Methods for Determinning Critical Values of Nonconservative Loads in Problems of Stability of Mechanical Systems
Authors: Radin V.P., Chirkov V.P., Shchugorev A.V., Shchugorev V.N. | Published: 28.10.2019 |
Published in issue: #10(715)/2019 | |
Category: Mechanical Engineering and Machine Science | Chapter: Machine Science | |
Keywords: non-conservative loads, stability of mechanical systems, stability research method, divergence and flutter |
Methods for determining critical values of nonconservative loads in stability problems of mechanical systems with distributed parameters are considered in this work. Based on a dynamic approach to stability problems, the method of direct integration of the linearized equation of perturbed motion is proposed, and the problem of determining critical loads is reduced to the problem of minimizing a complex function of several variables. As a second method, the method of decomposition of the solution of the equation of perturbed motion in the forms of natural oscillations is presented. The fundamentals of the application of the finite element method to the problems of stability under the action of non-conservative loads are also described. The methods are illustrated on classical problems: the stability of the cantilever rod under the action of potential and tracking forces and the stability of the pipeline section with flowing liquid. The accuracy and convergence of the latter two methods are analyzed depending on the number of members in the series and the number of finite elements.
References
[1] Nikolai E.L. Trudy po mekhanike [Works on mechanics]. Moscow, Gostekhizdat publ., 1955, pp. 357–406.
[2] Bolotin V.V. Nekonservativnye zadachi teorii uprugoy ustoychivosti [Nonconservative problems of the theory of elastic stability]. Moscow, Fizmatgiz publ, 1961. 339 p.
[3] Feodos’ev V.I. Izbrannye zadachi i voprosy po soprotivleniyu materialov [Selected problems and questions on the resistance of materials]. Moscow, Nauka publ., 1973. 400 p.
[4] Tsigler G. Osnovy teorii ustoychivosti konstruktsiy [Fundamentals of the theory of stability of structures]. Moscow, Mir publ., 1971. 192 p.
[5] Radin V.P., Samogin Yu.N., Chirkov V.P., Shchugorev A.V. Reshenie nekonservativnykh zadach teorii ustoychivosti [Solution of non-conservative problems of stability theory]. Moscow, Fizmatlit publ., 2017. 240 p.
[6] Kagan-Rozentsveyg L.M. Voprosy nekonservativnoy teorii ustoychivosti [Questions of the nonconservative theory of stability]. Sankt-Petersburg, SPbGASU publ., 2014. 174 p.
[7] Seyranian A.R., Elishakoff I. Modern problem of structural stability. New York, Springer-Verlag Wien, 2002. 394 p.
[8] Elishakoff I. Resolution of the 20th century conundrum in elastic stability. Florida Atlantic University, 2014. 334 p.
[9] Lagozinskii S.A., Sokolov A.I. Straight-line stability of the rods loaded by tracking forces. Problemy prikladnoi mekhaniki, dinamiki i prochnosti mashin. Sb. statei [Problems of applied mechanics, dynamics and strength of machines. Collected papers]. Moscow, Bauman Press, 2005, pp. 244–259 (in Russ.).
[10] Bigoni D., Noselli G. Experimental evidence of flutter and divergence instabilities induced by dry friction. Journal of the Mechanics and Physics of Solids, 2011, vol. 59, pp. 2208–2226, doi: 10.1016/j.jmps.2011.05.007
[11] Shvartsman B.S. Large deflections of a cantilever beam subjected to a follower force. Journal of Sound and Vibration, 2007, vol. 304, pp. 969–973, doi: 10.1016/j.jsv.2007.03.010
[12] Elishakoff I. Controversy associated with the so-called «follower forces»: critical overview. Applied Mechanics Reviews, 2005, vol. 58, pp. 117–142, doi: 10.1115/1.1849170
[13] Xiao Q.-X., Li X.-F. Flutter and vibration of elastically restrained nanowires under a nonconservative force. ZAMM Zeitschrift für Angewandte Mathematik und Mechanik, 2018, no. 9, pp. 1–15, doi: 10.1002/zamm.201700325
[14] Yang X., Yang T., Jin J. Dynamic stability of a beam-model viscoelastic pipe for conveying pulsative fluid. Acta Mechanica Solida Sinica, 2007, vol. 20, no. 4, pp. 350–356, doi: 10.1007/s10338-007-0741-x
[15] Olson L., Jamison D. Application of a general purpose finite element method to elastic pipes conveying fluid. Journal of Fluids and Structures, 1997, no. 11, pp. 207–222, doi: 10.1006/jfls.1996.0073
[16] Alshorbagy A.E., Eltaher M.A., Mahmoud F.F. Free vibration characteristics of a functionally graded beam by finite element method. Applied Mathematical Modelling, 2011, vol. 35, no. 1, pp. 412–425, doi: 10.1016/j.apm.2010.07.006