Forces of rails for electromagnetic railguns

In this paper, using electromagnetic theory, the equations for the total magnetic force on the rails are deduced. Besides this, the equations for the changes of the magnetic force along with time and the running position of the armature under jointed pulse operating current are presented. Then, the equations for changes of the bending stress and shear stress in the rail along with time and the running position of the armature are given. Changes of the magnetic force distribution between the two rails along with the running position of the armature and other parameters are investigated. The total electromagnetic forces on the rail under jointed pulse operating current are analyzed. Distribution and changes of the bending stresses in the rail are studied. A number of results are obtained. The results are useful for design and application of the railgun system.     

The driving torque of a rotating levitated cylindrical permanent magnet above a superconductor

Due to the Meissner effect, a permanent magnet is levitated, when released above a high temperature superconductor. When there is an inhomogeneous temperature field around the magnet, the magnet might start to oscillate with increasing amplitude until it remains in a continuous rotation. A mathematical model for the described effect is presented which couples heat transfer and electromagnetic forces with the equation of motion, yielding to a multiphysics task. In a detailed analysis it is found, that the torque which drives the rotation of the magnet, is explicitly given in terms of Bessel functions and the Fourier coefficients of order zero and one of the temperature field of the surrounding air.     

Waves and vibrations in strings with windings: Commemorating the 100th anniversary of the birth of Kh.A. Rakhmatulin

The equations of the propagation of transverse, twisting and longitudinal waves and vibrations are obtained, taking into account their interactions in musical strings with windings. Their solutions are obtained. The occurrence of transverse and twisting motions leads to the appearance of longitudinal motions, while the transverse and longitudinal components play the role of inducing forces for the twisting components. The contributions of the transverse, twisting and longitudinal components to the dynamic loading of the string are of the same order. The longitudinal-twisting vibrations occur both at natural frequencies and at frequencies of the transverse vibrations. Resonance phenomena between the individual modes of these vibrations are possible.     

Electrodynamic-mechanical boundary value problems and gauge transformations in rigid dielectrics with constitutive magnetoelectric coupling

Gauge transformations are generally applied to decouple Maxwell’s equations, introducing gauge invariant vector and scalar potentials in connection with suitable gauge differential equations. This step simplifies the analytical or numerical solution of electrodynamic boundary value problems. In dielectrics, the propagation of electromagnetic waves is usually investigated restricting the problems to simple isotropic non-functional materials. On the other hand, magnetoelectric (ME) solids are of particular interest, converting electrical to magnetic energy and vice versa. The constitutive behavior of those materials has been investigated extensively, however restricting considerations to static or quasi-static loading. The goal of this paper is to combine electrodynamics in terms of the classical Maxwell equations with the constitutive behavior of ME materials. The interacting mechanisms of ME energy conversion lead to a complex behavior and are supposed to give rise to interesting phenomena influencing wave dispersion, deflection and reflection. The coupled boundary value problem is comprehensively formulated first, including mechanical stress and strain fields as well as electromagnetically induced forces. Gauge transformations are presented to decouple the electrodynamic potential equations for anisotropic ME bodies, neglecting a mechanical compliance at that point. Weak formulations are derived as a basis for numerical discretization procedures like the finite element method and simple examples demonstrate the impact of ME coupling on the phase velocity of an electromagnetic wave.     

On soliton and periodic solutions of Maxwell–Bloch system for two-level medium with degeneracy

Maxwell–Bloch (MB) system describing the resonant propagation of electromagnetic pulses in either two-level media with degeneracy in angle moment projection or a three-level media with equal oscillator forces is considered. An inhomogeneous broadening of energy levels and a polarization of the wave are accounted. The equations are integrated by the binary Darboux transformations technique. Pulses corresponding to a transition between levels with the largest population difference are shown to be stable. The solution describing the propagation of pulses in the medium excited by a periodic wave is obtained. The hierarchy of infinitesimal symmetries is obtained by means of Darboux transformation.     

轴向运动导电导磁梁的磁弹性振动方程

针对磁场环境中轴向运动导电导磁梁磁弹性耦合振动的理论建模问题进行研究.基于Timoshenko(铁木辛柯)梁理论并考虑几何非线性因素,给出轴向运动弹性梁在横向双向振动下的形变势能、动能计算式以及电磁力和机械力的虚功表达式.应用Hamilton(哈密顿)变分原理,推得磁场中轴向运动Timoshenko梁的非线性磁弹性耦合振动方程,并给出了简化形式的Euler-Bernoulli(欧拉 伯努利)梁磁弹性振动方程.根据电磁理论和相应的电磁本构关系,得到载流导电弹性梁所受电磁力的表达式,基于磁偶极子-电流环路模型给出铁磁弹性梁所受磁体力和磁体力偶的表述形式.通过算例,分析了轴向运动导电弹性梁的奇点分布及其稳定性问题.     

