Abstract
Numerous specialized books and papers have been written about the subject of stability in mechanics. Most of these concentrate on methods for obtaining critical values of certain parameters and typically contain algorithms and graphs generated for describing important but very specific problems. In the present paper we take a step back and discuss the truly central notions regarding mechanical stability. Our intention is to treat the required concepts on a fairly elementary level, while simultaneously offering a bit of useful historical perspective. We also attempt to explain some common discrepancies between theoretical and practical results.
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References
Abed, E., Chou, Y., Guran, A., Tits, A.: Nonlinear stabilization and parametric optimization in the benchmark nonlinear control design problem, Proc. IEEE American Control Conf., pp. 4357–4359. Seattle, WA (1995)
Antman, S.S.: Nonlinear Problems of Elasticity. Springer-Verlag, New York (1995)
Atanakovic, T.M., Guran, A.: Theory of Elasticity for Scientists and Engineers Birkhauser. Boston (2000)
T.M., Atanackovic, Guran, A.: A generalized Greenhill problem with shear deformation, compressibility and imperfections. Theoretical and Applied Mechanics 25, 1–20 (1999)
Beletskii, V.V.: Resonance Phenomena at Rotation of Artificial and Natural Celestial Bodies. In: Giacaglia, G.E.O., (eds.) Satellite Dynamic, pp. 191–232. Springer, Berlin (1975)
Blekhman, I.: Selected Topics in Vibrational Mechanics.. World Scientific, Singapore, New Jersey, London,Hong Kong (2004)
Byskov, E.: Smooth postbuckling stresses by a modified finite element method. Int. Journal for Numerical Methods in Engineering. 28, 2877–888 (1989)
Dirichlet, J.P.L.: Über die stabilität des Gleichgewichts. J. reine angew. Math 32, 85–88 (1846)
Euler, L.: Methodus Inveniendi Lineas Curvas; Maximi Minimive Proprietate Gaudentes (Appendix. De Curvas Elasticas). Marcus Michaelem, Bousquet Lausanneand Geneva (1744)
Farshad, M., Guran, A.: Adaptive Material and Structures. In: Guran, A., Valasek, M. (eds.), 3rd International Congress on Mechatronic, Prague, Czech Technical University in Prague, pp 319–374 (2004)
Guran, A.: On the stability of an elastic imperfect column including axial compressibility. Journal of Applied Mathematics and Mechanics-ZAMM (Zeitschrift fur Angewandte Mathematik und Mechanik) 72(10), 481–485 (1992)
Guran, A.: Influence of various types of load-dependent supports on the stability of a compressible column model. Acta Mech. 97, 91–100 (1993)
Guran, A.: Shape control of smart structures by using fluidic actuators. In: The 3rd International IEEE Scientific Conference on Physics and Control (2007)
Guran, A.: Controlling the Chaos Using Fuzzy Estimation in a Gyrostat Satellite. In: Sanjuan, M.A.F., Grebogi, C. (eds.) Recent Progress in Controlling Chaos, World Scientific, Singapore, New Jersey, London, Hong Kong (2010)
Guran, A.: Buckling of a column as a benchmark nonlinear control design. In: Kurths, J., Fradkov, A., Chen, G. (eds.) The 3rd international IEEE scientific conference on physics and Control, University of Potsdam, Germany (2007)
Guran, A.: Ernst Mach and Peter Salcher: The development of nonlinear wave mechanics during the period 1850–1950. In: International Sysposium Peter Salcher and Ernst Mach: A Successful Teamwork, pp. 23–25. Rijeka, Croatia (2004)
Guran, A. et al.: Oscillations, bifurcations and chaos in Ziegler’s pendulum with eccentric load in a gravitational field. In: Proceedings of the 3rd International Conference on Complex Systems and Applications(ICCSA 2009), University of Le Havre Le Havre, Normandy France (2009)
