Different Combinations Of Boundary Conditions Research Articles

Free Vibration Analysis of Triclinic Nanobeams Based on the Differential Quadrature Method

In this work, the nonlocal strain gradient theory is applied to study the free vibration response of a Timoshenko beam made of triclinic material. The governing equations of the problem and the associated boundary conditions are obtained by means of the Hamiltonian principle, whereby the generalized differential quadrature (GDQ) method is implemented as numerical tool to solve the eigenvalue problem in a discrete form. Different combinations of boundary conditions are also considered, which include simply-supports, clamped supports and free edges. Starting with some pioneering works from the literature about isotropic nanobeams, a convergence analysis is first performed, and the accuracy of the proposed size-dependent anisotropic beam model is checked. A large parametric investigation studies the effect of the nonlocal, geometry, and strain gradient parameters, together with the boundary conditions, on the vibration response of the anisotropic nanobeams, as useful for practical engineering applications.

Hamiltonian system-based new analytic free vibration solutions of cylindrical shell panels

This paper deals with the classical challenging free vibration problems of non-Lévy-type cylindrical shell panels, i.e., those without two opposite edges simply supported, by a Hamiltonian system-based symplectic superposition method. The governing equations of a vibrating cylindrical panel are formulated within the Hamiltonian system framework such that the symplectic eigen problems are constructed, which yield analytic solutions of two types of fundamental problems. By the equivalence between the superposition of the fundamental problems and the original problem, new analytic frequency and mode shape solutions of the panels with four different combinations of boundary conditions are derived. Comprehensive benchmark results are tabulated and plotted, which are useful for validation of other numerical/approximate methods. The primary advantage of the developed approach that no pre-determination of solution forms is needed enables one to pursue more analytic solutions of intractable shell problems.

Решение комбинированных краевых задач для анизотропных тел вращения с массовыми силами

The aim of the work is to determine the stress-strain state of anisotropic bodies of revolution under the action of mass forces and external conditions of a physical and geometrical nature in a combined formulation. The task is ensured by the development of the method of boundary states, based on the concepts of spaces of internal and boundary states. The theory of the formation of the basis of the space of internal states, which includes displacements, deformations, stresses and mass forces, is constructed. A common basis is a combination of two bases. The first is formed for the case of plane deformation from the action of only mass forces, based on the application of the application of fundamental polynomials. Further, on its basis, according to the method of integral overlays, a basis of internal spatial states is induced. The second is the basis of internal states in the classical boundary value problem for transversely isotropic bodies. After orthogonalization of the common basis, where the inner energy of elastic deformation acts as the scalar product, the desired state is determined by a Fourier series, the coefficients of which are defined integrals. Checking the adequacy of the solution is carried out by comparing a given field of mass forces with that obtained, as well as by comparing the specified boundary conditions with the resulting solutions. Solutions of problems with different combinations of boundary conditions for a circular cylinder of rock with the corresponding conclusions are given. Presents a graphical visualization of the results.

Theoretical analysis of the frictional losses in magnetohydrodynamic microflows considering slippage

In this article, we study the frictional losses in magnetohydrodynamic (MHD) microflows by analyzing the Poiseuille number defined through the Darcy–Weisbach friction factor. We consider two-dimensional fully developed flow models characteristic of MHD micropumps including the Hartmann braking effect and the existence of slippage. Unlike the purely hydrodynamic case, in MHD flows the Poiseuille number depends not only on the aspect ratio but also on the physical properties of the fluid and the externally applied magnetic field. Three different combinations of boundary conditions (slip and no-slip) are investigated. Calculations show that the Poiseuille number is considerably reduced as the dimensionless slip length is increased, while it increases as Hartmann number does. The obtained results are consistent with previous models and are helpful for the design of magnetohydrodynamic microflow devices.

