Methods of Small and Large δ in the Nonlinear Dynamics – A Comparative Analysis

New asymptotic approaches for dynamical systems containing a power nonlinear term x ⁿ are proposed and analyzed. Two natural limiting cases are studied: n 1 + , 1 and n . In the firstcase, the 'small method' (SM)is used and its applicability for dynamical problems with the nonlinearterm sin as well as the usefulness of the SMfor the problem with small denominators are outlined. For n , a new asymptotic approach is proposed(conditionally we call it the 'large method' –LM). Error estimations lead to the followingconclusion: the LM may be used, even for smalln, whereas the SM has a narrow application area. Both of the discussed approaches overlap all values ofthe parameter n.     

Thin-Walled Beams: the Case of the Rectangular Cross-Section

In this paper we present an asymptotic analysis of the three-dimensional problem for a thin linearly elastic cantilever =×(0,l) with rectangular cross-section of sides and ², as goes to zero. Under suitable assumptions on the given loads, we show that the three-dimensional problem converges in a variational sense to the classical one-dimensional model for extension, flexure and torsion of thin-walled beams. Mathematics Subject Classifications (2000) 474K20, 74B10, 49J45.     

Non-gaussian diffusion modeling in composite porous media by homogenization: Tail effect

In the paper anomalous diffusion appearing in a porous medium composed of two porous components of considerably different diffusion characteristics is examined. The differences in diffusivities are supposed to result either from two medium types being present or from variations in pore size (double porosity media). The long-tail effect is predicted using the homogenization approach based on the application of multiple scale asymptotic developments. It is shown that, if the ratio of effective diffusion coefficients of two porous media is of the order of magnitude smaller or equal O( ²), where is a homogenization parameter, then the macroscopic behaviour of the composite may be affected by the presence of tail-effect. The results of the theoretical analysis were applied to a problem of diffusion in a bilaminate composite. Analytical calculations were performed to show the presence of the long-tail effect in two particular cases.Notations c _i the concentration of chemical species in water within the medium i - D _i the effective diffusion coefficient for the medium i - D _ij ^eff the macroscopic (or effective) diffusion tensor in the composite - ERV the elementary representative volume - h the thickness of the period - l a chracteristic length of the ERV or the periodic cell - L a characteristic macroscopic length - n the volumetric fraction of the material 2 - 1–n the volumetric fraction of the material 1 - N the unit vector normal to - t the time variable - x the macroscopic (or slow) space variable - y the microscopic (or fast) space variable - c _1c ,C _2c ,D _1c ,D _2c the characteristic quantities - T,T _1L ,T _2L ,T _1l ,T _2l the characteristic times - c ₁ ^* ,c ₂ ^* ,D ₁ ^* ,D ₂ ^* ,t ^* the non-dimensional variables - the homogenization parameter - ₁ the domain occupied by the material 1 - ₂ the domain occupied by the material 2 - the interface between the domains 1 and 2 - the total volume of the periodic cell - /x_i the gradient operator - the gradient operator     

Design of a Beam with Controllable Force Elements

A method for solving the problem of design of an intellectual structure formulated for the pair optimal position of actuators, optimal control of actuators is developed. In the method proposed, physical and logical objects are treated as equivalent.     

Permeability Effects of Turbulent Flow Through a Porous Insert in a Backward-Facing-Step Channel

In the present work, a k– model, based on the work of Lee and Howell (Proceedings of the ASME-JSME Thermal Engineering Hawaii, 1987), is rigorously derived based on time average of spatially averaged Navier–Stokes equations. The model is then employed to solve for a flow in a backward-facing step channel with a porous insert. The numerical solver is modified from the STREAM code (Lien and Leschziner, Comput. Meth. Appl. Mech. Eng. 114 (1994a) 123–148), and it has been validated against the experimental data of Seegmiller and Driver (AIAA Journal 23 (1985) 163–171). The code is then used to perform simulation for cases with a porous insert. The resistance of the porous insert can be altered by changing its permeability (), Forchheimers constant (F), or thickness (b). The goal is to examine the influence of each parameter on the resulting flow and turbulent kinetic energy (k) distributions. It is discovered that, by increasing the resistance of the insert, flow eventually enters a transitional regime towards relaminarization. This is due to the contribution of Darcys and Forchheimers terms in the governing equations, and modifying these two terms changes the levels of P_k and, hence, k and . Generally speaking, lowering or raising F results in a greater suppression of P_k than , causing the flow to relaminarize. Meanwhile, if the pore size is reasonably large to sustain turbulence within the porous media, increasing b reduces but does not eliminate the turbulent activity in the porous insert.     

