scholarly journals Methods of calculating the deflection of an orthotropic inhomogeneous plate on an elastic basis

2019 ◽  
pp. 106-109
Author(s):  
M. V. Lavrenyuk
Keyword(s):  
Iterative Process ◽  
Poisson Ratio ◽  
Elastic Equilibrium ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Shear Stresses ◽  
Orthotropic Plates ◽  
Winkler Model ◽  
The Difference ◽  
Elastic Basis

The problem of elastic equilibrium of an orthotropic nonhomogeneous rectangular plate on an elastic basis (one-parameter Winkler model) is considered, hingedly fixed from all sides. We use the Navier method for finding the deflection function at each step of the iterative process and perturbation methods and successive approximations as iterative methods for solving the problem. The suitability of the method of successive approximations and the method of perturbations for the numerical solution of the problem of determining the stress-strain state of such a plate, the limits of the applicability of these methods, their accuracy and convergence of the iterative process in solving the deformation problems of heterogeneous orthotropic plates have been analyzed. The dependence of the deflection on the mechanical and geometric parameters of the plate and the base is established. It was found that the Poisson ratio practically does not affect the stress state of the plate (when the Poisson ratio is changed two times, the difference between the intensities of the shear stresses does not exceed 10%), it is possible to consider it as a constant using the methods of successive approximations and disturbances. It is also established that the method of successive approximations and the method of perturbations has a limit on the nature of inhomogeneity, the convergence essentially depends on the nature of the heterogeneity.

TRANSFER OF LOADS FROM A FINITE NUMBER OF ELASTIC OVERLAYS WITH FINITE LENGTHS TO AN ELASTIC STRIP THROUGH ADHESIVE SHEAR LAYERS

2019 ◽  
Vol 53 (2 (249)) ◽  
pp. 109-118
Author(s):  
A.V. Kerobyan
Keyword(s):  
Finite Number ◽  
Integral Equations ◽  
Characteristic Parameter ◽  
Shear Layers ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Fredholm Integral Equations ◽  
Shear Stresses ◽  
Infinite Strip ◽  
Small Constant

This article deals with the problem of an elastic infinite strip, which is strengthened along its free boundary by a finite number of finite overlays with different elastic characteristics and small constant thicknesses. The interaction between the strip and the overlays is mediated by adhesive shear layers. The overlays are deformed under the action of horizontal forces. The problem of determination of unknown stresses acting between the strip and overlays are reduced to a system of Fredholm integral equations of the second kind for a finite number of unknown functions defined on different finite intervals. It is shown that in the certain domain of variation of the characteristic parameter of the problem this system of integral equations in Banach space may be solved by the method of successive approximations. Particular cases are discussed and the character and behaviour of unknown shear stresses are investigated.

scholarly journals The occurrence and properties of molecular vibrations with V ( x ) = ax 4

1945 ◽  
Vol 183 (994) ◽  
pp. 328-337 ◽  
Keyword(s):  
Harmonic Oscillator ◽  
Energy Levels ◽  
Wave Functions ◽  
Successive Approximations ◽  
Fourth Power ◽  
Method Of Successive Approximations ◽  
Group Iii ◽  
Modes Of Vibration ◽  
The Difference ◽  
Four Levels

