Uncoupled Equations Research Articles

A solution procedure based on the Ateb function for a two-degree-of-freedom oscillator

In this paper vibration of a two mass system with two degrees of freedom is considered. Two equal harmonic oscillators are coupled with a strong nonlinear viscoelastic connection. Mathematical model of the system is two coupled second-order strong nonlinear differential equations. Introducing new variables the system transforms into two uncoupled equations: one of them is linear and the other with a strong nonlinearity. In the paper a method for solving the strong nonlinear equation is developed. Based on the exact solution of a pure nonlinear differential equation, we assumed a perturbed version of the solution with time variable parameters. Due to the fact that the solution is periodical, the averaging procedure is introduced. As a special case vibrations of harmonic oscillators with fraction order nonlinear connection are considered. Depending on the order and coefficient of nonlinearities bounded and unbounded motion of masses is determined. Besides, the conditions for steady-state periodical solution are discussed. The procedure given in the paper is applied for investigation of the vibration of a vocal cord, which is modeled with two harmonic oscillators with strong nonlinear fraction order viscoelastic connection. Using the experimental data for the vocal cord the parameters for the steady-state solution which describes the flexural vibration of the vocal cord is analyzed. The influence of the order of nonlinearity on the amplitude and frequency of vibration of the vocal cord is obtained. The analytical results are close to those obtained experimentally.

Creep damage and crack initiation in P92–BNi2 brazed joint

Nickel-based brazed joints are widely used at high temperatures, and it is very important to ensure the creep strength. Using three-dimensional finite element method, this paper presents a study on the creep damage and creep crack initiation (CCI) in P92–BNi2 brazed compact tension (CT) joint specimen, and the effect of brazed residual stress on CCI has been investigated. Firstly, the as-brazed residual stress is simulated. Then the effect of residual stress on creep and damage is calculated by an uncoupled Norton equation and the ductility exhaustion approach. The results show that the CCI is a function of residual stress, load level, joint thickness and notch radius. The brazed residual stress has a great effect on creep damage, and it can lead to the initiation of creep crack lonely without any mechanical load. The combination of applied load not only accelerates the creep damage and decreases the CCI time, but also determines the creep crack propagation rate. The CCI time decreases with the increase of applied load and joint thickness. As the notch radius increases, the CCI time increases.

Simultaneous controllability of some uncoupled semilinear wave equations

We consider the exact controllability problem for some uncoupled semilinearwave equations with proportional, but different principal operators in a boundeddomain. The control is locally distributed,and its support satisfies the geometric control condition ofBardos-Lebeau-Rauch. First, we examine the case of a nonlinearity that is asymptotically linear;using a combination of the Bardos-Lebeau-Rauch observabilityresult for a single wave equation and a new unique continuation result for uncoupled wave equations, we solve the underlying linear control problem.The linear controllability result thus established,generalizes to higher space dimensions an earlier result of Harauxestablished in the one-dimensional setting. Then, applying a fixed point argument, we derive the controllability of the nonlinear problem. Afterwards, we use an iterative approach to prove a local controllability result when the nonlinearity is super-linear. Finally, we discuss some extensions of our results and some open problems.

An extended Galerkin representation for a transversely isotropic linearly elastic medium

An algorithm for splitting an equilibrium displacement equation system with bulk forces for a transversely isotropic linearly-elastic medium is described that leads to three uncoupled equations with certain canonical fourth-order differential operators in the three components of the displacement vector. It is shown that, in the special case of isotropy, the proposed algorithm is mathematically equivalent to the Galerkin representation, well-known in the theory of elasticity.

Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion

This paper investigates the stochastic averaging of slow-fast dynamical systems driven by fractional Brownian motion with the Hurst parameter $H$ in the interval $(\frac{1}{2},1)$. We establish an averaging principle by which the obtained simplified systems (the so-called averaged systems) will be applied to replace the original systems approximately through their solutions. Here, the solutions to averaged equations of slow variables which are unrelated to fast variables can converge to the solutions of slow variables to the original slow-fast dynamical systems in the sense of mean square. Therefore, the dimension reduction is realized since the solutions of uncoupled averaged equations can substitute that of coupled equations of the original slow-fast dynamical systems, namely, the asymptotic solutions dynamics will be obtained by the proposed stochastic averaging approach.

