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Review Article

A Global Dynamics Perspective for System Safety From Macro- to Nanomechanics: Analysis, Control, and Design Engineering

[+] Author and Article Information
Giuseppe Rega

Department of Structural and Geotechnical Engineering,
Sapienza University of Rome,
Via Antonio Gramsci 53,
Rome 00197, Italy
e-mail: giuseppe.rega@uniroma1.it

Stefano Lenci

Department of Civil and Building Engineering, and Architecture,
Polytechnic University of Marche,
via Brecce Bianche,
Ancona 60131, Italy
e-mail: lenci@univpm.it

1Corresponding author.

Manuscript received January 9, 2015; final manuscript received September 28, 2015; published online October 15, 2015. Assoc. Editor: Chin An Tan.

Appl. Mech. Rev 67(5), 050802 (Oct 15, 2015) (19 pages) Paper No: AMR-15-1006; doi: 10.1115/1.4031705 History: Received January 09, 2015; Revised September 28, 2015

The achievements occurred in nonlinear dynamics over the last 30 years entail a substantial change of perspective when dealing with vibration problems, since they are now deemed ready to meaningfully affect the analysis, control, and design of mechanical and structural systems. This paper aims at overviewing the matter, by highlighting and discussing the important, yet still overlooked, role that some relevant concepts and tools may play in engineering applications. Upon dwelling on such topical concepts as local and global dynamics, bifurcation and complexity, theoretical and practical stability, attractor robustness, basin erosion, and dynamical integrity, recent results obtained for a variety of systems and models of interest in applied mechanics and structural dynamics are overviewed in terms of analysis of nonlinear phenomena and their control. The global dynamics perspective permits to explain partial discrepancies between experimental and theoretical/numerical results based on merely local analyses and to implement effective dedicated control procedures. This is discussed for discrete systems and reduced order models of continuous systems, for applications ranging from macro- to micro/nanomechanics. Understanding of basic phenomena in nonlinear dynamics has now reached such a critical mass that it is time to exploit their potential to enhance the effectiveness and safety of systems in technological applications and to develop novel design criteria.