Attractors for the Two-Dimensional Navier–Stokes-α Model: An α-Dependence Study

The two-dimensional Navier–Stokes- model is considered on the torus and on the sphere. Upper and lower bounds for the dimension of the global attractors are given. The dependence of the dimension of the global attractors on is studied. Special attention is given for the limiting cases when 0, that is, when the Navier–Stokes- model tends to the Navier–Stokes equations, and to the case when .     

Ši’lnikov Chaos in the Generalized Lorenz Canonical Form of Dynamical Systems

This paper studies the generalized Lorenz canonical form of dynamical systems introduced by elikovský and Chen [International Journal of Bifurcation and Chaos 12(8), 2002, 1789]. It proves the existence of a heteroclinic orbit of the canonical form and the convergence of the corresponding series expansion. The ilnikov criterion along with some technical conditions guarantee that the canonical form has Smale horseshoes and horseshoe chaos. As a consequence, it also proves that both the classical Lorenz system and the Chen system have ilnikov chaos. When the system is changed into another ordinary differential equation through a nonsingular one-parameter linear transformation, the exact range of existence of ilnikov chaos with respect to the parameter can be specified. Numerical simulation verifies the theoretical results and analysis.     

The value of the critical exponent for reaction-diffusion equations in cones

Let D R ^N be a cone with vertex at the origin i.e., D = (0, )x where S ^N–1 and x D if and only if x = (r, ) with r=¦x¦, . We consider the initial boundary value problem: u _t = u+u ^p in D×(0, T), u=0 on Dx(0, T) with u(x, 0)=u ₀(x) 0. Let ₁ denote the smallest Dirichlet eigenvalue for the Laplace-Beltrami operator on and let ₊ denote the positive root of (+N–2) = ₁. Let p ^* = 1 + 2/(N + ₊). If 1 < p < p ^*, no positive global solution exists. If p>p ^*, positive global solutions do exist. Extensions are given to the same problem for u _t=+¦x¦ u ^p .This research was supported in part by the Air Force Office of Scientific Research under Grant # AFOSR 88-0031 and in part by NSF Grant DMS-8 822 788. The United States Government is authorized to reproduce and distribute reprints for governmental purposes not withstanding any copyright notation therein.     

The Experimental Approach to Effective Stress Law of Coal Mass by Effect of Methane

Effective stress law of all kinds of coal samples, including steam coal, fat coal, corking coal, thin coal and anthracite, under pore pressure of gas, is experimentally studied using a newly developed test machine. These samples are taken from Coal Mines in Wuda, Hebi, Yanzhou, Yangquan, Qingshui, and Gujiao in China. The experiment results show that, under pore pressure of gas, the tested coal samples comply with Biots effective stress law, where the Biots coefficient is not a constant, and is bilinear function of volumetric stress () and pore pressure (p), that is, We define four areas according to the numerical feature of , that is, functionless area of pore pressure, normal function area, fracturing function area, and quasi-soil function area. The effective stress law of coal mass introduced by this paper is a constitutive equation in the study of coupled solid and fluid. This has significance in the drainage and outburst of methane in coal seam.     

Asymptotic analysis of equilibrium states for rotating turbulent flows

The equilibrium states of homogeneous turbulence simultaneously subjected to a mean velocity gradient and a rotation are examined by using asymptotic analysis. The present work is concerned with the asymptotic behavior of quantities such as the turbulent kinetic energy and its dissipation rate associated with the fixed point (/kS)=0, whereS is the shear rate. The classical form of the model transport equation for (Hanjalic and Launder, 1972) is used. The present analysis shows that, asymptotically, the turbulent kinetic energy (a) undergoes a power-law decay with time for (P/)<1, (b) is independent of time for (P/)=1, (c) undergoes a power-law growth with time for 1<(P/)<(C ₂–1), and (d) is represented by an exponential law versus time for (P/)=(C ₂–1)/(C ₁–1) and (/kS)>0 whereP is the production rate. For the commonly used second-order models the equilibrium solutions forP/,II, andIII (whereII andIII are respectively the second and third invariants of the anisotropy tensor) depend on the rotation number when (P/kS)=(/kS)=0. The variation of (P/kS) andII versusR given by the second-order model of Yakhot and Orzag are compared with results of Rapid Distortion Theory corrected for decay (Townsend, 1970).     

