scholarly journals Calculation of deformations of a cantilever-frame planar truss model with an arbitrary number of panels

Vestnik MGSU ◽  
2020 ◽  
pp. 510-517
Author(s):  
Karina Buka-Vaivade ◽  
Mikhail N. Kirsanov ◽  
Dmitrijs O. Serdjuks
Keyword(s):  
Analytical Solution ◽  
Numerical Solutions ◽  
Engineering Practice ◽  
Symbolic Form ◽  
Concentrated Load ◽  
Structural Rigidity ◽  
Truss Model ◽  
Different Types ◽  
Homogeneous Linear ◽  
Independent Parameters

Introduction. By method of induction using three independent parameters (numbers of panels) formulas for deflection under different types of loading are derived. Curves based on the derived formulas are analyzed, and the asymptotic of solutions for the number of panels are sought. The frame is statically definable, symmetrical, with descending braces. The problem of deflection under the action of a load evenly distributed over the nodes of the upper chord, a concentrated load in the middle of the span, and the problem of shifting the mobile support is considered. Materials and methods. The calculation of forces in the truss bars is performed in symbolic form using the method of cutting nodes and operators of the Maple computer mathematics system. The deflection is determined by the Maxwell – Mohr formula. Operators of the Maple computer mathematics system are used for composing and solving homogeneous linear recurrent equations that satisfy sequences of coefficients of the required dependencies. The stiffness of all truss bars is assumed to be the same. Results. All the obtained dependencies have a polynomial form for the number of panels. To illustrate the obtained solutions and their qualitative analysis, curves of the deflection dependence on the number of panels are constructed. Conclusions. A scheme of a statically definable three-parameter truss is proposed that allows an analytical solution of the problem of deflection and displacement of the support. The obtained dependences can be used in engineering practice in problems of structural rigidity optimization and for evaluating the accuracy of numerical solutions.

Analtical Calculation of the Deflection of a Spatial Hinge-Rod Frame with an Arbitrary Number of Panels

2020 ◽  
pp. 68-77
Author(s):  
М. Н. Кирсанов
Keyword(s):  
Arbitrary Number ◽  
Numerical Solutions ◽  
Optimization Problems ◽  
Asymptotic Approximations ◽  
Induction Method ◽  
Symbolic Form ◽  
Regular Type ◽  
Proposed Model ◽  
Different Types ◽  
Independent Parameters

Постановка задачи. Ставится задача получить в символьном виде зависимость прогиба предлагаемой схемы статически определимой пространственной фермы регулярного типа от числа панелей при различных нагрузках, в том числе при нагрузке из плоскости фермы. Ферма имеет два независимых параметра, задающие ее пропорции. Результаты. Для нескольких видов нагружения по формуле Максвелла-Мора выведены аналитические зависимости прогибов конструкции от числа панелей, нагрузки и размеров. При обобщении серии частных решений с заданным числом панелей на произвольное число панелей совместно с операторами системы компьютерной математики Maple использован метод индукции. Получены асимптотические приближения решений. Выводы. Предложенная схема пространственной рамы с двумя независимыми числами панелей, задающими пропорции конструкции, допускает аналитическое решение задачи о прогибе при различных видах нагружения. Выведенные формулы могут быть использованы как тестовые для оценки приближенных численных решений и в задачах оптимизации. Statement of the problem. The task is to obtain in symbolic form the dependence of the deflection of the proposed scheme of a statically definable spatial truss of a regular type on the number of panels under various loads, including the load from the truss plane. A truss has two independent parameters that define its proportions. Results. For several types of loading according to the Maxwell - Mohr formula, analytical dependences of the deflections of the structure on the number of panels, load, and dimensions are derived. When generalizing a series of partial solutions with a given number of panels to an arbitrary number of panels, together with operators of the Maple computer mathematics system, the induction method is used. Asymptotic approximations of solutions are obtained. Conclusions. The proposed model of a spatial frame with two independent numbers of panels that define the proportions of the structure allows an analytical solution of the problem of deflection under different types of loading. The derived formulas can be used as test formulas for evaluating approximate numerical solutions and for optimization problems.