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References

Lenci, S. , Rega, G. , and Ruzziconi, L. , 2013, “ The Dynamical Integrity Concept for Interpreting/Predicting Experimental Behavior: From Macro- to Nano-Mechanics,” Phil. Trans. R. Soc. A, 371(1993), p. 20120423. [CrossRef]
Euler, L. , 1744, Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes, Sive Solutio Problematis Isoperimetrici Latissimo Sensu Accepti, Addentamentum 1: de Curvis Elasticis, Laussanae et Genevae, Apud Marcum-Michaelem, Bousquet et Socios, Geneva, Switzerland, pp. 245–310.
Lyapunov, A. M. , 1892, “ The General Problem of the Stability of Motion,” Ph.D. thesis, Moscow University, Moscow, Russia (English translation: 1992, Taylor & Francis, London).
Koiter, W. T. , 1945, “ Over de Stabiliteit van het Elastisch Evenwicht,” Ph.D. thesis, Delft University, Delft, The Netherlands (English translation: Koiter, W. T., 1967, “On the Stability of Elastic Equilibrium,” NASA Technical Translation, F-10, 833, U.S. Department of Commerce/National Bureau of Standards, N67–25033).
Guckenheimer, J. , and Holmes, P. , 1983, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York.
Wiggins, S. , 1990, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, New York.
Troger, H. , and Steindl, A. , 1991, Nonlinear Stability and Bifurcation Theory, Springer, Wien, Germany.
Kustnezov, Y. A. , 1995, Elements of Applied Bifurcation Theory, Springer, New York.
Scott, R. , 2001, In the Wake of Tacoma: Suspension Bridges and the Quest for Aerodynamic Stability, American Society of Civil Engineers, Reston, VA.
Gurung, C. B. , Yamaguchi, H. , and Yukino, T. , 2002, “ Identification of Large Amplitude Wind-Induced Vibration of Ice-Accreted Transmission Lines Based on Field Observed Data,” Eng. Struct., 24(2), pp. 179–188. [CrossRef]
Goncalves, P. B. , Silva, F. M. A. , and Del Prado, Z. J. G. N. , 2007, “ Global Stability Analysis of Parametrically Excited Cylindrical Shells Through the Evolution of Basin Boundaries,” Nonlinear Dyn., 50, pp. 121–145. [CrossRef]
Thompson, J. M. T. , 1989, “ Chaotic Phenomena Triggering the Escape From a Potential Well,” Proc. R. Soc. London A, 421(1861), pp. 195–225. [CrossRef]
Nayfeh, A. H. , and Balachandran, B. , 1995, Applied Nonlinear Dynamics, Wiley, New York.
Soliman, M. S. , and Thompson, J. M. T. , 1989, “ Integrity Measures Quantifying the Erosion of Smooth and Fractal Basins of Attraction,” J. Sound Vib., 135(3), pp. 453–475. [CrossRef]
Lansbury, A. N. , Thompson, J. M. T. , and Stewart, H. B. , 1992, “ Basin Erosion in the Twin-Well Duffing Oscillator: Two Distinct Bifurcation Scenarios,” Int. J. Bifurcation Chaos, 2(03), pp. 505–532. [CrossRef]
Lorenz, E. N. , 1963, “ Deterministic Nonperiodic Flow,” J. Atmos. Sci., 20(2), pp. 130–141. [CrossRef]
Rega, G. , and Lenci, S. , 2005, “ Identifying, Evaluating, and Controlling Dynamical Integrity Measures in Nonlinear Mechanical Oscillators,” Nonlinear Anal., 63, pp. 902–914. [CrossRef]
Gonçalves, P. B. , Silva, F. M. A. , and Del Prado, Z. J. G. N. , 2007, “ Transient and Steady State Stability of Cylindrical Shells Under Harmonic Axial Loads,” Int. J. Nonlinear Mech., 42(1), pp. 58–70. [CrossRef]
Thompson, J. M. T. , and Soliman, M. S. , 1990, “ Fractal Control Boundaries of Driven Oscillators and Their Relevance to Safe Engineering Design,” Proc. R. Soc. London A, 428(1874), pp. 1–13. [CrossRef]
Soliman, M. S. , and Goncalves, P. B. , 2003, “ Chaotic Behavior Resulting in Transient and Steady State Instabilities of Pressure-Loaded Shallow Spherical Shells,” J. Sound Vib., 259(3), pp. 497–512. [CrossRef]
