Infinite Bar Research Articles

An Analytic Solution for 2D Heat Conduction Problems with Space–Time-Dependent Dirichlet Boundary Conditions and Heat Sources

This study proposes a closed-form solution for the two-dimensional (2D) transient heat conduction in a rectangular cross-section of an infinite bar with space–time-dependent Dirichlet boundary conditions and heat sources. The main purpose of this study is to eliminate the limitations of the previous study and add heat sources to the heat conduction system. The restriction of the previous study is that the values of the boundary conditions and initial conditions at the four corners of the rectangular region should be zero. First, the boundary value problem of 2D heat conduction system is transformed into a dimensionless form. Second, the dimensionless temperature function is transformed so that the temperatures at the four endpoints of the boundary of the rectangular region become zero. Dividing the system into two one-dimensional (1D) subsystems and solving them by combining the proposed shifting function method with the eigenfunction expansion theorem, the complete solution in series form is obtained through the superposition of the subsystem solutions. Three examples are studied to illustrate the efficiency and reliability of the method. For convenience, the space–time-dependent functions used in the examples are considered separable in the space–time domain. The linear, parabolic, and sine functions are chosen as the space-dependent functions, and the sine, cosine, and exponential functions are chosen as the time-dependent functions. The solutions in the literature are used to verify the correctness of the solutions derived using the proposed method, and the results are completely consistent. The parameter influence of the time-dependent function of the boundary conditions and heat sources on the temperature variation is also investigated. The time-dependent function includes exponential type and harmonic type. For the exponential time-dependent function, a smaller decay constant of the time-dependent function leads to a greater temperature drop. For the harmonic time-dependent function, a higher frequency of the time-dependent function leads to a more frequent fluctuation of the temperature change.

An Analytic Solution for 2D Heat Conduction Problems with General Dirichlet Boundary Conditions

This paper proposed a closed-form solution for the 2D transient heat conduction in a rectangular cross-section of an infinite bar with the general Dirichlet boundary conditions. The boundary conditions at the four edges of the rectangular region are specified as the general case of space–time dependence. First, the physical system is decomposed into two one-dimensional subsystems, each of which can be solved by combining the proposed shifting function method with the eigenfunction expansion theorem. Therefore, through the superposition of the solutions of the two subsystems, the complete solution in the form of series can be obtained. Two numerical examples are used to investigate the analytic solution of the 2D heat conduction problems with space–time-dependent boundary conditions. The considered space–time-dependent functions are separable in the space–time domain for convenience. The space-dependent function is specified as a sine function and/or a parabolic function, and the time-dependent function is specified as an exponential function and/or a cosine function. In order to verify the correctness of the proposed method, the case of the space-dependent sinusoidal function and time-dependent exponential function is studied, and the consistency between the derived solution and the literature solution is verified. The parameter influence of the time-dependent function of the boundary conditions on the temperature variation is also investigated, and the time-dependent function includes harmonic type and exponential type.

Estrategias de control no lineal para la regulación de corriente y estabilización de voltaje DC para un Aerogenerador operando en una Microred

La generación a partir de energías renovables, en especial eólica, se encuentra ocupando cada vez más una mayor participación en el mercado energético a nivel mundial. Debido a la alta dinámica del viento, se presentan variaciones en la magnitud de voltaje y frecuencia del sistema que dificulta integrar estos generadores al sistema eléctrico. Varios trabajos proponen diferentes estrategias de control implementadas en sistemas electrónicos de potencia para solventar este problema. Estos sistemas primero rectifican el voltaje variable del generador creando un acople de corriente directa (DC) y luego convierten el resultado en voltaje con magnitud y frecuencia constante. Los controladores más usados son de tipo proporcional integral derivativo (PID), los cuales presentan un buen desempeño en zonas de operación lineal. Sin embargo, existen propuestas no lineales que presentan amplias ventajas frente a los PIDs, sobre todo en sistemas con dinámicas rápidas y no lineales. En el presente trabajo se implementan controladores PIDs lineales y se proponen dos estrategias de control no lineales, una con una estructura PID y otra con control de modo deslizante (SMC), para regular el suministro de corriente y voltaje en aerogeneradores conectados a un sistema electrónico de potencia. En los controladores propuestos se implementó una ganancia no lineal que se calcula en función del error del sistema y se comparó su desempeño con el de los controladores PIDs lineales. Los resultados mostraron mejoras notables en la velocidad de estabilización del sistema y reducción de oscilaciones ante variaciones repentinas de referencia y perturbaciones externas. Para probar el desempeño de los controladores se simuló una micro red enlazada a una barra infinita en Simulink.

