Abstract
Nonlinear dynamics of engineering systems has reached the stage of full maturity in which it makes sense to critically revisit its past and present in order to establish an historical perspective of reference and to identify novel objectives to be pursued. This paper makes a first step in this direction, focusing on the mechanics of machines, solids and structures, with applications in both classical and novel technological areas. This is accomplished first by identifying some main stages of scientific development over the last four decades, with the characterizing features as regards addressed systems, underlying mathematical tools and phenomenological aspects. These are substantiated in terms of topics and involved people with also an archival, and tentatively comprehensive, list of related activities. The second part of the study deals with Ali Nayfeh’s contributions as a scientist and a scholar, embedding his research achievements within the identified four stages of development and highlighting some qualifying methodological features of his activity as a book author. His legacy is framed within an overall scenario of expected future developments of nonlinear dynamics.
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References
Duffing, G.: Erzwungene Schwingungen bei veränderlicher Eigenfrequenz und ihre technische Bedeutung. Vieweg, Braunschweig (1918)
Kovacic, I., Brennan, M. (eds.): The Duffing Equation. Non-linear Oscillators and Their Behaviour. Wiley, Chichester (2010)
van der Pol, B.: On relaxation-oscillations. Philosophical Magazine 2, 978–992 (1926)
Poincaré, H.J.: Sur le problème des trois corps et les équations de la dynamique. Acta Mathematica 13, 1–270 (1890)
Lyapunov, A.M.: Problème général de la stabilité du mouvement. Annales de la Faculté des Sciences de Toulouse 9, 203–469 (1907). (Russian edition, Moscow 1892)
Birkhoff, G.D.: Dynamical Systems. American Mathematical Society, Providence, RI (1927)
Mira, C.: Some historical aspects of nonlinear dynamics: possible trends for the future. Int. J. Bifurcat. Chaos 7(9), 2145–2173 (1997)
Andronov, A.A., Witt, A.A., Khaïkin, S.E.: Theory of Oscillators. Pergamon, London (1966). (Russian edition, Moscow 1937)
Krylov, N.M., Bogoliubov, N.N.: Introduction to Nonlinear Mechanics. Princeton University Press, Princeton (1947). (Russian edition, Moscow 1937)
Holmes, P.: Ninety plus thirty years of nonlinear dynamics: More is different and less is More. Int. J. Bifurcat. Chaos 15(9), 2703–2716 (2005)
Aubin, D., Dahan Dalmedico, A.: Writing the history of dynamical systems and chaos: Longue durée and revolution, disciplines and cultures. Historia Mathematica 29, 273–339 (2002)
Shaw, S.W., Balachandran, B.: A review of nonlinear dynamics of mechanical systems in year 2008. J. Syst. Des. Dyn. 2(3), 611–640 (2008)
Stoker, J.J.: Nonlinear Vibrations in Mechanical and Electrical Systems. Interscience Publishers, New York (1950)
Bogoliubov, N.N., Mitropolski, Y.A.: Asymptotic Methods in the Theory of Nonlinear Oscillations. Gordon and Breach, New York (1961)
Minorski, N.: Nonlinear Oscillations. Van Nostrand, Princeton (1962)
Hayashi, C.: Nonlinear Oscillations in Physical Systems. McGraw-Hill, New York (1964)
Nayfeh, A.H.: Perturbation Methods. Wiley, New York (1973)
Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1979)
Ewan-Iwanowski, R.M.: Resonance Oscillations in Mechanical Systems. Elsevier, New York (1976)
Hagedorn, P.: Nonlinear Oscillations. Clarendon Press, Oxford (1981)