Lenci, S. , and Rega, G. , 2003, “ Optimal Control of Homoclinic Bifurcation: Theoretical Treatment and Practical Reduction of Safe Basin Erosion in the Helmholtz Oscillator,” J. Vib. Control, 9, pp. 281–315. [CrossRef]
Sun, J. Q. , 1994, “ Effect of Small Random Disturbance on the ‘Protection Thickness' of Attractors of Nonlinear Dynamic Systems,” Nonlinearity and Chaos in Engineering Dynamics, J. M. T. Thompson and S. R. Bishop , eds., Wiley, Chichester, UK, pp. 435–437.
Hsu, C. S. , 1987, Cell to Cell Mapping: A Method of Global Analysis for Nonlinear System, Springer, New York.
Hsu, C. S. , and Chiu, H. M. , 1987, “ Global Analysis of a System With Multiple Responses Including a Strange Attractor,” J. Sound Vib., 114(2), pp. 203–218. [CrossRef]
Sun, J. Q. , and Hsu, C. S. , 1991, “ Effects of Small Random Uncertainties on the Non-Linear Systems Studied by the Generalized Cell Mapping Methods,” J. Sound Vib., 147(2), pp. 185–201. [CrossRef]
Soliman, M. S. , and Thompson, J. M. T. , 1990, “ Stochastic Penetration of Smooth and Fractal Basin Boundaries Under Noise Excitation,” Dyn. Stab. Syst., 5(4), pp. 281–298.
Gan, C. B. , and He, S. M. , 2007, “ Studies on Structural Safety in Stochastically Excited Duffing Oscillator With Double Potential Wells,” Acta Mech. Sin., 23(5), pp. 577–583. [CrossRef]
de Souza, J. R., Jr. , and Rodrigues, M. L. , 2002, “ An Investigation Into Mechanisms of Loss of Safe Basins in a 2 DOF Nonlinear Oscillator,” J. Braz. Soc. Mech. Sci. Eng., 24, pp. 93–98.
Rega, G. , and Lenci, S. , 2008, “ Dynamical Integrity and Control of Nonlinear Mechanical Oscillators,” J. Vib. Control, 14, pp. 159–179. [CrossRef]
Lenci, S. , and Rega, G. , 2011, “ Load Carrying Capacity of Systems Within a Global Safety Perspective. Part II: Attractor/Basin Integrity Under Dynamical Excitations,” Int. J. Nonlinear Mech., 46(9), pp. 1240–1251. [CrossRef]
Lenci, S. , and Rega, G. , 2006, “ Control of Pull-In Dynamics in a Nonlinear Thermoelastic Electrically Actuated Microbeam,” J. Micromech. Microeng., 16(2), pp. 390–401. [CrossRef]
Rega, G. , and Settimi, V. , 2013, “ Bifurcation, Response Scenarios and Dynamical Integrity in a Single-Mode Model of Noncontact Atomic Force Microscopy,” Nonlinear Dyn., 73(1), pp. 101–123. [CrossRef]
Lenci, S. , and Rega, G. , 2008, “ Competing Dynamical Solutions in a Parametrically Excited Pendulum: Attractor Robustness and Basin Integrity,” ASME J. Comput. Nonlinear Dyn., 3(4), p. 041010. [CrossRef]
Goncalves, P. B. , Silva, F. M. A. , Rega, G. , and Lenci, S. , 2011, “ Global Dynamics and Integrity of a Two-Dof Model of a Parametrically Excited Cylindrical Shell,” Nonlinear Dyn., 63, pp. 61–82. [CrossRef]
Lenci, S. , Orlando, D. , Rega, G. , and Gonçalves, P. B. , 2012, “ Controlling Practical Stability and Safety of Mechanical Systems by Exploiting Chaos Properties,” Chaos, 22(4), p. 047502. [CrossRef] [PubMed]
Lenci, S. , Orlando, D. , Rega, G. , and Gonçalves, P. B. , 2012, “ Controlling Nonlinear Dynamics of Systems Liable to Unstable Interactive Buckling,” Procedia IUTAM, 5, pp. 108–123. [CrossRef]
Eason, R. P. , Dick, A. J. , and Nagarajaiah, S. , 2014, “ Numerical Investigation of Coexisting High and Low Amplitude Responses and Safe Basin Erosion for a Coupled Linear Oscillator and Nonlinear Absorber System,” J. Sound Vib., 333(15), pp. 3490–3504. [CrossRef]
Ruzziconi, L. , Younis, M. I. , and Lenci, S. , 2012, “ Multistability in an Electrically Actuated Carbon Nanotube: A Dynamical Integrity Perspective,” Nonlinear Dyn., 74(3), pp. 533–549. [CrossRef]
Silva, F. M. A. , and Goncalves, P. B. , 2015, “ The Influence of Uncertainties and Random Noise on the Dynamical Integrity of a System Liable to Unstable Buckling,” Nonlinear Dyn., 81(1), pp. 707–724. [CrossRef]
