两自由度振动系统的斜碰撞分析

研究斜碰撞振动系统动力学的一个关键问题是对系统在碰撞前后的状态进行合理描述和正确计算．针对两弹性体斜碰撞问题，基于瞬间碰撞假设，提出了采用步进冲量来分析和求解斜碰撞前后的状态关系；并以弹簧摆和振子组成的两自由度斜碰撞振动系统为例，具体介绍了该算法如何实现．用解析方法讨论了该系统在斜碰撞过程中可能出现的各种力学现象，将冲量步进算法得到的数值解与解析结果进行对比，取得了完全一致的结果．该数值方法能适应多种斜碰撞问题的计算．     

Method of calculation of interaction of wall flows

A flow of viscous compressible fluid in the neighborhood of the line of interaction of wall flows is considered. A method of calculating the line of interaction and the direction of the self-induced secondary flow is developed. Papers [1–3] are devoted to the simulation of a separation flow with singularities in the neighborhood of singular lines and points, where boundary-layer equations are invalid. However, the theories of local separation used at present have mainly been developed only for two-dimensional problems, while the models of viscous-inviscid interaction have restrictions in application for turbulent flows with developed separation. The interaction of three-dimensional wall turbulent flows is considered below. It is assumed that the thickness of the boundary layers and the scales of the interaction zones are small in comparison with the characteristic dimension of the system, while the line of discontinuity of the solutions of the three-dimensional boundary layer equations is the same as the line of interaction of the wall flows.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 53–59, March–April, 1987.The author is grateful to G. Yu. Stepanov and V. N. Ershov for their interest in my work and their valuable remarks.     

The uniqueness of the solution of boundary-value problems of a thermal model of discharge

The paper is devoted to a nonlinear analysis of superheating [1, 2] instability of an electric discharge stabilized by electrodes [3] in the framework of a thermal model [4] where the stability of the discharge relative to the long-wave and short-wave perturbations is proved in a linear approximation. Similar boundary-value problems arise in the theories of chemically and biologically reacting mixtures [5–7], thermal breakdown of dielectrics [8], thermal explosion [9], in the investigation of nonlinear waves in semiconductors and superconductors [10, 11], and in the investigation of Couette flow with variable viscosity [12]. The uniqueness of the one-dimensional steady solutions of the thermal model of discharge and the stability relative to the small spatial perturbations, respectively, for the exponential and step dependence of the electrical conductivity on the temperature are proved in [3, 13]. The uniqueness of the solutions in the one-dimensional case for the same electrode temperature and arbitrary dependences of the electrical and thermal conductivity on the temperature is established in paper [14]. In the present paper, the existence and uniqueness of steady solutions of the thermal model of discharge in a three-dimensional formulation for arbitrary fairly smooth electrical and thermal conductivity functions of the temperature in the case of isothermal isopotential electrodes are proved analytically.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 140–145, January–February, 1986.The author expresses his gratitude to A. G. Kulikovskii and A. A. Barmin for the formulation of the problem and their discussions.     

Homogenization of the equations of the Cosserat theory of elasticity of inhomogeneous bodies

The paper deals with the homogenization of a boundary value problem for an inhomogeneous body with Cosserat properties, which is referred to as the original problem. The homogenization process is understood as a method for representing the solution of the original problem in terms of the solution of precisely the same problem for a body with homogeneous properties. The problem for a body with homogeneous properties is called the accompanying problem, and the body itself, the accompanying homogeneous body. As a rule, a constructive homogenization procedure includes the following three stages: at the first stage, the properties of the inhomogeneous body are used to find the properties of the accompanying homogeneous body (efficient properties); at the second stage, the boundary value problem is solved for the accompanying body; at the third stage, the solution of the accompanying problem is used to find the solution of the original problem. This approach was implemented in mechanics of composite materials constructed of numerous representative elements. A significant contribution to the development of mechanics of composites is due to Rabotnov [1–3] and his students. Recently, the homogenization method has been widely used to solve problems for composites of regular structure by expanding the solution of the original problem in a power series in a small geometric parameter equal to the ratio of the characteristic dimension of the periodicity cell to the characteristic dimension of the entire body. The papers by Bakhvalov [4–6] and Pobedrya [7] were the first in the field. At present, there are numerous monographs partially or completely dealing with the method of a small geometric parameter [8–14]. Isolated problems for inhomogeneous bodies with nonperiodic dependence of their properties on the coordinates were considered by many authors. Most of such papers published before 1973 are collected in two vast bibliographic indices [15, 16]. General methods were considered, and many specific problems of the theory of elasticity of continuously inhomogeneous bodies were solved in Lomakin’s papers and his monograph [17]. The theory of torsion of inhomogeneous anisotropic rods was considered in [18]. In 1991, in his Doctoral dissertation, one of the authors of this paper proposed a version of the homogenization method based on an integral formula representing the solution of the original static problem of inhomogeneous elasticity via the solution of the accompanying problem [19, 20]. An integral formula for the dynamic problem of elasticity was published somewhat later [21]. This integral formula was used to develop a constructive method for the homogenization of the dynamic problem of inhomogeneous elasticity, which can be used in the case of both periodic and nonperiodic inhomogeneity of the properties [22]. The integral formula in the case of the Cosserat theory of elasticity was published in [23]. The present paper briefly presents constructive methods for homogenizing the problems of the Cosserat theory of elasticity based on the integral formula.     