Correspondence between bulk entanglement and boundary excitation spectra in two-dimensional gapped topological phases

We study the correspondence between boundary spectrum of non-chiral topological orders on an open manifold $\mathcal{M}$ with gapped boundaries and the entanglement spectrum in the bulk of gapped topological orders on a closed manifold. The closed manifold is bipartitioned into two subsystems, one of which has the same topology as $\mathcal{M}$. Specifically, we focus on the case of generalized string-net models and discuss the cases where $\mathcal{M}$ is a disk or a cylinder. When $\mathcal{M}$ has the topology of a cylinder, different combinations of boundary conditions of the cylinder will correspond to different entanglement cuts on the torus. When both boundaries are charge (smooth) boundaries, the entanglement spectrum can be identified with the boundary excitation distribution spectrum at infinite temperature and constant fugacities. Examples of toric code, $\mathbb{Z}_N$ theories, and the simple non-abelian case of doubled Fibonacci are demonstrated.

Analytical method for free-damped vibration analysis of viscoelastic shear deformable annular plates made of functionally graded materials

This contribution aims to study the free vibrations behavior of viscoelastic functionally graded (VFGM) annular plates using the first-order shear deformation theory. The Zener three-parameter model is employed to model the time-dependent behavior of the plate which is assumed viscoelastic in shear and elastic in bulk. The functionally graded material properties vary through the thickness of the plate with a power law function and exponential distribution. Hamilton’s principle is utilized to derive the governing equations which are three coupled partial differential equations with variable coefficients. The equations of motion are solved analytically using the perturbation technique. The natural frequencies of the plate are determined with different combinations of boundary conditions including simply supported, clamped and free. In addition, the effects of different viscoelastic and geometric parameters on the results are studied. The results are compared with those obtained from the literatures and the finite-element package. Investigation of the gradient index effect on the damping properties of VFGM plate shows that an increase in gradient index tends to decrease the damping behaviors of the VFGM plate.HighlightsFirst-order shear deformation theory is used for ANALYTICAL investigation of the vibrations and damping behaviors of VFGM annular plates with various combinations of boundary conditions.For more exact modeling of viscoelastic materials, we separated the bulk and shear behavior of the structure which is closer than to the behavior of a variety of viscoelastic structures.The plate properties are variable with time and space simultaneous.The perturbation technique in conjunction with a special variable change can convert the governing equations which are a system of coupled partial equations with variable coefficients to a system with constant coefficients so that each order of equations has an Cosed-form solution.A sensitivity analysis is performed and the results are compared with the three-dimensional theory of elasticity using FE method.

Symplectic superposition method-based new analytic bending solutions of cylindrical shell panels

Pursuing analytic bending solutions of cylindrical shell panels without two opposite simply supported edges is a classic but very difficult type of problems. The main challenge is on the mathematical complexity of the boundary value problems of the governing high-order partial differential equations. In this paper, the first endeavor is made on extending an up-to-date symplectic superposition method to bending of cylindrical panels, with focus on clamped panels and their variants. By introducing the problems into the Hamiltonian system (in physics) and the symplectic space (in mathematics), they come down to the symplectic eigen problems, which are analytically solved for fundamental analytic solutions of three types of subproblems, followed by superposition for final solutions. The new analytic solutions for the panels with four different combinations of boundary conditions are obtained, with comprehensive results tabulated to serve as benchmarks for future studies, all of which are well validated by the finite element method. The rigorous derivation by the present method without any assumptions/prior knowledge of solution forms may provide an exceptional route to more analytic solutions of some intractable shell problems.

Investigation of the influence of boundary conditions in the numerical solution of a vortex tube model

The applicability of various boundary conditions in the computational simulation of a Ranque–Hilsch vortex tube is investigated. A review of existing works on the effect of geometry and various thermodynamic parameters on the efficiency of the pipe is made. The substantiation of the possibility of introducing additional computational domains when moving the boundaries to study the influence of boundary conditions when modeling gas dynamics problems is given. To simulate the dynamics of a gas in a vortex tube, a mathematical model is written that includes the Navier–Stokes system of equations describing a compressible viscous fluid, which is closed by the equation of state of an ideal gas. Existing methods for calculating turbulent flows are considered. The applicability of various semi-empirical models of turbulence for modeling a vortex tube is described. The possibility of using the selected k−ε model and its description is argued. The boundary conditions characteristic of the vortex tube model are described, and the boundary conditions most combined in the simulation of gas dynamics problems are also shown. Presents a grid that takes into account the area formed by the removal of boundaries. The solution is based on the sonicFoam algorithm in the OpenFOAM package. Utilities of the postprocessor are used when preparing the model for calculations on a high-performance cluster and utilities for averaging the obtained physical quantities. The simulation results for different combinations of boundary conditions and models with remote boundaries are given. Comparison of the results obtained. It is shown that the geometrical dimensions have a strong influence on the operation of the pipe; the correct choice of boundary conditions makes it possible to obtain the values of physical quantities that are closest to the known experimental ones. Moving the boundaries away from direct exits provides an opportunity to more accurately estimate the effects that arise near the real boundaries of the vortex tube, especially affecting the magnitude of the Ranque–Hilsch effect.