Lenci, S. , and Rega, G. , 2011, “ Load Carrying Capacity of Systems Within a Global Safety Perspective. Part I: Robustness of Stable Equilibria Under Imperfections,” Int. J. Nonlinear Mech., 46(9), pp. 1232–1239. [CrossRef]
Gonçalves, P. B. , and Santee, D. , 2008, “ Influence of Uncertainties on the Dynamic Buckling Loads of Structures Liable to Asymmetric Post-Buckling Behavior,” Math. Probl. Eng., 2008, pp. 1–24. [CrossRef]
Orlando, D. , Gonçalves, P. B. , Rega, G. , and Lenci, S. , 2011, “ Influence of Modal Coupling on the Nonlinear Dynamics of Augusti's Model,” ASME J. Comput. Nonlinear. Dyn., 6(4), p. 041014. [CrossRef]
Thompson, J. M. T. , and Ueda, Y. , 1989, “ Basin Boundary Metamorphoses in the Canonical Escape Equation,” Dyn. Stab. Syst., 4, pp. 285–294. [CrossRef]
Bazant, Z. , and Cedolin, L. , 1991, Stability of Structures, Oxford University Press, Oxford, UK.
Lenci, S. , and Rega, G. , 2011, “ Forced Harmonic Vibration in a System With Negative Linear Stiffness and Linear Viscous Damping,” The Duffing Equation. Non-linear Oscillators and Their Behaviour, I. Kovacic and M. Brennan , eds., Wiley, Chichester, UK, pp. 219–276.
Xu, J. , Lu, Q. S. , and Huang, K. L. , 1996, “ Controlling Erosion of Safe Basin in Nonlinear Parametrically Excited Systems,” Acta Mech. Sin., 12(3), pp. 281–288. [CrossRef]
Thompson, J. M. T. , Rainey, R. C. T. , and Soliman, M. S. , 1990, “ Ship Stability Criteria Based on Chaotic Transients From Incursive Fractals,” Phil. Trans. R. Soc. A, 332(1), pp. 149–167. [CrossRef]
Soliman, M. S. , and Thompson, J. M. T. , 1991, “ Transient and Steady State Analysis of Capsize Phenomena,” Appl. Ocean Res., 13(2), pp. 82–92. [CrossRef]
Soliman, M. S. , and Thompson, J. M. T. , 1992, “ Global Dynamics Underlying Sharp Basin Erosion in Nonlinear Driven Oscillators,” Phys. Rev. A, 45(6), pp. 3425–3431. [CrossRef] [PubMed]
Infeld, E. , Lenkowska, T. , and Thompson, J. M. T. , 1993, “ Erosion of the Basin of Stability of a Floating Body as Caused by Dam Breaking,” Phys. Fluids, 5(10), pp. 2315–2316. [CrossRef]
Alsaleem, F. M. , Younis, M. I. , and Ruzziconi, L. , 2010, “ An Experimental and Theoretical Investigation of Dynamical Pull-In in MEMS Resonators Actuated Electrostatically,” J. Microelectromech. Syst., 19(4), pp. 794–806. [CrossRef]
Ruzziconi, L. , Younis, M. I. , and Lenci, S. , 2013, “ Dynamical Integrity for Interpreting Experimental Data and Ensuring Safety in Electrostatic MEMS,” IUTAM Symposium on Nonlinear Dynamics for Advanced Technologies and Engineering Design (IUTAM Bookseries), Vol. 32, M. Wiercigroch and G. Rega , eds., Springer, Berlin, pp. 249–261.
Ruzziconi, L. , Younis, M. I. , and Lenci, S. , 2013, “ An Electrically Actuated Imperfect Microbeam: Dynamical Integrity for Interpreting and Predicting the Device Response,” Meccanica, 48(7), pp. 1761–1775. [CrossRef]
Alsaleem, F. , and Younis, M. I. , 2011, “ Integrity Analysis of Electrically Actuated Resonators With Delayed Feedback Controller,” ASME J. Dyn. Syst. Meas. Control, 133(3), p. 031011. [CrossRef]
Ruzziconi, L. , Lenci, S. , and Younis, M. I. , 2013, “ An Imperfect Microbeam Under an Axial Load and Electric Excitation: Nonlinear Phenomena and Dynamical Integrity,” Int. J. Bifurcation Chaos, 23(02), p. 1350026. [CrossRef]
Lenci, S. , and Rega, G. , 1998, “ Controlling Nonlinear Dynamics in a Two-Well Impact System. I. Attractors and Bifurcation Scenario Under Symmetric Excitations,” Int. J. Bifurcation Chaos, 8(12), pp. 2387–2408. [CrossRef]
