Analytical Calculation of Deformations and Kinematic Analysis of a Flat Truss with an Arbitrary Number of Panels

2021 ◽  
pp. 113-122
Author(s):  
М. Н. Кирсанов ◽  
О. В. Воробьев
Keyword(s):  
Kinematic Analysis ◽  
Arbitrary Number ◽  
Numerical Solutions ◽  
Optimization Problems ◽  
Analytical Calculation ◽  
Middle Part ◽  
Asymptotic Approximations ◽  
Induction Method ◽  
Particular Solutions

Постановка задачи. Разыскиваются аналитические зависимости прогиба и смещения опоры плоской фермы решетчатого вида от числа панелей. Ферма имеет сдвоенную решетку, прямолинейный нижний и приподнятый в средней части верхний пояс. Результаты. Для двух видов нагружения по формуле Максвелла-Мора получены аналитические зависимости прогибов конструкции от нагрузки, размеров и числа панелей. Для обобщения серии частных решений с различным числом панелей ферм на произвольный случай использован метод индукции и аналитические возможности системы компьютерной математики Maple. Для некоторых решений получены асимптотические приближения. Показано распределение усилий в элементах фермы. Выводы. Полученные формулы могут быть использованы в задачах оптимизации и как тестовые для оценки приближенных численных решений. Выявлены случаи геометрической изменяемости фермы при числе панелей, кратном трем. Приведен алгоритм выявления соответствующего распределения возможных скоростей шарниров. Statement of the problem. Analytical dependences of the deflection and displacement of the support of a flat lattice truss on the number of panels are being sought. The truss has a double lattice, a rectilinear lower belt and an upper belt raised in the middle part. Results. For two types of loading, according to the Maxwell-Mohr formula, analytical dependences of the deflections of the structure on the load, dimensions and number of panels are obtained. To generalize a series of particular solutions for trusses with different numbers of panels for an arbitrary case, the induction method and the analytical capabilities of the Maple computer mathematics system were used. For some solutions, asymptotic approximations are obtained. The distribution of forces in the rods of the structure is shown. Conclusions. The obtained formulas can be used in optimization problems and as test ones for evaluating approximate numerical solutions. Cases of geometric variability of the truss with the number of panels being a multiple of three are revealed. An algorithm for identifying the corresponding distribution of possible velocities of the joints is presented.

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.

scholarly journals Analytical calculation of the deflection of an externally statically indeterminate lattice truss

2019 ◽  
Vol 265 ◽  
pp. 05027
Author(s):  
Mikhail Kirsanov ◽  
Evgeny Komerzan ◽  
Olesya Sviridenko
Keyword(s):  
Arbitrary Number ◽  
Analytical Calculation ◽  
Linear Recurrence ◽  
System Of Equations ◽  
Induction Method ◽  
Particular Solutions ◽  
Recurrence Equations ◽  
Nonmonotonic Character

A scheme of a statically definable truss with additional supports is proposed. Derive formulas for the dependence of the deflection of the truss against the number of panels for three types of symmetrical loads. It is shown that for definite numbers of panels the determinant of the system of equations for the equilibrium of nodes degenerates. This indicates an instant changeability of the structure. To generalize particular solutions to an arbitrary number of panels, the induction method is applied. For this purpose, in the computer mathematics system Maple linear recurrence equations are constructed for the terms of a sequence of coefficients from individual solutions. The graphs of the dependences obtained indicate a nonmonotonic character of the solutions found and the possibility of optimizing the design by choosing the number of panels.

scholarly journals Analytical calculation of deformations of a truss for a long span covering

Vestnik MGSU ◽  
2020 ◽  
pp. 1399-1406
Author(s):  
Mikhail N. Kirsanov
Keyword(s):  
Structural Optimization ◽  
Qualitative Analysis ◽  
Arbitrary Number ◽  
Numerical Solutions ◽  
Analytical Calculation ◽  
Linear Recurrence ◽  
Induction Method ◽  
Design Formula ◽  
Long Span ◽  
Final Design

Introduction. The method of induction based on the number of panels is employed to derive formulas designated for deflection of a square in plan hinged rod structure, which has supports on its sides and which consists of individual pyramidal rod elements. The truss is statically determinable and symmetrical. Some features of the solution are identified on the curves constructed according to derived formulas. Materials and methods. The analysis of forces arising in the rods of the covering is performed symbolically using the method of joint isolation and operators of the Maple symbolic math engine. The deflection in the centre of the covering is found by the Maxwell–Mohr’s formula. The rigidity of truss rods is assumed to be the same. The analysis of a sequence of analytical calculations of trusses, having different numbers of panels, is employed to identify coefficients, designated for deflection and reaction at the supports, in the final design formula. The induction method is employed for this purpose. Common members of sequences of coefficients are derived from the solution of linear recurrence equations made using Maple operators. Results. Solutions, obtained for three types of loads, are polynomial in terms of the number of panels. To illustrate the solutions and their qualitative analysis, curves describing the dependence of deflection on the number of panels are made. The author identified the quadratic asymptotics of the solution with respect to the number of panels. The quadratic asymptotics is linear in height. Conclusions. Formulas are obtained for calculating deflection and reactions of covering supports having an arbitrary number of panels and dimensions if exposed to three types of loads. The presence of extremum points on solution curves is shown. The dependences, identified by the author, are designated both for evaluating the accuracy of numerical solutions and for solving problems of structural optimization in terms of rigidity.

