UDK 539.3 Vestnik SibGAU. 2014, No. 3(55), P. 131–138
R. А. Sabirov
Siberian State Aerospace University named after academician M. F. Reshetnev 31, Krasnoyarsky Rabochy Av., Krasnoyarsk, 660014, Russian Federation E-mail: aaa@mail.sibsau.ru; rashidsab@mail.ru
The variational-differential method of calculation of rectilinear cores stability on axial inertial loadings is developed. The formulations of the boundary value problem of longitudinal cross bending movements are calculated: differential formulation with a final and differential approximation of the allowing equations and variational-differential one. The task is reduced to the generalized problem of own numbers – for an non-trivial vector it is required to identify its own number , here A is a matrix rigidity, B is a matrix of internal forces of inertia. In considering the differential formulation of the task the main particularity of the inertial loads is that the discrete matrix B gets null values on the main diagonal (the rows of matrix can degenerate). Another feature is associated with the approximation of the differential equations by the method of grids, which forms the matrix B asymmetrical about the main diagonal. The generalized problem has no decision like its feedback form where . Brining to the problem of eigenvalues and where and are the inverse matrix, E is the identity matrix, doesn't give any result. Therefore transition from the differential formulation of a task to the variation formulation with sampling by a variational and differential method is executed. The algorithm of formation of matrixes A and B is developed for this approach, which is based on uniform properties of variations of functional. Here the matrix is always symmetric to the main diagonal and is positively defined. Zeros on the main diagonal were presented because it is a feature of loading, however rows don't degenerate. The technique of the solution of a task is shown. Examples of calculation of own values and forms of stability loss are given. The сritical axial accelerations lose their stability and critical angular speeds for the cores rotating in a drum of the centrifuge when the core is fixed from both sides. Investigated the convergence of solutions from condensation of a finite-difference grid. Purpose: to develop a method of calculation of cores on inertial loadings.
calculation of cores, stability, variational and differential method.

1. Centrifugi. Tehnicheskie harakteristiki centrifugi CF-18. “Nauchno-issledovatel'skij ispytatel'nyj centr podgotovki kosmonavtov im. Ju.A. Gagarina” [Centrifuge. Specifications centrifuge CF-18. “Research and Testing Cosmonaut Training Center. Yury Gagarin”] (In Russ.) Available at: gctc.ru›print.php?id=131/ (accessed 20.08.14).

2. Centrifuga vysokoskorostnaja Avanti J-30I [High Speed Centrifuge Avanti J-30I] (In Russ.) Available at: promix.ru›catalog.htm?catalogID=1538 (accessed 2.08.14).

3. Jadernyj volchok [Nuclear dreidel] (In Russ.) Available at: URL: http://dn66.ru/fromnet/id/823-YAdernyiy-volchok.html/ (accessed 2012-10-01).

4. Timoshenko S. P. Istorija nauki o soprotivlenii materialov s kratkimi svedenijami iz istorii teorii uprugosti i teorii sooruzhenij [Science history about resistance of materials with short data from history of the theory of elasticity and the theory of constructions]. Moscow, Gos. izd-vo Tehniko-teoreticheskoj literatury, 1957, 536 p.

5. Timoshenko S. P. Ustojchivost' uprugih sistem [Stability of elastic systems]. Moscow – Leningrad, OGIZ GOSTEHIZDAT, 1946, 532 p.

6. Popov E. P. Nelinejnye zadachi statiki tonkih sterzhnej [Nonlinear problems of a statics of thin cores]. Leningrad – Moscow, OGIZ, 1948, 170 p.

7. Svetlickij V. A. Mehanika sterzhnej. Ch. 1. Statika [Mechanics of cores]. Moscow, Vyssh. Shk. Publ., 1987, 320 p.

8. Zaharov Ju. V., Ohotkin K. G. Nelinejnyj izgib tonkih uprugih sterzhnej [Nonlinear bend of thin elastic cores]. PMTF, 2002, Vol. 43, № 5, p. 124–131.

9. Timoshenko S. P., Gud'er Dzh. Teorija uprugosti [Elasticity theory]. Moscow, Nauka Publ., 1975, 576 p.

10. Samarskij A. A. Teorija raznostnyh shem [Theory of differential schemes]. Moscow, Nauka Publ., 1977, 656 p.

11. Molchanov I. N. Mashinnye metody reshenija prikladnyh zadach. Algebra, priblizhenie funkcij [Machine methods of the solution of applied tasks. Algebra, approach of functions]. Kiev, Nauk. dumka Publ., 1987, 288 p.

12. Lancosh K. Variacionnye principy mehaniki [Variation principles of mechanics]. Moscow, Mir Publ., 1965, 408 p.

13. Vasidzu K. Variacionnye metody v teorii uprugosti i plastichnosti [Variation methods in the elasticity and plasticity theory]. Moscow, Mir Publ., 1987, 542 p.

14. Matrosov A. V. Maple 6. Reshenie zadach vysshej matematiki i mehaniki [Solution of problems of the higher mathematics and mechanics]. St. Petersburg, BHV-Peterburg Publ., 2001, 528 p.

15. Samarskij A. A., Nikolaev E. S. Metody reshenija setochnyh uravnenij [Methods of the solution of the net equations]. Moscow, Nauka Publ., 1978, 592 p. 

Sabirov Rasheed Altavovich – Candidate of Engineering Science, associate professor, associate professor of the department of Technical Mechanics, Siberian State Aerospace University. E-mail: rashidsab@mail.ru