Electromagnetism on anisotropic fractal media

Basic equations of electromagnetic fields in anisotropic fractal media are obtained using a dimensional regularization approach. First, a formulation based on product measures is shown to satisfy the four basic identities of the vector calculus. This allows a generalization of the Green–Gauss and Stokes theorems as well as the charge conservation equation on anisotropic fractals. Then, pursuing the conceptual approach, we derive the Faraday and Ampère laws for such fractal media, which, along with two auxiliary null-divergence conditions, effectively give the modified Maxwell equations. Proceeding on a separate track, we employ a variational principle for electromagnetic fields, appropriately adapted to fractal media, so as to independently derive the same forms of these two laws. It is next found that the parabolic (for a conducting medium) and the hyperbolic (for a dielectric medium) equations involve modified gradient operators, while the Poynting vector has the same form as in the non-fractal case. Finally, Maxwell’s electromagnetic stress tensor is reformulated for fractal systems. In all the cases, the derived equations for fractal media depend explicitly on fractal dimensions in three different directions and reduce to conventional forms for continuous media with Euclidean geometries upon setting these each of dimensions equal to unity.     

Stress recovery procedure for solving boundary value problems in the mechanics of a deformable solid by the finite element method

A stress recovery procedure, based on the determination of the forces at the mesh points using a stiffness matrix obtained by the finite element method for the variational Lagrange equation, is described. The vectors of the forces reduced to the mesh points are constructed for the known stiffness matrices of the elements using the displacements at the mesh points found from the solution of the problem. On the other hand, these mesh point forces are determined in terms of the unknown forces distributed over the surface of an element and given shape functions. As a result, a system of Fredholm integral equations of the first kind is obtained, the solution of which gives these distributed forces. The stresses at the mesh points are determined for the values of these forces found on the surfaces of the finite element mesh (including at the mesh points) using the Cauchy relations, which relate the forces, stresses and the normal to the surface. The special features of the use of the stress recovery procedure are demonstrated for a plane problem in the linear theory of elasticity.     

Elasticity solution of multi-span beams with variable thickness under static loads

This paper studies the stress and displacement distributions of continuously varying thickness multi-span beams simply supported at two ends and under static loads. The intermediate supports of the beam may be elastic and/or rigid in one or two directions. On the basis of the two-dimensional plane elasticity theory, the general solution of stress function, which exactly satisfies the governing differential equations and the simply supported boundary conditions, is deduced. In the present analysis, the reaction forces of the intermediate supports are regarded as the unknown external forces acting on the lower surface of the beam under consideration. The unknown coefficients in the solutions are determined by using the Fourier sinusoidal series expansions to the boundary conditions on the upper and lower surfaces of the beam and using the linear relations between reaction forces and displacements of the beam at intermediate supports. The solution obtained is exact and excellent convergence has been confirmed. Comparing the numerical results obtained from the proposed method to those obtained from the Euler beam theory, the Timoshenko beam theory and those obtained from the commercial finite element software ANSYS, high accuracy of the present method is demonstrated.     

A stress recovery procedure for solving geometrically non-linear problems in the mechanics of a deformable solid by the finite element method

A stress recovery procedure is presented for non-linear and linearized problems, based on the determination of the forces at the mesh points using a stiffness matrix obtained by the finite element method for the Lagrange variational equation written in the initial configuration using an asymmetric Piola–Kirchhoff stress tensor. Vectors of the forces reduced to the mesh points are constructed using the displacements at the mesh points found by solving this equation and for the known stiffness matrices of the elements. On the other hand, these forces at the mesh points are defined in terms of unknown forces distributed over the surface of an element and given shape functions. As a result, a system of Fredholm integral equations of the first kind is obtained, the solution of which gives these distributed forces. The values of the Piola–Kirchhoff stress tensor of the first kind at the mesh points are determined using the values found for the distributed forces on the surfaces of the finite element mesh (including at the mesh points) using the Cauchy relations for the initial configuration. The linearized representation of this tensor enables all the derivatives of the increment in the strain vector with respect to the coordinates to be found without invoking the operation of differentiation. The particular features of the use of the stress recovery procedure are demonstrated for a plane problem in the non-linear theory of elasticity.     

