Open AccessCalculation of The Stability of the Form of Equilibrium of Discrete Systems
Author(s) -
Oleksandr Shkurupiy,
Pavlo Mytrofanov,
Vladislav Masiuk
Publication year - 2018
Publication title -
international journal of engineering and technology
Language(s) - English
Resource type - Journals
ISSN - 2227-524X
DOI - 10.14419/ijet.v7i3.2.14561
Subject(s) - stability (learning theory) , transcendental equation , mathematics , convergence (economics) , bifurcation , bisection method , matrix (chemical analysis) , nonlinear system , mathematical analysis , numerical analysis , computer science , physics , machine learning , materials science , quantum mechanics , economics , composite material , economic growth
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.