and technologically possible thicknesses. A mathematical model was developed to consider the shaping method based on the process of flanging. Theoretical studies were based on the principles of the plastic deformation theory of sheet materials. This was achieved by the following factors: approximate differential equations of force equilibrium; equations of constraint; plasticity conditions; and fundamental constitutive relations under given initial and boundary conditions. The process of flanging was simulated using the LS-DYNA software package with the following initial data of a conical billet made of 12Kh18N10T steel: cone angle 16.4°, thickness Sbillet = 0.3 mm. The aim was to eliminate errors in designing a tool for future implementation of the method on a manufactured die tooling, as well as to confirm the theoretical conclusions on the selection of technological parameters and achieve minimal thickness variation. The steps of computer modeling are presented, indicating the main process parameters such as material model, mechanical characteristics of the workpiece material, type of elements, kinematic loads, conditions of contact interaction of elements with each other, etc.