Analytical solution for the three-dimensional stress fields in anisotropic composite bimaterial corners

The paper presents a theoretical treatise for the three-dimensional elastic stress distributions close to anisotropic bimaterial corners according to which the 3D governing equations are successfully reduced to two uncoupled equations in the two-dimensional space. The former provides the solution of the corresponding plane notch problem, the latter provides the solution of the corresponding out-of-plane shear notch problem. With the new theory an analytical solution for the stress fields for bimaterial corners in thick anisotropic composite plates is presented and its degree of accuracy is discussed comparing theoretical results and numerical data from 3D FE analyses carried out on bimaterial composite plates and joints.

Analytical spectrum for a Hamiltonian of quantum dots with Rashba spin–orbit coupling

We determine the analytical solution for a Hamiltonian describing a confined charged particle in a quantum dot, including Rashba spin-orbit coupling and Zeeman splitting terms. The approach followed in this paper is straightforward and uses the symmetrization of the wave functionʼs components. The eigenvalue problem for the Hamiltonian in Bargmannʼs Hilbert space reduces to a system of coupled first-order differential equations. Then we exploit the symmetry in the system to obtain uncoupled second-order differential equations, which are found to be the Whittaker–Ince limit of the confluent Heun equations. Analytical expressions as well as numerical results are obtained for the spectrum. One of the main features of such models, namely, the level splitting, is present through the spectrum obtained in this paper.

Oscilaciones de la red de nitruros cúbicos

En el trabajo se estudia la dinámica de los nitruros del grupo III en la estructura blenda de zinc utilizando el modelo de cadenas lineales. Utilizando el hecho de que en direcciones de alta simetría las ramas longitudinales de desacoplan de las transversales se reportan las relaciones entre las constantes de fuerza del problema tridimensional (3D) y de la cadena lineal equivalente en estas direcciones. También mostramos cómo calcular las constantes de fuerza 3D del masivo (y consecuentemente la matriz dinámica en su totalidad) a partir de unos pocos puntos experimentales o teóricos. En particular se determinan las relaciones de dispersión y se calculan las constantes de fuerza de nitruros de Aluminio (AlN) y Galio (GaN).

Weak Solutions to Fokker–Planck Equations and Mean Field Games

We deal with systems of PDEs, arising in mean field games theory, where viscous Hamilton–Jacobi and Fokker–Planck equations are coupled in a forward-backward structure. We consider the case of local coupling, when the running cost depends on the pointwise value of the distribution density of the agents, in which case the smoothness of solutions is mostly unknown. We develop a complete weak theory, proving that those systems are well-posed in the class of weak solutions for monotone couplings under general growth conditions, and for superlinear convex Hamiltonians. As a key tool, we prove new results for Fokker–Planck equations under minimal assumptions on the drift, through a characterization of weak and renormalized solutions. The results obtained give new perspectives even for the case of uncoupled equations as far as the uniqueness of weak solutions is concerned.

Asymptotic solution for the problem of sound propagation in a sea with an underwater canyon.

The problem of the sound propagation in shallow-water waveguide with a seabottom featuring canyon-type inhomogeneity of a specific form is considered. The sound pressure in such waveguide is represented in a form of modal expansion and the equations for modal coefficients are derived. In case of a single-mode adiabatic propagation, it is possible to neglect the mode interaction and omit the coupling terms in these equations. The uncoupled equation for the mode amplitude admits an explicit analytical solution via the separation of variables.

SU(1,1)coherent states for Dirac–Kepler–Coulomb problem inD+1dimensions with scalar and vector potentials

We decouple the Dirac's radial equations in $D+1$ dimensions with Coulomb-type scalar and vector potentials through appropriate transformations. We study each of these uncoupled second-order equations in an algebraic way by using an $su(1,1)$ algebra realization. Based on the theory of irreducible representations, we find the energy spectrum and the radial eigenfunctions. We construct the Perelomov coherent states for the Sturmian basis, which is the basis for the unitary irreducible representation of the $su(1,1)$ Lie algebra. The physical radial coherent states for our problem are obtained by applying the inverse original transformations to the Sturmian coherent states.

