Two-dimensional problem of electrochemical metal machining

The two-dimensional problem of the electrochemical dimensional machining of a metal is investigated within the framework of the model of an ideal stationary process, which makes it possible to use the analogy with the problems of fluid flows with free surfaces. In the problem considered the cathode (machining tool) takes the form of two parallel semi-infinite rectangular electrodes. The blank (anode) is a half-plane whose boundary is perpendicular to the cathodes. Depending on the relationship between the physical and geometrical parameters of the problem, on the machined part (anode) a projection symmetrical about the center line between the cathodes may be formed. Additional mechanical machining of the part is then required. In order to exclude such solutions, a condition is obtained for the mathematical parameters which determine the solution of the problem in the auxiliary complex plane. General and particular limiting cases are considered. For the cases considered the calculation results are presented in the form of plots of the shape of the part machined.     

One problem of the theory of dimensional electrochemical machining of metals

A method is proposed for determining the shape of the anode-article boundary for a given shape of the cathode-tool in plane problems of the theory of dimensional electrochemical machining of metals. Under the assumptions used, the boundary of the anode-article is divided into the working zone, where metal dissolution occurs, and an adjacent zone, where the treatment (dissolution) is terminated. The initial problem is reduced to a problem of a fictitious plane-parallel potential flow of an ideal fluid with a nonlinear condition on the free surface. The point of separation of the fictitious flow from the solid boundary corresponds to the point separating these two zones of the anode boundary. The Brillouin-Will condition of smooth separation is imposed at the separation point to construct a closed system of equations determining the problem solution. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 3, pp. 214–220, May–June, 2009.     

Modeling of electrochemical machining with the use of a curvilinear electrode and a stepwise dependence of the current efficiency on the current density

An analytical solution of the problem of electrochemical machining of metals by a curvilinear cathode tool with allowance for a discontinuous function that describes the dependence of the current efficiency on the current density is obtained. According to the hydrodynamic interpretation, the original problem reduces to the problem of the theory of ideal fluid flows with a free surface. It is demonstrated that the use of the proposed dependence of the current efficiency on the current density ensures the existence of three domains on an unknown treated surface; these domains have different laws of the distribution of the charge fraction spent on metal dissolution. Results calculated for various particular cases are presented.     

Evolution of the Shape of the Anode Boundary under Electrochemical Dimensional Machining of Metals

This paper presents a method for calculating the anode boundary under unsteady conditions of electrochemical dimensional machining of metals. The plane quasistationary problem of determining the shape of the anode boundary for various machining times is considered.     

Simulation of precision electrochemical machining of metals by a segmented cathode

Bulge formation in the process of electrochemical machining by a flat tool electrode with an insulated area is studied. Based on a step function of the current efficiency, precision machining is simulated. The solution of a nonstationary problem shows that a limiting mode with the current density equal to its critical value is established in a finite time.     

Modeling of Electrochemical Copying in a Finite-Width Cell

The problem of modeling of electrochemical machining is reduced to the solution of the Schwartz problem on a parametrical rectangle with the use of theta-functions. Various conditions (non-equipotentiality of electrodes and inconstancy of current efficiency) at the boundary of a processed surface are considered. Nonstationary, quasistationary, stationary, and limit solutions are studied. Results of machining of surfaces by tool electrodes of various shapes are given. It is shown that machining mode parameters significantly affect the dissolved layer size necessary for obtaining high-precision copying.     

Al2O3陶瓷引弧微爆炸加工温度场模拟

建立了Al2O3陶瓷引弧微爆炸加工(micro-detonationofstrikingarcmachining,MDSAM)过程的 传热模型,基于有限元理论,利用ANSYS软件对加工过程中的温度场分布进行了模拟。结合材料性质,对模 拟和实验得到的蚀坑尺寸进行了比较,并分析了加工参数对温度场的影响。模拟结果表明,Al2O3陶瓷引弧 微爆炸加工时在给定的加工参数下的最高温度可达13435℃,且高温影响区范围很小,加工实验与模拟结果 符合较好。随着脉冲宽度和工作电流的增加,加工区域的温度以及蚀坑的半径和深度增大;随着喷嘴半径的 增大,加工区域的温度降低而蚀坑的径深比增大。模拟结果可为Al2O3 陶瓷引弧微爆炸加工过程中表面形 貌的预测、材料去除机理的揭示以及加工参数的选择等提供参考。     

Deviations Induced by Tool Sharpening in the Profile of Three Screw Pump Rotors

Three screw pumps represent an important family of positive displacement rotary pumps. They are quite easy to construct, even if the use of a shaped milling cutter for the rotor machining can cause some problems when the tool is not new. In fact the sharpening of the tool modifies the total geometry of the milling cutter by reducing its diameter. The shape of the single cutter does not change, but the cutter radial position is shifted towards the tool center. This causes deviations in the profile cut and the screws consequently need a long breaking-in. In this paper, the problem is analyzed from a quantitative point of view, using a method that allows us to determine the machining error as a function of the tool-geometry variation. Moreover, some alternative solutions to this problem are suggested.     

