scholarly journals STABILITY OF VIBRATIONS OF THE TRACTOR AS A TWO-MASS MODEL WITH TWO NONLINEARITIES OF THE TYPE OF DRY FRICTION IN RESONANCE MODES

Author(s):  
E. Kalinin ◽  
◽  
Y. Kolesnik ◽  
M. Myasushka

Purpose of the study is to assess the possibility of calculating the stability of tractor oscillations as a system with nonlinearities such as dry friction due to the inverse problem. Research methods. The methodological basis of the work is the generalization and analysis of known scientific results regarding the dynamics of two-mass systems in resonance modes and the use of a systematic approach. The analytical method and comparative analysis were used to form a scientific problem, determine the goal and formulate the research objectives. When creating empirical models, the main provisions of the theory of stability of systems, methodology of system analysis and research of operations were used. The results of the study. Oscillations of the system with harmonic excitation by its base are considered (for example, the movement of a tractor on an uneven supporting surface). Oscillations of this system are described by nonlinear differential equations. To solve this equation, instead of friction dampers with friction forces, linear dampers with corresponding drag coefficients are included in the system. By solving the obtained system of linear inhomogeneous differential equations for the steady-state mode of oscillation, the amplitudes of oscillations of masses and deformation of springs with certain stiffness are determined. To clarify the effect of friction forces on mass oscillations in resonance modes, the obtained expressions were analyzed. A diagram of stability of mass oscillations in resonance modes is obtained. Conclusions. It has been established that if the coefficients of relative friction have such values that the point that is determined by them lies within the region bounded by segments 1-2 and 2-3 and coordinate axes, then during oscillations in the low-frequency resonance mode, the friction forces do not limit the increase in amplitudes fluctuations of masses, but only reduce the rate of their growth. If the point, which is determined by the coefficients of relative friction, lies in the region 1-1'-2'-3 '3-2-1, then the springs have intermittent deformation, that is, during the period of oscillation, one mass of the system has stops relative to another mass, or the last has stops relative to the support surface, or both masses move part of the period as a whole with the support surface. At resonance with a high frequency, the friction forces limit the amplitudes of mass oscillations if the coefficients of relative friction have such values that the point that is determined by them does not lie in the region bounded by segments 4-5 and 5-6 and the coordinate axes. Sections 4-5 and 5-6 define the boundaries of vibration stability at resonance (lines of critical ratios of the coefficients of relative friction).

Filomat ◽  
2014 ◽  
Vol 28 (7) ◽  
pp. 1437-1444 ◽  
Author(s):  
Zengji Du ◽  
Jian Yin

By using Mawhin coincidence degree theory, we investigate the existence of solutions for a class of second order nonlinear differential equations with generalized Sturm-Liouville integral boundary conditions at resonance. The results extend some known conclusions of integral boundary value problem at resonance for nonlinear differential equations


2019 ◽  
Vol 29 (06) ◽  
pp. 1930015 ◽  
Author(s):  
S. Webber ◽  
M. R. Jeffrey

Dry-friction contacts in mechanical oscillators can be modeled using nonsmooth differential equations, and recent advances in dynamical theory are providing new insights into the stability and uniqueness of such oscillators. A classic model is that of spring-coupled masses undergoing stick-slip motion on a rough surface. Here, we present a phenomenon in which multiple masses transition from stick to slip almost simultaneously, but suffer a brief loss of determinacy in the process. The system evolution becomes many-valued, but quickly collapses back down to an infinitesimal set of outcomes, a sort of “micro-indeterminacy”. Though fleeting, the loss of determinacy means masses may each undergo different microscopic sequences of slipping events, before all masses ultimately slip. The microscopic loss of determinacy is visible in local changes in friction forces, and in creating a bistability of global stick-slip oscillations. If friction forces are coupled between the oscillators then the effect is more severe, as solutions are compressed instead onto two (or more) macroscopically different outcomes.


Author(s):  
E. Kalinin ◽  
◽  
S. Lebedev ◽  
Yu. Kozlov

Abstract Purpose of the study is to study the properties of frictional self-oscillations in systems with two degrees of freedom. As a research method, the asymptotic method of N.N. Bogolyubov and Y.A. Metropolitan. Research methods. The methodological basis of the work is the generalization and analysis of the known scientific results of the dynamics of systems in resonance modes and the use of a systematic approach. The analytical method and comparative analysis were used to form a scientific problem, goal and formulation of research objectives. When developing empirical models, the main provisions of the theory of stability of systems, methodology of system analysis and research of functions were used. The results of the study. A system with two degrees of freedom is considered, assuming that the friction function is approximated by a cubic polynomial in the sliding velocity, and friction is applied only to one of the masses. The exclusion of uniform rotation, corresponding to the third degree of freedom, leads to consideration not of the frictional moment, but the difference between the frictional moment and the moment of the moving forces. From the analysis of the results of the solutions of the equation, we can conclude that, with an accuracy up to the first approximation, inclusive, self-oscillations occur with constant frequencies equal to the natural frequencies of the system. This is consistent with the conclusions of other authors obtained using other methods. Stationary values of the amplitudes are found. The following four cases are possible: trivial solution corresponding to uniform rotation of the system without oscillations; single frequency oscillations with the first frequency; single frequency oscillations with a second frequency; two-frequency oscillatory mode. Conclusions. G. Boyadzhiev's method can be applied to study multi-mass self-oscillating systems and gives their general solution in the form of asymptotic expansions to any degree of accuracy. The obtained conditions for the stability of stationary regimes confirm the experimental results that in multi-mass systems, self-oscillations are possible only in the falling sections of the friction characteristics. The nature of the developing vibrations - their frequency and the ratio of the amplitudes of the constituent harmonics - is completely determined by the structure of the system, its elastic and inertial properties.


2021 ◽  
Vol 11 (2) ◽  
pp. 871-882
Author(s):  
Sattar Ramazanovich Alikulov

The article presents materials on the analysis of the process of cotton compaction in containers with a flexible shell, taking into account vertical pressures and friction forces using approximating functions and nonlinear differential equations.


Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3347-3354 ◽  
Author(s):  
Nematollah Kadkhoda ◽  
Michal Feckan ◽  
Yasser Khalili

In the present article, a direct approach, namely exp(-?)-expansion method, is used for obtaining analytical solutions of the Pochhammer-Chree equations which have a many of models. These solutions are expressed in exponential functions expressed by hyperbolic, trigonometric and rational functions with some parameters. Recently, many methods were attempted to find exact solutions of nonlinear partial differential equations, but it seems that the exp(-?)-expansion method appears to be efficient for finding exact solutions of many nonlinear differential equations.


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