scholarly journals Analytical calculation of the combined suspension truss

2019 ◽  
Vol 265 ◽  
pp. 05025
Author(s):  
Mikhail Kirsanov ◽  
Dmitriy Tinkov ◽  
Oleh Boiko
Keyword(s):  
Analytical Calculation ◽  
Analytical Dependence ◽  
System Of Equations ◽  
Vertical Load ◽  
Induction Method ◽  
Uniform Loading ◽  
Number Of Solutions ◽  
Particular Solutions ◽  
Common Term ◽  
Cutting Out

An algorithm is given for obtaining the formula for the dependence of the deflection of a regular planar truss of an arched type with a suspended lower belt on the number of panels. The cases of uniform loading of the nodes of the upper and lower belts by a vertical load are considered. To generalize a number of solutions obtained in the system of computer mathematics Maple to an arbitrary case, an induction method was applied. For this purpose, for a sequence of coefficients of the particular solutions found, a common term is determined which is a solution of the recurrence equation. The deflection was determined with the help of Mohr's integral, which depends on the forces in the rods. Forces in a statically determinate construction were performed by cutting out nodes from the solution of a system of equations written in a matrix form. The analytical dependence of displacement of the mobile support on the number of panels is found.

Kinematic analysis and deformation of a planar lattice with an arbitrary number of panels

2020 ◽  
pp. 15-19
Author(s):  
M.N. Kirsanov
Keyword(s):  
Exact Solution ◽  
Kinematic Analysis ◽  
Arbitrary Number ◽  
Design Parameters ◽  
Lattice Deformation ◽  
Planar Lattice ◽  
Force Loading

Formulae are obtained for calculating the deformations of a statically determinate lattice under the action of two types of loads in its plane, depending on the number of panels located along one side of the lattice. Two options for fixing the lattice are analyzed. Cases of kinematic variability of the structure are found. The distribution of forces in the rods of the lattice is shown. The dependences of the force loading of some rods on the design parameters are obtained. Keywords: truss, lattice, deformation, exact solution, deflection, induction, Maple system. [email protected]

scholarly journals ANALYTICAL SOLUTION OF THE FREQUENCY OF THE LOAD OSCILLATION AT AN ARBITRARY GIRDER NODE IN THE SYSTEM MAPLE

2019 ◽  
pp. 3-3
Author(s):  
Mikhail N. Kirsanov ◽  
Dmitriy V. Tinkov
Keyword(s):  
Analytical Solution ◽  
Arbitrary Number ◽  
Analytical Form ◽  
Labor Costs ◽  
Induction Method ◽  
Regular Type ◽  
Final Formula ◽  
The Common ◽  
Two Stages ◽  
Elastic Lattice

Introduction. We study the oscillations of a massive load on a planar statically definable symmetric truss of a regular type with parallel belts. Truss weight is not included. Free vertical oscillations are considered. The stiffness of the truss rods is assumed to be the same, the deformations are elastic. Lattice of the truss is double with descending braces and racks. New in the formulation and solution of the problem is the analytical form of the solution, which makes it possible in practice to easily evaluate the frequency characteristics of the structure depending on an arbitrary number of truss panels and the location of the load. Materials and methods. The operators and methods of the system of computer mathematics Maple are used. To determine the forces in the rods, the knotting method is used. The common terms of the sequence of coefficients of solutions for different numbers of panels are obtained from solving linear homogeneous recurrent equations of various order, obtained by special operators of the Maple system. Dependence on two arbitrary natural parameters is revealed in two stages. First, solutions for fixed load positions are found, then these solutions are summarized into one final formula for frequency. Results. By a series of individual solutions to the problem of load oscillation using the double induction method, it was possible to find common members of all sequences. The solution is polynomial in both natural parameters. Graphs constructed for particular cases, showed the adequacy of the approach. The discontinuous non-monotonic nature of the intermittent change depending on the number of truss panels and some other features of the solution are noted. Conclusions. It is shown that the induction method, previously applicable mainly to statics problems with one parameter (number of truss panels), is fully operational to the problems of the oscillations of system with two natural parameters. It should be noted that significant labor costs and a significant increase in the time symbolic transformations in such tasks

Formulation of a General Multiphase, Multicomponent Chemical Flood Model

1981 ◽  
Vol 21 (01) ◽  
pp. 63-76 ◽  
Author(s):  
Paul D. Fleming ◽  
Charles P. Thomas ◽  
William K. Winter
Keyword(s):  
Phase Equilibrium ◽  
Arbitrary Number ◽  
Numerical Solutions ◽  
Field Tests ◽  
Reservoir Rock ◽  
Phase System ◽  
Surfactant Flooding ◽  
Two Phase ◽  
Special Cases ◽  
Flood Model