It is shown that in certain modes of vibration of plane rings the potential energy for small displacements is proportional to the fourth power of the displacement, provided that there is free rotation about the bonds of the ring. This type of vibration is termed a ‘fourth-power vibration’. It is likely to occur in cyclobutane and its derivatives, in a number of halides having the formula X 2 Y 6 , and in the hydrides of group III elements. The energies and wave functions of the first four levels of a one-dimensional oscillator with V ( x ) = ax 4 have been derived by a method of successive approximations, and asymptotic formulae are given for the higher levels. The wave functions are qualitatively similar to those of a harmonic oscillator, but the energy levels differ considerably. A comparison is made between energy levels for oscillators with V ( x ) = a q | x q | and different values of q . The selection rule for dipole radiation from a fourth-pow er vibration is discussed. Overtones will be more numerous than in the spectrum of a harmonic oscillator. Estimates are made of the spectrum frequencies of fourth-power vibrations in actual molecules, with special reference to cyclobutane and diborane. For these two molecules there are observed infra-red frequencies of approximately the expected value. The isotope effect should provide a means of discriminating experimentally between harmonic and fourth-power vibrations. The contribution of a fourth-power vibration to any thermodynamic function will differ from that of a harmonic vibration with the same fundamental spectrum frequency. Figures are given for the specific heat, where the difference should be detectable experimentally. In the general case V ( x ) = a q | x q | the energy levels derived from the quantum theory lead to expressions for the thermodynamic functions which agree with the predictions of classical theory at high temperatures.

ON A PROBLEM FOR AN ELASTIC INFINITE SHEET STRENGTHENED BY TWO PARALLEL STRINGERS WITH FINITE LENGTHS THROUGH ADHESIVE SHEAR LAYERS

2020 ◽  
Vol 54 (3 (253)) ◽  
pp. 153-164
Author(s):  
Aghasi V. Kerobyan
Keyword(s):  
Banach Space ◽  
Integral Equations ◽  
Shear Layers ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Fredholm Integral Equations ◽  
Shear Stresses ◽  
Characteristic Parameters ◽  
System Of Integral Equations ◽  
Adhesive Layers

The article considers the problem for an elastic infinite sheet (plate), which is strengthened on two parallel finite parts of its upper surface by two parallel finite stringers with different elastic properties. The parallel stringers are located asymmetrically with respect to the horizontal axis of the sheet and deform under the action of horizontal forces. The interaction between the infinite sheet and stringers takes place through thin elastic adhesive layers. The problem of determining unknown shear stresses acting between the infinite sheet and stringers is reduced to a system of Fredholm integral equations of second kind with respect to unknown functions, which are specified on two parallel finite intervals. It is shown that in the certain domain of the change of the characteristic parameters of the problem this system of integral equations in Banach space can be solved by the method of successive approximations. Particular cases are considered, the character and behaviour of unknown shear stresses are investigated.

scholarly journals Curvilinear Squeeze Film Bearing Lubricated with a Dehaven Fluid or with Similar Fluids

2017 ◽  
Vol 22 (3) ◽  
pp. 697-715
Author(s):  
A. Walicka ◽  
P. Jurczak ◽  
J. Falicki
Keyword(s):  
Reynolds Equation ◽  
Equations Of Motion ◽  
Unified Model ◽  
Squeeze Film ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Shear Stresses ◽  
Strain Rates ◽  
Load Carrying Capacity ◽  
Load Carrying

AbstractIn the paper, the model of a DeHaven fluid and some other models of non-Newtonian fluids, in which the shear strain rates are known functions of the powers of shear stresses, are considered. It was demonstrated that these models for small values of material constants can be presented in a form similar to the form of a DeHaven fluid. This common form, called a unified model of the DeHaven fluid, was used to consider a curvilinear squeeze film bearing. The equations of motion of the unified model, given in a specific coordinate system are used to derive the Reynolds equation. The solution to the Reynolds equation is obtained by a method of successive approximations. As a result one obtains formulae expressing the pressure distribution and load-carrying capacity. The numerical examples of flows of the unified DeHaven fluid in gaps of two simple squeeze film bearings are presented.

Numerical Methods for Solving Physically Nonlinear Problems for Inhomogeneous Thick-Walled Shells

2017 ◽  
Vol 865 ◽  
pp. 325-330 ◽  
Author(s):  
Vladimir I. Andreev ◽  
Lyudmila S. Polyakova
Keyword(s):  
Numerical Method ◽  
Nonlinear Problems ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Deformation Diagram ◽  
Method Of Solution ◽  
Properties Of Materials ◽  
Physical Fields ◽  
Cylinders And Spheres ◽  
Numerical Method Of Solution

The paper proposes the numerical method of solution the problems of calculation the stress state in thick-walled cylinders and spheres from physically nonlinear inhomogeneous material. The urgency of solved problem due to the change of mechanical properties of materials under the influence of different physical fields (temperature, humidity, radiation, etc.). The deformation diagram describes the three-parameter formula. The numerical method used the method of successive approximations. The results of numerical calculation are compared with the test analytical solutions obtaining the authors with some restrictions on diagram parameters. The obtained results can be considered quite satisfactory.

