scholarly journals ANALYTICAL CALCULATION OF DEFLECTION OF A MULTI-LATTICE TRUSS WITH AN ARBITRARY NUMBER OF PANELS

2020 ◽  
Vol 16 (3) ◽  
pp. 23-29
Author(s):  
Mikhail Kirsanov
Keyword(s):  
Asymptotic Solution ◽  
Arbitrary Number ◽  
Analytical Calculation ◽  
Analytical Form ◽  
Linear Recurrence ◽  
Equilibrium Equations ◽  
Recurrence Equations ◽  
Cutting Out ◽  
Homogeneous Linear ◽  
System Of Equilibrium Equations

The scheme of a planar externally statically indeterminate truss with four supports is proposed. In analytical form, for several types of loads, the problem of forces in the rods and deflectionof the structure is solved, depending on the number of panels, the size and intensity of the load. The solution uses the Maple computer mathematics system. The deflectionat Midspan is determined using Maxwell – Mohr's formula, the forces in the rods – the method of cutting out nodes from the system of equilibrium equations for all nodes, which includes four reactions of the supports. By induction, a series of solutions for trusses with a consistently increasing number of panels is generalized to an arbitrary number of panels. For the elements of the sequences of coefficientare developed and are solved by homogeneous linear recurrence equations. The resulting formulas for the deflectio of the structure under various loads have the form of polynomials in the number of panels. A linear asymptotic solution for the number of panels is found. The kinematic degeneration of the structure and the distribution of node speeds corresponding to this case were found. The dependences of the reaction of supports and forces in the most compressed and stretched rods on the number of panels are determined.

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 Inductive analysis of the deflection of a beam truss allowing kinematic variation

2018 ◽  
Vol 239 ◽  
pp. 01012
Author(s):  
Mikhail Kirsanov ◽  
Evgeny Komerzan ◽  
Olesya Sviridenko
Keyword(s):  
Optimal Design ◽  
External Load ◽  
Analytical Calculation ◽  
System Of Equations ◽  
Number Of Solutions ◽  
Recurrence Equations ◽  
General Terms ◽  
Cutting Out ◽  
Selection Of ◽  
Inductive Analysis

A scheme of a statically determinate planar truss is proposed and an analytical calculation of its deflection and displacement of the mobile support are obtained. The forces in the rods from the external load, uniformly distributed over the nodes of the lower or upper belt, are determined by the method of cutting out nodes using the computer mathematic system Maple. In the generalization of a number of solutions of trusses with a different number of panels to the general case, the general terms of the sequence of coefficients in the formulas are found from solutions of linear homogeneous recurrence equations. To compose and solve these equations, Maple operators were used. In the process of calculation it was revealed that for even numbers of panels in half the span, the determinant of the system of equations degenerates. This corresponds to the kinematic degeneracy of the structure. The corresponding scheme of possible speeds of the truss is given. The displacement was determined by the Maxwell-Mohr’s formula. The graphs of the obtained dependences have appreciable jumps, which in principle can be used in the selection of optimal design sizes.

scholarly journals Lower estimate of the fundamental frequency of natural oscillations of a truss with an arbitrary number of panels

Vestnik MGSU ◽  
2019 ◽  
pp. 844-851
Author(s):  
Mikhail N. Kirsanov
Keyword(s):  
Fundamental Frequency ◽  
Arbitrary Number ◽  
Lower Estimate ◽  
Analytical Form ◽  
Equilibrium Equations ◽  
Rounding Errors ◽  
Natural Oscillations ◽  
Analytical Estimate ◽  
The Matrix ◽  
Analytical Result

Introduction: the paper deals with oscillations of a statically definable plane, truss with a double lattice of racks and descending braces with massive loads in the nodes of the lower chord. The weight of the truss rods is not taken into account. It is assumed that the freights are moved only vertically. The fundamental frequency of natural oscillations is estimated from the Dunkerley formula by the values of partial frequencies. Materials and methods: an analytical estimate is obtained by generalizing formulas obtained from a series of estimates for trusses with a consistently increasing number of panels. The stiffness of the truss was determined using the Mohr’s integral. The double lattice of the truss does not allow using the cross-section method; therefore, the forces in the rods were calculated (or estimated) in an analytical form using the method of cutting nodes with the compilation of a system of equilibrium equations simultaneously for all rods and three support reactions. The matrix of equilibrium equations was compiled in a software program written in the language of the Maple computer mathematics system based on the coordinates of the nodes and the values of the direction cosines of the forces. For a sequence of coefficients of the desired formula, linear homogeneous recurrent equations were found and solved by means of special operators of the Maple system. Results: the resulting formula estimating the relationship between the fundamental frequency and the panels number has the form of a sixth degree polynomial with coefficients depending on the parity of the number of panels. The analytical result is compared with the smallest frequency obtained numerically from the solution of the problem of oscillation of the cargo system. It is shown that the main frequency, depending on the truss height, has an extremum. Conclusions: the method of generalizing particular solutions using the Maple system operators allowed authors to obtain and analyze a formula for a lower estimate of the fundamental frequency of oscillation of a truss model with an arbitrary number of panels. The resulting estimate can be used as a test for numerically obtained solutions. The formula is especially efficient for systems with a large number of panels; as numerical methods for their calculation are time-consuming require considerable time and have a tendency for accumulating rounding errors.

