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 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 SPECTRUM OF OWN FREQUENCIES OF A SPATIAL SURFACING GIRDER

2021 ◽  
pp. 104-113
Author(s):  
M. N. Kirsanov
Keyword(s):  
Analytical Solution ◽  
Spatial Structure ◽  
Lower Estimate ◽  
Optimization Problems ◽  
Natural Oscillations ◽  
Analytical Estimate ◽  
Rod Cells ◽  
Mass Size ◽  
The Common ◽  
Oscillation Frequencies

Statement of the problem. The scheme of a statically definable girder of a spatial rectangular surfacing is discussed. The problem is to identify the formula for the dependence of the lower estimate of the first frequency of the natural oscillations of the structure by means of the Donkerley method on the number of panels. The truss has supports on the sides and consists of separate rod cells connected in pyramids. Results. Based on the analysis of the sequence of analytical solutions for the first frequency of girders with a different number of panels by induction, the coefficients in the desired formula are derived. The common members of the sequences of coefficients are found as solutions of homogeneous recurrent equations formed according to the results of the calculations using Maple operators. The resulting dependences are obtained in the form of polynomials by the number of panels. A comparison of the analytical solution with the numerical one is provided.Conclusions. An algorithm for deriving an analytical estimate of the fundamental frequency of oscillations of a spatial structure depending on the number of panels, mass, size, and elastic properties of the material is shown. The spectrum of oscillation frequencies of the structure is analyzed. The resulting dependences can be employed in seismic and structural optimization problems.

Spectrum of Own Frequencies of a Spatial Surfacing Grid

2021 ◽  
pp. 97-105
Author(s):  
М. Н. Кирсанов
Keyword(s):  
Analytical Solution ◽  
Spatial Structure ◽  
Lower Estimate ◽  
Optimization Problems ◽  
Natural Oscillations ◽  
Analytical Estimate ◽  
Rod Cells ◽  
Mass Size ◽  
The Common ◽  
Oscillation Frequencies

Постановка задачи. Рассматривается схема статически определимой фермы пространственного прямоугольного покрытия. Ставится задача найти формулу зависимости нижней оценки первой частоты собственных колебаний конструкции по методу Донкерлея от числа панелей. Ферма имеет опоры по сторонам и состоит из отдельных стержневых ячеек, соединенных в пирамиды. Результаты. Из анализа последовательности аналитических решений для первой частоты ферм с различным числом панелей методом индукции выводятся коэффициенты в искомой формуле. Общие члены последовательностей коэффициентов находятся как решения однородных рекуррентных уравнений, образованных по результатам расчетов с помощью операторов Maple . Найденные зависимости получены в виде полиномов по числу панелей. Дано сравнение аналитического решения с численным. Выводы. Приведен алгоритм вывода аналитической оценки основной частоты колебаний пространственной конструкции в зависимости от числа панелей, массы, размеров и упругих свойств материала. Проанализирован спектр частот колебаний сооружения. Найденные зависимости могут быть использованы в задачах сейсмостойкости и оптимизации конструкции. Statement of the problem. The scheme of a statically definable truss of a spatial rectangular covering is discussed. The problem is to identify the formula for the dependence of the lower estimate of the first frequency of the natural oscillations of the structure by means of the Donkerley method on the number of panels. The truss has supports on the sides and consists of separate rod cells connected in pyramids. Results. Based on the analysis of the sequence of analytical solutions for the first frequency of trusses with a different number of panels by induction, the coefficients in the desired formula are derived. The common members of the sequences of coefficients are found as solutions of homogeneous recurrent equations formed according to the results of the calculations using Maple operators. The resulting dependences are obtained in the form of polynomials by the number of panels. A comparison of the analytical solution with the numerical one is provided. Conclusions. An algorithm for deriving an analytical estimate of the fundamental frequency of oscillations of a spatial structure depending on the number of panels, mass, size, and elastic properties of the material is shown. The spectrum of oscillation frequencies of the structure is analyzed. The resulting dependences can be employed in seismic and structural optimization problems.