Method of estimating transients in induction machines

The present paper is concerned with induction motors with wound and squirrel-cage rotors. It is assumed that the magnetic field produced by the stator windings is constant in magnitude and rotates with constant angular velocity. Differential equations that describe relations between the electromagnetic torque and main electric and mechanical quantities of the induction motors under consideration are derived in detail, the geometry of the rotors of motors being fully taken into account. A special nonsingular transformation is applied to the initial systems of equations (with angular coordinates) to split them up and reduce to a lower-order (in fact, third-order) system. A local stability analysis of the resulting equations is carried out. The stable equilibrium states that correspond to the operating modes of induction motors are determined. Methods of speed control of induction motors with wound and squirrel-cage rotors are considered. The limit load problem on motors and the control problem of the speed of motors are discussed; these problems lead to the necessity to estimate the transients in induction motors. The method of estimating transients that occur due to changes of the operating parameters of motors is developed based on a modification of the nonlocal reduction method. Using this method in combination with Barbashin and Tabueva’s methods to apply to the obtained systems made it possible to find analytical estimates of the ultimate permissible loads on induction motors and the control ranges of the parameters of the system that correspond to additional external active and inductive resistances. Moreover, estimates for the region of the attraction of stable equilibrium states of systems that describe the dynamics of induction motors are obtained.     

Modeling of electrodynamic-mechanical coupling phenomena in smart magnetoelectroelastic materials

The coupling of electric, magnetic and mechanical phenomena may have various reasons. The famous Maxwell equations of electrodynamics describe the interaction of transient magnetic and electric fields. On the constitutive level of dielectric materials, coupling mechanisms are manyfold comprising piezoelectric, magnetostrictive or magnetoelectric effects. Electromagnetically induced specific forces acting at the boundary and within the domain of a dielectric body are, within a continuum mechanics framework, commonly denoted as Maxwell stresses. In transient electromagnetic fields, the Poynting vector gives another contribution to mechanical stresses. First, a system of transient partial differential equations is presented. Introducing scalar and vector potentials for the electromagnetic fields and representing the mechanical strain by displacement fields, seven coupled differential equations govern the boundary value problem, accounting for linear constitutive equations of magnetoelectroelasticity. To reduce the effort of numerical solution, the system of equations is partly decoupled applying generalized forms of Coulomb and Lorenz gauge transformations [1,2]. A weak formulation is given to establish a basis for a finite element solution. The influence of constitutive magnetoelectric coupling on electromagnetic wave propagation is finally demonstrated with a simple one-dimensional example. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

一类指定应力问题的变分原理与应用

针对有限元分析中对应力或内力有指定条件的问题,引入非弹性应变作为实现指定应力条件的附加未知量,在小变形条件下描述了指定应力条件应当满足的弹性力学控制方程;以位移和未知非弹性应变作为独立变量,建立了具有指定应力条件问题的势能变分原理和虚功方程;以位移、弹性应变、未知非弹性应变和应力为独立变量,建立了一个含四类变量的广义变...     

Equations of anisotropic elastodynamics as a symmetric hyperbolic system: Deriving the time-dependent fundamental solution

The dynamic system of anisotropic elasticity from three second order partial differential equations is written in the form of the time-dependent first order symmetric hyperbolic system with respect to displacement velocity and stress components. A new method of deriving the time-dependent fundamental solution of the obtained system is suggested in this paper. This method consists of the following. The Fourier transform image of the fundamental solution with respect to a space variable is presented as a power series expansion relative to the Fourier parameters. Then explicit formulae for the coefficients of these power series are derived successively. Using these formulae the computer calculation of fundamental solution components (displacement velocity and stress components arising from pulse point forces) has been made for general anisotropic media (orthorhombic and monoclinic) and the simulation of elastic waves has been obtained. These computational examples confirm the robustness of the suggested method.     

State of stress of nonuniformly heated plates loaded along the boundary surfaces : PMM vol. 43. no. 6, 1979, pp. 1065–1072

The general solution of ati elasticity theory problem for a constant thickness plate is constructed under the condition that a force and a nonuniformly heated plate are applied normally to the boundary planes. The solution is obtained as a result of applying the M.E. Vashchenko-Zakharchenko expansion formulas to the infinitely high-order differential equations obtained by A.I. Lur'e by a symbolic method [1,2], by a separate analysis of the symmetric and antisymmetric elasticity theory problems relative to the middle plane: 1) for constant temperature and given forces on the boundary planes; 2) for a given nonuniform heating and no forces. Simple formulas are presented to determine the state of stress in the case of a slowly varying external load and temperature of the unbounded plate. For a bounded plate the general solution for no forces on the boundary planes and heating resulted in the A.I. Lur'e solution [1].     

