Cauchy Form Research Articles

A generalized continuum theory with internal corner and surface contact interactions

We consider a classical derivation of a continuum theory, based on the fundamental balance laws of mass and momenta, for a body with internal corner and surface contact interactions. The balances of mass and linear and angular momentum are applied to the arbitrary parts of a continuum which supports non-classical internal corner and surface contact interactions. The form of the specific corner contact interaction force measured per unit length of the corner is derived. A generalized form of Cauchy’s stress theorem is obtained, which shows that the surface traction on an oriented surface depends in a specific way on both the oriented unit normal as well as the curvature of the surface. An explicit form of the surface-couple traction which acts on every oriented surface is obtained. Two fields in the continuum, which are denoted as stress and hyperstress fields, are shown to exist, and their role in representing the surface traction and the surface-couple traction is identified. Finally, the field equations for this theory are determined, and a fundamental power theorem is derived. In the absence of internal corner and surface-couple traction interactions, the equations of classical continuum mechanics are recovered. There is no appeal to any ‘principle of virtual power’ in this work.

Mathematical modeling of transients in the mine electric network using matrix-topological method

It is developed a mathematical model of the mine electric network, which allows us to analyze the transient and steady-state processes in normal and emergency modes. The model is a matrix differential equation in the form of Cauchy, compiled using matrix-topological method, which is different, taking into account the mutual influence of network components. The relevance of the analysis of transients in the mine electric network due to low precision of existing methods and the need to improve the technical and economic performance of the mining equipment. This mathematical model is applied in the form of a computer program with a graphical interface, which automatically calculates the matrix coefficients of the differential equation of condition of mine electric network and integrates its by hard-sustainable Gear-Nordsieck method. Using the proposed computer program by design organizations will increase the efficiency of the design process and increase the accuracy of determining the settings of protective devices.

Решение неоднородных дифференциально-алгебраических систем уравнений электрических цепей с помощью табличной записи уравнений и рядов Тейлора

Previously known method of solving systems of linear ordinary differential equations using Taylor Series is adapted for solving non-linear equations and non-homogeneous equations with any right part. Specific feature of non-linear problems is solving of linear equations many times at each time step. Proposed approach takes into account this feature. New recurrence formulae suitable for solving systems of linear algebraic equations by iterational methods were derived. These methods, including Conjugate Gradient Method, effectively use sparsity of matrixes of the systems and also proximity of initial approximation to the solution. Table form of equations of electrical circuits is used because it is simple and universal. This form allows not form matrix of the system of equations in apparent avatar and saves computation work and time. Table form of system of equations gives easy possibility to transform initial differential-algebraic equations of electrical circuits to the system of ordinary differential equations in Cauchy form.

Analysis of Nonlinear Stresses and Strains in a Thin Current-Carrying Elastic Plate

Two-dimensional magnetoelastic problems of a thin current-carrying plate under the interaction of an unsteady electromagnetic field and a mechanical field are studied. The nonlinear magnetoelastic kinetic equations, geometric equations, physical equations, electrodynamics equations, and expressions for the Lorentz force of a thin current-carrying plate under the action of a coupled field are given. The normal Cauchy form nonlinear differential equations, which include ten basic unknown functions in all, are obtained by the variable replacement method. Using the difference and quasi-linearization methods, the nonlinear magnetoelastic equations are reduced to a sequence of quasilinear differential equations, which can be solved by the discrete-orthogonalization method. Numerical solutions for the magnetoelastic stresses and strains in a thin current-carrying elastic plate are obtained by considering a specific example. The results that the stresses and strains in the thin current-carrying elastic plate change with variation of the electromagnetic parameters are discussed. The results show that the stress–strain state in thin plates can be controlled by changing the electromagnetic parameters. This provides a method of theoretical analysis and numerical calculation for changing the service conditions and intensity research of thin plates of engineering structures in the electromagnetic field