Kevorkian, J., Cole, J.D.: Perturbation Methods in Applied Mechanics. Springer, New York (1981)
Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields. Springer, New York (1983)
Sanders, J.A., Verhulst, F.: Averaging Methods in Nonlinear Dynamical Systems. Springer, New York (1985)
Schmidt, G., Tondl, A.: Non-Linear Vibrations. Akademie, Berlin (1986)
Rand, R.H., Armbruster, D.: Perturbation Methods, Bifurcation Theory, and Computer Algebra. Springer, New York (1987)
Wiggins, S.: Global Bifurcation and Chaos: Analytical Methods. Springer, New York (1988)
Szemplinska-Stupnicka, W.: The Behaviour of Nonlinear Vibrating Systems. Vols. I and II. Kluwer, Dordrecht (1990)
Nayfeh, A.H.: Method of Normal Forms. Wiley, New York (1993)
Nayfeh, A.H.: Nonlinear Interactions. Wiley, New York (2000)
Liao, S.J.: Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman & Hall/CRC Press, Boca Raton (2003)
Carr, J.: Applications of Centre Manifold Theory. Applied Mathematical Sciences, vol. 35, Springer, New York (1981)
Abraham, R., Shaw, C.: Dynamics: The Geometry of Behavior, vol. 1–4. Aerial Press, Santa Cruz (1982)
Arnold, V.I.: Geometrical Methods in the Theory of Ordinary Differential Equations. Springer, New York (1983)
Thompson, J.M.T., Stewart, H.B.: Nonlinear Dynamics and Chaos. Wiley, Chichester (1986)
Doedel, E.J., Oldeman, B.E.: AUTO-07P Continuation and Bifurcation Software for Ordinary Differential Equations (2012) (first edition, California Institute of Technology, Pasadena, CA 1986) http://indy.cs.concordia.ca/auto
Hsu, C.S.: Cell to Cell Mapping: A Method of Global Analysis for Nonlinear Systems. Springer, New York (1987)
Seydel, R.: From Equilibrium to Chaos: Practical Bifurcation and Stability Analysis. Elsevier, New York (1988)
Parker, T.S., Chua, L.O.: Practical Numerical Algorithms for Chaotic Systems. Springer, New York (1989)
Arrowsmith, D.K., Place, C.M.: An Introduction to Dynamical Systems. Cambridge University Press, Cambridge (1990)
Baker, G.L., Gollub, J.P.: Chaotic Dynamics. Cambridge University Press, Cambridge (1990)
Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (1990)
Rasband, S.N.: Chaotic Dynamics of Nonlinear Systems. Wiley, New York (1990)
Troger, H., Steindl, A.: Nonlinear Stability and Bifurcation Theory: An Introduction for Engineers and Applied Scientists. Springer, New York (1991)
Ott, E., Sauer, T., Yorke, J.A.: Coping with Chaos. Wiley, New York (1994)
Glendinning, P.: Stability, Instability and Chaos. Cambridge University Press, Cambridge (1994)
Strogatz, S.H.: Nonlinear Dynamics and Chaos. Addison-Wesley, New York (1994)
Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics. Wiley, New York (1995)
Kustnezov, Y.A.: Elements of Applied Bifurcation Theory. Springer, New York (1995)
Pfeiffer, F., Glocker, C.: Multibody Dynamics with Unilateral Constraints. Wiley, New York (1996)
Brogliato, B.: Nonsmooth Mechanics, 2nd edn. Springer, London (1999)
Blazejczyk-Okolewska, B., Czolczynski, K., Kapitaniak, T., Wojewoda, J.: Chaotic Mechanics in Systems with Impacts and Friction. World Scientific, Singapore (1999)
Leine, R., Nijmeijer, H.: Dynamics and Bifurcations of Non-Smooth Mechanical Systems. Springer, New York (2004)
Di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P.: Piecewise-smooth Dynamical Systems: Theory and Applications. Springer, New York (2007)
Insperger, T., Stépán, G.: Semi-Discretization Method for Time-Delay Systems. Springer, New York (2011)