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

scholarly journals Unified Fundamental Formulas for Static Analysis of Pin-Jointed Bar Assemblies

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 994
Author(s):  
Pei Zhang ◽  
Huiting Xiong ◽  
Junsheng Chen
Keyword(s):  
External Disturbance ◽  
Full Rank ◽  
Equilibrium Equations ◽  
Force Method ◽  
Axial Forces ◽  
The Matrix ◽  
Structural Responses ◽  
Internal Forces ◽  
Stress States ◽  
External Loads

The initial axial forces of members—whatever caused by prestress or external loads—may strongly change the mechanical properties of pin-jointed bar assemblies, to enhance, or even establish their structural stiffness. The structural responses under external disturbance cannot be calculated accurately if the influence of initial axial forces has not been considered appropriately. In this paper, an analytic theory considering the effect of initial internal forces is developed on the basis of linear elasticity hypothesis. The fundamental formulas proposed finally include generalized equilibrium equations and generalized compatibility equations, both of which have square coefficient matrices of full rank being transposed with each other. Generally, this method can be regarded as an extended version of a traditional force method considering the stiffening effect of initial internal forces. Compared with the matrix force method, it has a wider application scenario since few redundant simplifications are employed in the derivation of the formulas. In comparison with the displacement-based algorithm, the proposed method has the inherent advantages of the force method—the physical concepts of each item in equations are fairly explicit; and the combination coefficients of self-stress states and mechanisms are determined simultaneously in solving the structural responses. Thus, it is very helpful for us to essentially comprehend the principle that the pin-jointed bar assemblies resist the external loads.

Local Numerical Range for a Class of 2 ⊗ d Hermitian Operators

2010 ◽  
Vol 17 (04) ◽  
pp. 347-359 ◽  
Author(s):  
J. Jurkowski ◽  
A. Rutkowski ◽  
D. Chruściński
Keyword(s):  
Numerical Range ◽  
Matrix Representation ◽  
The Matrix ◽  
Analytical Result

Local numerical range is analyzed for a family of circulant observables and states of composite 2 ⊗ d systems. It is shown that for any 2 ⊗ d circulant operator [Formula: see text] there exists a basis giving rise to the matrix representation with real nonnegative off-diagonal elements. In this basis the problem of finding the extremum of [Formula: see text] on product vectors |x〉 ⊗ |y〉 ϵ ℂ2 ⊗ ℂd reduces to the corresponding problem in ℝ2 ⊗ ℝd. The final analytical result for d = 2 is presented.

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

2020 ◽  
Vol 16 (5) ◽  
pp. 351-360
Author(s):  
Mikhail N. Kirsanov
Keyword(s):  
Energy Method ◽  
Oscillation Frequency ◽  
Frequency Equation ◽  
Linear Recurrence ◽  
Natural Oscillations ◽  
Analytical Estimate ◽  
Mass System ◽  
Symbolic Form ◽  
Upper Estimate ◽  
Homogeneous Linear

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.

scholarly journals A Comparative Study of Optical Bistability in Three-Level EIT Configurations

2018 ◽  
Vol 28 (2) ◽  
pp. 127
Author(s):  
Doai Van Le ◽  
Phuong Thi Minh Le ◽  
Anh Tuan Nguyen ◽  
Son Hoai Doan ◽  
Khoa Xuan Dinh ◽  
...  
Keyword(s):  
Comparative Study ◽  
Optical Bistability ◽  
Electromagnetically Induced Transparency ◽  
Type System ◽  
Analytical Form ◽  
Physical Parameters ◽  
Threshold Intensity ◽  
Steady Regime ◽  
Analytical Result ◽  
Induced Transparency

We present a comparative study of optical bistability (OB) in three-level atomic configurations, including $\Lambda $-, cascade-, and V- types under the conditions of electromagnetically induced transparency (EIT). In the steady regime, the input-output intensity relations for the OB in each configuration have been derived in analytical form. The model allows one to construct a clear picture on how the threshold intensity, and other characteristics of the OB are continuously modified with respects to controllable parameters of the laser fields, cooperation parameter, and other physical parameters of atomic system. The results showed that the threshold intensity of OB in \(\Lambda \)-type system is much less than the other ones and the threshold intensity of OB in V-type system is the largest one. The analytical result is convenient to choose excitation configuration for experimental observations and related applications in photonic devices.