On a sphere performing linear and torsional oscillations in a viscous fluid

1988 ◽  
Vol 66 (7) ◽  
pp. 576-579
Author(s):  
G. T. Karahalios ◽  
C. Sfetsos
Keyword(s):  
Boundary Layer ◽  
Viscous Fluid ◽  
Secondary Flow ◽  
Small Amplitude ◽  
Equations Of Motion ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Time Varying ◽  
Torsional Oscillations

A sphere executes small-amplitude linear and torsional oscillations in a fluid at rest. The equations of motion of the fluid are solved by the method of successive approximations. Outside the boundary layer, a steady secondary flow is induced in addition to the time-varying motion.

scholarly journals Green’s Function In Free Axisymmetric Vibration Analysis Of Annular Thin Plates With Different Boundary Conditions

2015 ◽  
Vol 20 (4) ◽  
pp. 939-951
Author(s):  
K.K. Żur
Keyword(s):  
Power Series ◽  
Vibration Analysis ◽  
Free Vibration Analysis ◽  
Thin Plates ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Axisymmetric Vibration ◽  
Annular Plates ◽  
The Core ◽  
Analytical Frequency

Abstract Free vibration analysis of homogeneous and isotropic annular thin plates by using Green’s functions is considered. The formula of the influence function for uniform thin circular and annular plates is presented in closed-form. The limited independent solutions of differential Euler equation were expanded in the Neumann power series based on properties of integral equations. The analytical frequency equations as power series were obtained using the method of successive approximations. The natural axisymmetric frequencies for singularities when the core radius approaches zero are calculated. The results are compared with selected results presented in the literature.

Towards an Improved Parameter-Free Entrainment Model for Snow Avalanches

2021 ◽  
Author(s):  
Dieter Issler
Keyword(s):  
Snow Cover ◽  
Ad Hoc ◽  
Shear Stresses ◽  
Entrainment Rate ◽  
Snow Layer ◽  
Present Contribution ◽  
Weak Layer ◽  
Model Combining ◽  
The Difference ◽  
The 1960S

<p>On physical grounds, the rate of bed entrainment in gravity mass flows should be determined by the properties of the bed material and the dynamical variables of the flow. Due to the complexity of the process, most entrainment formulas proposed in the literature contain some ad-hoc parameter not tied to measurable snow properties. Among the very few models without free parameters are the Eglit–Grigorian–Yakimov (EGY) model of frontal entrainment from the 1960s and two formulas for basal entrainment, one from the 1970s due to Grigorian and Ostroumov (GO) and one (IJ) implemented in NGI’s flow code MoT-Voellmy. A common feature of these three approaches is their treating erosion as a shock and exploiting jump conditions for mass and momentum across the erosion front. The erosion or entrainment rate is determined by the difference between the avalanche-generated stress at the erosion front and the strength of the snow cover. The models differ with regard to how the shock is oriented and which momentum components are considered. The present contribution shows that each of the three models has some shortcomings: The EGY model is ambiguous if the avalanche pressure is too small to entrain the entire snow layer, the IJ model neglects normal stresses, and the GO model disregards shear stresses and acceleration of the eroded mass. As they stand, neither the GO nor the IJ model capture situations―observed experimentally by means of profiling radar―in which the snow cover is not eroded progressively but suddenly fails on a buried weak layer as the avalanche flows over it. We suggest a way to resolve the ambiguity in the EGY model and sketch a more comprehensive model combining all three approaches to capture gradual entrainment from the snow-cover surface together with erosion along a buried weak layer.</p>

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