scholarly journals ALGORITHM AND SOFTWARE FOR CALCULATING THE STABILITY OF THE FORM OF EQUILIBRIUM OF DISCRETE SYSTEMS

2019 ◽  
Vol 13 (3) ◽  
pp. 44-49
Author(s):  
A.A. SHKURUPIY ◽  
A.N. PASCHENKO ◽  
P.B. MYTROFANOV
Keyword(s):  
Discrete Systems ◽  
Stability Loss ◽  
Transcendental Equation ◽  
Loss Of Stability ◽  
Displacement Method ◽  
Software Complex ◽  
Complex Functions ◽  
Transcendental Functions ◽  
The Stability ◽  
Theoretical Mechanics

The paper presents an algorithm for calculating the stability of the form of equilibrium of the first kind of compressed discrete systems by the method of displacements in combination with themethods of iterations and bisection. The use of the displacement method in combination with the iteration and bisection methods makes it possible to effectively determine the minimum critical stress or strain at the first bifurcation and their corresponding form of loss of stability, both for statically determined and statically undetectable systems. This approach, using matrixforms, makes it possible to significantly simplify the calculations of the analytical condition for the loss of stability of compressed discrete systems (the stability loss equation), which has high orders, as well as to construct the form of loss of stability corresponding to a critical load, that is, to solve the problem of loss of stability of equilibrium. The calculation of the compressed discrete system on the stability of the form of equilibrium actually reduces to the solution of the difficultly described nonlinear transcendental equation, which is the equation of loss of stability. The difficulty lies in the absence of an analytical solution of such an equation due to the presence of complex functions of Zhukovsky, which have transcendental functions in their structure. Such solution can be performed only with the use of numerical methods. This algorithm for calculating the loss of equilibrium of the first kind of compressed discrete systems by displacement in combination with the methods of iteration and bisection is implemented in the software complex "Persist" for a PC in Windows OS. The program was approbated and implemented in theeducational process at the Department of Structural and Theoretical Mechanics of the Poltava National Technical Yuri Kondratyuk University during the training of specialists in engineering specialties.

scholarly journals Calculation of The Stability of the Form of Equilibrium of Discrete Systems

2018 ◽  
Vol 7 (3.2) ◽  
pp. 41
Author(s):  
Oleksandr Shkurupiy ◽  
Pavlo Mytrofanov ◽  
Vladislav Masiuk
Keyword(s):  
Critical Stress ◽  
Discrete Systems ◽  
Stability Loss ◽  
Transcendental Equation ◽  
Educational Process ◽  
Equilibrium Form ◽  
Software Complex ◽  
Transcendental Functions ◽  
The Stability ◽  
Theoretical Mechanics

The paper presents an algorithm for calculating the stability of the equilibrium form of the first kind of compressed discrete systems by the displacements method in combination with the methods of iterations and bisection. The use of the methods makes it possible to effectively determine the minimum critical stress or strain at the first bifurcation and their corresponding form of stability loss, both for statically determined and statically undetermined systems. This approach, using matrix forms, makes it possible to significantly simplify the calculations of the analytical condition for the stability loss of compressed discrete systems (the stability loss equation), which has high orders, as well as to construct the form of stability loss corresponding to a critical load, that is, to solve the problem of loss of equilibrium stability. The calculation actually leads to solving a nonlinear transcendental equation, which is the equation of stability loss. The difficulty lies in the absence of an analytical solution of such an equation due to the presence of complex of Zhukovsky functions, which have transcendental functions in their structure. Such solution can be performed only with the use of numerical methods. This algorithm for calculating the loss of equilibrium of the first kind of compressed discrete systems by displacement in combination with the methods of iteration and bisection is implemented in the software complex "Persist" for PC in Windows OS. The program was approbated and implemented in the educational process at the Department of Structural and Theoretical Mechanics of Poltava National Technical Yuri Kondratyuk University during the training of specialists in engineering specialties.  

scholarly journals COMPRESSED ELEMENTS WITH A VARIABLE IN LENGTH STIFFNESS EQUILIBRIUM FORM STABILITY DETERMINATION

2019 ◽  
Vol 2 (53) ◽  
pp. 30-36
Author(s):  
Oleksandr Shkurupiy ◽  
Pavlo Mytrofanov ◽  
Yuriy Davydenko ◽  
Muhlis Hajiyev
Keyword(s):  
Finite Element ◽  
Critical Load ◽  
Discrete Systems ◽  
Element Model ◽  
Stability Loss ◽  
Building Structures ◽  
Displacement Method ◽  
Equilibrium Form ◽  
Stability Calculation ◽  
Complex Building