scholarly journals Calculation of the deflection of an arched truss with suspended elements depending on the number of panels

2020 ◽  
Vol 16 (3) ◽  
pp. 179-184
Author(s):  
Mikhail N. Kirsanov
Keyword(s):  
Asymptotic Properties ◽  
Analytical Calculation ◽  
General System ◽  
Uniform Load ◽  
Equilibrium Equations ◽  
Rounding Errors ◽  
Symbolic Form ◽  
Total Load ◽  
Independent Parameters ◽  
System Of Equilibrium Equations

The aim of the work - to propose a scheme and analytical calculation of a statically definable planar truss with a suspended lower belt. Methods. The formula for the dependence of the deflection of the truss under the action of a uniform load on the lower belt on its size and the number of panels is derived in the computer mathematics system Maple. The forces in the rods are found from the solution of the general system of equilibrium equations of all nodes in symbolic form. The deflection is calculated using the Maxwell - Mohr's formula. Generalization of a number of formulas for deflection obtained by increasing the number of panels sequentially to an arbitrary number is performed by double induction using two independent parameters. In this case, special operators of the Maple system are used, allowing for a sequence of coefficients in the desired formula to create and solve recurrent equations that satisfy the elements of the sequences. Results. The obtained solutions have a polynomial form for the number of panels. Curves of deflection dependence on the number of panels are constructed and analyzed. Asymptotic properties of solutions are found in the case of a fixed span length of the structure and a given total load. The proposed scheme is a statically determinate structure with two independent parameters of regularity allows for the finding of a fairly simple analytical solution. The resulting formula is most effective in calculating systems with a large number of elements, where numerical methods tend to accumulate rounding errors.

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.

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.

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

Boundary Integral Equation Methods for a Refined Model of Elastic Plates

2006 ◽  
Vol 11 (6) ◽  
pp. 642-654
Author(s):  
Radu Mitric ◽  
Christian Constanda
Keyword(s):  
Integral Equation ◽  
Boundary Integral Equation ◽  
Boundary Integral ◽  
Normal Strain ◽  
Elastic Plates ◽  
Equilibrium Equations ◽  
Neumann Problems ◽  
Integral Equation Methods ◽  
Transverse Normal Strain ◽  
System Of Equilibrium Equations

A theory of bending of elastic plates is considered, in which the effect of transverse shear deformation and transverse normal strain are taken into account through a specific form of the displacement field. It is shown that the system of equilibrium equations is elliptic and that Betti and Somigliana formulae can be established, which permit the solution of the interior and exterior Dirichlet and Neumann problems by means of boundary integral equation methods.

scholarly journals The Replication Hypothesis along the Take-Off Run and a System of Equilibrium Equations at the Lift-Off of a Protobird

Aerospace ◽  
2019 ◽  
Vol 6 (2) ◽  
pp. 21
Author(s):  
Phillip Burgers
Keyword(s):  
Experimental Data ◽  
Time Span ◽  
Flapping Wings ◽  
Equilibrium Equations ◽  
Scaling Down ◽  
Lift Off ◽  
To Come ◽  
System Of Equilibrium Equations

An extant bird resorts to flapping and running along its take-off run to generate lift and thrust in order to reach the minimum required wing velocity speed required for lift-off. This paper introduces the replication hypothesis that posits that the variation of lift relative to the thrust generated by the flapping wings of an extant bird, along its take-off run, replicates the variation of lift relative to the thrust by the flapping wings of a protobird as it evolves towards sustained flight. The replication hypothesis combines experimental data from extant birds with evidence from the paleontological record of protobirds to come up with a physics-based model of its evolution towards sustained flight while scaling down the time span from millions of years to a few seconds. A second hypothesis states that the vertical and horizontal forces acting on a protobird when it first encounters lift-off are in equilibrium as the protobird exerts its maximum available power for flapping, equaling its lift with its weight, and its thrust with its drag.

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