Flutter of a beam in supersonic flow: truncated version of Timoshenko–Ehrenfest equation is sufficient
Source abstract
Abstract
This paper deals with flutter due to gas flow of a uniform and homogeneous beam with shear deformation and rotary inertia effects taken into account. It utilizes the truncated, consistent Timoshenko–Ehrenfest beam model in contrast to the original Timoshenko–Ehrenfest equations. Omission of the fourth order derivative term in the governing differential equation of the original equations of Timoshenko and Ehrenfest is shown to be more consistent than retaining it as is done in the original Timoshenko–Ehrenfest set. The original Timoshenko–Ehrenfest equations are exposed in almost all textbooks published after 1921. However, this paper shows that these original equations are unnecessarily overcomplicated, and simpler set is consistent. This truncation significantly simplifies the analytical and numerical analyses considerably and agrees with the principle of parsimony, i.e. the most acceptable explanation of a phenomenon is the simplest, involving the fewest entities and assumptions. The critical flutter velocities are compared with the results obtained by using the original set of Timoshenko–Ehrenfest equations.
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Research context
OpenAlex topics: Fluid Dynamics and Vibration Analysis · Vibration and Dynamic Analysis · Computational Fluid Dynamics and Aerodynamics
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References (32)
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1. Algazin, S.D., Kijko, I.A.: Aeroelastic Vibrations and Stability of Plates and Shells. Walter de Gruyter GmbH, Berlin (2015)10.1515/9783110338379 ↗
2. Alotta, G., Di Paola, M., Pinnola, F. P., Zingales, M.: A fractional nonlocal approach to nonlinear blood flow in small-lumen arterial vessels. Meccanica, pp. 1–16, (2020)10.1007/s11012-020-01144-y ↗
3. Ashley, H., Zartarian, G.: Piston theory-a new aerodynamic tool for the aeroelastician. J. Aeronaut. Sci. 23(12), 1109–1118 (1956)10.2514/8.3740 ↗
4. Bolotin, V.V.: Nonconservative Problems of the Theory of Elastic Stability. The Macmillan Company, New York (1963)
5. Bresse, J.A.C.: Cours de mécanique appliquée – Résistance des matériaux et stabilité desconstructions. Gauthier-Villars, Paris (1859).. ((in French))
6. Challamel, N., Elishakoff, I.: A brief history of first-order shear-deformable beam and plate models. Mech. Res. Commun. 102, 103389 (2019)10.1016/j.mechrescom.2019.06.005 ↗
7. Cottone, G., Di Paola, M., Zingales, M.: Fractional mechanical model for the dynamics of non-local continuum. In: Advances in numerical methods, pp. 389–423. Springer, Boston (2009)10.1007/978-0-387-76483-2_33 ↗
8. Di Paola, M., Zingales, M.: Long-range cohesive interactions of non-local continuum faced by fractional calculus. Int. J. Solids Struct. 45(21), 5642–5659 (2008)10.1016/j.ijsolstr.2008.06.004 ↗
9. Di Paola, M., Failla, G., Pirrotta, A., Sofi, A., Zingales, M.: The mechanically based non-local elasticity: an overview of main results and future challenges. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 371(1993), 20120433 (2013)10.1098/rsta.2012.0433 ↗
10. Dowell, E.H.: Aeroelasticity of Plates and Shells. Noordhoff International Publishing, Leyden (1975)
11. Dowell, E.H., Clark, R., Cox, D., Curtiss, H.C., Jr., Edwards, J.W., Hall, K.C., Peters, D.A., Scanlan, R., Simiu, E., Sisto, F., Strganac, T.W.: A Modern Course in Aeroelasticity. Kluwer Academic Publisher, Dordrecht (2004)
