Sufficient Global Stability Condition for a Model of the Synchronous Electric Motor under Nonlinear Load Moment

We study a model of the synchronous electric motor, which is described by a system of ordinary differential equations, including equations for electric currents in the windings of the rotor. The load moment is assumed to be a nonlinear function of the angular velocity of the rotor, allowing a linear estimate. The system of differential equations under consideration has a countable number of stationary solutions corresponding to the operating mode of uniform rotation of the rotor with the angular velocity equal to the angular velocity of rotation of the magnetic field in the stator. An effective sufficient condition is derived under which any motion of the rotor of the synchronous electric motor tends with time to uniform rotation.     

Fokker-planck equation for brownian motion of rotating spherical particles : PMM vol. 40, n 6, 1976, pp. 1127–1131

The Langevin equation to derive the Fokker-Planck equation is used for the Brownian motion of particles in translational motion. The Fokker-Planck equation for the Brownian motion of particles which have, in addition to the translational velocity also an angular velocity, has not, so far, been derived. This can apparently be explained by the fact that in the case of the rotational motion, the Langevin equation for the translational motion velocity vector must be supplemented by a corresponding equation for an angular velocity vector. The latter equation must contain, in addition to the systematic moment of reaction linearly dependent on the angular velocity of rotation itself, a random moment rapidly varying with time. Moreover, to ensure the compatibility of two differential vector equations within the system, additional relations which must be introduced, must connect not only the coefficients of the systematic reactions, but also the. random vectors varying rapidly with time.In [1],the Boltzmann's equation for a mixture of two gases was used to derive a Fokker-Planck equation for a translational motion of Brownian particles. The same method can be applied to the Brownian motion of spherical particles which have, in addition to the translational velocities, angular velocities of self-rotations. In this case there is no need to introduce additional relations connecting the random rapidly varying vectors.In the present paper we derive the Fokker-Planck equations for a new model of rotating spherical molecules which was used in [2].     

Continual mechanics of monodisperse suspensions. rheological equations of state for suspensions of moderate concentration : PMM vol. 37, n6, 1973, pp. 1059–1077

Rheological relationships linking mean and moment stresses and, also, the force and moment of interphase reaction in a macroscopic flow of small solid sphere suspension with the kinematic characteristics of the flow are derived. This makes it possible to close the system of equations of suspension hydrodynamics. Coefficients of viscosity and of moment viscosity of a suspension are obtained and calculated.The equations of conservation of mass, momentum and moment of momentum of suspension and of its phases, considered (from the macroscopic point of view) to be coexistent continuous media, were formulated in a general form in [1]. These equations contain unknown vectors and tensors which define the interaction between the considered continuous media and, also, stresses and moment stresses appearing when these are in motion. To close the equations of conservation it is necessary to express all these quantities in terms of unknown variables of these equations (mean concentration of suspension, pressure in the fluid phases, and phase velocities). This problem is the second of the fundamental problems of hydromechanics of suspensions indicated in [1].Here this problem is solved with the use of a kind of self-consistent field theory, which is essentially an extension and generalization of methods developed in [2 – 7]. Expressions for all of the quantities mentioned above are derived. They can be considered to be rheological equations of state for suspensions. Expressions for the various coefficients of these equations and their dependence on parameters of phases and on the flow frequency spectrum are also considered.     

Conditions for the existence and stability of the regime of type 0/1 for an excited unbalanced rotor

Stationary regimes of motion of an unbalanced rotor with harmonic base excitation are investigated. The equation of motion of the rotor can be transformed into integro-differential equations for the so called slow and fast motions of these regimes using the method of direct separation of motion. The existence and stability conditions of special regimes of motion then can be derived by introducing the approximate solutions of the fast equation into the slow equation. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the dynamical reduction of the Vlasov equation

The elimination of a fast-time scale from the Vlasov equation by Lie-transform methods is an important step in deriving a reduced Vlasov equation such as the drift-kinetic Vlasov equation or the gyrokinetic Vlasov equation. It is shown here that this dynamical reduction also leads to the introduction of polarization and magnetization effects in the reduced Maxwell equations, which ensure that the reduced Vlasov–Maxwell equations possess an exact energy–momentum conservation law.     