scholarly journals ANALYTICAL CALCULATION OF THE DEFLECTIONOF A SPATIAL HINGE-ROD FRAME WITH AN ARBITRARY NUMBER OF PANELS

2020 ◽  
pp. 65-75
Author(s):  
M. N. Kirsanov
Keyword(s):  
Arbitrary Number ◽  
Numerical Solutions ◽  
Optimization Problems ◽  
Analytical Calculation ◽  
Induction Method ◽  
Symbolic Form ◽  
Regular Type ◽  
Proposed Model ◽  
Different Types ◽  
Independent Parameters

Statement of the problem. The task is to obtain in symbolic form the dependence of the deflection of the proposed scheme of a statically definable spatial truss of a regular type on the number of panels under various loads, including the load from the truss plane. A truss has two independent parameters that define its proportions.Results. For several types of loading according to the Maxwell - Mohr formula, analytical dependences of the deflections of the structure on the number of panels, load, and dimensions are derived. When generalizing a series of partial solutions with a given number of panels to an arbitrary number of panels, together with operators of the Maple computer mathematics system, the induction method is used. Asymptotic approximations of solutions are obtained.Conclusions. The proposed model of a spatial frame with two independent numbers of panels that define the proportions of the structure allows an analytical solution of the problem of deflection under different types of loading. The derived formulas can be used as test formulas for evaluating approximate numerical solutions and for optimization problems.

Diffusion-Limited Cell Dehydration: Analytical and Numerical Solutions for a Planar Model

1999 ◽  
Author(s):  
Alexander V. Kasharin ◽  
Jens O. M. Karlsson
Keyword(s):  
Analytical Solution ◽  
Numerical Solutions ◽  
Planar System ◽  
One Dimensional ◽  
Diffusion Limited ◽  
Dimensional Diffusion ◽  
Constant Diffusivity ◽  
A Cell ◽  
Cell Dehydration ◽  
The One

Abstract The process of diffusion-limited cell dehydration is modeled for a planar system by writing the one-dimensional diffusion-equation for a cell with moving, semipermeable boundaries. For the simplifying case of isothermal dehydration with constant diffusivity, an approximate analytical solution is obtained by linearizing the governing partial differential equations. The general problem must be solved numerically. The Forward Time Center Space (FTCS) and Crank-Nicholson differencing schemes are implemented, and evaluated by comparison with the analytical solution. Putative stability criteria for the two algorithms are proposed based on numerical experiments, and the Crank-Nicholson method is shown to be accurate for a mesh with as few as six nodes.

scholarly journals Complex Variable Meshless Manifold Method for Elastic Dynamic Problems

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Hongfen Gao ◽  
Gaofeng Wei
Keyword(s):  
Analytical Solution ◽  
Complex Variable ◽  
Meshless Method ◽  
Numerical Solutions ◽  
Least Square ◽  
Dynamic Problems ◽  
Finite Covering ◽  
Moving Least Square ◽  
Elastic Dynamics ◽  
Good Agreement

Combining the finite covering technical and complex variable moving least square, the complex variable meshless manifold method can handle the discontinuous problem effectively. In this paper, the complex variable meshless method is applied to solve the problem of elastic dynamics, the complex variable meshless manifold method for dynamics is established, and the corresponding formula is derived. The numerical example shows that the numerical solutions are in good agreement with the analytical solution. The CVMMM for elastic dynamics and the discrete forms are correct and feasible. Compared with the traditional meshless manifold method, the CVMMM has higher accuracy in the same distribution of nodes.

scholarly journals Analytical Solution of 1D Advection-Dispersion Equation In Finite Groundwater Reservoir With Spatially Dependent Dispersivity And Two Inputs Sources With Source-Sink Term Impact

2021 ◽  
Author(s):  
Thomas TJOCK-MBAGA ◽  
Patrice Ele Abiama ◽  
Jean Marie Ema'a Ema'a ◽  
Germain Hubert Ben-Bolie
Keyword(s):  
Analytical Solution ◽  
Linear Function ◽  
Dispersion Equation ◽  
Source Term ◽  
Numerical Solutions ◽  
Integral Transform ◽  
Liouville Problem ◽  
Finite Domain ◽  
Advection Dispersion ◽  
Sink Term