Abstract A general multiphase, multicomponent chemical flood model has been formulated. The set of mass conservation laws for each component in an isothermal system is closed by assuming local thermodynamic (phase) equilibrium, Darcy's law for multiphase flow through porous media, and Fick's law of diffusion. For the special case of binary, two-phase flow of nonmixing incompressible fluids, the equations reduce to those of Buckley and Leverett. The Buckley-Leverett equations also may be obtained for significant fractions of both components in the phases if the two phases are sufficiently incompressible. To illustrate the usefulness of the approach, a simple chemical flood model for a ternary, two-phase system is obtained which can be applied to surfactant flooding, polymer flooding, caustic flooding, etc. Introduction Field tests of various forms of surfactant flooding currently are under way or planned at a number of locations throughout the country.1 The chemical systems used have become quite complicated, often containing up to six components (water, oil, surfactant, alcohol, salt, and polymer). The interactions of these components with each other and with the reservoir rock and fluids are complex and have been the subject of many laboratory investigations.2–22 To aid in organizing and understanding laboratory work, as well as providing a means of extrapolating laboratory results to field situations, a mathematical description of the process is needed. Although it seems certain that mathematical simulations of such processes are being performed, models aimed specifically at the process have been reported only recently in the literature.23–31 It is likely that many such simulations are being performed on variants of immiscible, miscible, and compositional models that do not account for all the facets of a micellar/polymer process. To help put the many factors of such a process in proper perspective, a generalized model has been formulated incorporating an arbitrary number of components and an arbitrary number of phases. The development assumes isothermal conditions and local phase equilibrium. Darcy's law32,33 is assumed to apply to the flow of separate phases, and Fick's law34 of diffusion is applied to components within a phase. The general development also provides for mass transfer of all components between phases, the adsorption of components by the porous medium, compressibility, gravity segregation effects, and pressure differences between phases. With the proper simplifying assumptions, the general model is shown to degenerate into more familiar special cases. Numerical solutions of special cases of interest are presented elsewhere.35

scholarly journals Quadratic Programming in Insurance

1974 ◽  
Vol 7 (3) ◽  
pp. 311-322 ◽  
Author(s):  
H. Schmitter ◽  
E. Straub
Keyword(s):  
Quadratic Programming ◽  
Numerical Solutions ◽  
Optimization Problems ◽  
Linear Equations ◽  
Quadratic Function ◽  
Existence Of A Solution ◽  
Open Questions ◽  
Cost Distribution ◽  
Linear Restrictions ◽  
Credibility Problem

Abstract and IntroductionQuadratic programming means maximizing or minimizing a quadratic function of one or more variables subject to linear restrictions i.e. linear equations and/or inequalities.Among the numerous insurance problems which can be formulated as quadratic programs we shall only discuss four, namely the Credibility, Retention, IBNR and the Cost Distribution problems.Generally, there is no explicite solution to quadratic optimization problems, only statements about the existence of a solution can be made or some algorithm may be recommended in order to get exact or approximate numerical solutions. Restricting ourselves to typical problems of the above mentioned type, however, enables us to give an explicit solution in terms of general formulae for quite a number of cases, such as the onedimensional credibility problem, the retention problem and—under relatively week assumptions— for the IBNR-problem.The results given here are by no means new. The only goal of this paper is to describe a few fundamental insurance problems from a common mathematical standpoint, namely that of quadratic programming and at the same time, to draw attention to a few special aspects and open questions in this field.

scholarly journals Mathematical modelling of the vitamin C clock reaction

2019 ◽  
Vol 6 (4) ◽  
pp. 181367
Author(s):  
R. Kerr ◽  
W. M. Thomson ◽  
D. J. Smith
Keyword(s):  
Vitamin C ◽  
Approximate Formula ◽  
Chemistry Education ◽  
Phase Plane ◽  
Numerical Solutions ◽  
Model System ◽  
Asymptotic Approximations ◽  
Nonlinear Ordinary Differential Equations ◽  
Household Chemicals ◽  
Clock Reaction

Chemical clock reactions are characterized by a relatively long induction period followed by a rapid ‘switchover’ during which the concentration of a clock chemical rises rapidly. In addition to their interest in chemistry education, these reactions are relevant to industrial and biochemical applications. A substrate-depletive, non-autocatalytic clock reaction involving household chemicals (vitamin C, iodine, hydrogen peroxide and starch) is modelled mathematically via a system of nonlinear ordinary differential equations. Following dimensional analysis, the model is analysed in the phase plane and via matched asymptotic expansions. Asymptotic approximations are found to agree closely with numerical solutions in the appropriate time regions. Asymptotic analysis also yields an approximate formula for the dependence of switchover time on initial concentrations and the rate of the slow reaction. This formula is tested via ‘kitchen sink chemistry’ experiments, and is found to enable a good fit to experimental series varying in initial concentrations of both iodine and vitamin C. The vitamin C clock reaction provides an accessible model system for mathematical chemistry.

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