Flow through a helically coiled annulus

In this paper the flow is studied of an incompressible viscous fluid through a helically coiled annulus, the torsion of its centre line taken into account. It has been shown that the torsion affects the secondary flow and contributes to the azimuthal component of velocity around the centre line. The symmetry of the secondary flow streamlines in the absence of torsion, is destroyed in its presence. Some stream lines penetrate from the upper half to the lower half, and if is further increased, a complete circulation around the centre line is obtained at low values of for all Reynolds numbers for which the analysis of this paper is valid, being the ratio of the torsion of the centre line to its curvature.Nomenclature A =constant - a outer radius of the annulus - b unit binormal vector to C - C helical centre line of the pipe - D rL - g 1000 - K Dean number=Re² - L 1+r sin - M (L ²+ ² r ²)^1/2 - n unit normal vector to C - P, P pressure and nondimensional pressure - p ₀, p pressures of O(1) and O() - Re Reynolds number=aW ₀/ - (r, , s), (r, , s) coordinates and nondimensional coordinates - nonorthogonal unit vectors along the coordinate directions - r ₀ radius of the projection of C - t unit tangent vector to C - V _r, V , V _s velocity components along the nonorthogonal directions - V_r, V, V _s nondimensional velocity components along - W ₀ average velocity in a straight annulus Greek symbols , curvature and nondimensional curvature of C - U, V, W lowest order terms for small in the velocity components along the orthogonal directions t - _r, , _s first approximations to V _r , V, V _s for small - =/=/ - kinematic viscosity - density of the fluid - , torsion and nondimensional torsion of C - , stream function and nondimensional stream function - nondimensional streamfunction for U, V - a inner radius of the annulus After this paper was accepted for publication, a paper entitled On the low-Reynolds number flow in a helical pipe, by C.Y. Wang, has appeared in J. Fluid. Mech., Vol 108, 1981, pp. 185–194. The results in Wangs paper are particular cases of this paper for =0, and are also contained in [9].     

RADAR ECHO DELAY IN THE(Ω,Aμν)-FIELD THEORY

The present work aims to consider the.fourth test of general relativity theory by Shapiro.using radar echo delay in Yu’s(Ω,A_μν)-field theory.     

Symbolic Sequences Generated by a Special Class of Discrete Systems With Discontinuity and Input

In this paper, we study the symbolic sequences generated by aclass of discrete systems. This class contains the double-modulators, a typical example of discretetime electronic systems with discontinuity and input. First we develop ageneral theory and then we apply it to some examples in order to obtainsets of inadmissible sequences.     

Stress analysis of the cylinder subjected to arbitrarily distributed axisymmetrical loads

Summary Stress analysis has been carried out for a finite cylinder subjected to arbitrarily distributed axisymmetrical surface loads. Direct stress _x in the axial direction is assumed to be of the form _x = ₀+r ₁ +r ₂ where ₀ to ₂ are functions of x. Using the equations of equilibrium and compatibility the other direct stresses and the shearing stress are expressed by ₁ and ₂. Fundamental equations governing ₁ and ₂ are introduced using the variational principle of complementary energy. From the results of the present analysis it is evident that the boundary conditions can be satisfied completely even for the case where the external forces are specified in complicated form, and that more accurate solutions can easily be obtained by introducing additional terms in _x.

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Spannungsanalyse für den Zylinder unter axialsymmetrischer Last in beliebiger Verteilung

Übersicht Für einen endlichen Zylinder unter axialsymmetrischer Oberflächenlast in beliebiger Verteilung werden die Spannungen ermittelt. Die Normalspannung in Axialrichtung wird in der Form _x = ₀+r ₁ +r ₂ angesetzt mit ₀, ₁, ₂ als Funktionen von x. Mit Hilfe der Gleichgewichtsund Verträglichkeitsbedingungen werden die anderen Normalspannungen und die Schubspannung durch ₁ und ₂ ausgedrückt. Über das Variationsprinzip für die Komplementärenergie werden die grundlegenden Gleichungen für ₁ und ₂ eingeführt. Die Ergebnisse zeigen, daß die Randbedingungen selbst für komplizierte Belastungsarten vollständig erfüllbar sind und mit zusätzlichen Termen in _x mühelos noch genauere Lösungen bestimmt werden können.

     

An experimental study on effects of solid concentration and ζ-potential on pseudoplastic flow of suspensions

The pseudoplastic flow of suspensions, alumina or styrene-acrylamide copolymer particles in water or an aqueous solution of glycerin has been studied by the step-shear-rate method. The relation between the shear rate,D, and the shear stress,, in the step-shear-rate measurements, where the state of dispersion was considered to be constant, was expressed as = AD ^1/2 +CD. The effective solid volume fraction,ø _F, andA were dependent on the shear rate and expressed byø _F =aD ^b andA = D . Combining the above relations, the steady flow curve was expressed by = D ^{1/2 +} + ₀ (1 – a D ^b/0.74)^–1.85 D, where ₀ is the viscosity of the medium.With an increase in solid volume fraction and a decreases in the absolute value of the-potential, the flow behavior of the suspensions changed from Newtonian ( = = b = 0), slightly pseudoplastic ( = b = 0), pseudoplastic ( = 0) to a Bingham-like behavior.The change in viscosity of the medium had an effect on the change in the effective volume fraction.     

