Principle Of Possible Displacements Research Articles

Mathematical Modeling of Mechanical Forces and Power Balance in Electromechanical Energy Converter

This article proposes a calculation method for mechanical (electromagnetic) forces arising in an electromechanical energy converter acting on current circuits in a magnetic field, or on capacitor plates in an electric one. Transformations were performed on the basis of the principle of possible displacements involving the apparatus of partial derivatives. It was found that the power converted into mechanical power is partially spent on changing the energy of the electromagnetic field, and the remaining power, determined by the co-energy, is converted into mechanical power. Expressions for the mechanical (electromagnetic) forces were obtained based on the power balance. The reliability of the obtained results was compared with the known results. Of these, one can observe the well-known 50/50 principle, which states that only part of the power associated with the movement of the circuits is converted into mechanical power, while the rest is intended for changing the energy of the magnetic field.

Influence of geometric parameters of crankshaft-planetary reducer on cyclograph of work of contact pairs

Background. Crank-planetary reducers (CPR) have a number of advantages: compactness, high load capacity, high structural rigidity, to name a few. At the same time, there is no regulated methodology for their kinematic and force calculation, which limits theirdevelopment and mass production. The creation of effective methods for calculating these reducers to improve their performance at the design stage is of great practical importance.Objective. The aim of this work is to develop an analytical method for calculating the angle of pressure and the contact force in contact pairs “satellite hole – pin of output link” of the CPR depending on the angle between the cranks of its crankshaft (V-shaped),number of satellites and number of pins. Based on the developed method, check the possibility of designing a CPR with arbitrary values of these parameters and investigate their impact on the cyclogram of contact pairs and the associated maximum value of contact force, taking into account the nonlinearity of materials’ deformation.Methods. At the first stage, based on the formulas of analytical geometry, the possibility of theoretical implementation of CPR with an arbitrary value of the angle between the crankshaft cranks, the number of satellites and the number of pins of the output link is shown.The values of pressure angles in the contact pairs “satellite hole – pin of output link” are analysed and on the basis of the received results the cyclogram of work of leading pairs is calculated. In the second stage, using the principle of possible displacements, the formula for calculating the contact force in the leading contact pairs is written as a function of the input shaft rotation angle. When calculating thecontact force, a nonlinear model of material deformation was used.Results. An analytical method for constructing a cyclogram of the contact pairs of a CPR and the contact forces calculation was developed depending on the geometric parameters of the reducer. The possibility of theoretical implementation of reducers with arbitrary value of the angle between the crankshaft cranks, the number of satellites and the number of output link’s pins is shown Conclusions. The dependence of the contact force is obtained as a rotation angle function, the number of satellites and the numberof source pins. With a constant number of simultaneously active leading contact pairs and, accordingly, a uniform load transfer between them, the law of change in the contact force per cycle is similar to parabolic. Otherwise, the change in the contact force per cycle is quite complex. The obtained results can be used in the refined calculation of the efficiency of the CPR and in making various design decisions at the design stage

NORMAL VIBRATIONS OF SAGGING CONDUCTORS OF OVERHEAD POWER LINES

The phenomenon of self-excitation of thermomechanical vibrations of current-carrying conductors, experimentally discovered by academician A.F. Ioffe, is of practical interest as a possible explanation of the phenomenon of galloping conductors of overhead power transmission lines (OHL) – low-frequency vibrations with frequencies of ~ 1 Hz and with amplitudes of the order of the static conductor sagging. To build the theoretical foundations of this phenomenon, as a special class of self-oscillating systems, it is necessary, first of all, a model of conductor vibrations in the OHL span. With regard to the most studied type of conductor vibrations, high-frequency aeolian vibration, excited by sign-alternating aerodynamic forces from the Karman vortex street, the classical model of a straight string is reasonably applied. However, to study low-frequency vibrations of the galloping type, it is necessary to take into account the effect of sagging of the conductor, the associated elastic tension and, in some cases, the nonlinear nature of the vibrations. The article presents two models for calculating the natural vibrations of sagging conductors (cables) within the framework of the technical theory of flexible threads, assuming the constancy of the tension force along the length. The first model describes linear oscillations of an elastic conductor in the sagging plane. For a system of equations with respect to the displacement components given in natural coordinates, an exact solution of the Sturm-Liouville problem with estimates of the frequency ranges arising is obtained. The second model describes nonlinear vibrations of an elastic conductor in the sagging plane and pendulum vibrations accompanied by an exit from it. The solution of the problem is based on the principle of possible displacements using the Ritz method. The structure of the frequency spectrum and the natural forms of transverse vibrations are studied. The developed models are intended for further investigation of thermomechanical vibrations of conductor and flexible cable systems.

