Equations Of Boundary Conditions Research Articles

A Mixed Layerwise-Differential Quadrature Method for 3-D Vibration and Buckling Analyses of Orthogonally Stiffened Composite Conical Shell

A layerwise-differential quadrature method (LW-DQM) is developed for the vibration analysis of a stiffened laminated conical shell. The circumferential stiffeners (rings) and meridional stiffeners (stringers) are treated as discrete elements. The motion equations are derived by applying the Hamilton’s principle. In order to accurately account for the thickness effects and the displacement field of stiffeners, the layerwise theory is used to discretize the equations of motion and the related boundary conditions through the thickness. Then, the equations of motion as well as the boundary condition equations are transformed into a set of algebraic equations applying the DQM in the meridional direction. The advantage of the proposed model is its applicability to thin and thick unstiffened and stiffened shells with arbitrary boundary conditions. In addition, the axial load and external pressure is applied to the shell as a ratio of the global buckling load and pressure. This study demonstrates the accuracy, stability, and the fast rate of convergence of the present method, for the buckling and vibration analyses of stiffened conical shells. The presented results are compared with those of other shell theories and a special case where the angle of conical shell approaches zero, i.e. a cylindrical shell, and excellent agreements are achieved.

Influence of Rotation on Vibration Behavior of a Functionally Graded Moderately Thick Cylindrical Nanoshell Considering Initial Hoop Tension

In this research, the effect of rotation on the free vibration is investigated for the size-dependent cylindrical functionally graded (FG) nanoshell by means of the modified couple stress theory (MCST). MCST is applied to make the design and the analysis of nano actuators and nano sensors more reliable. Here the equations of motion and boundary conditions are derived using minimum potential energy principle and first-order shear deformation theory (FSDT). The formulation consists of the Coriolis, centrifugal and initial hoop tension effects due to the rotation. The accuracy of the presented model is verified with literatures. The novelty of this study is the consideration of the rotation effects along with the satisfaction of various boundary conditions. Generalized differential quadrature method (GDQM) is employed to discretize the equations of motion. Then the investigation has been made into the influence of some factors such as the material length scale parameter, angular velocity, length to radius ratio, FG power index and boundary conditions on the critical speed and natural frequency of the rotating cylindrical FG nanoshell.

Dynamic boundary conditions for membranes whose surface energy depends on the mean and Gaussian curvatures

Membranes are an important subject of study in physical chemistry and biology. They can be considered as material surfaces with a surface energy depending on the curvature tensor. Usually, mathematical models developed in the literature consider the dependence of surface energy only on mean curvature with an added linear term for Gauss curvature. Therefore, for closed surfaces the Gauss curvature term can be eliminated because of the Gauss-Bonnet theorem. In [18], the dependence on the mean and Gaussian curvatures was considered in statics. The authors derived the shape equation as well as two scalar boundary conditions on the contact line. In this paper-thanks to the principle of virtual working-the equations of motion and boundary conditions governing the fluid membranes subject to general dynamical bending are derived. We obtain the dynamic 'shape equa-tion' (equation for the membrane surface) and the dynamic conditions on the contact line generalizing the classical Young-Dupr{\'e} condition.

A framework for biomechanics simulations using four-chamber cardiac models

Computational cardiac models have been extensively used to study different cardiac biomechanics; specifically, finite-element analysis has been one of the tools used to study the internal stresses and strains in the cardiac wall during the cardiac cycle. Cubic-Hermite finite element meshes have been used for simulating cardiac biomechanics due to their convergence characteristics and their ability to capture smooth geometries compactly–fewer elements are needed to build the cardiac geometry–compared to linear tetrahedral meshes. Such meshes have previously been used only with simple ventricular geometries with non-physiological boundary conditions due to challenges associated with creating cubic-Hermite meshes of the complex heart geometry. However, it is critical to accurately capture the different geometric characteristics of the heart and apply physiologically equivalent boundary conditions to replicate the in vivo heart motion. In this work, we created a four-chamber cardiac model utilizing cubic-Hermite elements and simulated a full cardiac cycle by coupling the 3D finite element model with a lumped circulation model. The myocardial fiber-orientations were interpolated within the mesh using the Log-Euclidean method to overcome the singularity associated with interpolation of orthogonal matrices. Physiologically equivalent rigid body constraints were applied to the nodes along the valve plane and the accuracy of the resulting simulations were validated using open source clinical data. We then simulated a complete cardiac cycle of a healthy heart and a heart with acute myocardial infarction. We compared the pumping functionality of the heart for both cases by calculating the ventricular work. We observed a 20% reduction in acute work done by the heart immediately after myocardial infarction. The myocardial wall displacements obtained from the four-chamber model are comparable to actual patient data, without requiring complicated non-physiological boundary conditions usually required in truncated ventricular heart models.