One of the most powerful modern methods of calculating complex building structures is the finite element method in theform of a displacement method for discrete systems, which involves the creation of a finite element model, that is, splittingthe structure into separate elements within each of which the functions of displacements and stresses are known. On the basisof the displacement method and the methods of iterations and half-division, an algorithm for stability calculation of the firstkind equilibrium form of compressed reinforced concrete columns with hinged fixing at the ends, considering the stiffnesschanging has been developed. The use of the above methods enables to determine the minimum critical load or stress at thefirst bifurcation and their stability loss corresponding form. The use of matrix forms contributes to simplification of high order stability loss equation. This approach enables to obtain the form of stability loss that corresponds to the critical load.

scholarly journals Scheme of soil stability losses in the laboratory tray

2020 ◽  
Vol 212 ◽  
pp. 02009
Author(s):  
Alexander Kremnev ◽  
Nikolay Vishnyakov ◽  
Victoria Ermachenko
Keyword(s):  
Stability Loss ◽  
Soil Stability ◽  
Loss Of Stability ◽  
Sliding Surface ◽  
Combined Action ◽  
Soil Foundation ◽  
The Stability ◽  
Question Today

The object of the research is the diagram of the stability loss of the soil foundation. Currently, there are still discussions about how the stability of the subgrade undergoes the combined action of vertical and horizontal loads. Most often, in the calculation, the sliding surface is taken to be circular. Whether this is true, especially for a heterogeneous or anisotropic base, is an important question today. In the work, chute tests were carried out, with modeling of the scheme of loss of stability of anisotropic and isotropic bases and results were obtained that confirm the formation of a circular cylindrical sliding surface.

Local Stability of Beams with a Flexible Wall with the Concentrated Force Action

2019 ◽  
Vol 974 ◽  
pp. 529-534
Author(s):  
Alexej I. Pritykin ◽  
Ilja E. Kirillov
Keyword(s):  
Local Stability ◽  
Stability Loss ◽  
Analysis Tool ◽  
Loss Of Stability ◽  
The Finite Element Method ◽  
Concentrated Force ◽  
Code Of Practice ◽  
Calculation Results ◽  
Flexible Wall ◽  
The Stability

The article considers the local stability of hinged supported beams with a flexible wall, supported by paired stiffeners on the supports and loaded with a concentrated force in the middle of the span. To prevent the loss of stability of the wall from compression, another edge was installed in the area of ​​application of force. The materials considered as beams were steel, aluminum, and stainless steel. In this work, the beam material is steel C345. The study was conducted by analyzing the requirements of the Code of Practice for beams with a flexible wall in terms of the stability loss caused by the two types of deformations - shear and bending. By means of small simplifications, the requirements of the Code of Practice have been transformed into empirical dependencies convenient for practical calculations for estimating the critical loads on the beam. The finite element method with ANSYS software was used as an effective analysis tool. It has been established that in some cases the cause of loss of stability is a shift, and in others - a bend. A criterion for changing the forms of buckling was also obtained. The calculation results for the obtained dependences are in satisfactory agreement with the FEM and experimental data.

scholarly journals Modeling of the criterion for predicting the loss of stability of discrete systems

2021 ◽  
Vol 9 (4) ◽  
pp. 96-100
Author(s):  
Vladimir Kulikov ◽  
Viktor Evstratov
Keyword(s):  
Life Cycle ◽  
Discrete System ◽  
Discrete Systems ◽  
Loss Of Stability ◽  
External Influences ◽  
Future State ◽  
Operation Process ◽  
Internal Properties ◽  
The Stability ◽  
External Forces

The paper proposes a method for determining the estimated parameter of the stability state of discrete systems exposed to external influences. As a rule, the loss of stability of the first and second kind leads to a problematic operation process throughout the life cycle, or even the destruction of the system. Hence the requirements of a certain rigidity to the designed and operated systems in order to ensure their geometric immutability. At the same time, in practice, there are no naturally deformable systems from external influences. The paper sets and solves the problem of determining the stability parameter, with the help of which, even before the stage of loss of stability, it is possible to predict the future state of a discrete system, i.e. to predict whether it (the system) has sufficient internal properties to return to a stable position at any exit from the preliminary state of equilibrium due to the influence of external forces.

scholarly journals Stability of Nonlinear Autonomous Quadratic Discrete Systems in the Critical Case

2010 ◽  
Vol 2010 ◽  
pp. 1-23 ◽  
Author(s):  
Josef Diblík ◽  
Denys Ya. Khusainov ◽  
Irina V. Grytsay ◽  
Zdenĕk Šmarda
Keyword(s):  
Lyapunov Functions ◽  
Critical Case ◽  
Autonomous Systems ◽  
Discrete Systems ◽  
Simple Eigenvalue ◽  
Discrete Equations ◽  
The Matrix ◽  
The Stability ◽  
Important Characterization

Many processes are mathematically simulated by systems of discrete equations with quadratic right-hand sides. Their stability is thought of as a very important characterization of the process. In this paper, the method of Lyapunov functions is used to derive classes of stable quadratic discrete autonomous systems in a critical case in the presence of a simple eigenvalueλ=1of the matrix of linear terms. In addition to the stability investigation, we also estimate stability domains.