Calculation of Vineyard Hail Protection Device

The kinematic and force analysis of an invented new mechanical system used to protect vineyards and other fruit-bearing plants from hail, which often occurs due to climatic changes, is determined. The system contains levers that support anti hail net. The offered mechanical scheme of system is simple and can be used for all type of vineyards. Moreover, the mechanical part of the system can be controlled remotely by a cell phone. In order to determine the mechanical parameters of the system the calculations are given based on the use of the principle of possible displacements, based on what theoretically is determined the necessary volume of weight of load that leads in motion of the system.

Manometric Tubular Springs Oscillatory Processes Modeling with Consideration of its Viscoelastic Properties

This article is dedicated to the operation and management of systems of machine-building and aviation enterprises, systems of production, transport, storage of oil and gas, issues of control of technological processes are of great importance. Control of technological processes is carried out by monitoring the pressure and other parameters. These measuring instruments must have high reliability and the necessary accuracy. In this connection, there is a sharp increase in interest in determining the dynamic parameters of the elements of measuring devices. The main elements of such devices are monomeric tubular springs (Bourdon tubes). The paper considers the natural and forced steady-state oscillations of a thin curved rod interacting with a liquid. Based on the principle of possible displacements, a resolving system of partial differential equations and the corresponding boundary conditions are obtained. The problem is solved numerically by the Godunov orthogonal run method, and the Muller method and the Eigen frequencies found are compared with the experimental results. As a result, for a given axial perturbation, it was possible to select such an effect, in the orthogonal direction, that the amplitude of the longitudinal vibrations of the rod at the first resonance decreased by 20 times. The described vibration damping effect is due to the interrelation of transverse and longitudinal vibrations and is fundamentally impossible in the case of a straight rod.

Variational Formulation of Linear Equations of Coupled Thermohydrodynamics and Heat Conductivity

We consider the coupled processes of thermohydrodynamics and heat conduction and construct a variational model of such coupled problems using four-dimensional space-time continuum where time is an equal coordinate along with spatial coordinates. In this consideration 3D subspace of generalized four-dimensional pseudo-continuum is associated with three-dimension deformed media. In the general case we consider irreversible processes and use the variational principle of possible displacements and the variational Sedov’s principle to construct variation model. It is assumed that the 4D pseudo-continuum considered below is transversely isotropic in the direction of the unit vector of time. Thus, an asymmetric 4D stress tensor preserves the symmetry properties with respect to spatial tangential stresses in 3D subspaces. It is shown that the proposed version of the model allows us to formulate the full range of consistent thermomechanical and thermodynamic physical relations, and the system of governing equations includes, as special cases, the equations of thermoelasticity, thermohydrodynamics, the linear Navier–Stokes equations for compressible and incompressible media, the equations of heat balance with the laws of thermal conductivity of Fourier, Maxwell–Cattaneo.