Experimental verification of the theory of thixotropy of viscoelastic polymer media

Previous studies [1, 2] were devoted to the formulation of a phenomenological theory of the reversible changes (thixotropy) in the physicomechanical properties of viscoelastic media, in particular, polymer systems. The basis of this theory is the idea that the reversible changes in the structure and properties of viscoelastic thixotropic media under the influence of mechanical action are due to a change in their relaxation spectrum. In this case the behavior of the mechanical properties is entirely and uniquely determined by two material functions which completely characterize the given material: the relaxation time (frequency) distribution function, which describes the behavior of the material in the linear region of deformation, and the thixotropy function, which determines the nature of the changes compression) of the relaxation spectrum on transition to the nonlinear region. The object of this approach is to reflect the effect of the change in supramolecular and molecular structures associated with the flow of polymer system, or in the bonds and orientation of the particles in disperse systems, on the viscoelastic and viscosity properties of the systems concerned. The essential validity of this approach was demonstrated in [3], where it was shown that at sufficiently high strain rates changes occur in the structure of the polymer leading to a change in mechanical properties. In [4] the basic structural functions were found. These, in conjunction with the proposed theory of thixotropy, characterize the behavior of a broad group of polymer systems in both the linear and the nonlinear regions of deformation. In this connection it should be noted that whereas for the linear region the accuracy of the laws obtained lies within the limits ± 100%, on transition to the nonlinear region the error may increase, Therefore, for the purposes of a rigorous quantitative verification of the theory we shall use not the universal functions obtained in [4], but the more precise characteristics of the specific material on which the experiments are performed, since those effects with respect to which it is desired to test the theory usually lie within the limits of 30% of the measured quantities.The authors thank G. V. Vinogradov for organizing and discussing their work.     

A study of influence of gravity on bulk behaviour of particulate solid

This paper examines the influence of gravity on the bulk responses of a granular solid. The loading scenarios in this study include confined compression, rod penetration into a granular medium and discharging through an orifice. Similar loading and flow conditions are likely to be encountered in the stress and deformation regimes that regoliths are subjected to in extraterrestrial exploration activities including in situ resource utilisation processes. Both spherical and non-spherical particles were studied using the discrete element method （DEM）. Whilst DEM is increasingly used to model granular solids, careful validations of the simulation outcomes are rather rare. Thus in addition to exploring the effect of gravity, this paper also compares DEM simulations with experiments under terrestrial condition to verify whether DEM can produce satisfactory predictions. The terrestrial experiments were conducted with great care and simulated closely using DEM. The key mechanical and geometrical properties for the particles were measured in laboratory tests for use in the DEM simulations. A series of DEM computations were then performed under reduced gravity to simulate these experiments under extraterrestrial environment. It was found that gravity has no noticeable effect on the force transmission in the confined compression case; the loading gradient in the rod penetration is linearly proportional to the gravity; the mass flow rate in silo discharge is proportional to square root of the gravity and the angle of repose increases with reducing gravity. These findings are in agreement with expectation and existing scientific evidence.     

