General Solution Technique Research Articles

Analytical solution of a cantilever functionally graded beam

The problem of a cantilever functionally graded beam subjected to different loads is studied. In terms of Airy stress function a general two-dimensional solution is presented for a cantilever functionally graded beam, assuming that all the elastic moduli of the material have the same variations along the beam-thickness direction. Explicit expressions of analytical solutions to some specific examples under different boundary conditions are obtained to demonstrate the usefulness of the proposed general solution technique. This solution will be useful in analyzing functionally graded beam with arbitrary variations of material properties and it can serve as a basis for establishing simplified functionally graded beam theories.

Metastable and resonance atom densities in a positive column: II. Application to light source modelling

Methods for the detailed description of new radiation and light sources based on low-pressure plasmas are required which also consider the spatial structure and the mechanisms of resonance radiation transport without considerable increase of computational effort. A general solution technique based on the reduction of the Holstein–Biberman integral equation to a system of linear equations is introduced to treat the radiation transport. This technique is applied in the framework of a self-consistent radially resolved model of the positive column of a cylindrical dc glow discharge at low pressure. A helium–xenon discharge which can be used as an efficient VUV-radiation source is studied as an example. The theoretical results for the radial density profiles of the resonance and metastable atoms of xenon are compared with absorption measurements providing atomic number densities for the levels of the 1s manifold. In addition, the solution technique for the Holstein–Biberman equation is validated by means of a comparison with Monte-Carlo simulations.

Elastic wave propagation in anisotropic spherical curved plates

Spherical plate-like structures are used in pressure vessels, spherical domes of power plants, and in many other industrial applications. For non-destructive evaluation of such spherical structures, the mechanics of elastic wave propagation in spherical curved plates must be understood. The current literature shows some valuable studies on Rayleigh surface wave propagation in isotropic solids with spherical boundaries. However, the guided wave propagation problem in an anisotropic spherical curved plate, which has not been studied before, is solved for the first time in this paper. The wave propagation, in both isotropic and anisotropic spherical curved plates, is investigated. The differential equations of motion and the stress-free boundary conditions on the inner and outer surfaces of a hollow sphere are approximately solved by a general solution technique. This solution technique was successfully utilized by the authors for solving the wave propagation problem in cylindrical plates, in their earlier works. Dispersion curves for spherical plates made of isotropic aluminum, steel, and anisotropic composite material are presented as well.

A Method for Solving Dynamically Accelerating Crack Problems in Linear Viscoelasticity

We develop a general solution technique for a dynamically accelerating crack in a linear viscoelastic material, based on a transform method developed by Slepyan (see [Models and Phenomena in Fracture Mechanics, Springer-Verlag, New York, 2002]) for solution of dynamic elastic fracture problems. We review the elastic fracture solution method and then treat the viscoelastic mode III fracture problem for crack tip speeds less than the short-time shear wave speed. The analysis includes an exact, closed-form expression for the stress intensity factor for an arbitrary time dependent crack face traction. As examples, we apply this solution method to the Achenbach--Chao and standard linear solid viscoelastic models.

Pipe-anchor discontinuity analysis utilizing power series solutions, Bessel functions, and Fourier series