Analysis of an orthotropic elastic plane problem of a half plane with an edge crack using a mapping function

It is a challenging problem to derive a solution for an orthotropic elastic plane problem using a mapping function. The complex variable method is employed to introduce a mapping function. Then, comparatively arbitrary configurations can be analyzed. The boundary condition equation is represented by two complex variables. This fact makes solving the present problem difficult. This difficulty is conquered by applying Cauchy integrals. Also, mapping functions represented by an infinite terms of fractional expressions are introduced to conquer the mathematical difficulties of the Cauchy integral in the process of the derivation of the stress function. One of two stress functions is derived by analytical continuation. The final exact stress functions are represented by an irrational mapping function in a closed form. Stress components are represented by one complex variable. Therefore, the calculation of the stress components is easier than that of an isotropic problem. Half planes with a notch or a mound expressed by an irrational mapping function can be solved using their mapping functions. Such a polygonal mapping function can be derived by Schwarz–Christoffel’s transformation. As a demonstration, a half plane with a vertical edge crack subjected to uniform tension is analyzed and the stress distributions are shown for two examples of two characteristic roots of the characteristic equation for the orthotropic elastic plane.

Исследование влияния рода граничных условий на интегральные характеристики стендового змеевикового теплообменного аппарата

A study of the influence of the boundary conditions type on such integral characteristics of a bench coiled heat exchanger as temperature and pressure fluctuations is presented in this paper. The change in the integral characteristics in the annular space of the heat exchanger was determined when specifying the equivalent boundary conditions of the first, second and third types. For the studied geometric model of the heat exchanger, numerical simulation was carried out in a wide range of Reynolds numbers taking into account the real thermal properties of the air. The fields of temperature, pressure and current density distribution in the annular space of the heat exchanger were obtained.

Design and Thermal Analysis of the Large Fire Door for AP1000 Nuclear Reactor

The large fire door is the key component to ensure the effectiveness of fire zone in AP1000 nuclear reactor. According to the fire design requirements and design criteria, the global structure of the large fire door is designed. Based on the designed structure, the thermal mathematical model of the large fire door is established. Based on the solid heat transfer theory, the multi-layer heat transfer theory and integrated heat transfer theory, the differential equations of heat conduction, initial conditions, and boundary conditions are determined. Thermal analysis for the fire door leaf and the closure is carried out by using the method of numerical simulation. Results show that: considering the thermal load, the whole structure of the large fire door can meet the fire resistance limit of 3 hours and the design is reasonable and feasible. This study provides theory basis for the design of the large fire door.

Dynamic Analysis of a Viscoelastic Nanobeam

Vibration of an axially loaded viscoelastic nanobeam is analyzed in this study. Viscoelasticity of the nanobeam is modeled as a Kelvin-Voigt material. Equation of motion and boundary conditions for viscoelastic nanobeam are provided with help of Eringen’s Nonlocal Elasticity Theory. Initial conditions are used in solution of governing equation of motion. Damping effect of the viscoelastic nanobeam structure is investigated. Nonlocal effect on natural frequency and damping of nanobeam and critical buckling load is obtained.

Modal characterization with nonlinear behaviors of a two-link flexible manipulator

This present work has made a noteworthy attempt to demonstrate brief modeling of N-link manipulator and subsequent modal characterization along with the determination of static deflection of a two-link flexible manipulator with a payload. In addition, investigation of nonlinear phenomena of dynamic responses under 3:1 internal resonance has also been accomplished considering geometric nonlinearities. An appropriate and realistic dynamic modeling of the two-link manipulator taking into account of inertia coupling and geometry compatibility between equations of motion and boundary conditions has been derived using the extended Hamilton’s principle. The effect of parametric variation on system eigenfrequencies is well tabulated, and the corresponding eigenspectrums are illustrated graphically. Further, the nonlinear phenomena of dynamic solutions have been demonstrated by using MMS of second order for its statutory effect onto the system instability for the existence of S-N bifurcations. The effect of nonlinearities and various design parameters on the dynamic responses and subsequent bifurcations for 3:1 internal resonance has also been demonstrated. The outcome of the present work enables new understanding into the design criterion and performance limitation of multi-link flexible robots.

Sound transmission loss of laminated composite sandwich structures with pyramidal truss cores

This paper presents a theoretical model that allows to calculate the acoustic transmission loss through laminated composite pyramidal sandwich structure consisting of two parallel plates connected by trusses. First-order shear deformation theory is adopted to model the vibration of such a structure, accounting for in-plane motion and coupling between extension and flexure of the different components, and taking into account elastic anisotropy. The interaction between the structure and the surrounding fluid is taken into account by imposing a velocity continuity condition at the interfaces. The displacement and stress fields are calculated in Fourier domain by solving the set of boundary condition equations at the connecting points. The theoretical predictions for the acoustic transmission through the structure show satisfactory agreement with experimental results of a standing wave tube experiment on specimens that were fabricated by cutting trusses from carbon fiber reinforced composite plates and snap-fitting them to plates made of the same material. Numerical simulations are used to verify, the influence of the stacking geometry and material parameters on the acoustic transmission for different frequencies and angles of incidence. Conclusions are presented that are helpful for practical design.