Mathematical simulation of hydrodynamic generators of oscillations

Unsteady turbulent swirled water flow in a channel in the presence of cavitation is calculated. It is shown that in the near-axial channel zone a fluctuating vapor region, or cavity, arises and variations in the cavity shape and dimensions lead to the onset of undamped pressure fluctuations. The amplitude-frequency characteristics of the oscillations are obtained in different channel cross-sections. The amplitude maximum position in the channel expansion zone is in agreement with the available experimental data. The dynamics of toroidal vortices formed in the hydrodynamic generator channel and in the expansion zone at its exit are established.     

Assessment of the performance of different classes of turbulence models in a wide range of non-equilibrium flows

The performance of thirteen benchmark turbulence models within the RANS framework has been assessed in classical non-equilibrium flows. Linear and non-linear eddy-viscosity schemes, Reynolds stress transport models and single- and two-time-scale approaches have been considered in the investigation. Among the test cases studied are homogeneous shear and normally strained flows, adverse-pressure-gradient, favourable-pressure-gradient and oscillatory boundary layer flows, fully developed oscillatory and ramp up pipe flows and steady and pulsated backward-facing-step flows. The main advantages and drawbacks of the models in each of the test cases are discussed. These discussions provide a reasonably wide understanding of the expected behaviour of the models for future applications in non-equilibrium flows, and also result in suggestions on how the effectiveness of existing models can be further improved.     

A study of influence of gravity on bulk behavior of particulate solid

This paper examines the influence of gravity on the bulk responses of a granular solid. The loading scenarios in this study include confined compression, rod penetration into a granular medium and discharging through an orifice. Similar loading and flow conditions are likely to be encountered in the stress and deformation regimes that regoliths are subjected to in extraterrestrial exploration activities including in situ resource utilisation processes. Both spherical and non-spherical particles were studied using the discrete element method (DEM). Whilst DEM is increasingly used to model granular solids, careful validations of the simulation outcomes are rather rare. Thus in addition to exploring the effect of gravity, this paper also compares DEM simulations with experiments under terrestrial condition to verify whether DEM can produce satisfactory predictions.The terrestrial experiments were conducted with great care and simulated closely using DEM. The key mechanical and geometrical properties for the particles were measured in laboratory tests for use in the DEM simulations. A series of DEM computations were then performed under reduced gravity to simulate these experiments under extraterrestrial environment. It was found that gravity has no noticeable effect on the force transmission in the confined compression case; the loading gradient in the rod penetration is linearly proportional to the gravity; the mass flow rate in silo discharge is proportional to square root of the gravity and the angle of repose increases with reducing gravity. These findings are in agreement with expectation and existing scientific evidence.     

One type of solutions of conduction of nervous pulses and the stability of action potential

This paper reports a type of laws which governs action poten-tial of nervous impulses,and it is discussed by general form-nonlinear dispersive process.We find that the nervous wave is aslowly varying amplitude solitary wave in the small dispersivecase.we prove that the solitary wave is not generated in the or-dinary dispersion,but a travelling wave with varying amplitudesmay be obtained.The stability of various possible action poten-tials and bifurcation in overdamped case are also discussed inthis paper.     

Degree of Stability of Equilibrium

ABSTRACT

A theoretical study is made of the hysteretic behavior of a portal frame under constant gravity and variable horizontal loading. Special account is taken of the variation of axial forces in the columns, the model simulating the plastic interaction of the overall shearing and bending in tall rectangular frames and battened columns. The assumed elastic-perfectly plastic behavior of columns makes a mathematical formulation feasible, and a detailed examination is given of the cyclic alternate displacement loading with the P-Δ-A effect taken into consideration. This example clarifies characteristic features of repeated loading such as shakedown, incremental collapse, and alternating plasticity under the combined action of bending moment and axial force existing in a frame member.     

Motion equations of the dynamic theory of consolidation from the point of view of the theory of transient pipe flows

An elastic fluid-saturated porous medium is modeled as a bundle of parallel cylindrical tubes aligned in a direction parallel to the fluid movement. The pore space is filled with viscous compressible liquid. A cell model and the theory of transient pipe flow are used to derive one-dimensional governing equations in such media. All macroscopic constants in these equations are defined by the individual material constants of the fluid and solid. The interaction force includes an additional term not found in Biot's theory.     