One of the paradigmatic classes of problems that frequently arise in piping stress analysis discipline is the effect of local stresses created by supports and restraints attachments. Over the past 20 years, concerns have been identified by both regulatory agencies in the nuclear power industry and others in the process and chemicals industries concerning the effect of various stiff clamping arrangements on the expected life of the pipe and its various piping components. In many of the commonly utilized geometries and arrangements of pipe clamps, the elasticity problem becomes the axisymmetric stress and deformation determination in a hollow cylinder (pipe) subjected to the appropriate boundary conditions and respective loads per se. One of the geometries that serve as a pipe anchor is comprised of two pipe clamps that are bolted tightly to the pipe and affixed to a modified shoe-type arrangement. The shoe is employed for the purpose of providing an immovable base that can be easily attached either by bolting or welding to a structural steel pipe rack. Over the past 50 years, the computational tools available to the piping analyst have changed dramatically and thereby have caused the implementation of solutions to the basic problems of elasticity to change likewise. The need to obtain closed form elasticity solutions, however, has always been a driving force in engineering. The employment of symbolic calculus that is currently available through numerous software packages makes closed form solutions very economical. This paper briefly traces the solutions over the past 50 years to a variety of axisymmetric stress problems involving hollow circular cylinders employing a Fourier series representation. In the present example, a properly chosen Fourier series represent the mathematical simulation of the imposed axial displacements on the outside diametrical surface. A general solution technique is introduced for the axisymmetric discontinuity stresses resulting from an anchor restraint on a selected of pipe geometry. These solutions can be economically implemented on today's symbolic calculus software packages with no loss in solution accuracy when compared to often more expensive techniques such as the finite element method. Verification of the axisymmetric solution technique is illustrated by the comparison of results for the closed form solutions versus those approximated by the finite element technique. Extensions of the general axisymmetric solution technique to other geometries and applied loads are also discussed while the numerical and graphical results are tendered.

A Real Options Approach to the Valuation of a Forestry Investment

The theory of real options is used to model the optimal tree harvesting decision. The value of the option to harvest is estimated using a dynamic programming approach and a general numerical solution technique suitable for any type of stochastic process for prices. The conditions under which the optimal harvest time is independent of price are examined. In addition, the impact of assuming future lumber prices are mean reverting is considered. It is found that option value and optimal cutting time are significantly different under the mean reversion assumption compared to geometric Brownian motion.

Characteristics of grating-assisted couplers

Characteristics of grating-assisted coupling between two parallel waveguides are analyzed. The influence of the grating parameters, such as groove depth, duty cycle, and refractive indices is considered. Chirped and parallel gratings as well as gratings with sinusoidal envelope periodicity are also addressed. The analysis is based on a unified coupled-mode formalism, with the transfer-matrix method as a general solution technique. It is shown how to modify the grating parameters to provide a specific spectral response (reflectivity and transmission coefficients). As an example, two parallel gratings are used to obtain a similar response to a single grating of double length. The location of the grating between the two waveguides as well as the light-wave injection direction are very important. The presented methods and effects may be useful for design and analysis in the fields of optical communications, sensing, and processing.

Thermoelastic stresses in a cylinder or disk with cubic anisotropy

The thermoelastic stresses in a crystal in the shape of a circular cylinder or disk are considered. The crystal is a cubically-orthotropic linear elastic solid, with three independent elastic properties. The cubic anisotropy renders the problem asymmetric, despite the axisymmetry of the geometry and thermal loading. This problem is motivated by a thermoelastic model used for certain crystal growth processes. Two simplifying assumptions are made here: (a) the problem is two-dimensional with plane strain or plain stress conditions, and (b) the elastic properties do not depend on the temperature. A new Fourier-type perturbation method is devised and an analytic asymptotic solution of a closed form is obtained, based on the weak cubic anisotropy of the crystal as a perturbation parameter. A general solution technique is described which yields the asymptotic solution up to a desired order. Numerical results are presented for typical parameter values.

Thermoelastic stresses in a crystal with weak anisotropy

The three-dimensional thermoelastic stresses in a circular cylindrical crystal are considered. The crystal is modelled as a cubically–orthotropic linear elastic solid, with three independent elastic properties. The cubic anisotropy renders the problem asymmetric, despite the axi-symmetry of the geometry and thermal loading. A general perturbation solution technique is described, valid for cubic crystals with weak anisotropy. The deviation of the elastic moduli tensor from that of an isotropic material is used as a perturbation parameter. Numerical results are presented for a GaAs boule grown in an LEC process, as a function of the anisotropy parameter.

