Expression For Potential Energy Research Articles

Toward Hamiltonian Adaptive QM/MM: Accurate Solvent Structures Using Many-Body Potentials.

Adaptive quantum mechanical (QM)/molecular mechanical (MM) methods enable efficient molecular simulations of chemistry in solution. Reactive subregions are modeled with an accurate QM potential energy expression while the rest of the system is described in a more approximate manner (MM). As solvent molecules diffuse in and out of the reactive region, they are gradually included into (and excluded from) the QM expression. It would be desirable to model such a system with a single adaptive Hamiltonian, but thus far this has resulted in distorted structures at the boundary between the two regions. Solving this long outstanding problem will allow microcanonical adaptive QM/MM simulations that can be used to obtain vibrational spectra and dynamical properties. The difficulty lies in the complex QM potential energy expression, with a many-body expansion that contains higher order terms. Here, we outline a Hamiltonian adaptive multiscale scheme within the framework of many-body potentials. The adaptive expressions are entirely general, and complementary to all standard (nonadaptive) QM/MM embedding schemes available. We demonstrate the merit of our approach on a molecular system defined by two different MM potentials (MM/MM'). For the long-range interactions a numerical scheme is used (particle mesh Ewald), which yields energy expressions that are many-body in nature. Our Hamiltonian approach is the first to provide both energy conservation and the correct solvent structure everywhere in this system.

Finite element analysis of machining processes of turbine disk of Inconel 718 high-temperature wrought alloy based on the theorem of minimum potential energy

Machining processes of Inconel 718 turbine disk including turning and broaching processes are, respectively, analyzed using the theorem of minimum potential energy. First, in order to provide the initial condition in terms of elastic strain energy for the machining processes, two-dimensional, axial symmetry assumptions are adopted and the simulation of multi-stage forging of disk is conducted and verified by the residual stresses test. Secondly, potential energy (PE) expression for the machining processes is developed. The predicted results of turning and broaching indicate that there exists an optimal value of removed material, and high level of strain energy signifies an unstable state of the structure, which could make the part deform to keep self-equilibrium.

The Boguslawski Melting Model

The anharmonic vibrator, whose expression of potential energy contains second and third powers of coordinates, is treated on the basis of dynamical procedure, which presents the state of motion by means of mean position and mean amplitude of vibration. The divergent statistical integral comes here not into consideration. The free energy is represented through mean atomic displacement and developed in power series, retaining fourth degree. The graphs show that at certain temperature, the minimum in free energy disappears, and the atom escapes from the potential pit. A simple atomic model that represents this phenomenon is proposed and the influence of model dimension and pressure on melting temperature will be presented.

A simple analytical model of the Coulomb cluster in a cylindrically symmetric parabolic trap

A cluster consisting of classical charged particles of the same sign in a cylindrically symmetric parabolic trap is considered in the approximation of a homogeneous distribution of particles over the cluster volume and with neglect of its discrete structure. Simple analytical expressions for the cluster size and potential energy in a trap of arbitrary anisotropy (the ratio of confining forces acting in the radial and axial directions) are obtained. The influence of possible inhomogeneity in the distribution of particles is estimated, and it is established that this factor weakly influences the potential energy of a cluster. The limiting case of a twodimensional cluster is considered. The adopted approximation is valid for clusters with a sufficiently large numbers of particles. Application of the model to relatively small clusters is possible by introducing correction factors into the analytical expressions obtained. These factors are determined based on approximation of the results of numerical simulations.

On the Local Available Energetics in a Moist Compressible Atmosphere

Abstract A new derivation of local available energetics for a fully compressible, nonhydrostatic, moist atmosphere is presented. The available energetics is defined relative to an arbitrary dry reference state in hydrostatic balance with stable stratification. By introducing the modified potential temperature, a positive-definite expression of the moist available potential energy (APE) is derived. The change of the moist APE must include the role of convection to function both as a source of latent heat and as an atmosphere dehumidifier. The sum of this moist APE and the available elastic energy (AEE) is the moist available energy. In the local energy cycle, the moist available energy is partly used to generate kinetic energy (KE) and partly used to lift the water vapor to the higher level where it precipitates, resulting in the increase of gravitational energy of moist species. The moist APE is converted into vertical KE through the buoyancy term; the vertical KE is converted into the AEE through the vertical perturbation pressure gradient term; and the AEE is converted into horizontal KE through the horizontal divergence/convergence term. In addition, there exist two adiabatic nonconservative processes, which act on the AEE and APE, respectively. A suitable choice of the reference state should make these two processes much less significant than the conversions between the available energy and KE. An alternative method is presented to construct such a reference state. Application to the idealized baroclinic atmosphere shows that this reference state is much more relevant to the local available energy analysis than the isothermal one.

