scholarly journals A Mathematical Model of the Plane-Parallel Movement of an Asymmetric Machine-and-Tractor Aggregate

Agriculture ◽  
2018 ◽  
Vol 8 (10) ◽  
pp. 151 ◽  
Author(s):  
Volodymyr Bulgakov ◽  
Simone Pascuzzi ◽  
Volodymyr Nadykto ◽  
Semjons Ivanovs
Keyword(s):  
Differential Equations ◽  
Center Of Mass ◽  
Longitudinal Axis ◽  
Design Parameters ◽  
Transverse Displacement ◽  
Parallel Movement ◽  
Axis Of Symmetry ◽  
Being There ◽  
Linear Differential ◽  
The Stability

Technological peculiarities of cultivation and harvesting of some agricultural crops make it necessary to use asymmetric machine-and-tractor aggregates. However, for the time being there is no sufficiently complete, analytical study of the steady movement of such machine-and-tractor aggregates. This necessitates the development of a theory of stable movement of the aggregates which would allow choosing their optimal kinematic and design parameters. On the basis of the results of mathematical simulation, a system of linear differential equations of the second order is obtained describing transverse displacement of the center of masses of the aggregating wheeled tractor and turning of its longitudinal axis of symmetry by some angle around the indicated center of mass, as well as the deviation angle of the rear-trailed harvester from the longitudinal axis of the tractor at any arbitrary moment of time. This system of differential equations can be applied for numerical calculations on the PC, which will make it possible to evaluate the stability of the movement of the asymmetric machine-and-tractor aggregate when it performs the technological process.

scholarly journals Influence of the position of the center of mass of a trailer category O1 on the stability of the road train

2022 ◽  
Vol 14 (2) ◽  
pp. 111-120
Author(s):  
Volodymyr Sakhno ◽  
◽  
Victor Poljakov ◽  
Svitlana Sharai ◽  
Iruna Tchovcha ◽  
...  
Keyword(s):  
Critical Speed ◽  
Center Of Mass ◽  
Design Parameters ◽  
Steering Wheel ◽  
Transverse Displacement ◽  
Maximum Displacement ◽  
The Road ◽  
Course Stability ◽  
The Stability ◽  
Oscillation Instability

In a number of operational properties of motor vehicle (ATZ) at the tendency of increase of speeds of movement the most important indicators of the kept quality, in any modes, are stability and controllability. The choice of constructive parameters of ATZ providing these properties increases active safety of operation and reduces probability of road accidents during the execution of transport operations. From the point of view of practical purposes at operation of ATZ not only the reason of infringement of stability becomes important, and reaction of ATZ to it and control actions of the driver which are ambiguous and unstable. Therefore, it is assumed that the stability and controllability of the ATZ movement should be provided by the design parameters of the machine itself. The result of the analysis of the course stability of the road train was the expression of the critical speed of rectilinear motion. According to the developed mathematical model, the critical velocity is determined. Calculations were made for a road train consisting of a VAZ-2107 car and the uniaxial trailer for different loads of the trailer and different location of its center of mass. According to the initial data inherent in the nominal load of the car and the maximum load of the trailer and the location of the center of mass of the trailer on the longitudinal axis and in the center of mass of the loading platform, the critical speed is about 36 m/s (129.6 km/h). In transient modes of movement, such as "entering the circle and moving in a circle", "jerk of the steering wheel", "shift", "snake", displacement of the center of mass of the trailer in both the longitudinal and transverse planes, the critical speed decreases, and more significantly reduction occurs when the transverse displacement of the center of mass. Thus, if at the maximum displacement of the center of mass of the trailer on the x-axis (x = -0.75 m) the rate of oscillation instability decreases by 36.4% (Gn = 350 kg), 38.4% (Gn = 500 kg) and 44.3% (Gn = 750 kg) in comparison with this speed in the absence of displacement, then at the maximum displacement along the y -axis in the rate of oscillation instability decreases by 45.4%, 55.2% and 63.6%, respectively. In the case of such a trailer loading, the center of mass of the trailer shifts along both the x-axis and the y-axis, there is a further decrease in both the critical speed of the road train and the rate of oscillation instability. This must be taken into account when loading the trailer.

