Dynamic stability of moderately thick beams and frames with the use of harmonic balance and perturbation methods

• Autorzy: Obara P. , Gilewski W.

Identyfikatory

DOI

10.1515/bpasts-2016-0083

• Języki publikacji: EN

Abstrakty

EN

The objective of this paper is to determine dynamic instability areas of moderately thick beams and frames. The effect of moderate thickness on resonance frequencies is considered, with transverse shear deformation and rotatory inertia taken into account. These relationships are investigated using the Timoshenko beam theory. Two methods, the harmonic balance method (HBM) and the perturbation method (PM) are used for analysis. This study also examines the influence of linear dumping on induced parametric vibration. Symbolic calculations are performed in the Mathematica programme environment.

Słowa kluczowe

EN

Bernoulli-Euler beam theory; Timoshenko beam theory; transverse shear deformation; rotatory inertia; natural frequency; critical force; areas of instability

PL

teoria wiązki Bernoulliego-Eulera; teoria wiązki Tymoszenko; moment bezwładności rotatora; częstotliwość; siła krytyczna; obszary niestabilności

• Wydawca: Polska Akademia Nauk, Wydział IV Nauk Technicznych
• Czasopismo: Bulletin of the Polish Academy of Sciences. Technical Sciences
• Rocznik: 2016
• Tom: Vol. 64, nr 4
• Strony: 739--750
• Opis fizyczny: Bibliogr. 32 poz., wykr., tab.

Twórcy

autor

Obara P.

• Faculty of Civil Engineering and Architecture, Kielce University of Technology, 7 Tysiąclecia Państwa Polskiego Ave., 25-314 Kielce, Poland

autor

Gilewski W.

• Faculty of Civil Engineering, Warsaw University of Technology, Armii Ludowej 16 Ave., 00-637 Warsaw, Poland