A turbulence model for the heat transfer near stagnation point of a circular cylinder

A one-equation low-Reynolds number turbulence model has been applied successfully to the flow and heat transfer over a circular cylinder in turbulent cross flow. The turbulence length-scale was found to be equal 3.7y up to a distance 0.05 and then constant equal to 0.185 up to the edge of the boundary layer (wherey is the distance from the surface and is the boundary layer thickness).The model predictions for heat transfer coefficient, skin friction factor, velocity and kinetic energy profiles were in good agreement with the data. The model was applied for Re 250,000 and Tu0.07.Nomenclature µ,C _D Constants in the turbulence kinetic energy equation - C ₁,C ₂ Constants in the turbulence length-scale equation - Skin friction coefficient atx - D Cylinder diameter - F Dimensionless flow streamwise velocityu/u _e - k Turbulence kinetic energy =1/2 the sum of the squared three fluctuating velocities - K Dimensionless turbulence kinetic energyk/u _e ^/2 - I Dimensionless temperature (T–T _w )/(T –T _w ) - l Turbulence length-scale - l _e Turbulence length-scale at outer region - Nu _D Nusselt number - p Pressure - Pr Prandtl number - Pr _t Turbulent Prandtl number - Pr _k Constant in the turbulence kinetic energy equation - R Cylinder radius - Re _D Reynolds number u D/µ - Re _x Reynolds number u x/µ - R _K Reynolds number of turbulence - T Mean temperature - T Mean temperature at ambient - T _s Mean temperature at surface - Tu Cross flow turbulence intensity, - u Mean flow streamwise velocity - u Fluctuating streamwise velocity - u _e Mean flow velocity at far field distance - u Mean flow velocity at ambient - u* Friction velocity - v Mean velocity normal to surface - V Dimensionless mean velocity normal to surface - x,x ₁ Distance along the surface - y Distance normal to surface - Dimensionless pressure gradient parameter - Boundary layer thickness atu=0.9995u _e - Transformed coordinate iny direction - Fluid molecular viscosity - _t Turbulent viscosity - _eff + _t - µ Fluid molecular viscosity at ambient - Kinematic viscosity/ - Density - Density at ambient - _w Wall shear stress - _w,0 Wall shear stress at zero free stream turbulence     

Fatigue crack propagaion under mixed mode loading

Mixed model fatigue crack propagation is analyzed in this paper, using a centre cracked plate geometry, loaded under un-iaxial cyclic tension. Based on maximum principal stress criterion, a modified Paris expression of fatigue crack growth rate is derived in terms of ΔK and crack angle β_α for an inclined crack. It is also shown that it is more convenient to express the Paris equation by means of crack length projected on the x -axis, α_x rather than the actual length, α itself. The crack trajectory due to cyclic loading is predicted, β is varied from 29° to 90°. Experimental data on Type L3 aluminium agree fairly well with predicted values when β_α exceeds 30°.     

On the thermodynamics of elastic-plastic materials with temperature-dependent moduli and yield stresses

Summary The subject of this article is the thermodynamics of perfect elastic-plastic materials undergoing unidimensional, but not necessarily isothermal, deformations. The first and second laws of thermodynamics are employed in a form in which only the following quantities appear: the temperature , the elastic strain _e, the plastic strain _p, the elastic modulus (gq), the yield strain (gq), the heat capacity (_e, _p,), the latent elastic heat _e(_e, _p, ), and the latent plastic heat _p(_e, _p, ). Relations among the response functions , , , _e, and _p are derived, and it is shown that a set of these relations gives a necessary and sufficient condition for compliance with the laws of thermodynamics. Some observations are made about the existence and uniqueness of energy and entropy as functions of state.Dedicated to Clifford Truesdell on the occasion of his 60th birthdayThis research was supported by the U.S. National Science Foundation.     

“Thermal layer” effect in MHD: Interaction of a parallel shock with a layer of decreased density

Interaction of a parallel fast MHD shock with a layer of decreased density is discussed using ideal MHD approach. This is an extrapolation of gas dynamic thermal layer effect on ideal MHD. Computer simulations show that a magnetic field of a moderate intensity ( 1) may change the character of the flow for intermediate Mach numbers (M 5) and a new raking regime may occur which is not observed in the absence of a magnetic field. Self similar precursor analogous to that in gas dynamics may develop in the case of highM and low density in the layer but magnetic forces essentially decrease its growth rate. This problem appears in connection with cosmical shock propagation where planetary magnetic tails play the role of the thermal layer, and it may also be observed in the laboratory when the shock is strong enough to heat the walls ahead of it.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.