BSU Bulletin. Mathematics, Informatics
Bibliographic description:
ON THE STABILITY OF ONE TYPE OF PERMANENT ROTATIONS OF A MECHANICAL SYSTEM WITH A HESSIAN PATICULAR INTEGRAL // BSU Bulletin. Mathematics, Informatics. - 2020. №3. . - С. 42-48.
Title:
ON THE STABILITY OF ONE TYPE OF PERMANENT ROTATIONS OF A MECHANICAL SYSTEM WITH A HESSIAN PATICULAR INTEGRAL
Financing:
Работа выполнена при поддержке РФФИ (проект N 19-08-00746)
Codes:
Annotation:
The paper considers a mechanical autonomous conservative system with a Hessian particular integral in the well-known problem of the rotation of a rigid body around a fixed point. This system is described by six ordinary differential equations. We have carried out a study of the Lyapunov stability of four types of stationary motions for which all components of the Poisson angles are nonzero. The proposed stationary movements are permanent rotations. The study is based on the first Lyapunov method. For this purpose, we derive a characteristic equation for the
differential equations of disturbed motion on the matrix of the right-hand side in the linear approximation. Obtaining algebraic expressions, their simplification and factorization are carried out by a system of analytical calculations on modern personal computers. The same characteristic equations are established for all permanent rotations. The analysis of the coefficients of the obtained equations made it possible to establish, in addition to zero and purely imaginary roots, the presence of two real solutions other than zero. In conservative systems, one of them will be positive. The calculations performed showed the instability of all the studied permanent rotations
differential equations of disturbed motion on the matrix of the right-hand side in the linear approximation. Obtaining algebraic expressions, their simplification and factorization are carried out by a system of analytical calculations on modern personal computers. The same characteristic equations are established for all permanent rotations. The analysis of the coefficients of the obtained equations made it possible to establish, in addition to zero and purely imaginary roots, the presence of two real solutions other than zero. In conservative systems, one of them will be positive. The calculations performed showed the instability of all the studied permanent rotations
Keywords:
mechanical autonomous conservative system; differential equations of motion; stationary motion; permanent rotation; general integral; particular integral; perturbed motion; characteristic equation; roots of an algebraic equation; necessary conditions of stability; instability of motion.
List of references:
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2. Golubev V. V. Lektsii po integrirovaniyu uravnenii dvizheniya tyazhelogo tverdogo tela okolo nepodvizhnoi tochki [Lectures on Integration of Equations of Motion for a Heavy Rigid Body about a Fixed Point]. Moscow: Regulyarnaya i khaoticheskaya dinamika Publ., 2002. 287 p.
3. Novikov M. A. O statsionarnykh dvizheniyakh tverdogo tela pri sushchestvovanii chastnogo integrala Gessa [On Steady-State Motions of a Rigid Body in the Case of the Existence of Hessian Particular Integral]. Mechanics of Solids. 2018. No. 3. Pp. 28–37.
4. Lyapunov A. M. Obshchaya zadacha ob ustoichivosti dvizheniya. Sobraniye sozhineniy [The General Problem of the Stability of Motion. Collected Works]. Moscow-Leningrad: USSR AS Publ., 1956. Vol. 2. Pp. 7–263.
5. Chetayev N. G. Ustoychivost Ustoichivost' dvizheniya. Raboty po analiticheskoi mekhanike [Stability of Motion. Works on Analytical Mechanics]. Moscow: USSR AS Publ., 1962. 535 p.
6. Kamenkov G. V. Ustoichivost dvizheniya, kolebaniya, aerodinamika [Stability of Motion, Vibrations, Aerodynamics]. Moscow: Nauka Publ., 1971. Vol. 1. 255 p.
7. Kamenkov G. V. Ustoichivost i kolebaniya nelineinykh sistem [Stability and Vibrations of Nonlinear systems]. Moscow: Nauka Publ., 1972. Vol. 2. 213 p.
8. Novikov M. A. Ob ustoichivosti statsionarnykh dvizhenii transportnykh sistem pri sushchestvovanii chastnogo integrala [On Stability of Steady-State Motions of Transport Systems in the Case of Existence of a Particular Integrals]. Sovremennye tekhnologii. Sistemnyi analiz. Modelirovanie. 2019. No. 4 (64). Pp. 57–64.
9. Novikov M. A. Ob ustoichivosti statsionarnogo dvizheniya mekhanicheskoi konservativnoi sistemy [On the Stability of Stationary Motions of a Mechanical Conservative System]. Vestnik Buryatskogo gosudarstvennogo universiteta. Matematika i informatika. 2018. No. 3. Pp. 22–39.
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