Vibration and Damping Analysis of Sandwich Cylindrical Shells by the GDQM

This paper focuses on the vibration and damping analysis of three-layered sandwich cylindrical shells with stiff composite face layers and a viscoelastic core. The equations of motion and boundary conditions governing the free vibration are derived by using the Hamilton’s principle. Then, generalized differential quadrature method is used to solve these equations to obtain the natural frequencies and modal loss factors. Results are validated against the ones that already exist in the literature and performed finite element method analyses of current study. In addition, a parametric study is performed for a sandwich shell with carbon fiber–reinforced plastic face layers and a frequency-dependent viscoelastic core. A 10-parameter fractional derivative model is used to represent the viscoelastic behavior of the core layer. The effects of system parameters, that is, layer thicknesses, the orientation angle of the face layers, and the subtended angle on the vibration and damping characteristics of open cylindrical shells, are investigated in detail. The vibration and damping analyzes of sandwich shells with frequency-dependent viscoelastic core are performed for the first time to the best of the authors’ knowledge.

Vibration analysis of size-dependent functionally graded micro-plates subjected to electrostatic and piezoelectric excitations

A size–dependent model based on the modified couple stress theory is developed to study free vibration, static deflection, and pull-in instability of a fully-clamped functionally graded (FG) micro-plate covered by piezoelectric layers subjected to both electrostatic and casimir forces. Considering the geometrically nonlinearities, the governing equations of motion and boundary conditions are derived by means of Hamilton's principle. The system of equations is solved numerically using finite element method. A Comprehensive Verification study is carried out to ensure the accuracy of the proposed model. Obtained results show that the volume fraction exponent, piezoelectric voltage, residual stresses, and MLSP considerably affect the natural frequency, maximum static deflection, and the electro-mechanical behavior of fully-clamped rectangular micro-plates. The results also reveal that the pull-in voltage could be maintained within a desirable range through proper utilizing of material properties, residual stresses, and piezoelectric voltages.

Exact solution for shear deformable model of free damped vibration of magnetorheological fluid sandwich beam

An exact procedure is proposed to analyze vibration suppression of magnetorheological fluid (MRF) sandwich beams based on the Timoshenko shear deformable beam model. The equations of motion and corresponding boundary conditions are extracted using Hamilton’s principle. Then, by introducing four innovative functions, five highly coupled equations of motion are divided into three coupled equations and two independent equations. Employing a systematic procedure, the displacement fields and consequently frequency characteristic equation of the beam are exactly derived. High accuracy and stability of the exact natural frequencies and corresponding loss factors are shown in comparison with those available in literature.

Bending of exponentially graded plates integrated with piezoelectric fiber-reinforced composite actuators resting on elastic foundations

This paper introduces analytical solution for exponentially graded (EG) plates attached with a single-layer of piezoelectric fiber reinforced composite (PFRC) actuator. The plate is undergoing sinusoidal electromechanical loads and resting on Pasternak-Winkler's elastic foundations. The displacement components are expressed via a higher-order shear deformation theory. The distributions of the electric potential are vary linearly along the thickness direction. The equations of equilibrium and boundary conditions are established by applying the principle of virtual displacements. Navier's method is used to obtain the problem solution. The influences of various parameters like voltage amount, side-to-thickness ratio, Young's modulus ratio and elastic foundations parameters are investigated. The current formulations are validated by comparison with published results.

Thermoelastic Coupling Vibration and Stability Analysis of Rotating Annular Sector Plates

This study investigates the thermoelastic coupling vibration and stability of rotating annular sector plates. Based on Hamilton’s principle and thermal conduction equation with deformation effect, the differential equation of transverse vibration for a rotating annular sector plate is established. The differential equation of vibration and corresponding boundary conditions are discretized by the differential quadrature method. Then, the thermoelastic coupling transverse vibrations under three different boundary conditions are calculated. The change curve of the first three order dimensionless complex frequencies of the rotating annular sector plate with the dimensionless angular speed are analyzed in the case of the thermoelastic coupling and uncoupling. The effects of the dimensionless angular speed, the ratio of inner to outer radius, the sector angle, and the dimensionless thermoelastic coupling coefficient on transverse vibration and stability of the annular sector plate are discussed. Finally, we obtained the type of instability and corresponding critical speed of the rotating annular sector plate in the case of the thermoelastic coupling and uncoupling.

