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In order to improve the accuracy of stiffness calculation and derive an accurate rotation angle function and a more accurate deflection function, bu using the chain rule, considering all the higher-order quantities, and combining the relations of higher-order quantities and geometric characteristics, the rotation angle function of the beam was accurately deduced, and the reason for the error of the traditional approximation method was revealed. Consequently a more accurate deflection function was obtained by using the precise rotation angle function. To further verify the reliability of the precise functions, simulation was performed by the method of finite elements. The results show that the value of the angle of rotation obtained by the precise rotation angle function satisfied the finite element simulation results. The accurate rotation angle function was more in line with the actual angle of the beam and was more suitable for practical engineering, which provides a theoretical basis for further approach of the precise deflection function. The calculation result of the accurate rotation angle function can be obtained quickly by related software, which could promote the update of the related software's solution method, and have a wide application prospect.

A set of fixed points in N-point gravitational lenses is studied in the paper. We use complex form of lens mapping to study fixed points. There are some merits of using a complex form over coordinate. In coordinate form gravitational lens is described by a system of two equations and in complex form is described by one equation. We transform complex equation of N-point gravitational lens into polynomial equation. It is convenient to study polynomial equation. Lens mapping presented as a linear combination of two mappings: complex analytical and identity. Analytical mapping is specified by deflection function. Fixed points are roots of deflection function. We show, that all fixed points of lens mapping appertain to the minimal convex polygon. Vertices of the polygon are points into which dimensionless point masses are. Method of construction of fixed points in N-point gravitational lens is shown. There are no fixed points in 1-point gravitational lens. We study properties of fixed points and their relation to the center of mass of the system. We obtained dependence of distribution of fixed points on center of mass. We analyzed different possibilities of distribution in N-point gravitational lens. Some cases, when fixed points merge with the center of mass are shown. We show a linear dependence of fixed point on center of mass in 2-point gravitational lens and we have built a model of this dependence. We obtained dependence of fixed point to center of mass in 3-point lens in case when masses form a triangle or line. In case of triangle, there are examples when fixed points merges. We study conditions, when there are no one-valued dependence of distribution of fixed points in case of 3-points gravitational lens and more complicated lens.

Relevant problems on modeling mechanisms with elastic links imply the improvement of existing formalisms and algorithms for dynamic analysis, synthesis, and optimal control of the considered class of systems. At the same time, modern research in this area is mainly focused on increasing the speed of computational algorithms without decreasing in the accuracy. If elastic multi-link dynamical systems that do not include closed kinematic chains can be comprehensively investigated using the approach without inversion of the mass matrix (generalized Newton-Euler method), then elastic mechanisms with closed kinematic chains should be studied using the methods with inversion of the mass matrix. The latter class includes the problem on finding the conditional minimum of the action functional in the sense of Ostrogradsky in the presence of holonomic (geometric) additional constraints. In this article, we analyze the problem on optimizing the motion of an elastic three-link tracking manipulator that consists in minimizing the function of deviation of the actuator from a predetermined circumferential trajectory. This problem is also reduced to finding the conditional minimum of the Ostrogradsky action functional in the presence of a holonomic additional constraint. [ABSTRACT FROM AUTHOR]

With the help of linear Volterra integral equation of the second kind the problem of the axisymmetric bending polar orthotropic annular disk of variable thickness, rotating around the normal axis with constant angular velocity w0 in a inhomogeneous thermal field is solved in the general form. Under the influence of centrifugal forces and thermal field the disk will experience stretch on its plane. The effect of the axisymmetric flow of incandescent gas or steam, directed normal by to the median plane of the disk, as well as the boundary moments and shear forces cause axisymmetric bending. Thus, the disc will experience the stretch and flexural bending at the same time. It is assumed that the temperature field in the disk is known and it is axisymmetric. The elastic constants – Youngʼs modulus and shear modulus – are linearly temperature dependent, and Poissonʼs coefficient are considered to be constant. Calculation of bending of the anisotropic thin disk is carried out according to the classical theory of bending of thin plates, based on Kirchhoff hypothesis. The problem of the axisymmetric bending polar-orthotropic annular disc of variable thickness is reduced to the integration of the ordinary differential equation of second order with variable coefficients for a rotation angle of a normal cell to the median plane of the disk. The resulting differential equation is reduced to a linear integral Volterra equation of the second kind. The general solution of the integral equation is written down the resolvent is used for this purpose. The conditions under which the integral equation has a unique continuous solution are given. Calculation formulas are given for the radial and tangential bending moments, transverse radial forces and radial deflection function through resolution function. Formula for the components of the radial, tangential and shear stresses, taking into account the simultaneous stretching and bending of anisotropic annular disc of variable thickness under the influence of applied loads is written.

The inelastic buckling behaviour of different rectangular thin isotropic plates having a free edge is studied. Various combinations of boundary conditions are subject to in-plane uniaxial compression and each rectangular plate is bounded by an unloaded free edge. The characteristic deflection function of each plate is formulated using a polynomial function in form of Taylor--Maclaurin series. A deformation plasticity approach is adopted and the buckling load equation is modified using a work principle technique. Buckling coefficients of the plates are calculated for various aspect ratios and moduli ratios. Findings obtained from the investigation are found to reasonably agree with data published in the literature. [ABSTRACT FROM AUTHOR]

The article is devoted to the study of the thin plate bending by the finite element method. The application of the finite element method to solving the problem of the plate bending leads to the necessity of studying the rectangular finite element of the plate. All deformation and statics characteristics of the plate are functions of the displacement in the direction of the normal to the middle surface of the plate, which is determined by the deflection function. In the article, the formation of the plate deflection function in explicit form is carried out. The ways for finding the deflection function by division of the variables in the equilibrium equation of the plate, through an incomplete fourth - degree polynomial and in the form of Hermite polynomials are presented. The article is focused mainly on mechanics, engineers and scientific employees of technical specialties.