Quadruple Trigonometrical Series Equations and Their Application to an Inclusion Problem in the Theory of Elasticity

Closed form solution of quadruple series equations involving cosine kernels has been obtained by reducing the series equations into triple Abel's type integral equations which in turn are reduced to a single integral equation. Making use of finite Hilbert transforms the solution of the single integral equation is obtained in closed form. This solution is used to solve an electrostatic problem. The results of this paper have also been used in a two-dimensional elastostatic problem under anti-plane shear and the effect of rigid line inclusions with thickness on the Griffith cracks has been examined. The expressions for shear stress and stress intensity factor at the tip of the crack are obtained. Finally, some numerical results for the stress intensity factor and shear stress distribution are obtained.     

非均匀半平面问题的弹性力学解

本文使用非均匀平面弹性力学的基本方程,通过富氏积分变换,求得了应力函数通解。在此基础上对弹性模量E(x)=E_oexp[βx]为指数型的非均匀半平面问题,具体求得了当边界上受任意载荷作用的精确解。最后经退化处理,还得到了有名的Boussnesq解,这说明本文的方法是成功的。     

Magnetic field effects on Newtonian and non-Newtonian ferrofluid flow past a circular cylinder

The changes in the flow properties under the action of electromagnetic body forces are investigated numerically for ferrofluid flow past a circular cylinder. Ferrofluid is modeled as both a Newtonian and a non-Newtonian Power-Law fluid. Magnetic forces are applied by placing magnets at different locations on the surface of the cylinder. The magnetostatic effects on the structure of the wake region, on drag reduction and on vortex formation length and frequency are shown and compared in terms of Reynolds number, interaction parameter, Power-Law index and magnet location. It is shown that the increase in the interaction parameter reduces drag for both Newtonian and non-Newtonian model. This decrease is observed to be higher for shear thinning and lower for shear thickening fluid compared to Newtonian case. It is also shown that vortex street formation in the wake region behind the cylinder may be delayed under high magnetic effects. The Strouhal number is higher for shear thinning case at both low and high Reynolds numbers, and lower for shear thickening case at high Reynolds numbers, compared to Newtonian fluid. The vortex formation frequency also decreases under the action of the magnetic field in all cases, however the vortex formation length increases. Placing the magnet towards the front region of the cylinder increases considerably the drag coefficient for both Newtonian and non-Newtonian model. This increase in drag coefficient is higher in the shear thinning fluid and lower in the shear thickening fluid compared to the Newtonian case.     

The local boundedness of the perturbed motions of an imperfect gyroscope in gimbals with dissipative and accelerating forces

An unbalanced dynamically symmetrical gyroscope in gimbals with constructive imperfections is considered in a central Newtonian field of forces. It is assumed that there is a moment of forces of viscous friction acting on the axis of rotation of one of the rings of the suspension and an accelerating (electromagnetic) moment applied to the axis of rotation of another ring. The equations of motion have a partial solution for which the basic plane of the frame is perpendicular to the direction from the specified fixed point of the frame to the centre of gravitation, the basic plane of the mantle is parallel to this direction and the rotor rotates with an arbitrary constant angular velocity.

The equations of perturbed motions of the reduced system with two degrees of freedom are obtained to within third-order terms at the corresponding position of equilibrium. In the domain of admissible values of the parameters F_o the characteristic equation of the system is considered and its coefficients are written down. A domain in F_o is specified in which complex conjugate pairs of the eigenvalues have small moduli of the real parts but the absolute values of the second- to fourth-order off-resonance mistuning between the imaginary parts are not small. For an imperfect gyroscope in gimbals with dissipative and accelerating forces the sufficient conditions of the local uniform boundedness of motions perturbed with respect to the specified partial solution are obtained in this domain. The conditions found provide the local uniform boundedness of solutions irrespective of the forms of higher than the third order in the equations of perturbed motions. These conditions are obtained in the form of constraints for the coefficients of the normal form and, finally, for the original parameters of the system and the real and imaginary parts of the eigenvalues. To provide a clear interpretation of the results, special cases when all but two parameters are fixed are analysed. The domains of local uniform boundedness are constructed in the two-dimensional domains F_o using a personal computer.