Stress fields at sharp angular corners in thick anisotropic composite plates

A new three-dimensional theory to be applied to thick anisotropic plates is developed, according to which the 3D governing equations of elasticity are successfully reduced to the simultaneous solution of two uncoupled equations in the two-dimensional space. With the new theory an analytical solution for the three-dimensional stress fields in anisotropic composite plates with V-notches is presented and its degree of accuracy is discussed comparing theoretical results and numerical data from 3D FE analyses.

Validity of the Weakly Nonlinear Solution of the Cauchy Problem for the Boussinesq‐Type Equation

We consider the initial‐value problem for the regularized Boussinesq‐type equation in the class of periodic functions. Validity of the weakly nonlinear solution, given in terms of two counterpropagating waves satisfying the uncoupled Ostrovsky equations, is examined. We prove analytically and illustrate numerically that the improved accuracy of the solution can be achieved at the timescales of the Ostrovsky equation if solutions of the linearized Ostrovsky equations are incorporated into the asymptotic solution. Compared to the previous literature, we show that the approximation error can be controlled in the energy space of periodic functions and the nonzero mean values of the periodic functions can be naturally incorporated in the justification analysis.

A layer-wise theory for laminated glass and photovoltaic panels

Laminated plates with glass skin layers and a core layer from soft polymers are widely used in the civil engineering. Photovoltaic panels currently available on the market are composed from stiff front and back layers and a solar cell layer embedded in a soft polymeric encapsulant. In this paper a layer-wise theory for the structural analysis of glass and photovoltaic laminates is developed. Starting from governing equations for individual layers, kinematical constraints and appropriate interaction forces, a twelfth order system of partial differential equations is derived. The primary variables in the theory include the Airy stress function, the deflection function and the vector of relative in-plane displacements of skin layers. For symmetric laminates a system of uncoupled differential equations with respect to scalar potentials is presented. Three of them correspond to the first order shear deformation plate. The new additional second order differential equation provides a correction function according to the layer-wise theory. Closed form analytical solutions for a plate strip are derived to illustrate the essential influence of this correction for laminates with soft core layer. The importance of additional boundary conditions is shown for examples of free and framed plate edges.

Minimal partitions and image classification using a gradient-free perimeter approximation

In this paper a new mathematically-founded method for the optimal partitioning of domains, with applications to the classification of greyscale and color images, is proposed.Since optimal partition problems are in general ill-posed, some regularization strategy is required.Here we regularize by a non-standard approximation of the total interface length, which does not involve the gradient of approximate characteristic functions, in contrast to the classical Modica-Mortola approximation. Instead, it involves a system of uncoupled linear partial differential equations and nevertheless shows $\Gamma$-convergence properties in appropriate function spaces.This approach leads to an alternating algorithm that ensures a decrease of the objective function at each iteration, and which always provides a partition, even during the iterations.The efficiency of this algorithm is illustrated by various numerical examples. Among them we consider binary and multilabel minimal partition problems including supervised or automatic image classification, inpainting, texture pattern identification and deblurring.

Dynamics of a Slender Spinning Membrane

A novel approach is introduced to conduct dynamic analysis of a spinning, high aspect ratio membrane. In this formulation, an inextensible, long, slender membrane is modeled using a discrete set of lumped masses. Lagranges equations are used to derive the highly coupled ordinary differential equations for in-plane, out-of-plane, and twisting motions for the spinning membrane. The generalized and uncoupled linear equations for small motion are used to compute the vibration mode frequencies which are compared to results from an uncoupled analysis of blade motion using rotor dynamics. Linearized behavior is shown to reduce to the linearized solutions for the spinning membrane blade developed by MacNeal. Numerical simulations along with 3-D animations are used to study the linear and nonlinear uncoupled dynamics of the spinning membrane.