Instability conditions due to structural nonlinearities in regenerative chatter

Chatter is an instability condition in machining processes characterized by nonlinear behavior, such as the presence of limit cycles, jump phenomenon, subcritical Hopf and period doubling bifurcations. Although the use of nonlinear techniques has provided a better understanding of chatter, neither a unifying model nor an exact solution has yet been developed due to the intricacy of the problem. This work proposes a weakly nonlinear model with square and cubic terms in both structural stiffness and regenerative terms, to represent self-excited vibrations in machining. An approximate solution is derived by using the method of multiple scales. In addition, a qualitative analysis of the effect of the nonlinear parameters on the stability of the system is performed. The structural cubic term gives a better representation of the nonlinear behavior, whereas the square term represents a distant attractor in the stability chart. Instability due to subcritical Hopf bifurcations is established in terms of the eigenvalues of the model in normal form. An important contribution of this analysis is the representation of hysteresis in terms of new lobes within the conventional stability limits, useful in restoring stability. This analysis leads to a further understanding of the nonlinear behavior of regenerative chatter.     

A modified Green–Lindsay thermoelasticity with strain rate to eliminate the discontinuity

Probing the mechanism of ultrafast thermoelastic processes is becoming increasingly important with the development of laser-assisted micro/nano machining. Although thermoelastic models containing temperature rate have been historically proposed, the strain rate has not been considered yet. In this work, a generalized thermoelastic model is theoretically established by introducing the strain rate in Green–Lindsay (GL) thermoelastic model with the aid of extended thermodynamics. Numerically, a semi-infinite one-dimensional problem is considered with traction free at one end and subjected to a temperature rise. The problem is solved using the Laplace transform method, and the transient responses, i.e. displacement, temperature and stresses are graphically depicted. Interestingly, it is found that the strain rate may eliminate the discontinuity of the displacement at the elastic and thermal wave front. Also, the present model is compared with Green–Naghdi (GN) models. It is found that the thermal wave speed of the present model is faster than GN model without energy dissipation, and slower than GN model with energy dissipation. In addition, the thermoelastic responses from the present model are the largest. The present model based upon GL model is free of the jump of GL model in the displacement distribution, and is safer in engineering practices than GN model. The present work will benefit the theoretical modeling and numerical prediction of thermoelastic process, especially for those under extreme fast heating.     

An explicit formulation for the evolution of nonlinear surface waves interacting with a submerged body

An explicit formulation to study nonlinear waves interacting with a submerged body in an ideal fluid of infinite depth is presented. The formulation allows one to decompose the nonlinear wave–body interaction problem into body and free‐surface problems. After the decomposition, the body problem satisfies a modified body boundary condition in an unbounded fluid domain, while the free‐surface problem satisfies modified nonlinear free‐surface boundary conditions. It is then shown that the nonlinear free‐surface problem can be further reduced to a closed system of two nonlinear evolution equations expanded in infinite series for the free‐surface elevation and the velocity potential at the free surface. For numerical experiments, the body problem is solved using a distribution of singularities along the body surface and the system of evolution equations, truncated at third order in wave steepness, is then solved using a pseudo‐spectral method based on the fast Fourier transform. A circular cylinder translating steadily near the free surface is considered and it is found that our numerical solutions show excellent agreement with the fully nonlinear solution using a boundary integral method. We further validate our solutions for a submerged circular cylinder oscillating vertically or fixed under incoming nonlinear waves with other analytical and numerical results. Copyright © 2007 John Wiley & Sons, Ltd.     

Using spline-approximation to solve problems of axisymmetric free vibration of thick-walled orthotropic cylinders

The three-dimensional theory of elasticity is used to study the free vibrations of an anisotropic hollow cylinder with different boundary conditions at the ends. The relevant problem is solved by a numerical-and-analytic method. Spline approximation and collocation is used to reduce the partial differential equations of elasticity to a boundary-value problem for a system of ordinary differential equations of high order for the radial coordinate, which is solved using the stable discrete-orthogonalization and incremental-search methods. The calculated results for an orthotropic inhomogeneous cylinder with boundary conditions of several types are presented Translated from Prikladnaya Mekhanika, Vol. 44, No. 10, pp. 74–85, October 2008.     

One scheme of electrochemical machining of metals by a curvilinear electrode tool

The nonlinear plane problem of the evolution of the shape of the metal surface (anode) during electrochemical machining by a curvilinear cathode of symmetric shape is solved. A condition is obtained which allows one to determine the position of the point of transition from the zone of anodic metal dissolution to the region in which machining stops.     