Lenci, S. , and Rega, G. , 2005, “ Computational Nonlinear Dynamics and Optimal Control/Anti-Control of a Rocking Block,” Multibody Dynamics, ECCOMAS Thematic Conference, Madrid, Spain, June 21–24.
de Freitas, M. S. T. , Viana, R. L. , and Grebogi, C. , 2003, “ Erosion of the Safe Basin for the Transversal Oscillations of a Suspension Bridge,” Chaos, Solitons Fractals, 18(4), pp. 829–841. [CrossRef]
Silva, F. M. A. , Gonçalves, P. B. , and del Prado, Z. J. G. N. , 2011, “ An Alternative Procedure for the Non-Linear Vibration Analysis of Fluid-Filled Cylindrical Shells,” Nonlinear Dyn, 66(3), pp. 303–333. [CrossRef]
Nayfeh, A. H. , and Mook, D. T. , 1979, Nonlinear Oscillations, Wiley, New York.
Wiercigroch, M. , 2010, “ A New Concept of Energy Extraction From Waves Via Parametric Pendulor,” UK Patent Application (pending).
Nandakumar, K. , Wiercigroch, M. , and Chatterjee, A. , 2012, “ Optimum Energy Extraction From Rotational Motion in a Parametrically Excited Pendulum,” Mech. Res. Commun., 43, pp. 7–14. [CrossRef]
Gendelman, O. , 2001, “ Transition of Energy to a Nonlinear Localized Mode in a Highly Asymmetric System of Two Oscillators,” Nonlinear Dyn., 25(1–3), pp. 237–253. [CrossRef]
Soliman, M. S. , 1994, “ Suppression of Steady State Bifurcations and Premature Fractal Basin Erosion in Nonlinear Systems Subjected to Combined External and Parametric Excitations,” Chaos, Solitons Fractals, 4(10), pp. 1871–1882. [CrossRef]
Rega, G. , and Lenci, S. , 2010, “ Recent Advances in Control of Complex Dynamics in Mechanical and Structural Systems,” Recent Progress in Controlling Chaos, M. A. F. Sanjuan and C. Grebogi , eds., World Scientific, Singapore, pp. 189–237.
Yagasaki, K. , 2010, “ New Control Methodology of Microcantilevers in Atomic Force Microscopy,” Phys. Lett. A, 375(1), pp. 23–28. [CrossRef]
Settimi, V. , Gottlieb, O. , and Rega, G. , 2015, “ Asymptotic Analysis of a Noncontact AFM Microcantilever Sensor With External Feedback Control,” Nonlinear Dyn., 79(4), pp. 2675–2698. [CrossRef]
Lenci, S. , and Rega, G. , 2004, “ A Unified Control Framework of the Nonregular Dynamics of Mechanical Oscillators,” J. Sound Vib., 278, pp. 1051–1080. [CrossRef]
Chacon, R. , 2005, Control of Homoclinic Chaos by Weak Periodic Perturbations (Series Nonlinear Science, Series A), Vol. 55, World Scientific, Singapore.
Yagasaki, K. , 2013, “ Nonlinear Dynamics and Bifurcations in External Feedback Control of Microcantilevers in Atomic Force Microscopy,” Commun. Nonlinear Sci. Numer. Simul., 18(10), pp. 2926–2943. [CrossRef]
Settimi, V. , and Rega, G. , “ Influence of a Locally-Tailored External Feedback Control on the Overall Dynamics of a Non-Contact AFM Model,” Int. J. Nonlinear Mech. (in press).
Lenci, S. , and Rega, G. , 2003, “ Optimal Numerical Control of Single-Well to Cross-Well Chaos Transition in Mechanical Systems,” Chaos, Solitons Fractals, 15(1), pp. 173–186. [CrossRef]
Settimi, V. , Rega, G. , and Lenci, S. , 2015, “ Analytical and Numerical Control of Global Bifurcations in a Noncontact Atomic Force Microcantilever,” IUTAM Symposium on Analytical Methods in Nonlinear Dynamics, Frankfurt, Germany, July 6–9, pp. 7–8.
Goncalves, P. B. , Orlando, D. , Rega, G. , and Lenci, S. , 2013, “ Nonlinear Dynamics and Instability as Important Design Concerns for a Guyed Mast,” IUTAM Symposium on Nonlinear Dynamics for Advanced Technologies and Engineering Design (IUTAM Bookseries), Vol. 32, M. Wiercigroch and G. Rega , eds., Springer, Berlin, pp. 223–234.
Orlando, D. , 2010, “ Nonlinear Dynamics, Instability and Control of Structural Systems With Modal Interaction,” Ph.D. thesis, Pontifícia Universidade Católica do Rio de Janeiro, PUC-Rio, Rio de Janeiro, Brazil (in Portuguese).
