Kinematically accurate separation of the large rotation into axial and transverse in problems of rotor dynamics
Authors: Sorokin F.D., Zhang Hao
Published in issue: #10(82)/2018
DOI: 10.18698/2308-6033-2018-10-1815
Category: Mechanics | Chapter: Dynamics, Strength of Machines, Instruments, and Equipment
In order to overcome the problem of «singular points», a method has been developed for the kinematically accurate separation of a large rotation into an axial (scalar) and a transverse (Euler vector). The proposal is based on the fact that in the problems of the rotor dynamics of machines consisting of shafts, gears, bearings, etc., the transverse rotation never reaches a value of 2𝜋 (a critical value for the Euler vector). The axial rotation is not limited in any way. Differential equations of the dynamics of rotational motion are also divided into axial (scalar) and transverse (vector). The above example shows that the constructed system of differential equations can be easily integrated by standard numerical methods to very large total rotations without any restrictions, provided that the transverse rotation is limited to 2𝜋. The control of the results is checked by observing the law of conservation of total energy
References
[1] Chelomei V.N., ed. Vibrations in technology. Volume 3. Oscillations of machines, structures and their elements. Moscow, Mashinostroenie Publ., 1980, 544 p.
[2] Branets V.N., Shmyglevsky I.P. Introduction to the theory of freeform inertial navigation systems. Moscow, Nauka Publ., 1992, 280 p.
[3] Bremer H. Elastic multibody dynamics: a direct Ritz approach. Dordrecht, Springer, 2008, 464 р.
[4] Zhuravlev V.F. Fundamentals of theoretical mechanics. 2nd ed., revised. Moscow, Publishing House of Physical and Mathematical Literature, 2001, 320 p.
[5] Popov V.V., Sorokin F.D., Ivannikov V.V. Trudy MAI — Proceedings of the MAI, 2017, no. 92. Available at: http://trudymai.ru/published.php?ID=76832
[6] Zhilin P.A. Vectors and second-rank tensors in three-dimensional space. St. Petersburg, Nestor Publ., 2001, 276 p.
[7] Zhilin P.A. Rational Mechanics of Continuous Media. St. Petersburg, Polytechnic University Publ., 2012, 584 p.
[8] Rankin C.C., Brogan F.A. An element independent corotational procedure for the treatment of large rotation. Journal of Pressure Vessel Technology-Transactions of the ASME, 1986, vol. 108 (2), pp. 165–174.
[9] Crisfield M.A. Nonlinear finite element analysis of solid and structures. 2nd edition. Chichester, John Wiley&Sons, 2012, 544 p.
[10] Eliseev V.V., Zinovieva Т.V. Mechanics of thin-walled structures. Theory of rods. St. Petersburg, Politechnical Uniersity Publ., 2008, 95 p.
[11] Geradin M., Cardona A. Flexible Multibody Dynamics. A Finite Element Approach. Chichester, John Wiley&Sons, 2001, 340 p.
[12] Felippa C.A. A Systematic Approach to the Element-Independent Corotational Dynamics of Finite Elements. Department of Aerospace Engineering Sciences and Center for Aerospace Structures. Boulder, University of Colorado, 2000, 42 p.
[13] Felippa C.A., Haugen B. A Unified Formulation of Small-Strain Corotational Finite Elements: I. Theory. Computer Methods in Applied Mechanics and Engineering, 2005, vol. 194, pp. 2285–2335.
[14] Dyakonov V.P. Kompyuternaya matematika. Teoriya i praktika [Computer mathematics. Theory and practice]. Moscow, Noledge/Knowledge Publ., 2001, 1296 p.