Moon, F.C.: Chaotic and Fractal Dynamics. An Introduction for Applied Scientists and Engineers. Wiley, New York (1992)
Abarbanel, H.D.I., Brown, R., Sidorowich, J.J., Tsimring, L.S.: The analysis of observed chaotic data in physical systems. Rev. Mod. Phys. 65(4), 1331–1392 (1993)
Holmes, P., Lumley, J.L., Berkooz, G.: Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press, Cambridge (1996)
Lauterborn, W., Kurz, T., Parlitz, U.: Experimental nonlinear physics. Int. J. Bifurcat. Chaos 7(9), 2003–2033 (1997)
Virgin, L.N.: Introduction to Experimental Nonlinear Dynamics. Cambridge University Press, New York (2000)
Ott, E., Grebogi, C., Yorke, J.A.: Controlling chaos. Phys. Rev. Lett. 64, 1196–1199 (1990)
Pyragas, K.: Continuous control of chaos by self-controlling feedback. Phys. Lett. 170A, 421–428 (1992)
Kapitaniak, T.: Controlling Chaos. Academic Press, Lodz (1996)
Fradkov, A.L., Pogromsky, A.Y.: Introduction to Control of Oscillations and Chaos. World Scientific, Singapore (1998)
Vakakis, A.F., Gendelman, O., Bergman, L.A., McFarland, D.M., Kerschen, G., Lee, Y.S.: Passive Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems: I and II. Springer, Berlin (2008)
Sanjuan, M.A.F., Grebogi, C. (eds.): Recent Progress in Controlling Chaos. World Scientific, Singapore (2010)
Wagg, D., Neild, S.: Nonlinear Vibration with Control. Springer, Dordrecht (2010)
Nayfeh, A.H., Pai, P.F.: Linear and Nonlinear Structural Mechanics. Wiley, Hoboken (2004)
Antman, S.: Nonlinear Problems of Elasticity. Springer, New York (2005), (first edition, 1995)
Amabili, M.: Nonlinear Vibrations and Stability of Shells and Plates. Cambridge University Press, New York (2008)
Younis, M.I.: MEMS Linear and Nonlinear Statics and Dynamics. Springer, New York (2011)
Lacarbonara, W.: Nonlinear Structural Mechanics. Springer, New York (2013)
Vakakis, A.F., Manevitch, L.I., Mikhlin, Y.V., Pilipchuk, V.N., Zevin, A.A.: Normal Modes and Localization in Nonlinear Systems. Wiley, New York (1996)
Landa, P.S.: Nonlinear Oscillations and Waves in Dynamical Systems. Springer, Dordrecht (1996)
Nesterenko, V.F.: Dynamics of Heterogeneous Materials. Springer, New York (2001)
Starosvetsky, Y., Jayaprakash, K.R., Hasan, M.A., Vakakis, A.F.: Topics on the Nonlinear Dynamics and Acoustics of Ordered Granular Media. World Scientific, Singapore (2017)
Manevitch, L.I., Kovaleva, A., Smirnov, V., Starosvetsky, Y.: Nonstationary Resonant Dynamics of Oscillatory Chains and Nanostructures. Springer Nature, Singapore (2017)
Andrianov, I., Gendelman, O., Manevitch, A., Mikhlin, Y. (eds): Problems of Nonlinear Mechanics and Physics of Materials. Advanced Structured Materials, vol. 94, Springer Nature, New York (2018)
Awrejcewicz, J., Krysko, V.A., Krysko, A.V.: Thermo-Dynamics of Plates and Shells. Springer, New York (2007)
Erturk, A., Inman, D.J.: Piezoelectric Energy Harvesting. Wiley, New York (2011)
Païdoussis, M.P.: Fluid-Structure Interactions: Slender Structures and Axial Flow, vol. 1, 2nd edn. Academic Press, New York (2013)
Dorfmann, L., Ogden, R.W.: Nonlinear Theory of Electroelastic and Magnetoelastic Interactions. Springer, New York (2014)
Dowell, E.: A Modern Course in Aeroelasticity (Fifth revised and enlarged edition). Springer, New York (2015)
Steindl, A., Troger, H.: Methods for dimension reduction and their application in nonlinear dynamics. Int. J. Solids Struct. 38(10–13), 2131–2147 (2001)
Rega, G., Troger, H.: Dimension reduction of dynamical systems: methods, models, applications. Nonlinear Dyn. 41, 1–15 (2005)