The solution of the variational problem of the construction of the contour of a two-regime pipe

This paper is devoted to the problem of the optimal shaping of the supersonic part of a pipe in those cases when the engine is designed to operate in two essentially distinct regimes. The different regimes may be determined, for example, by a difference in fuel consumption, in the configuration of the pipe, which may be a consequence of some adjustment in the conditions of the external medium, etc. We consider two cases. In one of these the pipe may be controlled by a central body moved into it, and in this case only two regimes are possible for such a pipe, i.e., when there is a central body and when the central body is absent. In the second, the flow parameters at the inlet of the pipe, which has no subsonic part, vary for some reason while the configuration of the pipe is fixed (for example, due to a change in fuel consumption or in the parameters at the entrance to the air intake, etc.). We obtain the necessary conditions determining the shape of the optimal contour, and to construct it we develop a numerical algorithm. We give examples of optimal pipes, the contours of which are constructed by the abovementioned algorithm and compare them with pipes chosen in accordance with other considerations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 96–103, July–August, 1973.The author wishes to thank A. N. Kraiko for drawing his attention to the problem.     

Theory of the combustion of small drops of metal

The article discusses the combustion of small drops of metal. It is postulated that the formation of an oxide in the liquid phase starts with the origin of a condensed phase and continues as the result of a reaction between the vapors of the drop and the oxidizer at the surface of the forming particles of the condensed phase. It is shown that the process of the formation of particles of condensed oxide in the gas, for very small drops, has an essentially unsteady-state character. Under these circumstances, a considerable fraction of the vaporization products of a drop does not succeed in condensing after the complete gasification of the drop and remains in the gaseous state.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 47–53, March–April, 1976.     

Calculation of autooscillations of the type of azimuthal waves occurring at loss of stability of flow of a viscous fluid between concentric cylinders rotating in opposite directions

Starting with the Navier-Stokes equation we use the Lyapunov-Schmidt method to investigate the nature of the loss of stability of Couette flow between cylinders as the Reynolds number passes through its critical value. We consider the rotation of the cylinders in opposite directions with the ratio of the angular velocities such that the role of the most dangerous disturbances passes over from rotationally symmetric to nonrotationally symmetric disturbances. Branching nonstationary secondary flows (autooscillations) are found in the form of azimuthal waves; the longitudinal wave number and the azimuthal wave number m are assumed given. The amplitude of autooscillations and the wave velocity are calculated for m = 1, and it is shown that depending on the value of both weak excitation of stable and strong excitation of unstable autooscillations are possible and the wave number for which the critical Reynolds number is a minimum corresponds to a stable wave regime in the supercritical region. The linear problem of the stability of the circular flow of a viscous fluid with respect to nonrotationally symmetric disturbances is discussed in [1–3]. Di Prima [1] solved the problem numerically by the Galerkin method when the gap is small and the cylinders rotate in the same direction. Di Prima's analysis is extended in [2] to cylinders rotating in opposite directions, and in [3] it is extended to gaps which are not small. The nonlinear stability problem is treated in [4], where for fixed = 3 and cylinders rotating in opposite directions the axisymmetric stationary secondary flow the Taylor vortex is calculated. The formation of azimuthal waves in the fluid between the cylinders was studied experimentally in detail by Coles [5].Translated from Zhurnal Prikladnoi Mekhanika i Tekhnicheskoi Fiziki, No. 2, pp. 68–75, March–April, 1976.     

Effect of viscoelastic properties of a liquid on the dynamics of small oscillations of a gas bubble

A series of papers has been devoted to questions of gas bubble dynamics in viscoeiastic liquids. Of these papers we mention [1–4]. The radial oscillations of a gas bubble in an incompressible viscoeiastic liquid have been studied numerically in [1, 2] using Oldroyd's model [5]. Anexact solution was found in [3], and independently in [4], for the equation of small density oscillations of a cavity in an Oldroyd medium when there is a periodic pressure change at infinity. The analysis of bubble oscillations in a viscoeiastic liquid is complicated by properties of limiting transitions in the rheological equation of the medium. These properties are of particular interest for the problem under investigation. These properties are discussed below, and characteristics of the small oscillations of a bubble in an Oldroyd medium are investigated on the basis of a numerical analysis of the exact solution obtained in [3].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 82–87, May–June, 1976.The authors are grateful to V. N. Nikolaevskii for useful advice and for discussing the results.