Approximate analytical solution to the stationary two-dimensional heat conduction problem on infinite bar with the source of heat

An approximate analytical solution to the boundary-value heat conduction problem for an infinite bar with a heat source was obtained with the use of the integral method of heat balance, by introducing a complementary required function and complementary boundary conditions. The boundary - value problem for a partial differential equation is reduced to an ordinary differential equation with respect to this function due to the complementary required function that characterizes the change in temperature along the axis of symmetry in the cross-section of the bar. The complementary boundary conditions determined by the initial differential equation and the given boundary conditions are found so that their satisfaction is equivalent to the solution of the initial equation of the boundary value problem at the boundary points. The fulfilment of the equation at the boundary points as well as the heat balance integral results in the fulfilment of the initial equation inside the domain. The approximate analytical solution obtained can be used to identify the amount of internal heat generated by various production processes (vibration and deformation loads, electromagnetic fields effects, etc.) in thermal and nuclear power plants, in the rocket and space industry and other industrial facilities.

Waves in one-dimensional quasicrystalline structures: dynamical trace mapping, scaling and self-similarity of the spectrum

Harmonic axial waves in quasiperiodic-generated structured rods are investigated. The focus is on infinite bars composed of repeated elementary cells designed by adopting generalised Fibonacci substitution rules, some of which represent examples of one-dimensional quasicrystals. Their dispersive features and stop/pass band spectra are computed and analysed by imposing Floquet–Bloch conditions and exploiting the invariance properties of the trace of the relevant transfer matrices. We show that for a family of generalised Fibonacci substitution rules, corresponding to the so-called precious means, an invariant function of the circular frequency, the Kohmoto’s invariant, governs self-similarity and scaling of the stop/pass band layout within defined ranges of frequencies at increasing generation index. Other parts of the spectrum are instead occupied by almost constant ultrawide band gaps. The Kohmoto’s invariant also explains the existence of particular frequencies, named canonical frequencies, associated with closed orbits on the geometrical three-dimensional representation of the invariant. The developed theory represents an important advancement towards the realisation of elastic quasicrystalline metamaterials.

Analytical method for forces between current carrying infinite bars

Purpose The calculation of forces between rectangular conductors with a uniform current density as can be found in the long coils as used in linear actuators and planar motors. Design/methodology/approach The proposed methodology relies on a single analytical equation that is used to compute all the force components. Findings Several comparisons with other available techniques are provided. All the comparisons confirm the validity of the proposed analytical method. Originality/value In the literature, other analytical methods for the force computation can be found. However, these often have limitations (e.g. only adjacent conductors or non-adjacent conductors). This paper adds a new compact expression that covers all the possible configurations at the same time. This new tool holds for all the analytical expression found in the literature.

Elastic wave propagation in bars of arbitrary cross section: a generalized Fourier expansion collocation method.