http://www.euromech.org/structure/conference-committees/enocc
Vakakis, A.F. (ed.): Advanced Nonlinear Strategies for Vibration Mitigation and System Identification. CISM Courses and Lectures, vol. 518, Springer, New York (2011)
Wagg, D.J., Virgin, L. (eds.): Exploiting Nonlinear Behavior in Structural Dynamics. CISM Courses and Lectures, vol. 536, Springer, New York (2012)
Kerschen, G. (ed.): Modal Analysis of Nonlinear Mechanical Systems. CISM Courses and Lectures, vol. 555, Springer, New York (2014)
Lenci, S., Rega, G. (eds.): Global Nonlinear Dynamics for Engineering Design and System Safety. CISM Courses and Lectures, vol. 588, Springer Nature, New York (2018)
Lacarbonara, W.: Obituary: Professor Ali H. Nayfeh 21 December 1933–27 March 2017. J. Sound Vib. 400, 680–683 (2017)
Younis, M.I.: In memoriam: Ali Hasan Nayfeh 1933–2017. Nonlinear Dyn. 88, 1535–1536 (2017)
Balachandran, B., Younis, M.I., Shen, I.Y.(Steve): In Memoriam: Ali Hasan Nayfeh. J. Vib. Acoust. 139, 040101-1 (2017)
Rega, G.: A tribute to Ali H. Nayfeh (1933-2017). In Kovacic, I., Lenci, S. (eds.) IUTAM Symposium on Exploiting Nonlinear Dynamics for Engineering Systems. IUTAM Book Series, Springer, Berlin (2019) (to appear)
Nayfeh, A.H.: On direct methods for constructing nonlinear normal modes of continuous systems. J. Vibr. Control 1, 389–430 (1995)
Nayfeh, A.H.: Resolving controversies in the application of the method of multiple scales and the generalized method of averaging. Nonlinear Dyn. 40(1), 61–102 (2005)
Rega, G.: Book review: Nonlinear interactions: analytical, computational, and experimental methods. Meccanica 35, 583–585 (2000)
Rega, G.: Book review: Linear and nonlinear structural mechanics. Meccanica 40, 221–222 (2005)
Nayfeh, A.H.: Introduction to Perturbation Techniques. Wiley, New York (1980)
Verhulst, F.: Profits and pitfalls of timescales in asymptotics. SIAM Rev. 57(2), 255–274 (2015)
Daqaq, M.F., Masana, R., Erturk, A., Quinn, D.D.: On the role of nonlinearities in vibratory energy harvesting: a critical review and discussion. Appl. Mech. Rev. 66(4), 040801 (2014)
Dou, S., Strachan, B.S., Shaw, S.W., Jensen, J.S.: Structural optimization for nonlinear dynamic response. Phil. Trans. R. Soc. A 373, 20140408 (2015)
Rega, G., Lenci, S.: A global dynamics perspective for system safety from macro- to nanomechanics: analysis, control and design engineering. Appl. Mech. Rev. 67(5), 050802 (2015)
Porter, M.A., Kevrekidis, P.G., Daraio, G.: Granular crystals: nonlinear dynamics meets materials engineering. Phys. Today 68(11), 44–50 (2015)
Kochmann, D.M., Bertoldi, K.: Exploiting microstructural instabilities in solids and structures: from metamaterials to structural transitions. Appl. Mech. Rev. 69(5), 050801-1-24 (2017)
Wiercigroch, M., Rega, G.: Introduction to NDATED. In: Wiercigroch, M., Rega, G. (eds.) IUTAM Symposium on Nonlinear Dynamics for Advanced Technologies and Engineering Design, IUTAM Book Series 32, pp. v–viii. Springer, Berlin (2013)
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The financial support of PRIN 2015 (No.2015JW9NJT), Italy, is gratefully acknowledged.
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Rega, G. Nonlinear dynamics in mechanics and engineering: 40 years of developments and Ali H. Nayfeh’s legacy. Nonlinear Dyn 99, 11–34 (2020). https://doi.org/10.1007/s11071-019-04833-w
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DOI: https://doi.org/10.1007/s11071-019-04833-w