Модель динамики колесной машины для комплекса натурно-математического моделирования

Introduction. The use of the method of full-scale-mathematical modeling in “real time” opens up wide opportunities associated with the analysis of the modes of operation of the “man – vehicle – environment” system, as well as the study of the loading of units and assemblies of vehicles. The existing research complexes of full-scale mathematical modeling are suitable for obtaining most of the indicators usually determined by full-scale tests. The difference lies in the ability to fully control the course of virtual testing, recording any parameters of the vehicle movement, taking into account the “human factor”, as well as complete safety of the experiment. Purpose of research. The purpose of this work is to create a mathematical model of the dynam-ics of a wheeled vehicle, suitable for use in such a complex of full-scale mathematical modeling and assessment of the load of transmission units in conditions close to real operation. Methodology and methods. The proposed model is based on the existing model of the dynamics of a wheeled vehicle developed at Bauman Moscow State Technical University. Within the framework of the model, the dynamics of a vehicle is described as a plane motion of a rigid body in a horizontal plane. The principle of possible displacements is applied to determine the normal reac-tions of the bearing surface. The interaction of the wheel with the ground in the plane of the support base is described using an approach based on the “friction ellipse” concept. To enable the driver and operator of the full-scale mathematical modeling complex to drive a virtual vehicle in “real time” mode, the mathematical model is supplemented with a control system that communicates between the control parameter set by the driver by pressing the accelerator and brake pedals and the control actions of the vehicle's transmission units, such as: an electric machine, an internal combustion en-gine, a hydrodynamic retarder and a brake system. The article presents a block diagram of the de-veloped control algorithm, as well as approbation of the system's operation in a complex of full-scale mathematical modeling. Results and scientific novelty. A mathematical model of the dynamics of a wheeled vehicle was developed. It opens up wide possibilities for studying the modes of operation of the “driver-vehicle-environment” system in “real time”, using a complex of full-scale mathematical modeling. Practical significance. A mathematical model of the dynamics of a wheeled vehicle was devel-oped. It is supplemented with an algorithm for the distribution of traction / braking torques between the transmission units, which provide a connection between the driver's pressing on the accelerator / brake pedal and the control parameters of each of the units.

Mathematical modeling of progressive failure of two-layer metal-plastic shells of revolution under pulse loading

A technique for numerical analysis of nonlinear dynamic deformation and progressive failure of multi-layered metal-plastic shells of revolution is developed with account for their strain-rate dependent strength characteristics. The kinematic deformation model of a layered package is based on the non-classical theory of shells. The geometric dependencies are formulated based on the quadratic version of the nonlinear theory of elasticity. The relationship between stress and strain tensors in a composite macrolayer is established on the basis of Hooke’s law for an orthotropic body combined with the theory of effective modules, while for metal macrolayer, within the flow theory with linear hardening. The process of progressive layer-by-layer failure is described in the framework of the degradation model of stiffness characteristics. The strain rate dependence of stiffness and strength characteristics of composite materials is accounted for. An energetically consistent system of equations of motion for a shell of revolution is constructed using the principle of possible displacements. A numerical method for solving the problem is based on an explicit variational-difference scheme. The adequacy of the proposed technique was considered on the problem of unsteady deformation of a cylindrical shell subjected to pulse pressure simulating an explosion in the shell center.

Stability of a nonlinear elastic plate under uniaxial loading with respect to finite perturbations

The stability of a plate made of nonlinear elastic material with respect to finite perturbations is considered. The analysis of the main process of plate deformation is reduced to the solution of the nonlinear boundary value problem with respect to finite perturbations. Solutions for perturbations of displacements are chosen in the form of eigenfunction series, that are solutions of the corresponding linearized problems and satisfy the geometric boundary conditions. After applying the principle of possible displacements, the question on the stability of the ground state of the nonlinear problem is reduced to the study of the stability of zero solution of an infinite system of ordinary differential equations with constant coefficients, the number of terms in which is specified by the elastic potential. For the obtained system of equations under certain restrictions on initial perturbations, a Lyapunov function is constructed. The dimension of the strange attractor of the dynamical system is found which allows to limit the number of terms in the Bubnov-Galerkin series.

