Novickov M. A. Ob ustoychivosti statsionarnykh dvizheniy transportnykh sistem pri sushchestvovanii chastnogo integrala [On the stability of steady-state motions of transport systems in the case of existence of a partial integral]. Sovremennye tekhnologii. Sistemnyi analiz. Modelirovanie [Modern Technologies. System Analysis. Modeling], 2019. Vol. 64, No. 4. Pp. 57–64. DOI: 10.26731/1813-9108.2019.4(64).57–64.
Many transport mechanical objects in course of investigations may be modeled by heavy solid bodies. It is more convenient to describe them using systems of ordinary differential equations. When considering investigated objects being at rest on the platform, in a car or some other moving transport vehicles, i.e. isolated from the influence of the dissipative forces, it is possible to consider the system as conservative. In the process of dynamic properties of model systems, it is possible to rely upon the properties of known conservative systems, preferably autonomous ones. There are first integrals of equations of motions in such systems. Among problem statements for conservative systems, the most popular is the problem about rotation of a solid body around a fixed point. In the most general form, there are first integrals known for it: integrals of total energy, integrals of the moment of impulse, Poisson’s integral. As far as the three properly studied cases of existence of the fourth general integral are concerned, the following main dynamic properties of systems are known: analytical solutions are written in the form of elliptic or hyperelliptic functions, with asymptotics of the solutions found and stationary motions distinguished. Investigations of their stability have been conducted for each particular case. Presently, the interest of researchers is drawn to to investigation of autonomous conservative systems with partial integral. Despite the fact that there are many systems with such integrals, the Hess partial integral is to be studied above all. The present paper considers the investigation of stability of steady-state motions of the solid body around a fixed point in the case of existence of the Hess partial integral. The state of rest is considered as a form of steady-state motions. The state of rest is the most usual for transport. The position of the mass center above the origin (the main axes of the body are chosen as the coordinate axes) corresponds to instability of the state of rest. This property follows from the existence of roots of the characteristic equation of disturbed motion with the positive real part. Sufficient conditions of stability are defined by Lyapunov’s second method by the constructing of fixed sign Lyapunov functions. In the case, when the mass center is lower than the coordinate axis, we have obtained coincidence of the sufficient stability conditions with the necessary ones. In this case, sufficient stability conditions are determined by linear terms of differential equations of motion. For the case of permanent rotation, we have conducted an investigation of necessary stability conditions in cases of degenerations of the characteristic equation composed on the basis of the matrix of the linear parts of differential equations of disturbed motion. It has been shown that degenerations arise in the following cases: 1) in satisfying the Appelrott equality, when there exists an additional partial Hess integral; 2) without any additional integral, in case when there exists some correspondence between static and dynamic parameters of the system; 3) in case when the above two cases are satisfied simultaneously. In all the cases considered above, there are no additional constraints imposed on the system’s parameters, besides the requirement to the inertia moments of the solid body. In the process of processing symbolic information, we have applied a system of analytical computations on personal computers.
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