scholarly journals SOME PROBLEMS OF OPTIMIZATION AND CONTROL OF THE NATURAL FREQUENCIES OF AN ELASTICALLY SUPPORTED RIGID BODY

2021 ◽  
Vol 3 (2) ◽  
pp. 88-102
Author(s):  
S. Bekshaev ◽  
Keyword(s):  
Rigid Body ◽  
Fundamental Frequency ◽  
Natural Frequencies ◽  
Optimization Problems ◽  
Free Vibrations ◽  
The Body ◽  
Natural Vibrations ◽  
Natural Oscillations ◽  
Optimal Position ◽  
Elastic Supports

The article analytically investigates the behavior of the frequencies and modes of natural vibrations of a rigid body, based on point elastic supports, when the position of the supports changes. It is assumed that the body is in plane motion and has two degrees of freedom. A linear description of body vibrations is accepted. The problems of determining such optimal positions of elastic supports at which the fundamental frequency of the structure reaches its maximum value are considered. Two groups of problems were studied. The first group concerns a body supported by only two supports. It was found that in the absence of restrictions on the position of the supports to maximize the fundamental natural frequency, these supports should be positioned so that the basic natural vibrations of the body are translational. Simple analytical conditions are formulated that must be satisfied by the corresponding positions of the supports. In real practical situations, these positions may be unreachable due to the presence of various kinds of restrictions due to design requirements. In this paper, optimization problems are considered taking into account a number of restrictions on the position of supports, typical for practice, expressed analytically by equations and inequalities. For each of the considered types of constraints, results are obtained that determine the optimal positions of the supports and the corresponding maximum values of the main natural frequencies. The approach applied allows us to consider other types of restrictions, which are not considered in the article. In the second group of problems for a body resting on an arbitrary number of supports, the optimal position of an additional elastic support introduced in order to maximize the fundamental frequency in fixed positions and the stiffness coefficients of the remaining supports was sought. It was found that this position depends on the value of the stiffness coefficient of the introduced support. Results are obtained that qualitatively and quantitatively characterize this position and the corresponding frequencies and modes of natural oscillations, including taking into account practically established limitations. The research method uses a qualitative approach, systematically based on the well-known Rayleigh theorem on the effect of imposing constraints on the free vibrations of an elastic structure.

scholarly journals NUMERICAL AND ANALYTICAL DIAGRAM OF A DISTRIBUTED SIMULATION OF DYNAMIC SYSTEMS

World Science ◽  
2019 ◽  
Vol 1 (3(43)) ◽  
pp. 4-9 ◽  
Author(s):  
Shvachych G. G. ◽  
Pobochii I. A. ◽  
Barteniev H. M. ◽  
Tkachenko O. G. ◽  
Tseluiko N. V.
Keyword(s):  
Distributed Simulation ◽  
A Priori ◽  
Rounding Errors ◽  
Computing Systems ◽  
Multiple Processors ◽  
The Matrix ◽  
Numerical Analytical Method ◽  
Priori Information ◽  
Structure Calculations ◽  
Single Processor

The work is dedicated to the construction of numerical-analytical method of designing efficient algorithms for the solution of problems in economics and engineering. Using a priori information about the smoothness of the solution, great attention is paid to the construction of high-accuracy solutions. The proposed approach eliminates recurrent structure calculations unknown vectors decisions, which leads to the accumulation of rounding errors. Parallel form of the algorithm is the maximum, and therefore has the shortest possible time the implementation on parallel computing systems. Most conventional algorithms for solving these problems (sweep techniques, decomposition of the matrix into a product of two diagonal matrices, doubling, etc.) when multiple processors work typically no faster than if a single processor. The reason for this is substantial sequence computations of these algorithms.

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