Numerical analysis for free vibration of hybrid laminated composite plates for different boundary conditions

This study aimed to develop a high-order shear deformation theory to predict the free vibration of hybrid cross-ply laminated plates under different boundary conditions. The equations of motion for laminated hybrid rectangular plates are derived and obtained by using Hamilton\'s principle. The closed-form solutions of anti-symmetric cross-ply and angle-ply laminates are obtained by using Navier\'s solution. To assess the validity of our method, we used the finite element method. Firstly, the analytical and the numerical implementations were validated for an antisymmetric cross-ply square laminated with available results in the literature. Then, the effects of side-to-thickness ratio, aspect ratio, lamination schemes, and material properties on the fundamental frequencies for different combinations of boundary conditions of hybrid composite plates are investigated. The comparison of the analytical solutions with the corresponding finite element simulations shows the good accuracy of the proposed analytical closed form solution in predicting the fundamental frequencies of hybrid cross-ply laminated plates under different boundary conditions.

A Study on Vibration of Tapered Non-Homogeneous Rectangular Plate with Structural Parameters

Effects of structural parameters on the vibration of a tapered non-homogeneous rectangular plate with different combinations of boundary conditions are discussed. Tapering in the plate is assumed to be sinusoidal in the <i>x</i>-direction. Here, temperature variation and non-homogeneity in the plate material are also considered sinusoidal in the <i>x</i>-direction. The Rayleigh-Ritz method is used to calculate the frequency parameter for the first two modes of vibration for different values of the structural parameters, i.e. the taper parameter, thermal gradient, aspect ratio and non-homogeneity constant. Results are obtained for three boundary conditions, i.e. clamped boundary (C-C-C-C), simply supported boundary (SS-SS-SS-SS) and clamped-simply supported boundary (CSS-C-SS). Numerical values of the frequency parameter are given in a compact tabular form.

Analytical solution for modal analysis of Euler-Bernoulli and Timoshenko beam with an arbitrary varying cross-section

In this article, the free vibrations of Euler-Bernoulli and Timoshenko beams with arbitrary varying cross-section are investigated analytically using the perturbation technique. The governing equations are linear differential equations with variable coefficients and the Wentzel, Kramers, Brillouin approximation is adopted for solving these eigenvalue equations and determining the natural frequencies and mode shapes. This method relates the solution of equations with the solving of some successive algebraic equations. A parametric study is performed and the effects of different profiles and different combinations of boundary conditions on the natural frequencies are investigated. To confirm the reliability of the present method, the analytical results are checked with those obtained from the finite elements method and other literatures which are found to be in a good agreement. The calculations show that the presented procedure is very effective to find the modal characteristics of the varying cross-sections beams.

Vibration analysis of a thin functionally graded plate having an out of plane material inhomogeneity resting on Winkler–Pasternak foundation under different combinations of boundary conditions