Guran, A. et al.:On the stability of a spinning viscoelastic column. Mechanics Based Design of Structures and Machines 19(4), 437–455 (1991)
Guran, A. et al.: Studies in spatial motion of a gyro on an elastic foundation. Mechanics Based Design of Structures and Machines An International Journal 21(2), 185–199 (1993)
Guran, A., Atanackovic, T.M.: Lecture Notes on Theory of Elasticity. St. Petersburg University Press, Petersburg (2002) (in Russian)
Guran, A., Atanackovic, T.M.: Fluid conveying pipe with shear and compressibility. Eur. J. Mech., A/Solids 17, 121–137 (1997)
Guran, A., Ahmadi, G.: Postbuckling behavior and imperfection sensivity of an elastic compressible rod whose flexural resistance changes with load. In: Second International Conference on Application of Mathematics in Technical and Natural Sciences., Sozopol, Bulgaria (2010)
Guran, A., Bajaj, A., Perkins, N., DEluterio, G., Pierre, C.: Stability of Gyroscopic Systems. World Scientific, Singapore, New Jersey, London, Hong Kong (1999)
Guran A., Khoshnood M. Control of an inverted double pendulum with eccentric fluid load by means of computed torque method. In: Proceedings of EUROMECH Colloquium 515 Advanced Applications and Perspectives of Multibody System Dynamics. pp. 36–37. Blagoevgrad (2010)
Guran, A., Sperling, L.: (Guest Editors) Wissenschaftliche Zeitschrift fur Grundlagen und Anwendungen der Technischen Mechanik, 24. Special issue In memoriam Friedrich P.J. Rimrott (2004)
Guran, A., Tadjbakhsh, I.G.: A mechanical actuator to suppress vibration. In: Hagood, W., (ed.) Proc. Smart Structures and Materials (SPIE), Smart Structures and Intelligent Systems. pp. 1113–1114 (1917), (1993)
Guran, A., Yousefghahari, B.: Biomechanical study of spine. XIII Mediterranean Congress of Rheumatology; Clinical and Experimental Rheumatology 27, 726 (2009)
Guran, A., Yousefghahari, B.: A nonlinear multibody system dynamics model to predict the impact response of human chest during cardiopulmonary resuscitations. In: Proceedings of EUROMECH Colloquium 515 Advanced Applications and Perspectives of Multibody System Dynamics, pp. 70–71. Blagoevgrad (2010)
Guran, A., Yousefghahari, B.: Shape-control of carbon nanotubes by means of nanofluidic actuators. In: Proc. Workshop on Biomedical applications of functionalized carbon nanotubes, Schloss Eckberg, Dresden Germany (2010)
Knops, R.J., Wilkens, E.W.: Theory of Elastic Stability, Handbuch der Physik, Bd. VIa/3.. Springer, Berlin (1973)
Koiter, W.T.: On the stability of elastic equilibrium. Delft in Trans NASA, Dissertation, 10, 833 (1945)
Koiter W.T. (1963) Elastic stability and postbuckling behavior. In: Langer R.E. (ed.), Proc Symp Non-Lrnear Probl. Univ. of Wisconsin Press, pp. 257–275
Koiter, W.T.: The non-linear buckling problem of complete sphercal shells under uniform external pressure. Proc. Konik, Ned. Akad. Wetensch. Ser B 72, 40 (1969)
W.T. Koiter (1976) Current trends in the theory of buckling, Buckling Structures, pp. 1–16
Lagrange, J.L.: Mécanique Analutique. Courier, Paris (1788)
Lebedev, L.P., Cloud, M.J.: The Calculus of Variations, Optimal Control, and Functional Analysis. World Scientific, Singapore, New Jersey, London, Hong Kong (2003)
Lebedev, L.P., Cloud, M.J.: Tensor Analysis. World Scientific, Singapore, New Jersey, London, Hong Kong (2003)
Lebedev, L.P., Cloud, M.J.: Introduction to Mathematical Elasticity. World Scientific, Singapore, New Jersey, London, Hong Kong (2009)