The Analysis of Electric Machines Electro-Mechanic Processes

The rapid development of modeling of various electromechanic energy transformers (power transformers, electric machines) over the past years created the necessity to develop the systematic method of quality and quantity model analysis. Typical analysis of model includes the following stages: the calculation of transient processes, the calculation of established processes, the evaluation of static stability, and the parametric sensitiveness defining. Nowadays, only the analysis of transient processes is performed continuously, i.e. in the state cognate with the physical nature of a process. Other stages are studied in discrete time. The variety of mathematical apparatus used on each stage complicates the analysis and requires significant expenses. In this article, the author describes the method of solving all four tasks on the basis of a unified mathematic apparatus - general theory of differential equations based on a unified algorithm. The completion of first three stages is illustrated on example of the induction machine, the mathematical model of which is arranged to the normal Cauchy form in order to avoid inversion of coefficient matrix on each integration step. By using the affine coordinate system the physical meaning of stator variables has been preserved and the rank of model equation system has been reduced. Stationary electromechanic process is defined by solv ing the twopoint Tperiodic boundary problem leaded do wn on period to Cauchy problem with initial conditions clarification by Newton iterations. The response matrix for the initial conditions, that is the basic element of the Newton formula, is defined from the analytical solving of homogeneous system of linear variation equations. The transient electro� mechanic process will result from the integration of state equations defining on time longer than period. The static stability of the process is defined by the multipliers of response matrix to the initial conditions. The matrix should be calculated for variables values of the last step in the integration process on period. The suggested method can be freely generalized for the analysis of complicated electromechanic schemes. Keywords: energy transformer, power transformer, electric machine, model analysis, unified mathematic apparatus, induction machine, electromagnetic proc ess, Tperiodic boundary problem, Cauchy problem.

Gradual diffusive capture: slow death by many mosquito bites

We study the dynamics of a single diffusing particle (a ‘man’) with diffusivity DM that is attacked by another diffusing particle (a ‘mosquito’) with fixed diffusivity Dm. Each time the mosquito meets and bites the man, the diffusivity of the man is reduced by a fixed amount, while the diffusivity of the mosquito is unchanged. The mosquito is also displaced by a small distance ±a with respect to the man after each encounter. The man is defined as dead when DM reaches zero. At the moment when the man dies, his probability distribution of displacements x is given by a Cauchy form, which asymptotically decays as x−2, while the distribution of times t when the man dies decays asymptotically as t−3/2, which has the same form as the one-dimensional first-passage probability.

Linearization of a heat-transfer system model with approximation of transport time delay

A method is proposed for linearizing the nonlinear model of a heat-transfer facility the state variables of which at equilibrium points are determined by numerically solving the initial bilinear system of differential equations for a stationary position of the control valve equipped with a constant-speed electric drive. The considerable transport time delay resulting from the distributed design of the heat-transfer system secondary circuit is approximated by a limited number of first-order inertial sections for obtaining a mathematical model in the Cauchy form. The proposed linearization method is tested on an operating hot-water supply heat-transfer system, and the study results are presented in the form of transient curves.

Solving analysis and synthesis problems for a spatially two-dimensional distributed object represented with an infinite system of differential equations

We prove theorems that define an algorithm for passing from differential equations with partial derivatives with respect to two spatial variables and time to an infinite-dimensional system of ordinary differential equations in Cauchy form. We study the convergence of resulting solutions and show that it is possible to pass from an infinite system in Cauchy form to a finite one, which opens up the possibilities to use state space methods for controller design in distributed systems. Based on the quadratic quality criterion, we design a controller for the case when controlling influences are applied at the boundaries of the control object. We obtain the solution of this system analysis problem in the form of Fourier series with respect to spatial variables based on orthogonal systems of trigonometric functions and Bessel functions.

A Very Smooth Ride in a Rough Sea

It has been known for some time that a 3D incompressible Euler flow that has initially a barely smooth velocity field nonetheless has Lagrangian fluid particle trajectories that are analytic in time for at least a finite time (Ph. Serfati C.R. Acad. Sci. S\'erie I 320, 175-180 (1995); A. Shnirelman arXiv:1205.5837 (2012)). Here an elementary derivation is given, based on Cauchy's form of the Euler equations in Lagrangian coordinates. This form implies simple recurrence relations among the time-Taylor coefficients of the Lagrangian map, used here to derive bounds for the C^{1,\gamma} H\"older norms of the coefficients and infer temporal analyticity of Lagrangian trajectories when the initial velocity is C^{1,\gamma}.

Complex Refractive Indices of Thin Films of Secondary Organic Materials by Spectroscopic Ellipsometry from 220 to 1200 nm

The complex refractive indices of three different types of secondary organic material (SOM) were obtained for 220 to 1200 nm using a variable angle spectroscopic ellipsometer. Aerosol particles were produced in a flow tube reactor by ozonolysis of volatile organic compounds, including the monoterpenes α-pinene and limonene and the aromatic catechol (benzene-1,2-diol). Optically reflective thin films of SOM were grown by electrostatic precipitation of the aerosol particles onto silicon substrates. The ellipsometry analysis showed that both the real and imaginary components of the refractive indices decreased with increasing wavelength. The real part n(λ) could be parametrized by the three-term form of Cauchy's equation, as follows: n(λ) = B + C/λ(2) + D/λ(4) where λ is the wavelength and B, C, and D are fitting parameters. The real refractive indices of the three SOMs ranged from 1.53 to 1.58, 1.49-1.52, and 1.48-1.50 at 310, 550, and 1000 nm, respectively. The catechol-derived SOM absorbed light in the ultraviolet (UV) range. By comparison, the UV absorption of the monoterpene-derived SOMs was negligible. On the basis of the measured refractive indices, optical properties were modeled for a typical atmospheric particle population. The results suggest that the wavelength dependence of the refractive indices can vary the Angstrom exponent by up to 0.1 across the range 310 to 550 nm. The modeled single-scattering albedo can likewise vary from 0.97 to 0.85 at 310 nm (UV-B). Variability in the optical properties of different types of SOMs can imply important differences in the relative effects of atmospheric particles on tropospheric photochemistry, as well as possible inaccuracies in some satellite-retrieved properties such as optical depth and mode diameter.