The problem of elastic wave propagation in an infinite bar of arbitrary cross section is studied via a generalized version of the Fourier expansion collocation method. In the current formulation, the exact three dimensional solution to Navier's equation in cylindrical coordinates is used to obtain the boundary traction vector as a periodic, piecewise continuous/differentiable function of the angular coordinate. Traction free conditions are then met by setting the Fourier coefficients of the boundary traction vector to zero without approximating the bounding surface by multi-sided polygons as in the method presented by Nagaya. The method is derived for a general cross section with no axial planes of symmetry. Using the general formulation it is shown that the symmetric and asymmetric modes decouple for cross sections having one axial plane of symmetry. An efficient algorithm for computing dispersion curves based on the current method is presented and used to obtain the fundamental longitudinal and flexural wave speeds for a bar of elliptical cross section. The results are compared to those obtained by previous researchers using exact and approximate treatments.

3D Elastic Solutions for Laterally Loaded Discs: Generalised Brazilian and Point Load Tests

This paper investigates the application of a double Fourier series technique to the construction of an elastic stress field in a cylindrical bar subject to lateral boundary loads. The lateral loads, including the constant load boundary conditions, are represented by two Fourier series: one on the perimeter of the circular section (r 0, θ) and the other on the longitudinal curved surface parallel to the bar axis (z). The technique invokes acceptable potential functions of the Papkovich–Neuber displacement field, satisfying the governing partial differential equations, to assign appropriate odd and even trigonometric Fourier terms in cylindrical coordinates (r, θ, z). The generic solution decomposes the problem of interest to a state of stress caused by two independent boundary conditions along the z axis and θ-polar angle, both superimposed on a solution for which these potentials are the product of the trigonometric terms of the independent variables (θ, z). Constants appearing in the resultant second-order partial differential equations are determined from the generally mixed (tractions and/or displacements) boundary conditions. While the solutions are satisfied exactly at the ends of an infinite bar, they are satisfied weakly on average, in the light of Saint Venant’s approximation at the two ends of a finite bar. The application of the proposed analysis is verified against available elastic solutions for axisymmetric and non-axisymmetric engineering problems such as the indirect Brazilian Tensile Strength and Point Load Strength tests.

Torsional travelling wave reflection in the bar with an artificial defect

Reflection and propagation of a torsional wave in an infinite bar with an artificial transverse defect are investigated. Dependence of the reflected wave on the damage parameters is obtained. The solution of the inverse problem allows us to determine the damaged region coordinate and the parameter, including damage depth and length, based on the characteristics of falling and reflected waves at the sighting point.

Mixed Finite Element Method for a Degenerate Convex Variational Problem from Topology Optimization

The optimal design task of this paper seeks the distribution of two materials of prescribed amounts for maximal torsion stiffness of an infinite bar of a given cross section. This example of relaxation in topology optimization leads to a degenerate convex minimization problem $E\left( v \right):= \int_\Omega \varphi_0\left( \left\lvert\nabla v\right\rvert \right)\operatorname{dx} - \int_\Omega fv\operatorname{dx}\text{for } v \in V:=H^1_0( \Omega )$ with possibly multiple primal solutions u, but with unique stress $\sigma:=\varphi_0'\left( \left\lvert\nabla u\right\rvert \right)\operatorname{sign}\nabla u.$ The mixed finite element method is motivated by the smoothness of the stress variable $\sigma \in H^1_{\operatorname{loc}}( \Omega ; \mathbb R^2)$, while the primal variables are uncontrollable and possibly nonunique. The corresponding nonlinear mixed finite element method is introduced, analyzed, and implemented. The striking result of this paper is a sharp a posteriori error estimation in the dual formulation, while the a posteriori error analysis in the primal problem suffers from the reliability-efficiency gap. An empirical comparison of that primal formulation with the new mixed discretization schemes is intended for uniform and adaptive mesh refinements.