Orlando, D. , Gonçalves, P. B. , Rega, G. , and Lenci, S. , 2011, “ Non-Linear Dynamics and Sensitivity to Imperfections in Augusti's Model,” J. Mech. Mater. Struct., 6(7–8), pp. 1065–1078. [CrossRef]
Lenci, S. , and Rega, G. , 1998, “ A Procedure for Reducing the Chaotic Response Region in an Impact Mechanical System,” Nonlinear Dyn., 15(4), pp. 391–409. [CrossRef]
Rega, G. , and Lenci, S. , 2003, “ Bifurcations and Chaos in Single-D.o.f. Mechanical Systems: Exploiting Nonlinear Dynamics Properties for Their Control,” Recent Research Developments in Structural Dynamics, A. Luongo , ed., Transworld Research Network, Trivandrum, India, pp. 331–369.
Lenci, S. , and Rega, G. , 2003, “ Optimal Control of Nonregular Dynamics in a Duffing Oscillator,” Nonlinear Dyn., 33(1), pp. 71–86. [CrossRef]
Rega, G. , Lenci, S. , and Thompson, J. M. T. , 2010, “ Controlling Chaos: The OGY Method, Its Use in Mechanics, and an Alternative Unified Framework for Control of Non-Regular Dynamics,” Nonlinear Dynamics and Chaos: Advances and Perspectives, M. Thiel , J. Kurths , C. Romano , A. Moura , and G. Károlyi , eds., Springer, Berlin, pp. 211–269.
Lenci, S. , and Rega, G. , 2004, “ Global Optimal Control and System-Dependent Solutions in the Hardening Helmholtz–Duffing Oscillator,” Chaos, Solitons Fractals, 21(5), pp. 1031–1046. [CrossRef]
Virgin, L. N. , 2000, Introduction to Experimental Nonlinear Dynamics: A Case Study in Mechanical Vibration, Cambridge University Press, Cambridge, UK.
Amabili, M. , 2008, Nonlinear Vibrations and Stability of Shells and Plates, Cambridge University Press, Cambridge, UK.
Ruzziconi, L. , Younis, M. I. , and Lenci, S. , 2013, “ Parameter Identification of an Electrically Actuated Imperfect Microbeam,” Int. J. Nonlinear Mech., 57, pp. 208–219. [CrossRef]
Younis, M. I. , 2011, MEMS Linear and Nonlinear Statics and Dynamics, Springer, New York.
Lenci, S. , and Rega, G. , 2011, “ Experimental Versus Theoretical Robustness of Rotating Solutions in a Parametrically Excited Pendulum: A Dynamical Integrity Perspective,” Physica D, 240, pp. 814–824. [CrossRef]
Lenci, S. , Pavlovskaia, E. , Rega, G. , and Wiercigroch, M. , 2008, “ Rotating Solutions and Stability of Parametric Pendulum by Perturbation Method,” J. Sound Vib., 310, pp. 243–259. [CrossRef]
Pavlovskaia, E. , Horton, B. , Wiercigroch, M. , Lenci, S. , and Rega, G. , 2012, “ Approximate Rotational Solutions of Pendulum Under Combined Vertical and Horizontal Excitation,” Int. J. Bifurcation Chaos, 22(5), p. 1250100. [CrossRef]
Lenci, S. , Brocchini, M. , and Lorenzoni, C. , 2012, “ Experimental Rotations of a Pendulum on Water Waves,” ASME J. Comput. Nonlinear Dyn., 7(1), p. 011007. [CrossRef]
Ruzziconi, L. , Bataineh, A. M. , Younis, M. I. , Cui, W. , and Lenci, S. , 2013, “ Nonlinear Dynamics of an Electrically Actuated Imperfect Microbeam Resonator: Experimental Investigation and Reduced-Order Model,” J. Micromech. Microeng., 23(7), p. 075012. [CrossRef]
Szemplinska-Stupnicka, W. , 1992, “ Cross-Well Chaos and Escape Phenomena in Driven Oscillators,” Nonlinear Dyn., 3(3), pp. 225–243. [CrossRef]
Thompson, J. M. T. , 1997, “ Designing Against Capsize in Beam Seas: Recent Advances and New Insights,” ASME Appl. Mech. Rev., 50(5), pp. 307–324. [CrossRef]
Wu, X. , Tao, L. , and Li, Y. , 2004, “ The Safe Basin Erosion of a Ship in Waves With Single Degree of Freedom,” 15th Australasian Fluid Mechanical Conference, Sydney, Australia, Dec. 13–17, Paper No. AFMC00029.
Hornstein, S. , and Gottlieb, O. , 2008, “ Nonlinear Dynamics, Stability and Control of the Scan Process in Noncontacting Atomic Force Microscopy,” Nonlinear Dyn., 54, pp. 93–122. [CrossRef]