Toeplitz Matrix Method and Volterra-Hammerstien Integral Equation with a Generalized Singular Kernel

In this work, the existence of a unique solution of Volterra-Hammerstein integral equation of the second kind (V-HIESK) is discussed. The Volterra integral term (VIT) is considered in time with a continuous kernel, while the Fredholm integral term (FIT) is considered in position with a generalized singular kernel. Using a numerical technique, V-HIESK is reduced to a nonlinear system of Fredholm integral equations (SFIEs). Using Toeplitz matrix method we have a nonlinear algebraic system of equations. Also, many important theorems related to the existence and uniqueness of the produced algebraic system are derived. Finally, some numerical examples when the kernel takes the logarithmic, Carleman, and Cauchy forms, are considered.

Synthesis of optimal digital controller of flocculant dosing

Purpose. The task of automatic process control of the slime water thickening and flotation tailings clarification is the stabilization of thicken product density within the given range and keeping up the solids content in the overflow not above the permissible level with minimum use of the flocculants. In existing systems for automatic control the flocculant dosing is carried out according to the solids content in the device input (the principle of open-loop control). This leads to the excess consumption of the flocculants and increase the dispersion density of the overflow. To perform the synthesis of the optimal digital controller in order to minimize the deviations from the master control and ensure the specified quality of the transition process. Over controlling value should not exceed 5 %. To perform the system operation modeling in order to determine the quality of transient processes. Methodology. Synthesis of the optimal digital controller is based on the method of dynamic programming. Findings. A mathematical model of the object control is represented in the normal form of Cauchy and further in the form of differential equations. The optimum period of quantization as the function from specified error of control and the output coordinate change is calculated. The differential equation of Bellman is obtained and the condition for minimization of the quality functional. Bellman function is represented as a quadratic form from the variables of the system condition. In order to limit possible control, the weight coefficients of the functional are calculated based on maximum permitted values of the system condition variables and the control actions during the transient process. Practical value. Using the modeling of ACS of the flocculant dosing it was established that the over controlling amount is 3.5%, the transient process life 5.6 sec, the transient process is aperiodical, non-static control, which meets the requirements imposed on the ACS.

Mathematical problems of calculating the multi-zone models of a cycle of internal combustion engine with spark ignition

The article describes the features of mathematical models of internal combustion engine cycles which do not allow to put ordinary differential equations (ODE) of these models in the Cauchy form. The authors proposed a stable algorithm of numerical integration of ODE systems that are not solved with respect to derivatives, using Hermite splines and methods of nonlinear programming.

СИНТЕЗ ОПТИМАЛЬНОГО ЦИФРОВОГО РЕГУЛЯТОРА ДОЗИРОВАНИЯ ФЛОКУЛЯНТА

Purpose. The task of automatic process control of the slime water thickening and flotation tailings clarification is the stabilization of thicken product density within the given range and keeping up the solids content in the overflow not above the permissible level with minimum use of the flocculants. In existing systems for automatic control the flocculant dosing is carried out according to the solids content in the device input (the principle of open-loop control). This leads to the excess consumption of the flocculants and increase the dispersion density of the overflow. To perform the synthesis of the optimal digital controller in order to minimize the deviations from the master control and ensure the specified quality of the transition process. Over controlling value should not exceed 5 %. To perform the system operation modeling in order to determine the quality of transient processes. Methodology. Synthesis of the optimal digital controller is based on the method of dynamic programming. Findings. A mathematical model of the object control is represented in the normal form of Cauchy and further in the form of differential equations. The optimum period of quantization as the function from specified error of control and the output coordinate change is calculated. The differential equation of Bellman is obtained and the condition for minimization of the quality functional. Bellman function is represented as a quadratic form from the variables of the system condition. In order to limit possible control, the weight coefficients of the functional are calculated based on maximum permitted values of the system condition variables and the control actions during the transient process. Practical value. Using the modeling of ACS of the flocculant dosing it was established that the over controlling amount is 3.5 %, the transient process life 5.6 sec, the transient process is aperiodical, non-static control, which meets the requirements imposed on the ACS.