Abstract This study derives an analytical solution of a one-dimensional (1D) advection-dispersion equation (ADE) for solute transport with two contaminant sources that takes into account the source term. For a heterogeneous medium, groundwater velocity is considered as a linear function while the dispersion as a nth-power of linear function of space and analytical solutions are obtained for and . The solution in a heterogeneous finite domain with unsteady coefficients is obtained using the Generalized Integral Transform Technique (GITT) with a new regular Sturm-Liouville Problem (SLP). The solutions are validated with the numerical solutions obtained using MATLAB pedpe solver and the existing solution from the proposed solutions. We exanimated the influence of the source term, the heterogeneity parameters and the unsteady coefficient on the solute concentration distribution. The results show that the source term produces a solute build-up while the heterogeneity level decreases the concentration level in the medium. As an illustration, model predictions are used to estimate the time histories of the radiological doses of uranium at different distances from the sources boundary in order to understand the potential radiological impact on the general public.

scholarly journals Analytical Solution of a Circular Opening considering Nonuniform Pressure and Its Engineering Application

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Peng Wu ◽  
Yanlong Chen ◽  
Liang Chen ◽  
Xianbiao Mao ◽  
Wei Zhang
Keyword(s):  
Analytical Solution ◽  
Plastic Zone ◽  
Young’S Modulus ◽  
Lateral Pressure ◽  
Young's Modulus ◽  
Pressure Coefficient ◽  
Surrounding Rock ◽  
Engineering Practice ◽  
Circular Opening ◽  
Lateral Pressure Coefficient

Based on the Mohr–Coulomb criterion, a new analytical solution of a circular opening under nonuniform pressure was presented, which considered rock dilatancy effect and elastic-brittle-plastic failure characteristics. In the plastic zone, the attenuation of Young’s modulus was considered using a radius-dependent model (RDM), and solution of the radius and radial displacement of plastic zone was obtained. The results show that many factors have important impact on the response of the surrounding rock, including lateral pressure coefficient, dilation coefficient, buried depth, and Young’s modulus attenuation. Under nonuniform pressure condition, the distribution of plastic zone and deformation around the opening show obvious nonuniform characteristic: with the increasing of lateral pressure coefficient, the range of plastic zone and deformation decrease gradually at side, while they increase at roof and floor, and the location of the maximum value of support and surrounding rock response curve transfers from side to roof. Based on the analytical results and engineering practice, an optimization method of support design was proposed for the circular opening under nonuniform pressure.

scholarly journals Application of hybrid asymptotic methods and modern software for mathematical models developing for nonlinear dynamics of structures with variables in time parameters

2021 ◽  
pp. 66-70
Author(s):  
S. Homeniuk ◽  
S. Grebenyuk ◽  
D. Gristchak
Keyword(s):  
Nonlinear Dynamics ◽  
Numerical Methods ◽  
Nonlinear Systems ◽  
Analytical Solution ◽  
Mathematical Models ◽  
Nonlinear Dynamic ◽  
Numerical Solutions ◽  
Hybrid Approach ◽  
Time Dependent ◽  
Forced Oscillations

The relevance. The aerospace domain requires studies of mathematical models of nonlinear dynamic structures with time-varying parameters. The aim of the work. To obtain an approximate analytical solution of nonlinear forced oscillations of the designed models with time-dependent parameters. The research methods. A hybrid approach based on perturbation methods, phase integrals, Galorkin orthogonalization criterion is used to obtain solutions. Results. Nonlocal investigation of nonlinear systems behavior is done using results of analytical and numerical methods and developed software. Despite the existence of sufficiently powerful numerical software systems, qualitative analysis of nonlinear systems with variable parameters requires improved mathematical models based on effective analytical, including approximate, solutions, which using numerical methods allow to provide a reliable analysis of the studied structures at the stage designing. An approximate solution in analytical form is obtained with constant coefficients that depend on the initial conditions. Conclusions. The approximate analytical results and direct numerical solutions of the basic equation were compared which showed a sufficient correlation of the obtained analytical solution. The proposed algorithm and program for visualization of a nonlinear dynamic process could be implemented in nonlinear dynamics problems of systems with time-dependent parameters.

