Arbitrary Number Of Panels Research Articles

Formulas for Calculating the Deflections of a Triangular Truss of Spatial Cover

Statement of the problem. The scheme of triangular in terms of dome type cover is proposed. The construction is statically determinate. The formulas for the dependence of the deflection on the number of panels and sizes are derived by generalizing a series of individual solutions by induction. Materials and methods. The forces in the coating rods are performed by cutting nodes in symbolic form using operators of the Maple symbolic mathematics system. The unknown systems of equilibrium equations in projections on the coordinate axis include the reactions of vertical supports located on the sides of the truss. One of the corners of the truss also has a spherical support, one is cylindrical. The Maxwell—Mohr formula is used to calculate the deflection of the vertex. The analysis of sequences of coefficients in solutions for individual trusses with different numbers of panels yields expressions for common terms included in the desired calculation formula. Results. Formulas for the dependence of the deflections of the truss on the number of panels for a vertical load evenly distributed over the nodes of the truss and a horizontal wind load applied to one of the sides of the structure are obtained. The solutions have a simple polynomial form. The curves of the dependence of the horizontal displacement of the dome top on the number of panels reveal a minimum. Asymptotics of the solutions is identified. Conclusions. A scheme of a statically determinate symmetric spatial dome is developed and its mathematical model is constructed, allowing analytical solutions with an arbitrary number of panels. The identified dependencies can be used both to evaluate the accuracy of numerical solutions and to find optimal combinations of structural dimensions in terms of rigidity.

Analytical estimation of the natural frequency of vibrations of a flat lattice

A formula is derived for calculating the first natural frequency of vibrations of a statically determined regular lattice with an arbitrary number of panels and masses uniformly distributed over the nodes. Comparison with the minimum frequency from the numerical solution of the problem of the vibration frequency spectrum of the structure shows the high accuracy of the derived assessment. It is shown that for some number of panels, the design allows for kinematic variability. The influence of the lattice attachment method on the natural frequency is studied. Keywords: truss, lattice, vibration frequency, induction, Maple, Dunkerley estimate, attachment method.c216@ya.ru

The natural frequency of a mast with an arbitrary number of panels

Introduction. The article addresses a spatial model of a statically definable mast truss consisting of four identical planar trusses with a crosswise grid system and a base with four supports at the corners. The authors solve the problem of deriving the analytical dependence between the bottom vibration frequency of the mast truss and the number of panels, mass, linear dimensions of its construction and properties of the material.
 
 Materials and methods. To calculate the values of forces, arising in the rods of a mast truss with an arbitrary number of panels, and analyze the obtained results, the induction method and operators of the Maple computer system for mathematics were used. The problem of deriving the analytical dependence between the bottom frequency of vibrations of the mast truss and its parameters is solved using the Dunkerley method, which generates the bottom estimate of the natural frequency. The rigidity of the truss structure is calculated according to the Maxwell – Mohr formula. To calculate the common members of sequences of coefficients, homogeneous linear recurrent equations are derived and solved in the frequency formula.
 
 Results. A formula is obtained for estimating the first frequency of natural vibrations of a truss. The formula coefficients have the form of polynomials of no higher than the fourth order. The accuracy of the calculation formula, obtained using the Dunkerley method, is estimated by the comparison with the first frequency, obtained through the numerical calculation of the entire spectrum of natural frequencies.
 
 Conclusions. The analysis of the analytical results and their comparison with the numerical ones shows high accuracy of the derived formula. The authors have identified a dependence, whereby an increase in the number of mast truss panels boosts the accuracy of the bottom estimate of the natural frequency.

Аналитический расчет деформаций и кинематический анализ плоской фермы с произвольным числом панелей

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.

The analysis of dependence of the vibration frequency of a space cantilever truss on the number of panels