Modeling of isothermal processes in porous materials on the basis of the concept of an ensemble of pores

A class of models of porous media based on the concept of an ensemble of pores with a certain distribution of the main geometric parameters (for example, the pore size) is considered. The cases of pores saturated with single-and two-phase multicomponent liquid mixtures are investigated. The properties of equilibrium states of the mixture are derived from the minimum free energy condition and the transfer laws from the decreasing free energy condition. The hydrodynamic connectivity of the pores is described by two types of kernels: one describes the spatial connectivity and the other the connectivity in an elementary macrovolume. Analytically and numerically, the one-dimensional problems of establishment of a steady-state regime, propagation of a passive admixture, and two-phase flow (an analog of the Buckley-Leverett problem) are investigated. A relationship between the models in question and relaxational filtration models is demonstrated. A simple model of capillary hysteresis related with the non-monotonicity of the pore area to volume ratio function is proposed.     

工程陶瓷引弧微爆炸加工冲击力建模及实验分析

介绍了引弧微爆炸加工过程中冲击力产生的原因,利用单因素实验确定影响冲击力的主要工艺参 数,利用正交实验和回归分析的方法建立了冲击力指数型经验模型,得出工艺参数对冲击力的影响规律。实 验结果表明:工作电流、工作气压和微爆炸发生器喷嘴直径是影响冲击力大小的主要工艺参数,冲击力随着工 作电流的增大而减小,随着工作气压和喷嘴直径的增大而增大,而工作脉宽和工作距离对冲击力大小影响不 显著;建立的冲击力模型与实验结果吻合良好。研究结果可以为引弧微爆炸加工过程的控制提供参考依据。     

Steady-state conditions of a nonisothermal film with a heat-insulated free boundary

The equilibrium shapes of a nonisothermal liquid film with a heat-insulated free surface for large Marangoni numbers are investigated in the long-wave approximation using a combination of analytical and numerical methods. It is proved that the two-dimensional problem of the equilibrium of a strip-shaped film has a steady-state solution for an arbitrary large temperature gradient on the boundaries of the strip. An increase in this gradient leads to an abrupt thinning of the film near the heated boundary, which can result in instability and rupture of the film. In the equilibrium problem for a film fixed on a circular contour, the nonuniform distribution of the heat flux on the contour was found to have a significant influence on the free-surface shape. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 4, pp. 59–73, July–August, 2008.     

Form of a free surface during steady flow of a capillary fluid in a rectangular channel

The two-dimensional problem of the form of a free surface of an ideal incompressible fluid during steady flow from a rectangular channel through a thin slot with simultaneous uniform delivery of fluid through the side walls is examined. Forces of gravity and surface tension are taken into account. The nonlinear problem of the simultaneous determination of the free surface and velocity field of the fluid is solved by the iteration method. Convergence of the iterations to the solution of the problem for small values of the parameters is investigated. The solution of the linearized problem is obtained in a closed form for a small depth of the discharge and small width of the channel, which is compared with the solution of the problem in a complete formulation. Graphs of the free surface of the fluid for different values of the parameters, obtained as a result of numerical solution of the nonlinear problem, are presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 67–75, January–February, 1977.     

Unsteady motion of an elliptic cylinder under a free surface

A problem of generation of nonlinear unsteady waves on the surface of an ideal liquid in an infinitely deep fluid due to the motion of a submerged elliptic cylinder is considered. The initial formulation of the problem is reduced to an integrodifferential system of equations for the function defining the free surface shape and for the normal and tangential components of velocity on the free surface. The small-time asymptotics of the solution is constructed for the case of cylinder motion with a constant acceleration from the state at rest.     

A moving boundary hyperbolic problem for a stress impact in a bar of rate-type material

In this paper we present some results obtained by studying the mathematical model describing a moving boundary hyperbolic problem related to a time dependent stress impact in a bar of Maxwell-like material. Due to the impact a shock front propagates with a finite speed. Here our interest is to underline the influence of the dissipative term on the propagation of the shock front.

In the framework of the similarity analysis we are able to reduce the moving boundary hyperbolic problem to a free boundary value problem for an ordinary differential system. It is then possible, by applying two numerical transformation methods, to solve the free boundary value problem numerically. The influence of the dissipative term is evident: the free boundary (that defines the shock front propagation) is an increasing function of the dissipative coefficient.     

Free oscillations of a fluid in a cylindrical container with arbitrary axisymmetric bottom and elastic elements on the free surface of the fluid

We consider the problem of free oscillations of an ideal fluid in a container that has the form of a right circular cylinder with arbitrary axisymmetric bottom in the case where the unperturbed free surface of the fluid is covered by an elastic membrane or plate. Using the expansion in eigenfunctions of an auxiliary spectral problem with a parameter in boundary conditions and the method of decomposition of the domain of meridional cross-section of a container, we obtain an analytic solution of the problem. Individual examples of mechanical systems are considered, for which we construct solutions by using the proposed algorithm, analyze these solutions, and compute the frequencies and forms of oscillations.