A statistical analysis of the effect of electric vehicle battery charging on distribution system harmonic voltages

This paper presents a statistical method for predicting the effect that widespread electric vehicle (EV) battery charging will have on power distribution system harmonic voltage levels. The method uses a statistical model for nonlinear load currents to generate the probabilities of specific harmonic voltage levels. The statistical model for the harmonic currents produced by a concentration of EVs accounts for partial harmonics cancellation introduced by uncertainty and variation in charger start-time and initial battery state-of-charge. A general solution technique is presented along with several examples using data from a commercially-available EV charger and an actual power distribution system. The results show that there is a definite threshold penetration below which EV charging has negligible impact on the number of buses whose voltage total harmonic distortion (THD/sub V/) exceeds 5%. During the late evening of a summer day, the authors' example distribution system can accommodate EV penetration levels as high as 20%. A similar analysis of the system in the spring or fail indicates that the system can accommodate a 15% EV penetration before THD/sub V/ exceeds 5% at an unacceptable number of buses.

A statistical method for predicting the net harmonic currents generated by a concentration of electric vehicle battery chargers

This paper presents a method for predicting the net harmonic currents produced by a large number of electric vehicle (EV) battery chargers. The problem is stochastically formulated in order to account for randomness in individual charger start-time and battery state-of-charge. The authors introduce a model that allows for partial harmonics cancellation due to diversity in magnitudes and phase angles. A general solution technique is presented along with an example using data from a commercially available EV charger. Their results show that a limiting distribution of 7-10 chargers is adequate for accurately predicting harmonic injection currents using the central limit theorem. They also show that the expected values of net harmonic currents are considerably less than the peak values that would have been realized if the same number of chargers were operated in unison.

Fast solution techniques for a class of optimal trajectory planning problems with applications to automated spray coating

Optimal trajectory planning problems are often formulated as constrained variational problems. In general, solutions to variational problems are determined by appropriately discretizing the underlying objective functional and solving the resulting nonlinear differential equation(s) and/or nonlinear programming problem(s) numerically. These general solution techniques often require a significant amount of time to be computed, and therefore are of limited value when optimal trajectories need to be frequently computed and/or recomputed. In this paper, a realistic class of optimal trajectory planning problems is defined for which the existence of fast numerical solution techniques are demonstrated. To illustrate the practicality of this class of trajectory planning problems and the proposed solution techniques, three optimal trajectory planning problems for spray coating applications are formulated and solved. Based on the proposed discretization technique, it is shown that these problems can be reduced to either a linear program or a quadratic program, which are readily solved.

On the convergence of traffic measurement and queueing analysis: a statistical-matching and queueing (SMAQ) tool

The analytical tool developed by the authors provides a general solution technique for integration of traffic measurement and queueing analysis. The frequency-domain approach is used to combine the advanced techniques in two areas: signal processing and queueing analysis. Essentially, signal processing techniques are used to obtain the steady-state and second-order statistics of a traffic stream. We propose a new programming method for the construction of a special class of Markov chains to statistically match with each given traffic stream (or superposition of heterogeneous traffic streams). The analytical queueing solutions can therefore be obtained by the folding-algorithm based on Markov chain input modeling. Comprehensive numerical examples show the great potential of the statistical-matching and queueing (SMAQ) tool to solve measurement-based traffic management issues.

A simple approach to obtain tight upper bounds for the asymptotic queueing behavior of statistical multiplexers with heterogeneous traffic

A statistical multiplexer with infinite buffer capacity and a finite number of independent (not necessarily identical) 2-state MMBP (Markov Modulated Bernoulli Process) traffic sources is considered in this paper, which is modeled as a discrete-time single-server queueing system with correlated arrivals. A simple analytic approach is presented to obtain an upper bound for the tail distribution of the buffer contents. This upper bound is good and even tight in many cases, as shown by the numerical results. Compared to a previously reported exact and general solution technique, our approximate analytic approach is very easy to use and not limited by the system size, and costs nearly no CPU-time. So it is useful in practice. This analytic method can also be easily extended to general m-state Markov modulated arrival processes (e.g., m-state MMBP's).