Introduction. The first (lowest) frequency of natural vibrations of a structure is one of its most important dynamic characteristics. Analytical solutions supplement numerical ones; they can be efficiently used to perform a rapid assessment of properties of structures, to analyze and optimize constructions and to test numerical results. A space cantilever truss consisting of three planar trusses with a rectangular grid is considered in the article. The objective is to find the analytical dependence between the frequency of natural vibrations of a structure and the number of panels. It is assumed that the truss mass is distributed among the joints. Only the vertical mass displacement is taken into account.
 Materials and methods. Forces, arising in cantilever rods, are calculated by the Maple software as symbolic expressions, and the method of joint isolation is used here. The stiffness matrix is identified using the Mohr integral. Rods are assumed to be elastic, they have identical stiffness. The lower value of the vibration frequency is determined using the Dunkerley method. The final calculation formula used to identify the value of the vibration frequency is derived using the method of induction applied to a series of analytical solutions developed for trusses with a consistently increasing number of panels. When common members of sequences are found, genfunc operators of the Maple system are used. The analytical solution is compared with the numerical solution in terms of the first frequency using the analysis of the system spectrum featuring many degrees of freedom. The eigenvalues of the characteristic matrix are identified using the Eigenvalues operator from the Linear Algebra package.
 Results. The comparison between the analytical values and the numerical solution shows that the Dunkerley method ensures the accuracy varying from 20 % for a small number of panels to 3 % if the number of panels exceeds ten. The size of the structure, the weight and stiffness of rods have little effect on the accuracy of the obtained values.
 Conclusions. The lowest value obtained using the Dunkerley method in the form of a fairly compact formula has good accuracy, its application to a space structure with an arbitrary number of panels has a polynomial form equal to the number of panels, and it can be used in practical calculations.

Analytical evaluation of the fundamental frequency of natural vibrations of the spatial coverage

The scheme of statically definable truss of spatial coverage is proposed. The formula for the dependence of the vibration frequency on the number of panels is derived. The Dunkerley lower bound and the induction method are used to generalize particular solutions to the case of an arbitrary number of panels. The calculation of the forces in the rods by cutting out the nodes and the analytical transformations to obtain the desired dependence are performed in the Maple computer mathematics system. The solution is compared with the numerical one obtained by solving the problem on the eigenvalues of the characteristic matrix for a system with many degrees of freedom. It is shown that the estimation accuracy depends on the number of panels.

Analytical assessment of the frequency of natural vibrations of a truss with an arbitrary number of panels

The aim of the work is to derive a formula for the dependence of the first frequency of the natural oscillations of a planar statically determinate beam truss with parallel belts on the number of panels, sizes and masses concentrated in the nodes of the lower truss belt. Truss has a triangular lattice with vertical racks. The solution uses Maple computer math system operators. Methods. The basis for the upper estimate of the desired oscillation frequency of a regular truss is the energy method. As a form of deflection of the truss taken deflection from the action of a uniformly distributed load. Only vertical mass movements are assumed. The amplitude values of the deflection of the truss is calculated by the Maxwell - Mohrs formula. The forces in the rods are determined in symbolic form by the method of cutting nodes. The dependence of the solution on the number of panels is obtained by an inductive generalization of a series of solutions for trusses with a successively increasing number of panels. For sequences of coefficients of the desired formula, fourth-order homogeneous linear recurrence equations are compiled and solved. Results. The solution is compared with the numerical one, obtained from the analysis of the entire spectrum of natural frequencies of oscillations of the mass system located at the nodes of the truss. The frequency equation is compiled and solved using Eigenvalue search operators in the Maple system. It is shown that the obtained analytical estimate differs from the numerical solution by a fraction of a percent. Moreover, with an increase in the number of panels, the error of the energy method decreases monotonically. A simpler lower bound for the oscillation frequency according to the Dunkerley method is presented. The accuracy of the lower estimate is much lower than the upper estimate, depending on the size and number of panels.

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

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.

Аналитический расчет прогиба пространственной шарнирно-стержневой рамы с произвольным числом панелей

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.

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

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. c216@ya.ru

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

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.

АНАЛИЗ ЧАСТОТ КОЛЕБАНИЙ ГРУЗА В ЗАВИСИМОСТИ ОТ ЕГО ПОЛОЖЕНИЯ В УЗЛАХ ПЛОСКОЙ ФЕРМЫ

A beam statically determinate truss with an arbitrary number of panels, in one of the nodes of which there is a massive load, is considered. The mass of the rods of the truss is neglected. A formula is derived for the dependence of the vibration frequency of the load on the hinge number in which it is located. The rigidity of the truss in the frequency equation is determined by the Maxwell-Mohr's formula. Forces in rods are determined in symbolic form by cutting out nodes in a program written in the computer mathematics system Maple. By a method of double induction (by the number of the node where the load is located and the number of panels), a series of particular solutions is generalized to an arbitrary case. A recurrence equation that is satisfied by a sequence of coefficients of particular solutions is given by a special operator of the Maple system.

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

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.