Energy Considerations for Mechanical Fractional-Order Elements

This paper considers the energy aspects of fractional-order elements defined by the equation: force is proportional to the fractional-order derivative of displacement, with order varying from zero to two. In contrast to the typically conservative assumption of classical physics that leads to the potential and kinetic energy expressions, a number of important nonconservative differences are exposed. Firstly, the considerations must be time-based rather than displacement or momentum based variables. Time based equations for energy behavior of fractional elements are presented and example applications are considered. The effect of fractional order on the energy input and energy return of these systems is shown. Importantly, it is shown that the history, or initialization, has a significant effect on energy response. Finally, compact expressions for the work or energy, are developed.

A fifth-order model for shells which combines bending, stretching and transverse shearing deduced from three-dimensional elasticity

We approximate the displacement field in a shell by a fifth-order Taylor–Young expansion in thickness. The model is derived from the truncation of the potential energy at fifth order. The equilibrium equations imply local constraints on the through-thickness derivatives of the zero-order displacement field. This leads to an analytical expression for the two-dimensional potential energy of a shell in terms of the zero-order displacement field and its derivatives that include non-standard transverse shearing and normal stress energy. Then we derive the equation of equilibrium and the boundary conditions.

Entropy and Its Correlations with Other Related Quantities

In order to find more correlations between entropy and other related quantities, an analogical analysis is conducted between thermal science and other branches of physics. Potential energy in various forms is the product of a conserved extensive quantity (for example, mass or electric charge) and an intensive quantity which is its potential (for example, gravitational potential or electrical voltage), while energy in specific form is a dissipative quantity during irreversible transfer process (for example mechanical or electrical energy will be dissipated as thermal energy). However, it has been shown that heat or thermal energy, like mass or electric charge, is conserved during heat transfer processes. When a heat transfer process is for object heating or cooling, the potential of internal energy U is the temperature T and its potential “energy” is UT/2 (called entransy and it is the simplified expression of thermomass potential energy); when a heat transfer process is for heat-work conversion, the potential of internal energy U is (1 − T0/T), and the available potential energy of a system in reversible heat interaction with the environment is U − U0 − T0(S − S0), then T0/T and T0(S − S0) are the unavailable potential and the unavailable potential energy of a system respectively. Hence, entropy is related to the unavailable potential energy per unit environmental temperature for heat-work conversion during reversible heat interaction between the system and its environment. Entropy transfer, like other forms of potential energy transfer, is the product of the heat and its potential, the reciprocal of temperature, although it is in form of the quotient of the heat and the temperature. Thus, the physical essence of entropy transfer is the unavailable potential energy transfer per unit environmental temperature. Entropy is a non-conserved, extensive, state quantity of a system, and entropy generation in an irreversible heat transfer process is proportional to the destruction of available potential energy.

110 建物・エレベーター・ロープの連成振動解析のための動的モデルに関する研究

This study is concerned with the dynamic modeling and simulation for the elevator rope vibration coupled with building vibrations excited by external forces such as earthquake and gusty wind. To this end, the extended Hamilton's principle was employed to cope with time-varying dynamic problem. The kinetic and potential energy expressions for the main and compensating ropes and building were derived and inserted into the Hamilton's principle. As a result, the dynamic model suitable for numerical simulation was derived and implemented by MATLAB code. Numerical simulations were carried out under various scenarios of elevator operation. Simulation results show that the vibration of the compensating rope becomes concern when the cage is moving up. To verify the theoretical prediction, an elevator simulator consisting of building and elevator rope is being developed.

On the equivalence of the Badawi–Bessis–Bessis, Wei and Tietz potential energy functions for diatomic molecules

It is found that the general potential energy expression introduced by Badawi et al for several empirical potential energy functions that is conventionally defined in terms of five parameters, actually only has four independent parameters. It is demonstrated exactly that the Badawi–Bessis–Bessis potential is equivalent to the well-known Tietz potential model for diatomic molecules. When parameter a in the Badawi–Bessis–Bessis potential function has the values 0, +1 and −1, the Badawi–Bessis–Bessis potential becomes the standard Morse, Rosen–Morse and Manning–Rosen potentials, respectively.