In the present research article, classical plate theory has been adopted to analyze functionally graded material plate, having out of plane material inhomogeneity, resting on Winkler–Pasternak foundation under different combinations of boundary conditions. The material properties of the functionally graded material plate vary according to power law in the thickness direction. Rayleigh–Ritz method in conjugation with polynomial displacement functions has been used to develop a computationally efficient mathematical model to study free vibration characteristics of the plate. Convergence of frequency parameters (nondimensional natural frequencies) has been attained by increasing the number of polynomials of displacement function. The frequency parameters of the functionally graded material plate obtained by proposed method are compared with the open literature to validate the present model. Firstly, the present model is used to calculate first six natural frequencies of the functionally graded plate under all possible combinations of boundary conditions for the constant value of stiffness of Winkler and Pasternak foundation moduli. Further, the effects of density, aspect ratio, power law exponent, Young’s modulus on frequency parameters of the functionally graded plate resting on Winkler–Pasternak foundation under specific boundary conditions viz. CCCC (all edges clamped), SSSS (all edges simply supported), CFFF (cantilever), SCSF (simply supported-clamped-free) are studied extensively. Furthermore, effect of stiffness of elastic foundation moduli (kp and kw) on frequency parameters are analyzed. It has been observed that effects of aspect ratios, boundary conditions, Young’s modulus and density on frequency parameters are significant at lower value of the power law exponent. It has also been noted from present investigation that Pasternak foundation modulus has greater effect on frequency parameters as compared to the Winkler foundation modulus. Most of the results presented in this paper are novel and may be used for the validation purpose by researchers. Three dimensional mode shapes for the functionally graded plate resting on elastic foundation have also been presented in this article.

An analytical approach for free vibrations analysis of viscoelastic circular and annular plates using FSDT

ABSTRACTIn this paper, free vibrations analysis of asymmetric viscoelastic annular and circular plates are studied using the first order shear deformation theory. The viscoelastic behavior is assumed standard linear solid in shear and elastic in bulk. Hamilton's principle is employed to derive governing equations which are five coupled partial differential equations with variable coefficients. Using the perturbation technique and the Fourier series, the equations of motion are solved analytically. Determination of the natural frequencies of the plate with different combinations of boundary conditions including simply supported, clamped, and free are considered. In addition, the sensitivity of the results to the geometrical and mechanical parameters is investigated. The results are compared with those obtained from the literature, classical plate theory, and finite elements package.

Free vibration and wave power reflection in Mindlin rectangular plates via exact wave propagation approach

Reflection, propagation and energy analysis are crucially important in designing structures, especially plates. A thick plate is considered based on first order shear deformation theory. Wave Propagation Method (WPM) is employed to exactly derive resonant frequencies and wave power reflection from different classical boundary conditions. Firstly, the frequency results are compared with other literature to validate the exact proposed wave solution in the present work. Then, wave analysis and benchmark results for natural frequencies are presented for six different combinations of boundary conditions. The results indicate that the wave power reflection of thick rectangular plates is quite complicated and an incident wave of a specific type gives rise to other types of waves except for simply supported boundary conditions where the reflected wave power does not depend on the system parameters.

Novel Weak Form Quadrature Element Method for Free Vibration Analysis of Hybrid Nonlocal Euler-Bernoulli Beams with General Boundary Conditions

A novel weak form quadrature element method (QEM) is presented for free vibration analysis of hybrid nonlocal Euler-Bernoulli beams with general boundary conditions. For demonstrations, the stiffness and mass matrices of a beam element with Gauss-Lobatto-Legendre (GLL) nodes are explicitly given by using the nodal quadrature method together with the differential quadrature (DQ) law. Convergence studies are performed and comparisons are made with exact solutions to show the excellent behavior of the proposed beam element. C ase studies on hybrid nonlocal Euler-Bernoulli beams with different length scale parameters have been conducted. Accurate frequencies of the beams with different combinations of boundary conditions are obtained and presented .

A novel weak form three-dimensional quadrature element solution for vibrations of elastic solids with different boundary conditions

Three-dimensional (3D) vibration behavior of elastic parallelepipeds, including beams, plates, and solids, is critical for a wide range of engineering applications. However, obtaining accurate 3D solutions of parallelepipeds is a relatively challenging task. In this paper, a novel and general 3D weak form quadrature element method (QEM) is presented for solutions of vibrations of parallelepipeds with different combinations of boundary conditions. The element stiffness and mass matrices are explicitly derived via the numerical integration together with the differential quadrature (DQ) law. A number of case studies on beams, thin and thick plates, and 3D solids with different combinations of boundary conditions have been conducted. The natural frequencies and mode shapes were in excellent agreement with existing results and data obtained by the finite element method with a very fine mesh. It is seen that the proposed 3D quadrature element is simple in formulations, computationally efficient and capable of capturing the 3D vibration behavior of parallelepipeds with high precision. In addition, some new frequencies and mode shapes are provided to augment the archived reference frequencies and mode shapes.