Lagrange-Maxwell方程与发电机组磁饱和参数共振

推广了Lagrange-Maxwell方程,使之适用于分析发电机组磁饱和振动问题．应用机电分析动力学方法和电机学理论,找到了考虑磁饱和时发电机气隙磁场能,建立了发电机组电磁激发振动的非线性常微分方程组．发现在磁饱和状态下发电机电磁干扰力包括倍频参数激励成分．应用平均法求得系统主参数共振时的解,分析了发电机组电磁参数对共振特性的影响,揭示了一些新现象．     

Hamiltonian restriction of Vlasov equations to rotating isopycnic and isentropic two-layer equations

A direct link between a Vlasov equation and the equations of motion of a rotating fluid with an effective pressure depending only on a pseudo-density is illustrated. In this direct link, the resulting fluid equations necessarily appear in flux conservative form when there are no topographical and rotational terms. In contrast, multilayer isopycnic and isentropic equations used in atmosphere and ocean dynamics, in the absence of topographical and rotational terms, cannot be brought into a conservative flux form, and, hence, cannot be derived directly from the Vlasov equations. Another route is explored, therefore: deriving the Hamiltonian formulation of the two-layer isopycnic and isentropic equations as a restriction from a Hamiltonian formulation of two decoupled Vlasov equations. The work is motivated by our search for energy-preserving or even Hamiltonian (kinetic) numerical schemes.     

Weak solutions of the Vlasov–Poisson initial boundary value problem

This paper deals with existence results for a Vlasov-Poisson system, equipped with an absorbing-type law for the Vlasov equation and a Dirichlet-type boundary condition for the Poisson part. Using the ideas of Lions and Perthame [21], we prove the existence of a weak solution having good L^p estimates for moment and electric field, by a good control on the higher moments of the initial data. As an application, we establish a homogenization result in the Hilbertian framework for this type of problem in non-homogeneous media, following the work by Alexandre and Hamdache [2] for general kinetic equations, and Cioranescu and Mural [11] for the Laplace problem.     

Nonlinear dynamics and synchronization of coupled electromechanical systems with multiple functions

This paper deals with the nonlinear dynamics and synchronization of coupled electromechanical systems with multiple functions, described by an electrical Duffing oscillator magnetically coupled to linear mechanical oscillators. Firstly, the amplitudes of the sub- and super-harmonic oscillations for the resonant states are obtained and discussed using the multiple time scales method. The equations of motion are solved numerically using the Runge–Kutta algorithm. It is found that chaotic and periodic orbit coexist in the electromechanical system depending on the set of initial conditions. Secondly, the problem of synchronization dynamics of two coupled electromechanical systems both in the regular and chaotic states is also investigated, and estimation of the coupling coefficient under which synchronization and no-synchronization take place is made.     

An initial-boundary value problem of physiological significance for equations of nerve conduction

A model of nerve conduction that has gained wide acceptance in the biophysical community is that for the giant axon of the squid Loligo of Hodgkin and Huxley [5]. This nonlinear parabolic partial differential equation (PDE), when considered as part of a particular initial-boundary value problem (IBVP), models the electrical activity of an axon under conditions found both in nature and in the laboratory. This IBVP for the Hodgkin-Huxley equations is proved to be well posed in the sense of Hadamard [3], and a priori bounds on the solution are derived.     

On the rotordynamic stability of two-pole synchronous electric machinery considering different load cases

This paper discusses stationary rotordynamics of synchronous electric machinery, considering different load cases. The model comprises the electrical operation in a rigid network and in an isolated condition. The mechanical part is modelled as a Laval Rotor (Jeffcott Rotor) with a noncircular shaft, accounting for both static- and dynamic rotor eccentricities. The results show, that the machine's electrical operation may influence the occurence of mechanical vibrations significantly and therefore demonstrate the importance of analysing the electromechanical interaction. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rotor Stator Interaction with Many Degrees of Freedom

A many-degrees-of-freedom model of rotor stator contact is presented. The stability analysis of the synchronous motion of the simple JEFFCOTT rotor contacting a single-mass stator is extended to systems with many degrees of freedom. The stationary synchronous motion as well as its stability are validated by direct numerical integration of the differential equations. The resulting dynamics is exemplified in two examples, which may be quite different than the dynamics of the simple JEFFCOTT rotor. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Counting the Number of Eigenvalues in the Right Half-Plane for Spectral Problems Connected with Hyperbolic Systems. II. Differential Equations

This article is an immediate continuation of [1]. Solution of the Lyapunov equation leads to a boundary value problem for the first-order hyperbolic equations in two variables with data on the boundary of the unit square. In general, the problems of this kind are not normally solvable. We prove that the boundary value problems in question possess the Fredholm property under some conditions.     

On some nonlinear evolution equations with the strong dissipation,II

In this paper we continue the existence theories of classical solutions of nonlinear evolution equations with the strong dissipation studied in a previous paper [5]. In particular, we give sufficient conditions under which some of the equations have global solutions and at the same time we find steady state solutions of these equations which are exponentially stable as t → ∞. In the application, we improve the existence results to the equations which describe a local statement of balance of momentum for materials for which the stress is related to strain and strain rate through some constitutive equation (cf. Greenberg et al. [6], Greenberg [7], Davis [2], Clements [1], etc.).     