Electromagnetic interaction of moving conductors with magnets: kinematic solutions for infinitely long square and cylindrical geometries

AbstractThe electromagnetic drag force on a point dipole near a moving conductor caused by the induced electric currents is investigated by numerical and analytical computations. Our focus is on prototypical configurations for Lorentz force velocimetry, i.e. velocity measurement from the electromagnetic drag force on the dipole. We examine the particular cases of conducting infinite bars of square or round cross‐section, which are moving with constant velocity in the field of arbitrary oriented magnetic dipole. In addition, we study the laminar liquid‐metal flow in a square duct. The motion of the conductor is prescribed. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A Fractional Model of Continuum Mechanics

Although there has been renewed interest in the use of fractional models in many application areas, in reality fractional analysis has a long and distinguished history and can be traced back to the likes of Leibniz (Letter to L’Hospital, 1695), Liouville (J. Ec. Polytech. 13:71, 1832), and Riemann (Gesammelte Werke, p. 62, 1876). Recent publications (Podlubny in Math. Sci. Eng. 198, 1999; Sabatier et al. in Advances in fractional calculus: theoretical developments and applications in physics and engineering, Springer, Berlin, 2007; Das in Functional fractional calculus for system identification and controls, Springer, Berlin, 2007) demonstrate that fractional derivative models have found widespread applications in science and engineering. Late fundamental considerations have led to the introduction of fractional calculus in continuum mechanics in an attempt to develop non-local constitutive relations (Lazopoulos in Mech. Res. Commun. 33:753–757, 2006). Attempts have also been made to model microscopic forces using fractional derivatives (Vazquez in Nonlinear waves: classical and quantum aspects, pp. 129–133, 2004). Our approach in this paper differs from previous theoretical work, in that we develop a general framework directly from the classical continuum mechanics, by defining the laws of motion and the stresses using fractional derivatives. The timeliness and relevance of this work is justified by the surge in interest in applications of fractional order models to biological, physical and economic systems. The aim of the present paper is to lay the foundations for a new non-local model of continuum mechanics based on fractional order derivatives which we will refer to as the fractional model of continuum mechanics. Following the theoretical development, we apply this framework to two one-dimensional model problems: the deformation of an infinite bar subjected to a self-equilibrated load distribution, and the propagation of longitudinal waves in a thin finite bar.

Iterated bar complexes ofE–infinity algebras and homology theories

We proved in a previous article that the bar complex of an E-infinity algebra inherits a natural E-infinity algebra structure. As a consequence, a well-defined iterated bar construction B^n(A) can be associated to any algebra over an E-infinity operad. In the case of a commutative algebra A, our iterated bar construction reduces to the standard iterated bar complex of A. The first purpose of this paper is to give a direct effective definition of the iterated bar complexes of E-infinity algebras. We use this effective definition to prove that the n-fold bar complex B^n(A) admits an extension to categories of algebras over E_n-operads. Then we prove that the n-fold bar complex determines the homology theory associated to a category of algebras over E_n-operads. For n infinite, we obtain an isomorphism between the homology of an infinite bar construction and the usual Gamma-homology with trivial coefficients.

Cut‐off Frequencies Induced by the Length of Strain Gauges Measuring Impact Events

Abstract: The finite length of strain gauges may induce filtering effects when measuring impact events. In this study, we are interested in quantifying these effects. Precisely, we determined the cut‐off frequencies of strain gauges cemented on visco‐elastic bars and measuring impact‐induced strain waves. This study shows that the cut‐off frequencies increase with the bar’s wave velocity and decrease with the bar’s diameter. The asymptotic value, corresponding to an infinite bar diameter, is reached rapidly (bar diameter ≈ 15 mm). Moreover, we showed that the mode cut‐off frequencies are more severe (lower) than the gauge length cut‐off frequencies for bar diameters greater than 8 mm.