Ott, E. , Grebogi, C. , and Yorke, J. A. , 1990, “ Controlling Chaos,” Phys. Rev. Lett. E, 64(11), pp. 1196–1199. [CrossRef]
Pyragas, K. , 1992, “ Continuous Control of Chaos by Self-Controlling Feedback,” Phys. Lett. A, 170(6), pp. 421–428. [CrossRef]
Settimi, V. , 2013, “ Bifurcation Scenarios, Dynamical Integrity and Control of a Noncontact Atomic Force Microscope,” Ph.D. thesis, Sapienza University of Rome, Rome, Italy.
Settimi, V. , and Rega, G. , 2015, “ Dynamical Integrity of Noncontact AFM With External Feedback Control,” Euromech Colloquium on Stability and Control of Nonlinear Vibrating Systems, Sperlonga, Italy, May 24–28, pp. 51–52.
Eason, R. , and Dick, A. J. , 2014, “ A Parallelized Multi-Degrees-of-Freedom Cell Map Method,” Nonlinear Dyn., 77(3), pp. 467–479. [CrossRef]
Belardinelli, P. , and Lenci, S. , “ A First Parallel Programming Approach in Basins of Attraction Computation,” Int. J. Nonlinear Mech. (submitted).
Gan, C. B. , Lu, Q. S. , and Huang, K. L. , 1998, “ Nonstationary Effects on Safe Basins of a Forced Softening Duffing Oscillator,” Acta Mech. Solida Sin., 11, pp. 253–260.
Hong, L. , and Sun, J. Q. , 2006, “ Bifurcations of Forced Oscillators With Fuzzy Uncertainties by the Generalized Cell Mapping Method,” Chaos, Solitons Fractals, 27(4), pp. 895–904. [CrossRef]
Gan, C. B. , 2005, “ Noise-Induced Chaos and Basin Erosion in Softening Duffing Oscillator,” Chaos, Solitons Fractals, 25(5), pp. 1069–1081. [CrossRef]
Silva, F. M. A. , Gonçalves, P. B. , and del Prado, Z. J. G. N. , 2013, “ Influence of Physical and Geometrical System Parameters Uncertainties on the Nonlinear Oscillations of Cylindrical Shells,” J. Braz. Soc. Mech. Sci. Eng., 34, pp. 622–632.
Brazão, A. F. , Silva, F. M. A. , del Prado, Z. J. G. N. , and Gonçalves, P. B. , 2014, “ Influence of Physical Parameters and Geometrical Imperfection Uncertainties on the Nonlinear Vibrations of Axially Excited Cylindrical Shells,” 9th International Conference on Structural Dynamics, Porto, Portugal, June 30–July 2, pp. 2043–2048.
van Campen, D. H. , van de Vorst, E. L. B. , van der Spek, J. A. W. , and de Kraker, A. , 1995, “ Dynamics of a Multi-Dof Beam System With Discontinuous Support,” Nonlinear Dyn., 8(4), pp. 453–466. [CrossRef]
Kreuzer, E. , and Lagemann, B. , 1996, “ Cell Mapping for Multi-Degree-of-Freedom-Systems Parallel Computing in Nonlinear Dynamics,” Chaos, Solitons Fractals, 7(10), pp. 1683–1691. [CrossRef]
Marszal, M. , Jankowski, K. , Perlikowski, P. , and Kapitaniak, T. , 2014, “ Bifurcations of Oscillatory and Rotational Solutions of Double Pendulum With Parametric Vertical Excitation,” Math. Probl. Eng., 2014, p. 892793. [CrossRef]
Carvalho, E. C. , Gonçalves, P. B. , Rega, G. , and del Prado, Z. J. G. N. , 2013, “ Influence of Axial Loads on the Nonplanar Vibrations of Cantilever Beams,” Shock Vib., 20(6), pp. 1073–1092. [CrossRef]
Carvalho, E. C. , Gonçalves, P. B. , Rega, G. , and del Prado, Z. J. G. N. , 2014, “ Nonlinear Nonplanar Vibration of a Functionally Graded Box Beam,” Meccanica, 49(8), pp. 1795–1819. [CrossRef]
Xu, W. , Sun, C. , Sun, J. , and He, Q. , 2013, “ Development and Study on Cell Mapping Methods,” Adv. Mech., 43, pp. 91–100.
Wiercigroch, M. , and Pavlovskaia, E. , 2008, “ Non-Linear Dynamics of Engineering Systems,” Int. J. Nonlinear Mech., 43(6), pp. 459–461. [CrossRef]
Wiercigroch, M. , and Rega, G. , 2013, “ Introduction to NDATED,” IUTAM Symposium on Nonlinear Dynamics for Advanced Technologies and Engineering Design (IUTAM Bookseries), Vol. 32, M. Wiercigroch and G. Rega , eds., Springer, Berlin, pp. v–viii.