Dispersion and damping analysis of orthotropic composite plate in mid frequency based on symplectic method

A semi-analytical method is used to investigate the dispersion and damping characteristics for rectangular thin orthotropic composite plates in mid frequency analysis. A symplectic eigenproblem is formed whose solution gives analytically the dispersion equation and the wave shape for any combination of simple boundary conditions along the plate edges. The damping capacity of each wave component is obtained using the strain energy theory. In the numerical examples, the dispersion and damping characteristics are calculated and analyzed for orthotropic composite laminated plates with three different combinations of boundary conditions. Comparison of the present results with the Rayleigh-Ritz results and the finite element results validates the effectiveness of the present method. Based on the damping analysis in the wave domain, the thresholds of mid frequency and high frequency vibration characteristics of the composite plate are illustrated. The relationship between the wave components in the present method and the modal modes in the modal method is also explained.

Large deformation analysis of single-crystalline nanoplates with cubic anisotropy

In this study, considering surface and bulk residual stresses, size- and orientation-dependent nonlinear bending, large amplitude vibration and postbuckling of single-crystalline nanoplates with cubic anisotropy are studied. Based on the first-order shear deformation plate theory, von-Karman geometrical nonlinearity and Gurtin-Murdoch theory, the model is formulated. In order to discretize the governing equations along with different combinations of boundary conditions, the generalized differential quadrature method is used and then a direct iterative method (Picard) is employed to solve the nonlinear system of equations. The presented results reveal that when thickness of the nanoplate increases, especially at high orientation and anisotropicity, ignoring shear deformation could cause some noticeable errors in the response of nonlinear bending, large amplitude vibration and postbuckling of nanoplates. Moreover, although orientation has a very direct effect on large amplitude vibration of nanoplates, it is observed that at a specific amplitude, the natural frequencies are independent from the orientation. The results of this study show that geometrical nonlinearity could be ignored in analyzing large amplitude vibration and postbuckling of highly anisotropic nanoplates in the [110] direction.

An analytical solution for axisymmetric buckling of annular plates based on perturbation technique

This article investigates the axisymmetric buckling of annular plates under in-plane loading applied at the inner and outer boundaries. The first order shear deformation theory is used to represent the displacement field and the governing equations are derived from the principle of virtual work. Unlike the circular plate in which the prebuckling stress components are constant, for an annulus, these stress components are functions of the radial distance from the plate center. As a consequence, the resulting eigenproblem, which is in the form of coupled differential equations with variable coefficients, does not have a closed form solution. In this article, an asymptotic approximation based on the perturbation technique is introduced to solve this problem. A parametric analysis is performed and the effects of different combinations of boundary conditions at the inner and outer edges, load ratios, inner to outer radius ratios and thicknesses are investigated. The obtained results are in a good agreement with the numerical data and also with the other references.

Computational homogenization of mesoscale gradient viscoplasticity

Variationally consistent selective homogenization is applied to a class of gradient-enhanced dissipative materials which is adopted for the mesoscale modeling. The adopted standard first order homogenization assumption results in the classical equilibrium equation for a local continuum on the macroscale, while the internal variables “live” on the mesoscale only. Among the issues considered in the paper, we note (i) the variationally consistent setting of homogenization of gradient theory on the mesoscale, (ii) the variational basis of the SVE-problem (SVE = Statistical Volume Element) in the time-incremental setting. The SVE-problem is formulated for the classical boundary conditions (Dirichlet and Neumann) pertinent to the standard momentum balance as well as the “micro-momentum” balance. The weak format of the SVE-problem constitutes the stationarity condition of an incremental SVE-potential, which represents an extension of the situation for a single-phase continuum model. The macroscale stress in a given time-increment is derivable from an incremental “macroscale pseudo-elastic strain energy”. Moreover, the saddle-point properties of the SVE-potential are shown to form the basis for establishing upper and lower bounds on the pseudo-elastic strain energy. Bingham viscoplasticity with gradient-enhanced hardening is chosen as the prototype model problem for the numerical evaluation. The computed stress–strain response relations confirm the theoretical predictions of the influence of different combinations of boundary conditions on the SVE.