Study on the bearing resistance of axially compressed L-shaped stainless steel core plate wall based on the stability loss

2021 ◽  
Vol 249 ◽  
pp. 113264
Author(s):  
Xing-Ping Shu ◽  
Huai-Bing Wang ◽  
Yi Li ◽  
Zhi-Shen Yuan ◽  
Ke Li
Keyword(s):  
Stainless Steel ◽  
Stability Loss ◽  
Steel Core ◽  
The Stability

Nonstationary Response of a Two-Degrees-of-Freedom Nonlinear Ship Model Under Modulated Excitation

1995 ◽  
Author(s):  
Ruigui Pan ◽  
Huw G. Davies
Keyword(s):  
Degrees Of Freedom ◽  
Stability Boundary ◽  
Period Doubling ◽  
Approximate Solutions ◽  
Loss Of Stability ◽  
Two Degrees Of Freedom ◽  
Ship Model ◽  
Nonstationary Response ◽  
Period 2 ◽  
The Stability

Abstract Nonstationary response of a two-degrees-of-freedom system with quadratic coupling under a time varying modulated amplitude sinusoidal excitation is studied. The nonlinearly coupled pitch and roll ship model is based on Nayfeh, Mook and Marshall’s work for the case of stationary excitation. The ship model has a 2:1 internal resonance and is excited near the resonance of the pitch mode. The modulated excitation (F0 + F1 cos ωt) cosQt is used to model a narrow band sea-wave excitation. The response demonstrates a variety of bifurcations, loss of stability, and chaos phenomena that are not present in the stationary case. We consider here the periodically modulated response. Chaotic response of the system is discussed in a separate paper. Several approximate solutions, under both small and large modulating amplitudes F1, are obtained and compared with the exact one. The stability of an exact solution with one mode having zero amplitude is studied. Loss of stability in this case involves either a rapid transition from one of two stable (in the stationary sense) branches to another, or a period doubling bifurcation. From Floquet theory, various stability boundary diagrams are obtained in F1 and F0 parameter space which can be used to predict the various transition phenomena and the period-2 bifurcations. The study shows that both the modulation parameters F1 and ω (the modulating frequency) have great effect on the stability boundaries. Because of the modulation, the stable area is greatly expanded, and the stationary bifurcation point can be exceeded without loss of stability. Decreasing ω can make the stability boundary very complicated. For very small ω the response can make periodic transitions between the two (pseudo) stable solutions.

scholarly journals Computer methods for stability analysis of the Roesser type model of 2D continuous-discrete linear systems

2012 ◽  
Vol 22 (2) ◽  
pp. 401-408 ◽  
Author(s):  
Mikołaj Busłowicz ◽  
Andrzej Ruszewski
Keyword(s):  
Stability Analysis ◽  
Asymptotic Stability ◽  
Linear Systems ◽  
Type Model ◽  
Stability Tests ◽  
Numerical Examples ◽  
Computer Methods ◽  
Discrete Linear Systems ◽  
Complex Functions ◽  
The Stability

Computer methods for stability analysis of the Roesser type model of 2D continuous-discrete linear systemsAsymptotic stability of models of 2D continuous-discrete linear systems is considered. Computer methods for investigation of the asymptotic stability of the Roesser type model are given. The methods require computation of eigenvalue-loci of complex matrices or evaluation of complex functions. The effectiveness of the stability tests is demonstrated on numerical examples.

Zeros of a Class of Transcendental Equation with Application to Bifurcation of DDE

2016 ◽  
Vol 26 (04) ◽  
pp. 1650062 ◽  
Author(s):  
Kit Ian Kou ◽  
Yijun Lou ◽  
Yong-Hui Xia
Keyword(s):  
Small Parameter ◽  
Climate Changes ◽  
Transcendental Equation ◽  
Bifurcation Parameter ◽  
Parameter Perturbation ◽  
Distribution Of Zeros ◽  
World Model ◽  
The Real ◽  
Stability And Bifurcations ◽  
The Stability

Zeros of a class of transcendental equation with small parameter [Formula: see text] are considered in this paper. There have been many works in the literature considering the distribution of zeros of the transcendental equation by choosing the delay [Formula: see text] as bifurcation parameter. Different from standard consideration, we choose [Formula: see text] as bifurcation parameter (not the delay [Formula: see text]) to discuss the distribution of zeros of such transcendental equation. After mathematical analysis, the obtained results are successfully applied to the bifurcation analysis in a biological model in the real word phenomenon. In the real world model, the effect of climate changes can be seen as the small parameter perturbation, which can induce bifurcations and instability. We present two methods to analyze the stability and bifurcations.

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