Controllability and stabilization of programmed motions of an automobile-type transport robot

A non-linear model of the motion of an automobile-type transport robot (TR) with absolutely rigid wheels, a steering device and actuators based on DC motors, is considered. Such a model for TR motion is a non-holonomic electromechanical system and, if the dynamics of the actuators and the steering device (forces of elasticity and attenuation in its elements) is ignored, corresponds to the model of automobile motion devised by Lineikin [1]. Non-linear canonical transformations of the state and control space coordinates are constructed which reduce the initial equations of motion of the TR to a simpler canonical form, convenient for the analysis and synthesis of control systems for the TR. These transformations are used to find the conditions for the controllability of the TR as a controlled object. Algorithms are given for constructing programmed controls and programmed motions of the TR. Stabilizing control laws are synthesized that make the programmed motions of the TR asymptotically stable and guarantee that the transients will have preassigned properties     

On the three-dimensional finite Larmor radius approximation: The case of electrons in a fixed background of ions

This paper is concerned with the analysis of a mathematical model arising in plasma physics, more specifically in fusion research. It directly follows, Han-Kwan (2010) [18], where the three-dimensional analysis of a Vlasov–Poisson equation with finite Larmor radius scaling was led, corresponding to the case of ions with massless electrons whose density follows a linearized Maxwell–Boltzmann law. We now consider the case of electrons in a background of fixed ions, which was only sketched in Han-Kwan (2010) [18]. Unfortunately, there is evidence that the formal limit is false in general. Nevertheless, we formally derive from the Vlasov–Poisson equation a fluid system for particular monokinetic data. We prove the local in time existence of analytic solutions and rigorously study the limit (when the inverse of the intensity of the magnetic field and the Debye length vanish) to a new anisotropic fluid system. This is achieved thanks to Cauchy–Kovalevskaya type techniques, as introduced by Caflisch (1990) [7] and Grenier (1996) [14]. We finally show that this approach fails in Sobolev regularity, due to multi-fluid instabilities.     

Control of large velocities in the two-dimensional gyrokinetic approximation

Consider the plane motion of a plasma subject to a magnetic field orthogonal to the plane. The equation on the density obtained in the gyrokinetic limit (as |B| tends to infinity), the so-called drift equation, lets appear a defect measure μ corresponding to a possible lack of energy at large velocities [F. Golse, L. Saint-Raymond, The guiding center approximation for the Vlasov–Poisson System, J. Math. Pures Appl. 78 (1999) 791–817]. In the present paper, it is proved that μ is symmetric and so is not involved in the drift equation. Moreover, for sufficiently smooth initial data, μ is actually equal to zero.     

Further development of the mathematical model of a snakeboard

This paper gives the further development for the mathematical model of a derivative of a skateboard known as the snakeboard. As against to the model, proposed by Lewis et al. [1] and investigated by various methods in [1–13], our model takes into account an opportunity that platforms of a snakeboard can rotate independently from each other. This assumption has been made earlier only by Golubev [13]. Equations of motion of the model are derived in the Gibbs-Appell form. Analytical and numerical investigations of these equations are fulfilled assuming harmonic excitations of the rotor and platforms angles. The basic snakeboard gaits are analyzed and shown to result from certain resonances in the rotor and platforms angle frequencies. All the obtained theoretical results are confirmed by numerical experiments.      

Dual integral equations related to the kontorovich-lebedev transform : PMM vol. 38, n 6, 1974, pp. 1090–1097

We study the dual integral equations related to the Kontorovich-Lebedev integral transforms arising in the course of solution of the problems of mathematical physics, in particular of the mixed boundary value problems for the wedge-shaped regions. We show that the solutions of these equations can be expressed in quadratures, using the auxilliary functions satisfying the integral Fredholm equation of second kind with a symmetric kernel.At present, the dual equations investigated in most detail are those connected with the Fourier and Hankel integral transforms. The results obtained and their applications are given in [1–3]. A large number of papers also deal with the theory and applications of the dual integral equations connected with the Mehler-Fock integral transform and its generalizations [4–11]., The dual integral transforms considered in the present paper belong to a more complex class than those listed above, and so far, no effective solution has been obtained for them. The only relevant results known to the authors are those in [12, 13]. In [12] a method of solving the equations (1.2) is given for a single particular value of the parameter γ = π/2, while in [13] the dual equations of the type under consideration are reduced to a solution of an infinite system of linear algebraic equations.