A nonlinear two-node superelement for use in flexible multibody systems

A nonlinear two-node superelement is proposed for the modeling of flexible complex-shaped links for use in multibody simulations. Assuming that the elastic deformations with respect to a corotational reference frame remain small, substructuring methods may be used to obtain reduced mass and stiffness matrices from a linear finite element model. These matrices are used in the derivation of potential and kinetic energy expressions of the nonlinear two-node superelement. By evaluating Lagrange’s equations, expressions for the internal and external forces acting on the superelement can be obtained. The inertia forces of the superelement are derived in terms of absolute nodal velocities and accelerations, which greatly simplifies the dynamic formulation. Three examples are included. The first two examples are used to validate the method by comparing the results with those obtained from nonlinear beam element solutions. We consider a benchmark simulation of the spin-up motion of a flexible beam with uniform cross-section and a similar simulation in which the beam is simultaneously excited in the out-of-plane direction. Results from both examples show good agreement with simulation results obtained using nonlinear finite beam elements. In a third example, the method is applied to an unbalanced rotating shaft, illustrating the potential of the proposed methodology for a more complex geometry.

Modeling intermolecular potential of He–F2 dimer from symmetry-adapted perturbation theory using multi-gene genetic programming

Any molecular dynamical calculation requires a precise knowledge of interaction potential as an input. In an appropriate form, such that the potential, with respect to the coordinates, can be evaluated easily and accurately at arbitrary geometries (in our study parameters for geometry are R and θ), a good potential energy expression can offer the exact intermolecular behavior of systems. There are many methods to create mathematical expressions for the potential energy. In this study for the first time, we utilized the Multi-gene Genetic Programming (MGGP) method to generate a potential energy model for the He–F2 system. The MGGP method is one of the most powerful methods used for non-linear regression problems. A dataset of size 714 created by the SAPT 2008 program is used to generate models of MGGP. The results obtained show the power of MGGP for producing an efficient nonlinear regression model, in terms of accuracy and complexity.

Cyclic J-integral using the Linear Matching Method

The extended version of the latest Linear Matching Method (LMM) has the capability to evaluate the stable cyclic response, which produces cyclic stresses, residual stresses and plastic strain ranges for the low cycle fatigue assessment with cyclic load history. The objective of this study is to calculate ΔJ through the LMM and suggest future development directions. The derivation of the ΔJ based on the potential energy expression for a single edge cracked plate subjected to cyclic uniaxial loading condition using LMM is presented. To extend the analysis so that it can be incorporated to other plasticity models, material Ramberg–Osgood hardening constants are also adopted. The results of the proposed model have been compared to the ones obtained from Reference Stress Method (RSM) for a single edge cracked plate and they indicate that the estimates provide a relatively easy method for estimating ΔJ for describing the crack growth rate behaviour by considering the complete accumulated cycle effects.

ELASTIC RESTRAINED DISTORTIONAL BUCKLING OF STEEL-CONCRETE COMPOSITE BEAMS BASED ON ELASTICALLY SUPPORTED COLUMN METHOD

Restrained distortional buckling (RDB) is different from overall buckling and distortional buckling, and it usually occurs in the hogging moment region of composite beams. It may govern the member design for some cases, and should be taken into consideration. The present paper investigates the elastic RDB of composite beams based on elastically supported column method. First, Svensson's elastically supported column model is improved by including the participation of the web as part of the buckling column. The corresponding theoretical expressions for the improved elastically support column subjected to a varying axial force are derived by the potential energy expression and equilibrium differential equation, respectively. Then, the precision of several elastically supported column methods for the RDB problem is discussed, based on the finite element method (FEM). The results show that elastically supported columns subjected to a negative uniform moment or a triangular moment have a good agreement with FEM results, but they are not valid for the cases of nonlinear moment distributions. In order to consider the complicated load conditions of composite beams, the equivalent moment assumption is introduced and proved to be suitable for the RDB problem. However, there is an applicable range of the length of the hogging moment region. Thus, a critical length formula of equivalent moment assumption of RDB is put forward and verified by FEM. Furthermore, a three-step simplified method is presented which can solve the elastic RDB problem of continuous composite beams. The proposed method, in which values of elastically supported column methods can be tabulated previously, is simple and suitable for design purposes.

Two-dimensional model for the combined bending, stretching and shearing of shells: A general approach and application to laminated cylindrical shells derived from three-dimensional elasticity

For laminated shells, the displacement field can be approximated in each layer by a third-order Taylor–Young expansion in thickness. Then we are motivated to consider a simplified theory based on the thickness-wise expansion of the potential energy truncated at third-order in thickness. The equilibrium equations imply local constraints on the through-thickness derivatives of the zero-order displacement field in each layer. These lead to an analytical expression for the two-dimensional potential energy of cylindrical shells in terms of the zero-order displacement field, and its derivatives, that includes non-standard transverse shearing and normal stress energy. As a consequence this potential energy satisfies the stability condition of Legendre–Hadamard which is necessary for the existence of a minimizer.