Calculation of bending plates by finite element method in stresses

The finite element models for plate bending problems are constructed on the basis of the approximations of moments fields. The bending and twisting moments are approximated in the finite element area by piecewise constant functions. The solution is based on the functional of the additional energy. The algebraic equations of equilibrium of nodes of grid of finite elements are formed using the principle of possible displacements and are included in the functional with the help of Lagrange multipliers. The calculations of square clamped and hinged supported plates on the action of uniformly distributed load are performed. It is shown that the presented method of calculating bent plates by the finite element method in stresses has the property of convergence of the displacements from above.

Nonlinear Oscillations of Elastic Solar Panels of a Spacecraft at Finite Turn by Roll

Non-stationary rotational motion and nonlinear oscillations in the roll plane of a spacecraft with two elastic panels of solar batteries are considered. The spacecraft and sections of solar panels are considered to be non-deformable; they are interconnected by elastic-hinged joints that allow large angles of rotation. The spacecraft makes a turn about its axis as a solid body. Motion of elastic system under consideration is described in a connected coordinate system. Equations of motion are obtained on the basis of principle of possible displacements. Examples of numerical integration of a system of nonlinear differential equations with the analysis of convergence are given.

Nonlinear dynamics in problems of stability of complex media

The problem of stability of nonlinear viscoelastic bodies with respect to finite perturbations is considered in this article. The analysis of the basic process of deformation of a viscoelastic medium is reduced to solution of a nonlinear boundary value problem with variable coefficients. Solutions for perturbations of displacements are in the form of series in eigenfunctions. Using the principle of possible displacements the question on stability of the ground state of a variational nonlinear problem is reduced to investigation of stability of the zero solution of an infinite system of ordinary differential equations with constant coefficients. For the resulting system of equations we construct a function, which under certain restrictions on the initial perturbations is a Lyapunov function. The dimension of the strange attractor of the dynamical system which allows to limit the number of terms in a Bubnov-Galerkin set is found.

Numerical modelling of the stability of loaded shells of revolution containing fluid flows

A mixed finite-element algorithm is proposed to study the dynamic behavior of loaded shells of revolution containing a stationary or moving compressible fluid. The behavior of the fluid is described by potential theory, whose equations are reduced to integral form using the Galerkin method. The dynamics of the shell is analyzed with the use of the variational principle of possible displacements, which includes the linearized Bernoulli equation for calculating the hydrodynamic pressure exerted on the shell by the fluid. The solution of the problem reduces to the calculation and analysis of the eigenvalues of the coupled system of equations. As an example, the effect of hydrostatic pressure on the dynamic behavior of shells of revolution containing a moving fluid is studied under various boundary conditions.

Methods for calculation of free vibrations of a cylindrical shell with attached rigid body

We consider a mechanical system consisting of a circular cylindrical shell and an absolutely rigid body attached to one of the ends of the shell. Using the principle of possible displacements, we construct a mathematical model for the equilibrium state of the considered system under loads of general form. Using this model, we formulate an eigenvalue boundary-value problem that describes the free vibrations of the “shell-body” system and propose its approximate solution. In the case where the shell is replaced by an equivalent Timoshenko beam, we construct an exact solution of the problem under consideration. We also give an estimate for the effect of the rigid body on the vibrations of the system and investigate the accuracy of the beam approximation of flexural vibrations of the shell.

Application of the principle of possible displacements to problems of the dynamics of a “shell-fluid” system

We state in general form the principle of possible displacements for a “shell-fluid” mechanical system, on the basis of which it is possible to solve dynamic problems taking account of a geometrically nonlinear process of deformation of the shell and nonpotential motions of a viscous fluid. It is shown that this principle yields the equations of motion of the shell and fluid as components of this system, confirming the reliability of the principle. The conditions of force contact are taken into account as a load term in the equations of motion of the shell. Bibliography: 5 titles.

Experimental and theoretical analysis of the dynamic characteristics of rotors of electromechanical energy accumulators

The natural frequencies and forms of rotor vibrations are determined with the aid of the semianalytical finite element method. The mathematical statement of the problem is based on the principle of possible displacements and on the reltations of the three-dimensional theory of elasticity.