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 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 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 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

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.

scholarly journals Assessment of parting rock weak zones under the joint and downward mining of coal seams

2018 ◽  
Vol 66 ◽  
pp. 03001 ◽  
Author(s):  
Volodymyr Bondarenko ◽  
Iryna Kovalevska ◽  
Hennadii Symanovych ◽  
Mykhailo Barabash ◽  
Vasyl Snihur
Keyword(s):  
Stress Component ◽  
Stability Loss ◽  
Coal Seams ◽  
Security Systems ◽  
Relationship Patterns ◽  
Instrumental Observations ◽  
Set Up ◽  
The Stability ◽  
The Relationship ◽  
Weak Zones

The aim of the forecasting effort is to identify troublesome zones of stability loss by a parting lengthwise of the extraction panel under the joint and downward mining of coal seams. Analyses have been carried out of active stress component curves for a 3-D model computational experiment compared with the strength characteristic of each lithotype of a parting. An algorithm has been developed for the stability assessment of a parting lengthwise along the extraction panel. The relationship patterns have been estimated between the sizes of the parting rocks discontinuity zones and the main geomechanical parameters. A scientifically grounded basis has been created for the detection of the parting rock weak zones lengthwise along the extraction panel for the calculation of the mounting and security systems of the development works. A complex of underground instrumental observations was made, which was used to set up a correspondence of patterns to indicate the variation in rock pressure manifestation intensity and the tendencies for changes in the parting structure. All of this confirms the adequacy of the techniques for parting state forecasting, which is recommended for use in the engineering documentation for the joint and downward mining of coal seams.

scholarly journals Computation of Thin-Walled Cross-Section Resistance to Local Buckling with the Use of the Critical Plate Method

2016 ◽  
Vol 62 (2) ◽  
pp. 229-264 ◽  
Author(s):  
A. Szychowski
Keyword(s):  
Cross Section ◽  
Critical Stress ◽  
Analytical Calculation ◽  
Local Buckling ◽  
Stability Loss ◽  
Longitudinal Stress ◽  
Plate Method ◽  
Thin Walled ◽  
The Cross ◽  
Elastically Restrained

Abstract Thin-walled bars currently applied in metal construction engineering belong to a group of members, the cross-section res i stance of which is affected by the phenomena of local or distortional stability loss. This results from the fact that the cross-section of such a bar consists of slender-plate elements. The study presents the method of calculating the resistance of the cross-section susceptible to local buckling which is based on the loss of stability of the weakest plate (wall). The “Critical Plate” (CP) was identified by comparing critical stress in cross-section component plates under a given stress condition. Then, the CP showing the lowest critical stress was modelled, depending on boundary conditions, as an internal or cantilever element elastically restrained in the restraining plate (RP). Longitudinal stress distribution was accounted for by means of a constant, linear or non-linear (acc. the second degree parabola) function. For the critical buckling stress, as calculated above, the local critical resistance of the cross-section was determined, which sets a limit on the validity of the Vlasov theory. In order to determine the design ultimate resistance of the cross-section, the effective width theory was applied, while taking into consideration the assumptions specified in the study. The application of the Critical Plate Method (CPM) was presented in the examples. Analytical calculation results were compared with selected experimental findings. It was demonstrated that taking into consideration the CP elastic restraint and longitudinal stress variation results in a more accurate representation of thin-walled element behaviour in the engineering computational model.

Domains and Types of Aeroelastic Instability of Slender Beam

2007 ◽  
Author(s):  
Jirˇi´ Na´prstek
Keyword(s):  
Stability Boundary ◽  
Stability Loss ◽  
Stability Domain ◽  
Theoretical Explanation ◽  
Point Of View ◽  
System Response ◽  
Physical Point ◽  
Model Response ◽  
Gyroscopic Forces ◽  
The Stability

Slender structures exposed to a cross air flow are prone to vibrations of several types resulting from aeroelastic interaction of a flowing medium and a moving structure. Aeroelastic forces are the origin of nonconservative and gyroscopic forces influencing the stability of a system response. Conditions of a dynamic stability loss and a detailed analysis of a stability domain has been done using a linear mathematical model. Response properties of a system located on a stability boundary together with tendencies in its neighborhood are presented and interpreted from physical point of view. Results can be used for an explanation of several effects observed experimentally but remaining without theoretical explanation until now.

Rectangular Cross Section Beams Calculation on the Stability of a Flat Bending Shape Taking into Account the Initial Imperfections

2019 ◽  
Vol 974 ◽  
pp. 551-555 ◽  
Author(s):  
I.M. Zotov ◽  
Anastasia P. Lapina ◽  
Anton S. Chepurnenko ◽  
B.M. Yazyev
Keyword(s):  
Cross Section ◽  
Basic Equation ◽  
Applied Load ◽  
Rectangular Cross Section ◽  
Twist Angle ◽  
Stability Loss ◽  
Lateral Buckling ◽  
Initial Imperfections ◽  
Beam Stability ◽  
The Stability

The article presents the derivation of the resolving equation for the calculation of lateral buckling of rectangular beams. When deriving the basic equation, the initial imperfections of the beam are taken into account, which are specified in the form of the eccentricity of the applied load, the initial deflection in the plane of least stiffness and the initial twist angle. The influence of initial imperfections on the process of beam stability loss is investigated.

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