A new proof of the Routh-Hurwitz stability criterion using the second method of Liapunov

1962 ◽  
Vol 58 (4) ◽  
pp. 694-702 ◽  
Author(s):  
P. C. Parks
Keyword(s):  
Differential Equations ◽  
Direct Proof ◽  
Real Constant ◽  
Integral Calculus ◽  
Linear Ordinary Differential Equations ◽  
Hurwitz Stability ◽  
Constant Coefficients ◽  
Linear Differential ◽  
The Stability ◽  
Complex Integral

ABSTRACTThe second method of Liapunov is a useful technique for investigating the stability of linear and non-linear ordinary differential equations. It is well known that the second method of Liapunov, when applied to linear differential equations with real constant coefficients, gives rise to sets of necessary and sufficient stability conditions which are alternatives to the well-known Routh-Hurwitz conditions. In this paper a direct proof of the Routh-Hurwitz conditions themselves is given using Liapunov's second method. The new proof is ‘elementary’ in that it depends on the fundamental concept of stability associated with Liapunov's second method, and not on theorems in the complex integral calculus which are required in the usual proofs. A useful by-product of this new proof is a method of determining the coefficients of a linear differential equation with real constant coefficients in terms of its Hurwitz determinants.

The Analysis of the Systems Stability of Linear Differential Equations Based on the Transformation of Difference Schemes

2019 ◽  
Vol 20 (9) ◽  
pp. 542-549 ◽  
Author(s):  
S. G. Bulanov
Keyword(s):  
Stability Analysis ◽  
Asymptotic Stability ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Computer Analysis ◽  
Linear Differential Equations ◽  
System Stability ◽  
The Difference ◽  
Linear Differential ◽  
The Stability

The approach to the analysis of Lyapunov systems stability of linear ordinary differential equations based on multiplicative transformations of difference schemes of numerical integration is presented. As a result of transformations, the stability criteria in the form of necessary and sufficient conditions are formed. The criteria are invariant with respect to the right side of the system and do not require its transformation with respect to the difference scheme, the length of the gap and the step of the solution. A distinctive feature of the criteria is that they do not use the methods of the qualitative theory of differential equations. In particular, for the case of systems with a constant matrix of the coefficients it is not necessary to construct a characteristic polynomial and estimate the values of the characteristic numbers. When analyzing the system stability with variable matrix coefficients, it is not necessary to calculate the characteristic indicators. The varieties of criteria in an additive form are obtained, the stability analysis based on them being equivalent to the stability assessment based on the criteria in a multiplicative form. Under the conditions of a linear system stability (asymptotic stability) of differential equations, the criteria of the systems stability (asymptotic stability) of linear differential equations with a nonlinear additive are obtained. For the systems of nonlinear ordinary differential equations the scheme of stability analysis based on linearization is presented, which is directly related to the solution under study. The scheme is constructed under the assumption that the solution stability of the system of a general form is equivalent to the stability of the linearized system in a sufficiently small neighborhood of the perturbation of the initial data. The matrix form of the criteria allows implementing them in the form of a cyclic program. The computer analysis is performed in real time and allows coming to an unambiguous conclusion about the nature of the system stability under study. On the basis of a numerical experiment, the acceptable range of the step variation of the difference method and the interval length of the difference solution within the boundaries of the reliability of the stability analysis is established. The approach based on the computer analysis of the systems stability of linear differential equations is rendered. Computer testing has shown the feasibility of using this approach in practice.

scholarly journals Application of M-Matrices for the Study of Mathematical Models of Living Systems

2018 ◽  
Vol 13 (1) ◽  
pp. 208-237 ◽  
Author(s):  
N.V. Pertsev ◽  
B.Yu. Pichugin ◽  
A.N. Pichugina
Keyword(s):  
Differential Equations ◽  
Stability Problem ◽  
Distributed Delay ◽  
Linearization Method ◽  
Living Systems ◽  
High Dimensional ◽  
Linear Differential ◽  
The Stability ◽  
Several Delays ◽  
The Right

Some results are presented of application of M-matrices to the study the stability problem of the equilibriums of differential equations used in models of living systems. The models studied are described by differential equations with several delays, including distributed delay, and by high-dimensional systems of differential equations. To study the stability of the equilibriums the linearization method is used. Emerging systems of linear differential equations have a specific structure of the right-hand parts, which allows to effectively use the properties of M-matrices. As examples, the results of studies of models arising in immunology, epidemiology and ecology are presented.

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