Green's function solution for thermal wave equation in finite bodies

The classical diffusion theory is based on the assumption of local thermal equilibrium. For conduction in thin films or at low temperature, the classical theory of heat conduction breaks down. Various investigations have shown that a wave-type conduction equation adequately describes the thermal energy transport. This paper describes a general solution technique when the wave nature of thermal energy transport is dominant. The solution for temperature distribution is derived for finite bodies. The definition of Green's functions for a wave-type conduction equation is presented and a general form of the Green's function solution method for finite bodies is introduced.

A General Solution Technique for Piezothermoelasticity of Hexagonal Solids of Class 6mm in Cartesian Coordinates

AbstractPiezoelastic materials have attracted considerable recent attention because of their potential use in “intelligent” structural systems. In this paper, we propose a general solution technique for three‐dimensional problems of piezothermoelasticity for hexagonal solids of class 6mm in cartesian coordinates. Potential functions are introduced to represent the coupled elastic and electric fields. Simple uncoupled governing equations for the potential functions are derived from he stress equations of is analyzed. Numerical calculations for the stresses and the electric potential are carried out for a cadmium selenide body exhibiting class 6mm symmetry. Stress distributions are compared with those obtained for the similar thermoelastic problem without the piezo‐effect.

Simulation applied to a three hundred degree polynomial

Abstract A simulation technique called multi stage Monte Carlo optimization (MSMCO) is used to approximate the roots to a 300 degree polynomial. MSMCO consists of repeated random searches in an ever-narrowing focused region centred about the “best answer so far”. This process continues until a good approximate root is produced. Then it is factored out of the polynomial equation and the search for the next root begins in a similar fashion. Multi stage is a general purpose solution technique that can work on a wide variety of problems. Its best potential for application is for the approximate solutions to difficult problems.

Chaotic Vibrations of Beams: Numerical Solution of Partial Differential Equations

The objective of this paper is to examine the utility of direct, numerical solution procedures, such as finite difference or finite element methods, for partial differential equations in chaotic dynamics. In the past, procedures for solving such equations to detect chaos have utilized Galerkin approximations which reduce the partial differential equations to a set of truncated, nonlinear ordinary differential equations. This paper will demonstrate that a finite difference solution is equivalent to a Galerkin solution, and that the finite difference method is more powerful in that it may be applied to problems for which the Galerkin approximations would be difficult, if not impossible to use. In particular, a nonlinear partial differential equation which models a slender, Euler-Bernoulli beam in compression issolvedto investigate chaotic motions under periodic transverse forcing. The equation, cast as a system of firstorder partial differential equations is directly solved by an explicit finite difference scheme. The numerical solutions are shown to be the same as the solutions of an ordinary differential equation approximating the first mode vibration of the buckled beam. Then rigid stops of finite length are incorporated into the model to demonstrate a problem in which the Galerkin procedure is not applicable. The finite difference method, however, can be used to study the stop problem with prescribed restrictions over a selected subdomain of the beam. Results obtained are briefly discussed. The end result is a more general solution technique applicable to problems in chaotic dynamics.

General solution technique for transient thermoelasticity of transversely isotropic solids in cylindrical coordinates

The present solution technique consists of five fundamental solutions which are developed from the Goodier's thermoelastic potential function, the generalized Boussinesq solutions and the Michell function. Considering the relations among the material constants of transverse isotropy, the present solution technique can be classified into two cases. One of them can be reduced to the three solution techniques above which are specifically for isotropic solids only

General solution technique for transient thermoelasticity of transversely isotropic solids in Cartesian coordinates

In the present paper we propose a new general solution technique for transient thermoelastic problems of transversely isotropic solids in Cartesian coordinates. The solution technique consists of five fundamental solutions. By considering the relations among the material constants of transverse isotropy, the solution technique is classified into five groups. One among those corresponds to Goodier's thermoelastic potential function as well as the generalized Boussinesq solutions and the Michell function. For an application of the solution technique, an inverse problem of transient thermoelasticity in a transversely isotropic semi-infinite solid is analyzed.