Bibliografia

• [1] V.V. Bołotin, Dynamic Stability of Elastic Systems, Moscow, 1956, [in Russian].
• [2] A.C. Volmir, Stability of Elastic Systems, Moscow, 1963, [in Russian].
• [3] M. Życzkowski, Strength of Structural Elements. Part 3: Stability of Bars and Bar Structures, Polish Scientific Publishers, Warsaw, 1991.
• [4] S.P. Timoshenko and J.M. Gere, Theory of Elastic Stability, McGraw-Hill, New York, 1961.
• [5] C.E. Majorana, “Dynamic stability of elastic structures: a finite element approach”, Computers & Structures 69, 11–25 (1998).
• [6] W. Sochacki, “The dynamic stability of a simply supported beam with additional discrete elements”, Journal of Sound and Vibration 314, 180–193 (2008).
• [7] B. Sahoo, L.N. Panda, and G. Pohit, „Nonlinear dynamics of an Euler-Bernoulli beam with parametric and internal resonances”, International Conference on Design and Manufacturing, Procedia Engineering 64, 727–736 (2013).
• [8] Y.Q. Huang, H.W. Lu, J.Y. Fu, A.R. Liu, and M. Gu, “Dynamic stability of Euler beams under axial unsteady wind force”, Mathematical Problems in Engineering 2014, (2014).
• [9] S.P. Timoshenko, “On the correction for shear of the differential equation for transverse vibrations of prismatic bars”, Philosophical Magazine 41, 744–746 (1921).
• [10] S.P. Timoshenko, “On the transverse vibrations of bars of uniform cross–section”, Philosophical Magazine 43, 125–131 (1922).
• [11] M. Sabuncu and K. Evran, “The dynamic stability of a rotating asymmetric cross-section Timoshenko beam subjected to lateral parametric excitation”, Finite Elements in Analysis and Design 42, 454–469 (2006).
• [12] M. Pirmoradian, M. Keshmiri, and H. Karimpour, “On the parametric excitation of a Timoshenko beam due to intermittent passage of moving masses: instability and resonance analysis”, Acta Mechanica 226, 1241–1253, (2015).
• [13] W.R. Chen, “Dynamic stability of linear parametrically excited twisted Timoshenko beams under periodic axial loads”, Acta Mechanica 216, 207–223 (2011).
• [14] W.R. Chen, “Parametric studies on bending vibration of axially-loaded twisted Timoshenko beams with locally distributed Kelvin–Voigt damping”, International Journal of Mechanical Sciences 88, 61–70 (2014).
• [15] B.J. Ryu, K. Katayama, and Y. Sugiyama, “Dynamic stability of Timoshenko columns subjected to subtangential forces”, Computers & Structures 68 (5), 499–512 (1998).
• [16] J.H. Kim and Y.S. Choo, “Dynamic stability of a free-free Timoshenko beam subjected to a pulsating follower force”, Journal of Sound and Vibration 216 (4), 623–636 (1998).
• [17] P.A. Djondjorov and V.M. Vassilev, “On the dynamic stability of a cantilever under tangential follower force according to Timoshenko beam theory”, Journal of Sound and Vibration 311, 1431–1437 (2008).
• [18] B. Nayak, S.K., Dwivedy, and K.S.R.K. Murth, “Dynamic stability of magnetorheological elastome based adaptive sandwich beam with conductive skinsusing FEM and the harmonic balanc method”, International Journal of Mechanical Sciences 77, 205–216 (2013).
• [19] E. Magnucka-Blandzi, “Dynamic stability and static stress state of a sandwich beam with a metal foam core using three modified Timoshenko hypotheses”, Mechanics of Advanced Materials and Structures 18, 147–158 (2011).
• [20] S.C. Mohanty, R.R. Dash, and T. Rout, “Static and dynamic stability analysis of a functionally graded Timoshenko beam”, International Journal of Structural Stability and Dynamics 12 (4), (2012).
• [21] W.R. Chen and C.S. Chen, “Parametric instability of twisted Timoshenko beams with localized damage”, International Journal of Mechanical Sciences 100, 298–311 (2015).
• [22] K.H. Kim and J.H. Kim, “Effect of a crack on the dynamic stability of a free–free beam subjected to a follower force”, Journal of Sound and Vibration 233 (1), 119–135 (2000).
• [23] K. Dems and J. Turant, “Structural damage identification using frequency and modal changes”, Bull. Pol. Ac.: Tech. 59 (1), 27–32 (2011).
• [24] A. Gomuliński and M. Witkowski, Structural Mechanics. A Course for Advanced Readers, Publishing House of Warsaw University of Technology, Warsaw, 1993, [in Polish].
• [25] H.P. Lee, T.H. Tan, and G.S.B. Leng, “Parametric instability of spinning pretwisted beams subjected to sinusoidal compressive axial loads”, Computers & Structures 66, 745–764 (1998).
• [26] X.D. Yang, Y.Q. Tang, L.Q. Chen, and C.W. Lim, “Dynamic stability of axially accelerating Timoshenko beam: Averaging method”, European Journal of Mechanics A/Solids 29, 81–90 (2010).
• [27] Z. Song, Z. Chen, W. Li, and Y. Chai, “Dynamic stability analysis of beams with shear deformation and rotary inertia subjected to periodic axial forces by using discrete singular convolution method”, Journal of Engineering Mechanics 142 (3), 1–18 (2016).
• [28] M. Ghomeshi Bozorg and M. Keshmiri, “Stability analysis of nonlinear time varying system of beam-moving mass considering friction interaction”, Indian Journal of Science and Technology 6 (11), 5459–5468 (2013).
• [29] E. Hetmaniok, D. Słota, T. Trawiński, and R. Wituła, “An analytical technique for solving general linear integral equations of the second kind and its application in analysis of flash lamp control circuit”, Bull. Pol. Ac.: Tech. 62 (3), 413–421 (2014).
• [30] W.J. Cunningham, Introduction to Nonlinear Analysis, McGraw-Hill, New York, 1959.
• [31] M. Siwczyński and M. Jaraczewski, “The Poincare theorem in linear circuit synthesis”, Bull. Pol. Ac.: Tech. (1), 49–58 (2007).
• [32] W. Gilewski and A. Gomuliński, “Physical shape functions: a new concept in finite elements”, Finite Elements News 3, 20–23 (1990).

Uwagi

PL

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę.