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 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 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

RATIONAL RADIAL DISTORTION MODELS OF CAMERA LENSES WITH ANALYTICAL SOLUTION FOR DISTORTION CORRECTION

2004 ◽  
Vol 01 (02) ◽  
pp. 135-147 ◽  
Author(s):  
LILI MA ◽  
YANGQUAN CHEN ◽  
KEVIN L. MOORE
Keyword(s):  
Analytical Solution ◽  
Polynomial Approximation ◽  
Distortion Correction ◽  
Camera Model ◽  
Distortion Model ◽  
Radial Distortion ◽  
New Class ◽  
Comparable Accuracy ◽  
The Common ◽  
Distortion Parameters

The common approach to radial distortion is by the means of polynomial approximation, which introduces distortion-specific parameters into the camera model and requires estimation of these distortion parameters. The task of estimating radial distortion is to find a radial distortion model that allows easy undistortion as well as satisfactory accuracy. This paper presents a new class of rational radial distortion models with easy analytical undistortion formulae. Experimental results are presented to show that with this class of rational radial distortion models, satisfactory and comparable accuracy can be achieved.

AN IMPROVED POPULATION BASED OPTIMIZATION SOLUTION METHOD BY COMBINING LOCAL AND GLOBAL SEARCHING

2007 ◽  
Vol 16 (05) ◽  
pp. 907-915
Author(s):  
WEI JIANG ◽  
XIAO-LONG WANG ◽  
XIU-LI PANG
Keyword(s):  
Local Search ◽  
Objective Function ◽  
Optimization Problems ◽  
Population Based ◽  
Search Algorithms ◽  
Global Search ◽  
Local Optimum ◽  
Hard Problems ◽  
The Common ◽  
Optimum Solution

Optimization Solution Task is a typical and important task in many applications. Many optimization problems have been proved to be NP-hard problems, which cannot be solved by some predefined mathematic formulae. In this case, computer aided method is very helpful. While some local search algorithms are easily to fall into a local optimum solution. On contrast, the population based methods, such as Genetic Algorithms, Artificial Immune System, Autonomy Oriented Computing, are global search algorithms. However, they are not good at the local search. In this paper, an improved method is proposed by combining the local and global search ability, so as to improve the performance in terms of the convergence speed and the convergence reliability. We construct a generic form to deal with the common objective function space or the objective function with the partial derivative. In addition, we present an n-hold method in population based evolution method. The experiments indicate that our approach can effectively improve the convergence reliability, which is much concerned in some applications with the expensive executing expense.

Optimal Supersonic Ejector Designs

1986 ◽  
Vol 108 (4) ◽  
pp. 414-420 ◽  
Author(s):  
J. C. Dutton ◽  
B. F. Carroll
Keyword(s):  
Flow Model ◽  
Optimization Problems ◽  
Pressure Ratio ◽  
Area Ratio ◽  
Stagnation Temperature ◽  
Entrainment Ratio ◽  
One Dimensional ◽  
Supersonic Ejector ◽  
Constant Area ◽  
The Common

A technique based on a one-dimensional constant area flow model has been developed for solving a large class of supersonic ejector optimization problems. In particular, the method determines the primary nozzle Mach number and ejector area ratio which optimizes either the entrainment ratio, compression ratio, or stagnation pressure ratio given values for the other two variables and the primary and secondary gas properties and stagnation temperatures. Design curves for the common case of diatomic primary and secondary gases of equal molecular weight and stagnation temperature are also presented and discussed.

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 THE PROBLEM OF THE OSCILLATION OF THE ELASTIC LAYER BOUNDED BY RIGID BOUHDARIES

2018 ◽  
pp. 42-48
Author(s):  
Seitmuratov Angisin ◽  
Tileubay Sarsenkul ◽  
Toxanova Sveta ◽  
Ibragimova Nuraim ◽  
Doszhanov Bayanalui ◽  
...  
Keyword(s):  
Cylindrical Shell ◽  
Phase Velocity ◽  
Elastic Layer ◽  
Direct Relation ◽  
Anisotropic Layer ◽  
Harmonic Oscillations ◽  
Natural Oscillations ◽  
Integral Transformations ◽  
Oscillation Frequencies ◽  
Rotational Shear

In the case of harmonic oscillations of a cylindrical shell, the phase velocity is expressed in terms of the frequency of natural oscillations freely supported along the edges of the shell, and therefore, the study of waves in plane and circular elements has the most direct relation to the problem of determining its own forms and oscillation frequencies shells finite length. Below let us consider some problems of oscillation of an elastic layer bounded by rigid boundaries under the influence of a normal or rotational shear stress. The solutions of the problems under consideration are obtained by using integral transformations by the coordinate. Key words: harmonic oscillations, cylindrical shells, phase velocity, frequency, eigenvibrations, Bessel function, wave, anisotropic, layer.

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