0
Research Papers

Stability of Gyroscopic Circulatory Systems

[+] Author and Article Information
Firdaus E. Udwadia

Professor
Aerospace and Mechanical Engineering,
Civil Engineering, Information and Operations
Management, and Mathematics,
430K Olin Hall,
University of Southern California,
Los Angeles, CA 90089-1453
e-mail: feuUSC@gmail.com

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received August 26, 2018; final manuscript received October 24, 2018; published online November 16, 2018. Assoc. Editor: George Haller.

J. Appl. Mech 86(2), 021002 (Nov 16, 2018) (6 pages) Paper No: JAM-18-1495; doi: 10.1115/1.4041825 History: Received August 26, 2018; Revised October 24, 2018

This paper presents results related to the stability of gyroscopic systems in the presence of circulatory forces. It is shown that when the potential, gyroscopic, and circulatory matrices commute, the system is unstable. This central result is shown to be a generalization of that obtained by Lakhadanov, which was restricted to potential systems all of whose frequencies of vibration are identical. The generalization is useful in stability analysis of large scale multidegree-of-freedom real life systems, which rarely have all their frequencies identical, thereby expanding the compass of applicability of stability results for such systems. Comparisons with results in the literature on the stability of such systems are made, and the weakness of results that deal with only general statements about stability is exposed. It is shown that the commutation conditions given herein provide definitive stability results in situations where the well-known Bottema–Karapetyan–Lakhadanov result is inapplicable.

Copyright © 2019 by ASME
Your Session has timed out. Please sign back in to continue.

References

Arrowsmith, D. K. , and Place, C. M. , 1998, Dynamical Systems, Differential Equations, Maps and Chaotic Behavior, Chapman and Hall, New York, pp. 77–81.
Perko, L. , 1991, Differential Equations and Dynamical Systems, Springer Verlag, New York, pp. 120–127.
Kirillov, O. N. , 2013, Nonconservative Stability Problems of Modern Physics, Walter de Gruyter, Berlin.
Kelvin, B. W. T. , and Tait, P. G. , 1879, Treatise on Natural Philosophy, Cambridge University Press, New York.
Zajac, E. E. , 1964, “ The Kelvin-Tait-Chetaev Theorem and Extensions,” J. Astronaut. Sci., 11(2), pp. 46–49.
Huseyin, K. , Hagedorn, P. , and Teschner, W. , 1983, “ On the Stability of Linear Conservative Gyroscopic Systems,” J. Appl. Math. Phys., 34(6), pp. 807–815.
Merkin, D. R. , 1974, Gyroscopic Systems, Nauka, Moscow, Russia.
Bulatovic, R. M. , 1999, “ On the Stability of Linear Circulatory Systems,” Z. Angew. Math. Phys., 50, pp. 669–674. [CrossRef]
Udwadia, F. E. , 2017, “ Stability of Dynamical Systems With Circulatory Forces: Generalization of the Merkin Theorem,” Am. Inst. Aeronaut. Astronaut., 55(9), pp. 2853–2858. [CrossRef]
Beletsky, V. V. , 1995, “ Some Stability Problems in Applied Mechanics,” Appl. Math. Comput., 70(2–3), pp. 117–141.
Zhuravlev, V. P. , and Klimov, D. M. , 2010, “ Theory of the Shimmy Phenomenon,” Mech. Solids, 45(3), pp. 324–330. [CrossRef]
Hagedorn, P. , Eckstein, M. , Heffel, E. , and Wagner, A. , 2014, “ Self-Excited Vibrations and Damping in Circulatory Systems,” ASME J. Appl. Mech., 81(10), p. 101008. [CrossRef]
Spelsberg-Korspeter, G. , Hochlenert, D. , and Hagedorn, P. , 2011, “ Self-Excitation Mechanisms in Paper Calendars Formulated as a Stability Problem,” Tech. Mech., 31(1), pp. 15–24.
Dowell, E. H. , 2011, “ Can Solar Sails Flutter?,” AIAA J., 49(6), pp. 1305–1307. [CrossRef]
Ziegler, H. , 1953, “ Linear Elastic Stability, a Critical Analysis of Methods—Part 1,” Z. Angew. Math. Phys., 4(2), pp. 89–121. [CrossRef]
Ziegler, H. , 1953, “ Linear Elastic Stability, a Critical Analysis of Methods—Part 2,” Z. Angew. Math. Phys., 4(3), pp. 167–185. [CrossRef]
Horn, R. A. , and Johnson, C. R. , 1991, Matrix Analysis, Cambridge University Press, Cambridge, UK, p. 103.
Lakhadanov, V. M. , 1975, “ On Stabilization of Potential Systems,” J. Appl. Math. Mech., 39(1), pp. 45–50. [CrossRef]
Karapetyan, A. V. , 1975, “ About the Stability of Nonconservative Systems,” Vestn. Mosk. Univ. Ser. 1: Mat. Mekh., 4, pp. 109–113.
Bottema, O. , 1955, “ On the Stability of the Equilibrium of a Linear Mechanical System,” Z. Angew. Math. Phys., 6(2), pp. 97–104. [CrossRef]
Casti, J. , 1992, Reality Rules: I, John Wiley, New York, pp. 63–64.
Hagedorn, P. , and Hochlenert, D. , 2015, Technische Schwingungslehre, Verlag-Europa-Lehrmittel, Vollmer GmbH & Co. KG, Haan, Germany, pp. 155–157.

Figures

Tables

Errata

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In