This study investigates the inelastic stability of a thin flat rectangular isotropic plate subjected to uniform uniaxial compressive loads using Taylor-Maclaurin series formulated deflection function. The plate has clamped and simply supported edges in both characteristic directions (CCSS boundary conditions). The governing equation is derived using a deformation plasticity theory and a work principle. Values of the plate buckling coefficient are calculated for aspect ratios from 0.1 to 2.0 at intervals of 0.1 . The results compared favourably with the elastic stability values and the percentage differences ranged from − 0.353% to − 7.427% . Therefore, the theoretical approach proposed in this study is recommended for the inelastic stability analysis of thin flat rectangular isotropic plates under uniform in-plane compression.

In this article, the deflection functions of nanobeams are found by employing the well-known Green's function method. Based on Eringen's nonlocal elasticity theory, stress gradient model is employed to formulate the problem. Different boundary conditions, including, clamped–clamped, clamped-free, simply supported-simply supported and clamped-simply supported are studied. The beams are assumed to be under arbitrary distributed or concentrated loads. By presenting some new Green's integral formulations, three different ways of treating the nonlocal terms are suggested. The specificity of inhomogeneous boundary conditions for the stress gradient nonlocal model is found. A particular property is a possible loss of the symmetry of Green's functions for nonlocal boundary conditions. To validate the proposed results, findings are compared with the available ones in the literature. [ABSTRACT FROM AUTHOR]

In the article research of the third boundary value problem for the two - dimensional equation of free membrane vibrations is presented. The solution of the original differential operator is found in the form of a combination of the linearly independent system of orthonormal eigenfunctions on the given interval. Using the spectral decomposition for sufficiently smooth function, one can obtain the exact analytical representation of the deflection function for investigated problem in two - dimensional environs. The deflection function characterizes a membrane deviation from the equilibrium position.

The article presents the calculation of rectangular plates by a variational method. For considered rectangular plate the research was conducted by the equilibrium conditions of the elementary strip isolated from the plate, by the method that was used in the works of V.Z.Vlasov. To illustrate the above variational method were given specific examples of the calculation of a square plate, hinged along the contour and loaded equally by distributed loading of given intensity, as well as a square plate, rigidly clamped along the contour. A comparative analysis of the results was carried out.

Goal. Determination of deflections plates taking into account genetic inhomogeneity. Methods. Used ratio of layers of elastic modulus as a factor by which the influence of the surrounding is taken into account. Results. The problem to determine the deflection plates under the influence of the Scientific novelty. A function of the influence of the , using which you can calculate the stiffness of the cylindrical plate and solve the problem of bending considering genetic inhomogeneity. Practical significance.An analytical solution for the determination of hardness of the cylindrical plate "D *", taking into account environmental effects with various changes of the material properties.Biharmonic bending equation is written for a homogeneous isotropic plate with the replacement of "D" to "D*". To determine the deflection and forces also need to replace the "D" to "D *" in certain expressions, which is convenient for calculation. [ABSTRACT FROM AUTHOR]

The Firsov inverse method was used to retrieve potential energy for helium-helium system from deflection function. Using a combination of accurate simulated data for large deflection function and a Lennard-Jones type potential for smaller values, it was possible to recover the short-range potential in excellent agreement with the theoretical results. Errors in the deflection function ranged from 1 to 10% with a collision energy from 2 × 10−3 to 2 eV. Inverted potential were obtained with a precision from 2 to 8%. This study explores the possibility to use Firsov approach for small collision energy, unlike the previous work on the subject. The method was proved to be robust, stable against experimental error, and very easy to be implemented numerically. © 2012 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

Abstract: High-energy scattering of fast charged particles in a continuous potential field of the bent crystal atomic planes is discussed. It is shown that this problem represents a scattering of charged particles in the central field. The deflection functions of a particle in such field are obtained. The reflection possibility for positively and negatively charged particles by the bent crystal atomic planes at the angles approaching to twice value of the plane channeling critical angle is demonstrated. Conditions at which such effect occurs practically for all particles of the beam are obtained. The requirements to realize these effects are discussed. [Copyright &y& Elsevier]

A nondestructive method is presented to determine simultaneously the refractive index profile and the cross section geometry of optical fiber preforms. An improved formula for calculating the optical path length from the deflection function is derived. The preform is rotated through 180° and the deflection angle data collected, then the back projection technique and linear interpolation algorithm are used in the computed tomographic two-dimensional (2-D) profile reconstruction for preforms of nonuniform cross section. To minimize the computing time, the spatial Nyquist frequency is analytically used to estimate the angle sampling number of views and the numbers of points per scan in advance. Numerical simulations and experimental results show that using 180° angle of view deflection data, accurate preform index distribution reconstructions are obtained. [ABSTRACT FROM PUBLISHER]