Vibration of Conducting Two-Layer Sandwich Homogeneous Elastic Beams in Transverse Magnetic Fields

AbstractA theory governing the flexural vibration of a conducting two-layer sandwich homogeneous elastic beam in a transverse magnetic field is presented. The physics driving this problem derives from an energy dissipation mechanism through press-fit joints in structural laminates. Recent advances made in the mechanics of sandwich-layered structures have shown that by simulating an environment of nonuniform interface pressure, structural vibration can be attenuated significantly. Equations of mathematical physics governing the stresses and the structural vibration are derived via a laminated beam theory employing the Newtonian form of Cauchy’s stress equations. By restricting mathematical analysis to the case of cantilever architecture, a closed-form polynomial expression is derived for the system response. In particular, the effects of magnetoelasticity, material conductivity, and interfacial pressure gradient on the response characteristics are demonstrated for design analysis and engineering applicatio...

Bulk-mediated diffusion on a planar surface: Full solution

We consider the effective surface motion of a particle that intermittently unbinds from a planar surface and performs bulk excursions. Based on a random-walk approach, we derive the diffusion equations for surface and bulk diffusion including the surface-bulk coupling. From these exact dynamic equations, we analytically obtain the propagator of the effective surface motion. This approach allows us to deduce a superdiffusive, Cauchy-type behavior on the surface, together with exact cutoffs limiting the Cauchy form. Moreover, we study the long-time dynamics for the surface motion.

О напряжённом состоянии кусочно-однородного упругого прямоугольника с двумя симметричными трещинами при антиплоской деформации.

This paper presents a boundary value problem and formulation for the analysis of linear elastic fracture mechanics problems involving piecewise homogeneous bimaterials. Investigation and mathematical works in this paper is focused on the stress- strain state of plates with two symmetric central cracks that are made of two bonded dissimilar materials which behave as a piecewise homogeneous elastic plate. Two cracks are on the bondness line of the two segments of plate that have two different shear modulus G1 and G2, and equal length and height. The antiplane distributed shear loading act on the edges of plate. Using sinusoidal Fourier transformation the equation governing this boundary value problem converts to a singular integral equation (S.I.E) of the Cauchy form, which can be solved with the aid of Gaussian numerical-analytical solution for singular integral equations. Consequently the dislocation field around the cracks boundaries and the tearing stresses of plate and the Stress Intensity Factor (S.I.F) equations at the tips of cracks are derived.

Importance functions in calculation of fuel burnup in a VVER

Equations that describe fuel burnup in a VVER are given. Equations for the neutron flux density and the content of fission products are presented in the canonical Cauchy form. Such form of representation of equations lends itself well for their use in solving problems related to optimization of the process of fuel burnup in a nuclear reactor. Also given are equations for the importance functions of neutrons and fission products that correspond to the basic system of equation for phase variables.

Truncated Lévy flights and generalized Cauchy processes

A continuous Markovian model for truncated Levy random walks is proposed. It generalizes the approach developed previously by Lubashevsky et al. Phys. Rev. E 79, 011110 (2009); 80, 031148 (2009), Eur. Phys. J. B 78, 207 (2010) allowing for nonlinear friction in wondering particle motion and saturation of the noise intensity depending on the particle velocity. Both the effects have own reason to be considered and individually give rise to truncated Levy random walks as shown in the paper. The nonlinear Langevin equation governing the particle motion was solved numerically using an order 1.5 strong stochastic Runge-Kutta method and the obtained numerical data were employed to calculate the geometric mean of the particle displacement during a certain time interval and to construct its distribution function. It is demonstrated that the time dependence of the geometric mean comprises three fragments following one another as the time scale increases that can be categorized as the ballistic regime, the Levy type regime (superballistic, quasiballistic, or superdiffusive one), and the standard motion of Brownian particles. For the intermediate Levy type part the distribution of the particle displacement is found to be of the generalized Cauchy form with cutoff. Besides, the properties of the random walks at hand are shown to be determined mainly by a certain ratio of the friction coefficient and the noise intensity rather then their characteristics individually.

Numerical Solution for Volterra-Ferdholm Integral Equation with A Generalized Singular Kernel

In this paper, the existence of a unique solution of Volterra-Fredholm integral equation of the second kind (V-FIESK) is discussed. The Volterra integral term (VIT) is considered in time with a continuous kernel, while the Fredholm integral term (FIT) is considered in position with a generalized singular kernel. Using a numerical technique,� V-FIESK� is reduced to a system of Fredholm integral equations ( SFIEs ). Using Toeplitz matrix method and Product Nystr�m method we have a linear algebraic system of equations ( LAS ). Finally, some numerical examples when the kernel takes the logarithmic, Carleman, Cauchy and Hilbert forms, are considered.