Figures

Grahic Jump Location
Fig. 1

(a) Asymmetrically constrained inverted pendulum under coexisting static axial load p and lateral dynamic excitation q1 (P = p; Q = q + q1 sin(ωt)) [40]. (b) Integrity surface for a fixed value of static imperfection q, with the robustness (for varying p) and erosion (for varying q1) profiles highlighting meaningful excitation coupling effects [30].

Grahic Jump Location
Fig. 2

(a) Model of guyed mast with equal (k1 = k2 = k3) springs, under horizontal excitation F at the base in anyone of the three symmetry planes [35]. (b) Comparison of GIM erosion profiles for perfect and imperfect models, under either harmonic or control periodic (harmonic + one superharmonic) excitation [35], with also the homoclinic bifurcation thresholds (vertical lines) triggering the erosion.

Grahic Jump Location
Fig. 3

Parametrically excited pendulum (sole vertical support motion). (a) Response chart in the excitation parameters plane (ω, p): numerical boundaries of the region of existence of period 1 rotations (lines) and experimentally observed rotations (triangles). (b) Relevant IF and LIM profiles for h = 0.015 and ω = 1.3 (subfigures taken in part from Ref. [1]).

Grahic Jump Location
Fig. 4

MEMS capacitive accelerometer with harmonic excitation close to primary resonance (Ωres = 192.5 Hz). (a) Response chart, with theoretical regions of existence of nonresonant/resonant attractors and of inevitable escape (delimited by solid lines), along with thresholds of experimental escape (dots) against level curves (dashed) of IF. (b) IF (solid) and LIM (dashed) profiles of nonresonant (left) and resonant (right) attractors at VAC = 15 V (lines with squares) and VAC = 30 V (lines with triangles) (subfigures taken in part from Ref. [1]).

Grahic Jump Location
Fig. 5

Noncontact AFM. (a) Theoretical global (solid) and local (dashed) stability boundaries obtained separately with increasing parametric (the upper in the left part of the picture) and external (the lower in the left part of the picture) excitation in a frequency region encompassing fundamental (primary) and principal (subharmonic) resonances [32]. (b) Around fundamental resonance of parametric excitation, comparison between theoretical, global (bd) and local (ni), stability thresholds, and practical stability thresholds, the latter corresponding to level curves of possibly acceptable residual integrity depending on a priori defined design targets [32].

Grahic Jump Location
Fig. 6

Noncontact AFM. Bounded resonant (PIH) and nonresonant (PIL) solutions, and global escape threshold for uncontrolled (a) and controlled (b) systems under scan excitation [71]. The latter also accounts for the occurrence of TR and T bifurcations, besides the SN and PD ones also occurring for the former.

Grahic Jump Location
Fig. 7

Noncontact AFM. Global escape thresholds for controlled (the thin deep tongues) and uncontrolled systems under scan excitation [71]. Dark gray area is the stability region of both controlled and uncontrolled system and light gray area is stable only for uncontrolled system.

Grahic Jump Location
Fig. 8

Schematic safety chart for a system subjected to a harmonic excitation

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