scholarly journals Numerical Modeling on Hydraulic Fracturing in Coal-Rock Mass for Enhancing Gas Drainage

2018 ◽  
Vol 2018 ◽  
pp. 1-16 ◽  
Author(s):  
Zhigang Yuan ◽  
Yaohua Shao
Keyword(s):  
Mathematical Model ◽  
Rock Mass ◽  
Hydraulic Fracturing ◽  
Gas Flow ◽  
Numerical Solutions ◽  
Fracture Initiation ◽  
Engineering Practice ◽  
Gas Drainage ◽  
Fracturing Process ◽  
Coal Rock

The mechanism of how hydraulic fracturing influences gas drainage in coal-rock mass is still not clear due to its complex mechanism. In this work, statistical distributions are firstly introduced to describe heterogeneity of coal-rock mass; a novel simultaneously coupled mathematical model, which can describe the fully coupled process including seepage-damage coupling during hydraulic fracturing process and subsequent gas flow during gas drainage process, is established; its numerical implementation procedure is coded into a Matlab program to calculate the damage variables, and it partly uses COMSOL solver to obtain numerical solutions of governing equations with damage-flow coupling; the mathematical model and its implementation are validated for initial damage pressure and mode of a single solid model without considering flow-damage coupling, as well as fracture initiation pressure and influence of heterogeneity on damage evolution of hydraulic fracturing considering flow-damage coupling; and finally, based on an engineering practice of hydraulic fracturing with two boreholes, the mechanism of how hydraulic fracturing influences gas drainage is investigated, numerical simulation results indicate that coal-rock mass pore-fissure structure has been improved, and there would exist a gas migration channel with characteristics of higher porosity and lower stresses, which demonstrates significant effects and mechanism of hydraulic fracturing on improving coal-rock permeability and enhancing gas drainage. The research results provide a guide for operation of hydraulic fracturing and optimal layout of gas drainage boreholes.

Analytical-Solution Based Corner Correction for Transient Thermal Measurement

2015 ◽  
Vol 137 (11) ◽  
Author(s):  
H. Jiang ◽  
W. Chen ◽  
Q. Zhang ◽  
L. He
Keyword(s):  
Heat Transfer ◽  
Analytical Solution ◽  
Data Processing ◽  
Numerical Solutions ◽  
Thermal Measurement ◽  
Time Step ◽  
Experimental Data Processing ◽  
Numerical Solution Methods ◽  
Transient Thermal ◽  
Analytical Approaches

The one-dimensional (1D) conduction analytical approaches for a semi-infinite domain, widely adopted in the data processing of transient thermal experiments, can lead to large errors, especially near a corner of solid domain. The problems could be addressed by adopting 2D/3D numerical solutions (finite element analysis (FEA) or computational fluid dynamics (CFD)) of the solid field. In addition to needing the access to a conduction solver and extra computing effort, the numerical field solution based processing methods often require extra experimental efforts to obtain full thermal boundary conditions around corners. On a more fundamental note, it would be highly preferable that the experimental data processing is completely free of any numerical solutions and associated discretization errors, not least because it is often the case that the main purposes of many experimental measurements are exactly to validate the numerical solution methods themselves. In the present work, an analytical-solution based method is developed to enable the correction of the 2D conduction errors in a corner region without using any conduction solvers. The new approach is based on the recognition that a temperature time trace in a 2D corner situation is the result of the accumulated heat conductions in both the normal and lateral directions. An equivalent semi-infinite 1D conduction temperature trace for a correct heat transfer coefficient (HTC) can then be generated by reconstructing and removing the lateral conduction component at each time step. It is demonstrated that this simple correction technique enables the use of the standard 1D conduction analysis to get the correct HTC completely analytically without any aid of CFD or FEA solutions. In addition to a transient infrared (IR) thermal measurement case, two numerical test cases of practical interest with turbine blade tip heat transfer and film cooling are used for validation and demonstration. It has been consistently shown that the errors of the conventional 1D conduction analysis in the near corner regions can be greatly reduced by the new corner correction method.

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