12. Elishakoff, I.: An equation which is simpler and more consistent than bresse-timoshenko equations. In: Gilat, R., Sills-Banks, L. (eds.) Advances in Mathematical Modeling and Experimental Methods for Materials and Structures, pp. 249–254. Springer Verlag, Berlin (2010)
13. Elishakoff, I.: Handbook on Timoshenko-Ehrenfest Beam and Uflyand-Mindlin Plate Theories. World-Scientific, Singapore (2020a)
14. Elishakoff, I.: Who developed the so-called timoshenko beam theory. Math. Mech. Solids 25(1), 97–116 (2020b)10.1177/1081286519856931 ↗
15. Elishakoff, I., Lubliner, E.: Random vibration of a structure via classical and nonclassical theories. In: Eggwertz, S., Lind, N.C. (eds.) Probabilistic Methods in the Mechanics of Solids and Structures, pp. 455–467. Springer, Berlin (1985)10.1007/978-3-642-82419-7_43 ↗
16. Elishakoff, I., Kaplunov, J., Nolde, E.: Celebrating the centenary of Timoshenko’s study of effects of shear deformation and rotary inertia. Appl. Mech. Rev. 67, 060802 (2015)10.1115/1.4031965 ↗
17. Hurwitz, A.: Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitz. Math. Ann. 46(2), 276–284 (1895). ((German))10.1007/BF01446812 ↗
18. Ilyushin, A.A.: Law of plane sections in aerodynamics of high supersonic speed. J. Appl. Math. Mech. 20(6), 733–735 (1956). ((Russian))
19. Rayleigh Lord (J. W. S. Strutt), The Theory of Sound, London: Macmillan, 1877–1878 (see also Dover, New York, 1945).
20. Paola, M.D., Failla, G., Zingales, M.: Mechanically based nonlocal Euler-Bernoulli beam model. J. Nanomech. Micromech. 4(1), A4013002 (2014)10.1061/(ASCE)NM.2153-5477.0000077 ↗
21. Pielorz, A.: Discrete-continuous models in the analysis of low structures subject to kinematic Excitations caused by transversal waves. J. Theor. Appl. Mech. 34(3), 547–566 (1996)
22. qian y-j., yang x-d., zhang w., liang f., yang t-z. ren y.: flutter mechanism of Timoshenko beams in supersonic flow, J. Aerosp. Eng. Vol. 32(4), 2019.10.1061/(ASCE)AS.1943-5525.0001025 ↗
23. Routh, E.J.: A Treatise on the Stability of a Given State of Motion, Particularly Steady Motion. Macmillan and co., London (1877)
24. Strand7, Theoretical Manual – Theoretical background to the Strand7 finite element analysis system, (2004)
25. Timoshenko S.P.: A Course of Elasticity Theory. Part 2: Rods and Plates, St. Petersburg: A.E. Collins Publishers, 1916 (in Russian), (2nd Edition, Kiev: “Naukova Dumka” Publishers, pp. 337–338, 341, 1972).
26. Timoshenko, S.P.: On the differential equation for the flexural vibrations of prismatical rods. Glasnik Hrvatskoga Prirodaslovnoga Društva 32, 55–57 (1920)
27. Timoshenko, S.P.: On the Correction for Shear of the Differential Equation for Transverse Vibrations of Prismatic Bar. Philosophical Magazine Series 41/245(6), 744–746 (1921)10.1080/14786442108636264 ↗
28. Timoshenko, S.P.: Vibration Problems in Engineering, p. 231. Constable and Company, London (1928)
29. Timoshenko, S.P.: Vibration Problems in Engineering, 2nd edn. Van Nostrand Reinhold Company, New York (1937)
30. Timoshenko, S.P., Young, D.H.: Vibration Problems in Engineering, 3rd edn. Van Nostrand Reinhold Company, New York (1955)
31. Timoshenko, S.P., Young, D.H., Weaver, W., Jr.: Vibration Problems in Engineering. Wiley, Fourth Edition, New York (1974)
32. Weaver, W., Jr., Timoshenko, S.P., Young, D.H.: Vibration Problems in Engineering. Wiley, Fifth Edition, New York (1990)