Free vibration analysis of a rigidly fixed viscothermoelastic hollow sphere

In this paper an exact analysis of homogenous rigidly fixed vibrations of viscothermoelastic hollow sphere is presented. The basic governing partial differential equations have been reduced to ordinary differential equations by using Helmholtz decomposition equations. The uncoupled equation is taken for first class vibrations and remains independent of temperature variations, while coupled system of equations are taken for second class vibrations. Matrix Frobenious method of extended power series has been applied in the coupled system of differential equations to get displacements and temperature. Numerical results have been presented, giving lowest frequency, dissipation factor, displacements and temperature change.

Exact Solution for Thermal Buckling of Functionally Graded Plates Resting on Elastic Foundations with Various Boundary Conditions

In this article, we investigate the buckling analysis of plates that are made of functionally graded materials (FGMs) resting on two-parameter Pasternak's foundations under thermal loads. Three different thermal loads were considered, i.e., uniform temperature rise (UTR), linear and non-linear temperature distributions (LTD and NTD) through the thickness. The mechanical and thermal properties of functionally graded material (FGM) vary continuously along the plate thickness according to a simple power law distribution. Employing an analytical approach, the five coupled governing stability equations, which are derived based on first-order shear deformation plate theory, are converted into two uncoupled partial differential equations (PDEs). Considering the Levy-type solution, these two PDEs are reduced to two ordinary differential equations (ODEs) with variable coefficients. Then, the ODEs are solved using an exact analytical solution, which is called the power series Frobenius method. The appropriate convergence study and comparison with previously published related articles was employed to verify the accuracy of the proposed method. After such verifications, the effects of parameters such as the plate aspect ratio, side-to-thickness ratio, gradient index, and elastic foundation stiffnesses on the critical buckling temperature difference are illustrated and explained. The critical buckling temperatures of functionally graded rectangular plates with six various boundary conditions are reported for the first time and can serve as benchmark results for researchers to validate their numerical and analytical methods in the future.

The Role of Neck Evolution in the Synthesis of Superheavy Element 112

The coupled and uncoupled Langevin equations in two-dimensional collective space are used to study the neck dynamics of the mass asymmetric system 48Ca+238U at the above barrier energy. The results show that the coupling between the neck and radial degrees of freedom delays the transition from dinucleus to mononucleus, and correspondingly increases the lifetime of the dinucleus system. The lifetime of the dinucleus for the asymmetric system48Ca+238U is about 11.6 and 13.0 × 10−22 s, obtained with uncoupled and coupled Langevin equations. We calculate the evaporation residue (ER) cross sections for the 3n and 4n evaporation channels in the 48Ca+238U reaction leading to the formation of 283112 and 282112 isotopes in three different approaches, i.e., coupled, uncoupled and frozen approximation, and compare them with the experimental data. It is found that the results of the uncoupled and frozen approximation are in close similarity, while the coupling between the radial and neck degrees of freedom reduces the ER cross section by about 30%, compared with the case of frozen approximation.

A numerical framework for diffusion-controlled bimolecular-reactive systems to enforce maximum principles and the non-negative constraint

We present a novel computational framework for diffusive–reactive systems that satisfies the non-negative constraint and maximum principles on general computational grids. The governing equations for the concentration of reactants and product are written in terms of tensorial diffusion–reaction equations. We restrict our studies to fast irreversible bimolecular reactions. If one assumes that the reaction is diffusion-limited and all chemical species have the same diffusion coefficient, one can employ a linear transformation to rewrite the governing equations in terms of invariants, which are unaffected by the reaction. This results in two uncoupled tensorial diffusion equations in terms of these invariants, which are solved using a novel non-negative solver for tensorial diffusion-type equations. The concentrations of the reactants and the product are then calculated from invariants using algebraic manipulations. The novel aspect of the proposed computational framework is that it will always produce physically meaningful non-negative values for the concentrations of all chemical species. Several representative numerical examples are presented to illustrate the robustness, convergence, and the numerical performance of the proposed computational framework. We will also compare the proposed framework with other popular formulations. In particular, we will show that the Galerkin formulation (which is the standard single-field formulation) does not produce reliable solutions, and the reason can be attributed to the fact that the single-field formulation does not guarantee non-negative solutions. We will also show that the clipping procedure (which produces non-negative solutions but is considered as a variational crime) does not give accurate results when compared with the proposed computational framework.