Lebedev, L.P., Cloud, M.J., Eremeyev, V.A.: Tensor Analysis with Applications in Mechanics. World Scientific, Singapore, New Jersey, London, Hong Kong (2010)
Lebedev, L.P., Vorovich, I.I.: Functional Analysis in Mechanics. Springer, New York (2002)
Lebedev, L.P., Vorovich, I.I., Gladwell, G.M.L.: Functional Analysis: Applications in Mechanics and Inverse Problems. Kluwer Academic Publishers (2/e 2002) (1996)
Lyapunov, A.M.: On the stability of ellipsoidal figures of equilibrium of a rotating fluid. Master’s thesis, Sankt-Petersbourgue, Akademy of Sciences, pp. 109 (1884) (in Russian)
Lyapunov, A.M.: The General Problem of Motion Stability, Doctoral dissertation, p. 250. Kharkov Russia (1892) (in Russian)
Merkin, D.R.: Introduction to Stability of Motion. Springer, New York (1996)
Movchan, A.A.: On the direct Liapunov method in the stability problems for elastic systems. Prikl. Mat. Mekh. 23(3) (1959) (in Russian)
Movchan, A.A.: On stability of motion of continuous bodies. Lagrange’s theorem and its inverse. Inzhenernyj sbornik, vol. 29 (1960) (in Russian)
Olesen, J.F., Byskov, E.: Accurate detemination of asymptotic postbuckling stresses by the finite element method. Computer and Structures 15, 157–163 (1982)
Palassopoulos, G.V.: Effect of stochastic imperfections on the buckling strength of imperfectionsensitive structures". In: Fifth World Congress on Computational Mechanics, Austria, Vienna (2002)
Plaut, R.H., Guran, A.: Buckling of plates with stiffening elastically restrained edges. Journal of Engineering Mechanics–ASCE 120(2), 408–411 (1994)
Roorda, J.: Stability of structures with small imperfections. Journal of Engineering Mechanics–ASCE, pp 87–106 (1965)
Roorda, J., Chilver, A.H.: Frame buckling: an illustration of the perturbation technique. Int. J. Non-Linear Mechanics 5, 235–246 (1970)
Seyranian, A.P., Mailybeav, A.A.: Multiparameter Stability Theory with Mechanical Applications. World Scientific, Singapore, New Jersey, London, Hong Kong (2003)
Tabarrok, B., Rimrott, F.P.J.: Variational Methods and Complementary Formulations in Dynamics. Kluwer Academic Publishers, Netherland (1994)
Tovstik, P.E., Smirnov, A.L.: Asymptotic Methods in the Buckling Theory of Elastic Shells. World Scientific, Singapore, New Jersey, London, Hong Kong (2001)
Vorovich, I.I.: Some question of shell stability “in the large". Dokl. Akad. Nauk SSSR 122, 37–40 (1958)
Vorovich, I.I.: The problem of non-uniqueness and stability in the non-linear mechanics of continuum mechanics. In: Applied Mechanics, Proc. 13th Int. Congr. Theor. Appl. Mech, pp. 340–357. Springer, Berlin (1973)
Vorovich, I.I.: Nonlinear Theory of Shallow Shells. Springer, New York, 1999, Translation from Russian edition by Nauka, Moscow (1989)
Vorovich, I.I., Lebedev, L.P.: Some issues of continuum mechanics and mathematical problems in the theory of thin-walled structures. International Applied Mechanics (Prikladnaya Mekhanika) 38(4), 387–398 (2002)
Yakubovich, V.A., Leonov, G.A., Gelig, A.Kh.: Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities. World Scientific, Singapore, New Jersey, London, Hong Kong (2004)
Yurkevich, V., Guran, A.: Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities (Book Reviews). IEEE Transaction on Automatic Control 50(4), 542–543 (2005)
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Guran, A., Lebedev, L.P. (2011). Basic Concepts in the Stability Theory of Thin-Walled Structures. In: Altenbach, H., Eremeyev, V. (eds) Shell-like Structures. Advanced Structured Materials, vol 15. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21855-2_11
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