Finite element procedures for nonlinear structures in moving coordinates. Part II: Infinite beam under moving harmonic loads

This paper presents a numerical approach to the stationary solution of infinite Euler–Bernoulli beams posed on Winkler foundations under moving harmonic loads. The procedure proposed in Part 1 [Nguyen V-H, Duhamel D. Finite element procedures for nonlinear structures in moving coordinates. Part I: infinite bar under moving axial loads. Comput Struct 2006;84(21):1368–80], which has been applied to consider the longitudinal vibration of rods under constant amplitude moving loads in moving coordinates, is enhanced herein for the case of moving loads with time-dependent amplitudes. Firstly, the separation of variables is used to distinguish the convection component from the amplitude component of the displacement function. Then, the stationary condition is applied to the convection component to obtain a dynamic formulation in the moving coordinates. Numerical examples are computed with a linear structure to validate the proposed method. Finally, nonlinear elastic foundation problems are presented.

A convergent adaptive finite element method for an optimal design problem

The optimal design problem for maximal torsion stiffness of an infinite bar of given geometry and unknown distribution of two materials of prescribed amounts is one model example in topology optimisation. It eventually leads to a degenerate convex minimisation problem. The numerical analysis is therefore delicate for possibly multiple primal variables u but unique derivatives σ : = DW(D u). Even fine a posteriori error estimates still suffer from the reliability-efficiency gap. However, it motivates a simple edge-based adaptive mesh-refining algorithm (AFEM) that is not a priori guaranteed to refine everywhere. Its convergence proof is therefore based on energy estimates and some refined convexity control. Numerical experiments illustrate even nearly optimal convergence rates of the proposed AFEM.

Proper dynamics of phase interface in an infinite elastic bar with variable cross section

Proper dynamics of phase interface in an infinite elastic bar with variable cross section

Dynamics of a free phase boundary in an infinite bar with variable cross‐sectional area

AbstractThe dynamic processes in an elastic bar composed of a material, which is capable of undergoing phase transitions, are under consideration. We use a model of an elastic body with non‐convex strain energy potential. The admissible motions of the phase boundary along the bar are sought for. It is assumed that the phase boundary moves at a variable speed. The motion is caused by a non‐equilibrium initial condition for the phase boundary position. We obtain an analytical non‐self‐similar solution of the problem and investigate its properties. The results obtained in the framework of this full dynamic approach are compared with the results obtained in the framework of the kinetic (quasi‐static) approach.

Analysis of breather state in thin bar by using collective coordinate

Considering Peierls-Nabarro (P-N) force and viscous effect of material, the dynamic behavior of one-dimensional infinite metallic thin bar subjected to axially periodic load is investigated. Governing equation, which is sine-Gordon type equation, is derived. By means of collective-coordinates, the partial equation can be reduced to ordinary differential dynamical system to describe motion of breather. Nonlinear dynamic analysis shows that the amplitude and frequency of P-N force would influence positions of hyperbolic saddle points and change subharmonic bifurcation point, while the path to chaos through odd subharmonic bifurcations remains. Several examples are taken to indicate the effects of amplitude and period of P-N force on the dynamical response of the bar. The simulation states that the area of chaos is half-infinite. This area increases along with enhancement of the amplitude of P-N force. And the frequency of P-N force has similar influence on the system.

A numerical investigation of dispersion in Hopkinson Pressure Bar experiments

The majority of published research on numerical modelling of Split Hopkinson Pressure Bars (SHPB) focuses on the response of the specimen, with very little attention given to the dispersive wave propagation in the bars themselves. This investigation focussed on the numerical modelling of wave propagation in cylindrical pressure bars. The Eigen modes of an infinitely long cylindrical bar were compared to the vibrational modes predicted by the analytical solution of Pochammer and Chree. There was very good agreement between the numerical and analytical predictions for the phase velocity-frequency relationship derived from the Eigen modes. Eigen analysis of a finite length bar showed several differences. The most notable of these was that the infinite bar Eigen modes could be characterised by a single wavelength whereas most finite bar Eigen modes required two or more wavelengths. The response of a finite bar to an evenly distributed oscillating pressure pulse was modelled using Abaqus/Explicit. The numerical predictions compared well with the analytical predictions, for frequencies below 100 kHz, but large discrepancies arose at higher frequencies.