Analysis of the natural frequencies of oscillations of a planar truss with an arbitrary number of panels

Introduction. Analytical solutions for problems of structural mechanics are not only an alternative approach to solving problems of strength, reliability and dynamics of structures, but also the possibility for simple performance evaluations and optimization of structures. Frequency analysis of planar trusses, most often used in construction and engineering, is an important part of the study of structures. Objectives - development of a three-parameter induction algorithm for deriving the analytical dependence of the natural oscillation frequencies of the truss on the number of panels. Materials and methods. A flat, statically definable truss with one additional external link and double braces has been considered. The inertia properties of the truss are modeled by point masses located in the nodes of the lower straight truss belt. Each mass is assumed to have only one vertical degree of freedom. The stiffness of all truss rods is assumed to be the same. The task is to obtain analytical dependences of the oscillation frequencies of the proposed truss model on the number of panels. The derivation of the desired formulas is performed by the method of induction in three stages - according to the numbers of rows and columns of the compliance matrix, calculated using the Maxwell - Mohr formula and the number of panels. To find common members of the obtained sequences of coefficients, an apparatus was used to compile and solve the recurrent equations of the Maple computer mathematics system. The task of determining frequencies has been reduced to the eigenvalue problem of a bisymmetric matrix. Results. For the elements of the compliance matrix, general formulas have been found, according to which the frequency equations are compiled and solved. It is shown that in the frequency spectra of trusses with different numbers of panels there is always one common frequency (middle frequency) located in the middle of the spectrum. An expression is found for the maximum value of the average oscillation frequency as a function of the height of the truss. Conclusions. The proposed truss scheme, despite its external static indeterminacy and the lattice, which does not allow for the calculation of forces by such methods as the method of cutting nodes and the cross section method, allows analytical solutions for the natural frequencies of loads in the nodes. The obtained formulas have a rather simple form, and some general properties, such as frequency coincidences for different numbers of panels and the presence of an analytically calculated maximum of the average frequency function of the truss height, make this solution convenient for practical structural evaluations.

Расчетная модель плоской фермы рамного типа с произвольным числом панелей

Расчетная модель плоской фермы рамного типа с произвольным числом панелей

Analytical calculation of the deflection of a beam truss with double bracing

A statically determinate planar truss has rectilinear belts and a triangular lattice consisting of double braces. The four links make it seemingly statically indeterminate. The derivation of the formula for the dependence of the deflection on its dimensions and the number of panels is given. Forces in rods are determined in symbolic form by cutting out nodes from the solution of a system of linear equations in the system of computer mathematics Maple. To determine the deflection, the Maxwell - Mohr’s formula is used. Rods (except all rigid support) are assumed to be elastic with the same rigidity. The generalization of individual solutions to an arbitrary number of panels is done by induction. Operators of the Maple system from the calculation data yield linear homogeneous recurrence equations for the coefficients of the desired formula. The solutions of these equations give the general terms of the obtained sequences. Formulas for three types of loads are obtained and compared (the uniform loading of the nodes of the lower and upper belts and the concentrated force in the middle of the span). Curves of the dependence of deflection on the number of panels have weakly expressed minima. The dependencies of the forces in the most compressed and stretched rods on the number of panels are derived. Also given are asymptotic estimates for solutions in accordance with the number of panels in fixed spans of the construction and at a given total load.

Формула для определения прогиба решетчатой фермы с произвольным числом панелей

Формула для определения прогиба решетчатой фермы с произвольным числом панелей

The exact formulas for calculating deflection and forces in the rods truss "Molodechno" with an arbitrary number of panels

The exact formulas for calculating deflection and forces in the rods truss "Molodechno" with an arbitrary number of panels

Superelement Stiffness Matrix for Space Trusses

The techniques of discrete field analysis are used to analyze a class of longitudinally oriented space trusses. These trusses consist of three planar trusses which make up the faces of the space truss and are combined to form a triangular cross section with common chords. Three types of configurations are considered: (1) X‐braced; (2) Warren (with verticals); and (3) Vierendeel. The approach used here is to treat a truss segment having regular repetitive properties over an arbitrary number of panels as a superelement. Difference equations for each of the three truss configurations are derived and deflections found using finite fourier series. Based on these solution forms, stiffness coefficients are derived. The results are exact in the classical sense and enables a space truss with a large number of joints to be accurately reduced to a single member defined by beam type stiffness coefficients. This superelement can be used in conjunction with other elements to analyze complex truss systems such as guyed towers and space truss roof systems.