MATHEMATICAL MODELING SINGULARLY PERTURBED PROCESSES OF WATER SOFTENING ON SODIUM-CATIONITE FILTERS

Mathematical model of the process of water softening using ion exchange pre-treatment of waters to desalination, with a view to removal of scale forming components, such as calcium and magnesium, are formed in the paper. In this process, no additional chemicals, except for brines formed during desalination, are required for regeneration of ion-exchanger in operation cycles. An asymptotic approximation of a solution of a corresponding model problem is constructed. Theoretical description and modelling assumptions included the set of differential equations of mass balance, initial, boundary and operational conditions. The paper deals with the development of a computer model for description and prediction of the performance of ion exchange columns.

Study of an Axial-Flux Modulated Superconducting Magnetic Gear

An axial-flux modulated superconducting magnetic gear has been specifically studied using superconducting bulk magnet replacing conventional permanent magnet. A mathematical model of the superconducting bulk magnets utilized as the magnetic source is constructed from the perspective of superconducting mechanism. After that, a physical analytical model based on Laplace's equation, Poisson's equation, equations of boundary conditions, and the principle of axial-flux modulated superconducting magnetic gear is presented to predict the air-gap magnetic density. Finally, a three-dimensional (3-D) model of the proposed superconducting magnetic gear is built and analyzed via finite-element method (FEM). The calculation results by the proposed analytical model are verified with those issued from 3-D FEM.

The First Exit Time Stochastic Theory Applied to Estimate the Life-Time of a Complicated System

We develop a first exit time methodology to model the life time process of a complicated system. We assume that the functionality level of a complicated system follows a stochastic process during time and the end of the functionality of the system comes when the functionality function reaches a zero level. After solving several technical details including the Fokker-Planck equation for the appropriate boundary conditions we estimate the transition probability density function and then the first exit time probability density of the functionality of the system reaching a barrier during time. The formula we arrive is essential for complicated system forms. A simpler case has the form called as Inverse Gaussian and was first proposed independently by Schrodinger and Smoluchowsky in the same journal issue (1915) to express the probability density of a simple first exit time process hitting a linear barrier. Applications to the health state of biological systems (the human population and the Mediterranean flies) and to the functionality life time of cars are done.

Static analysis of sandwich plates embedded with shape memory alloy wires using active strain energy tuning method

In this study, static analysis of sandwich plates with flexible core and composite face sheets embedded with shape memory alloy (SMA) wires in the face sheets is conducted. Third-order plate theory is used to model face sheets, and quadratic and cubic functions are assumed for transverse and in-plane displacements of the flexible core. Continuity conditions of transverse shear stresses at the interfaces between face sheets and core and the zero transverse shear stresses conditions on the upper and lower face sheets are satisfied. Also, transverse flexibility and transverse normal strain and stress of the orthotropic core are noted. The SMA wires are placed in the laminated composite face sheets. Constitutive equation of the SMA wires is employed for modeling their behavior. Nonlinear equations of motion and boundary conditions of sandwich plates are derived using Hamilton’s principle. The nonlinear governing equations of the simply supported sandwich plates under uniform and sinusoidal transverse loads are solved based on Navier’s method. The effect of activating embedded SMA wires on bending behavior of composite sandwich plates are investigated using the active strain energy tuning method. The Liang’s constitutive model is employed to obtain the SMA wires recovery force. The obtained results show that activating SMA wires causes a decrease in the deflection and stresses in these plates.

Influence of homogenization schemes on vibration of functionally graded curved microbeams

The influence of various homogenization models is studied on the free vibration behavior of a functionally graded material (FGM) curved microbeam which is modeled using the modified strain gradient theory of elasticity as well as the first-order shear deformation theory. Different homogenization models (such as Voigt, Reuss, Hashin-Shtrikman bounds, and cubic local representative volume elements (LRVE) schemes) are used to predict the effective material properties of a two-phase particle composite as a function of the volume fraction of particles which are continuously varying along the thickness of a functionally graded microbeam. Employing Hamilton’s principle, the governing equations of motion and boundary conditions are derived. Finally, the numerical results are presented to determine the effect of homogenization models on the vibrational behavior of FGM curved microbeams corresponding to different continuum models for over a wide range of material composition, opening angle, dimensionless length scale parameter and aspect ratio. It is shown that the common used Voigt model overestimates frequencies and the Mori-Tanaka model also overestimates frequencies but provides smaller error in comparison with the Voigt model. In terms of theoretical analysis, LRVE model creates a good compromise between estimation accuracy and the easy of implement, especially for the FGM with a relative high stiffness ratio. The difference of various homogenization models can be raised when considering the decrement in aspect ratio or the increment of opening angle and mode number.