Two-dimensional model of order h5 for the combined bending, stretching, transverse shearing and transverse normal stress effect of homogeneous plates derived from three-dimensional elasticity

For homogeneous plates, the highest order term of transverse shear and normal stresses is of second order in thickness. To take this effect into account, we show that the thickness-wise expansion of the potential energy must be truncated at least from fifth order in thickness. The equilibrium equations imply local constraints on the through-thickness derivatives of the zeroth-order displacement field. These lead to an analytical expression for two-dimensional potential energy in terms of the zeroth-order displacement field and its derivatives, which include non-standard shearing and transverse normal energies and coupled stretching–shearing, bending–shearing and stretching–transverse normal energies. As a consequence, this potential energy satisfies the stability condition of Legendre–Hadamard, which is necessary for the existence of a minimizer.

Vector-based model of elastic bonds for simulation of granular solids

A model (further referred to as the V model) for the simulation of granular solids, such as rocks, ceramics, concrete, nanocomposites, and agglomerates, composed of bonded particles (rigid bodies), is proposed. It is assumed that the bonds, usually representing some additional gluelike material connecting particles, cause both forces and torques acting on the particles. Vectors rigidly connected with the particles are used to describe the deformation of a single bond. The expression for potential energy of the bond and corresponding expressions for forces and torques are derived. Formulas connecting parameters of the model with longitudinal, shear, bending, and torsional stiffnesses of the bond are obtained. It is shown that the model makes it possible to describe any values of the bond stiffnesses exactly; that is, the model is applicable for the bonds with arbitrary length/thickness ratio. Two different calibration procedures depending on bond length/thickness ratio are proposed. It is shown that parameters of the model can be chosen so that under small deformations the bond is equivalent to either a Bernoulli-Euler beam or a Timoshenko beam or short cylinder connecting particles. Simple analytical expressions, relating parameters of the V model with geometrical and mechanical characteristics of the bond, are derived. Two simple examples of computer simulation of thin granular structures using the V model are given.

Some thoughts on a surprising result concerning the lateral-torsional buckling of monosymmetric I-section beams

This technical note addresses a surprising result concerning the lateral-torsional buckling of monosymmetric I-section beams: amongst all bending moment diagrams caused by any combination of applied end moments and transverse loads acting at the shear centre, the lowest critical moment does not necessarily correspond to uniform bending, which is at odds with the “intuitive expectation” of most engineers that such bending moment distribution is the most unfavourable with respect to lateral-torsional buckling. It is shown, by means of several illustrative examples, that this may not be the case for I-section beams with unequal flanges (monosymmeric cross-section – symmetry with respect to the minor axis). Moreover, the critical inspection of the terms (i) associated with the cross-section monosymmetry and (ii) appearing in the beam potential energy expression and differential equilibrium equation provides the mechanical/mathematical explanation for this surprising behavioural feature.

Energy expressions and free vibration analysis of a rotating Timoshenko beam featuring bending–bending-torsion coupling

In this study, free vibration analysis of a uniform, rotating, cantilever Timoshenko beam featuring bending–bending-torsion coupling is performed. To the best of the authors’ knowledge, there is no explicit formulation in open literature for rotating Timoshenko beams featuring bending–bending-torsion coupling. Therefore, in this study, derivation of the kinetic and the potential energy expressions for the mentioned beam model is carried out in a detailed way by using several explanatory tables and figures. The parameters for the hub radius, rotational speed, rotary inertia, shear deformation and bending–bending-torsion coupling are incorporated into the energy expressions. The governing differential equations of motion are obtained by applying the Hamilton’s principle to the derived energy expressions and solved using an efficient mathematical technique, called the differential transform method. The natural frequencies are calculated, and comparisons are made with the results in open literature. Consequently, it is observed that there is a good agreement between the results, which validates the accuracy of the derived formulation and the built beam model.

Stiffness Matrix of Nonlinear FEM Equilibrium Equation

For structural stability analysis, either tangent stiffness matrix or secant stiffness matrix is used depending on different situations. In this study, the general mathematic relationship between structural secant and tangent stiffness matrices is developed in detail based on Taylor series expression of the total potential energy. The result is important to the analysis of structural nonlinear stability. Moreover, it can be used not only in finite element method but also Rayleigh-Ritz method, Galerkin method, etc.