Investigation of power-type variational principles in liquid-filled system

doi:10.1007/s10483-015-2004-6

Abstract

Abstract: Starting from the basic equations of hydrodynamics, the maximum powertype variational principle of the hydrodynamics of viscous fluids was established by Weizang CHIEN in 1984. Through long-term research, it is clarified that the maximum power-type variational principle coincides with the Jourdian principle, which is one of the common principles for analytical mechanics. In the paper, the power-type variational principle is extended to rigid-body dynamics, elasto-dynamics, and rigid-elastic-liquid coupling dynamics. The governing equations of the rigid-elastic-liquid coupling dynamics in the liquid-filled system are obtained by deriving the stationary value conditions. The results show that, with the power-type variational principles studied directly in the state space, some transformations in the time domain space may be omitted in the establishing process, and the rigid-elastic-liquid coupling dynamics can be easily numerically modeled. Moreover, the analysis of the coupling dynamics in the liquid-filled system in this paper agrees well with the numerical analyses of the coupling dynamics in the liquid-filled system offered in the literatures.

Key words: elasto-dynamics, hydrodynamics, coupling dynamics, rigid-body dynamics, power-type variational principle

2010 MSC Number:

Haiyan SONG, Lifu LIANG. Investigation of power-type variational principles in liquid-filled system. Applied Mathematics and Mechanics (English Edition), 2015, 36(12): 1651-1662.

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References

[1] Cheng, C. J. Chien Wei-zang's contributions to mechanics and applied mathematics(in Chinese). Advances in Mechanics, 40, 480-494(2010)
[2] Reissner, E. On a variational theorem in elasticity. Journal of Mathematics and Physics, 29, 90-98(1950)
[3] Washizu, K. Variational Method in Elasticity and Plastisity, Pergamon Press, New York(1982)
[4] Theodore, H. H. Derivation of element stiffness matrices by assumed stress distributions. AIAA Journal, 2, 1333-1336(1964)
[5] Chien, W. Z. Variational prin.ciples and generalized variational principles in hydrodynamics of viscous fluids. Applied Mathematics and Mechanics(English Edition), 5(3), 1281-1296(1984) DOI 10.1007/BF01904951
[6] Hu, H. C. On some variational principles in the theory of elasticity and the theory of plasticity. Scientia Sinica, 4, 33-54(1955)
[7] Ma, X. R., Wang, B. L., and Gou, X. Y. Dynamics of Spacecraft(in Chinese), Science Press, Beijing, 30-80(2001)
[8] Moiseyev, N. N. and Rumyantsev, V. V. Dynamic Stability of Bodies Containing Fluid, SpringerVerlag, New York(1968)
[9] Wang, Z. L. and Liu, Y. Z. Dynamics of Liquid-Filled System(in Chinese), Science Press, Beijing(2002)
[10] Liang, L. F. and Shi, Z. F. On the inverse problem in calculus of variations. Applied Mathematics and Mechanics(English Edition), 15(9), 815-830(1994) DOI 10.1007/BF02451631
[11] Liang, L. F. and Shi, Z. F. Variational principles and generalized variational principles in hydrodynamics of viscous fluids(in Chinese). Chinese Journal of Applied Mechanics, 10, 119-123(1993)
[12] Liang, L. F., Liu, S. Q., and Zhou, J. S. Quasi-variational principles of single flexible body dynamics and their applications. Science in China, Series G:Physics, Mechanics and Astronomy, 52, 775-787(2009)
[13] Liang, L. F. and Song, H. Y. Non-linear and non-conservative quasi-variational principle of flexible body dynamics and application in spacecraft dynamics. Science in China, Series G:Physics, Mechanics and Astronomy, 56, 2192-2199(2013)
[14] Zienkiewicz, O. C. and Taylor, R. L. The Finite Element Method, Fifth Edition Volume 1:The Basis, Butterworth-Heinemann, Oxford(2000)
[15] Paul, G. H. Conceptual Physics, Addison-Wesley, New Jersey(2010)