Dynamical Integrity: A Novel Paradigm for Evaluating Load Carrying Capacity

The chapter offers an overview of the effects of the research advancements in nonlinear dynamics on the evaluation of system safety. The achievements developed over the last 30 years entailed a substantial change of perspective. After recalling the outstanding contributions due to Euler and Koiter, we focus on Thompson’s intuition of global safety. This concept represents a paramount enhancement, full of theoretical and practical implications. Its relevance as a novel paradigm for evaluating the load carrying capacity of a system is highlighted. Making reference to a variety of different case studies, we emphasize that global safety has induced a deep development in the analysis, control, and design of mechanical and structural systems. Recent results are presented, and the possibility to implement effective dedicated control procedures based on global safety concepts is explored. We stress the importance of global safety for valorizing all the potential of the system and guaranteeing superior targets. The very general character of the dynamical integrity approach to design is highlighted.

[1]  Laura Ruzziconi,et al.  The dynamical integrity concept for interpreting/ predicting experimental behaviour: from macro- to nano-mechanics , 2013, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[2]  Y. Kuznetsov.  Elements of Applied Bifurcation Theory , 2023, Applied Mathematical Sciences.

[3]  Marian Wiercigroch,et al.  Approximate analytical solutions for oscillatory and rotational motion of a parametric pendulum , 2006 .

[4]  Remco I. Leine.  The historical development of classical stability concepts: Lagrange, Poisson and Lyapunov stability , 2009 .

[5]  Mohammad I. Younis,et al.  Interpreting and Predicting Experimental Responses of Micro- and Nano-Devices via Dynamical Integrity , 2018, Global Nonlinear Dynamics for Engineering Design and System Safety.

[6]  J. M. T. Thompson,et al.  The transient capsize diagram ― A new method of quantifying stability in waves , 1991 .

[7]  Mohammad I. Younis,et al.  Multistability in an electrically actuated carbon nanotube: a dynamical integrity perspective , 2013 .

[8]  D. H. van Campen,et al.  Dynamics of a multi-DOF beam system with discontinuous support , 1995 .

[9]  Maurizio Brocchini,et al.  Experimental Rotations of a Pendulum on Water Waves , 2012 .

[10]  Ling Hong,et al.  Bifurcations of a Forced Duffing oscillator in the Presence of Fuzzy Noise by the Generalized Cell Mapping Method , 2006, Int. J. Bifurc. Chaos.

[11]  J. M. T. Thompson,et al.  Transient and steady state analysis of capsize phenomena , 1991 .

[12]  J. Thompson.  Designing Against Capsize in Beam Seas: Recent Advances and New Insights , 1997 .

[13]  Wanda Szemplińska-Stupnicka,et al.  The analytical predictive criteria for chaos and escape in nonlinear oscillators: A survey , 1995 .

[14]  Stefano Lenci,et al.  Forced Harmonic Vibration in a Duffing Oscillator with Negative Linear Stiffness and Linear Viscous Damping , 2011 .

[15]  Edwin Kreuzer,et al.  Cell mappings for multi-degree-of-freedom-systems — Parallel computing in nonlinear dynamics☆ , 1996 .

[16]  H. Troger,et al.  Nonlinear stability and bifurcation theory , 1991 .

[17]  Stefano Lenci,et al.  Global dynamics and integrity of a two-dof model of a parametrically excited cylindrical shell , 2011 .

[18]  Stefano Lenci,et al.  Load carrying capacity of systems within a global safety perspective. Part II. Attractor/basin integrity under dynamic excitations , 2011 .

[19]  Z. Bažant,et al.  Stability Of Structures , 1991 .

[20]  Stefano Lenci,et al.  Control of pull-in dynamics in a nonlinear thermoelastic electrically actuated microbeam , 2006 .

[21]  J. M. T. Thompson,et al.  Fractal Control Boundaries of Driven Oscillators and Their Relevance to Safe Engineering Design , 1991 .

[22]  Stefano Lenci,et al.  A first parallel programming approach in basins of attraction computation , 2016 .

[23]  Ling Hong,et al.  Global Analysis of Nonlinear Dynamical Systems , 2019 .

[24]  Jian-Qiao Sun,et al.  Effects of small random uncertainties on non-linear systems studied by the generalized cell mapping method , 1991 .

[25]  Stefano Lenci,et al.  RECENT ADVANCES IN CONTROL OF COMPLEX DYNAMICS IN MECHANICAL AND STRUCTURAL SYSTEMS , 2010 .

[26]  G. Housner.  The behavior of inverted pendulum structures during earthquakes , 1963 .

[27]  Jian-Qiao Sun,et al.  Control of Nonlinear Dynamic Systems with the Cell Mapping Method , 2012, EVOLVE.

[28]  Stefano Lenci,et al.  A Global Dynamics Perspective for System Safety From Macro- to Nanomechanics: Analysis, Control, and Design Engineering , 2015 .

[29]  Giuseppe Rega,et al.  Exploiting Global Dynamics of a Noncontact Atomic Force Microcantilever to Enhance Its Dynamical Robustness via Numerical Control , 2016, Int. J. Bifurc. Chaos.

[30]  A. Nayfeh,et al.  Applied nonlinear dynamics : analytical, computational, and experimental methods , 1995 .

[31]  Marian Wiercigroch,et al.  IUTAM Symposium on Nonlinear Dynamics for Advanced Technologies and Engineering Design , 2013 .

[32]  Michael F. Shlesinger,et al.  Chaotic and Fractal Dynamics: An Introduction for Applied Scientists and Engineers , 1993 .

[33]  Mohammad I. Younis,et al.  Dynamical Integrity for Interpreting Experimental Data and Ensuring Safety in Electrostatic MEMS , 2013 .

[34]  Paulo B. Gonçalves,et al.  Influence of Uncertainties on the Dynamic Buckling Loads of Structures Liable to Asymmetric Postbuckling Behavior , 2008 .

[35]  Stefano Lenci,et al.  Seamless variation of isometric and anisometric dynamical integrity measures in basins's erosion , 2018, Commun. Nonlinear Sci. Numer. Simul..

[36]  Angelo Luongo,et al.  Stability, Bifurcation and Postcritical Behaviour of Elastic Structures , 1992 .

[37]  Stefano Lenci,et al.  A Procedure for Reducing the Chaotic Response Region in an Impact Mechanical System , 1998 .

[38]  Weili Cui,et al.  Nonlinear dynamics of an electrically actuated imperfect microbeam resonator: experimental investigation and reduced-order modeling , 2013 .

[39]  Stefano Lenci,et al.  A dynamical systems approach to the overturning of rocking blocks , 2006 .

[40]  M. Younis,et al.  Global investigation of the nonlinear dynamics of carbon nanotubes , 2017 .

[41]  M. Novak.  Aeroelastic Galloping of Prismatic Bodies , 1969 .

[42]  John Milne.  EXPERIMENTS IN OBSERVATIONAL SEISMOLOGY , 1881 .

[43]  Giles W Hunt,et al.  A general theory of elastic stability , 1973 .

[44]  Stefano Lenci,et al.  Controlling Chaos: The OGY Method, Its Use in Mechanics, and an Alternative Unified Framework for Control of Non-regular Dynamics , 2010 .

[45]  S. Lenci,et al.  Optimal control and anti-control of the nonlinear dynamics of a rigid block , 2006, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[46]  Soliman,et al.  Global dynamics underlying sharp basin erosion in nonlinear driven oscillators. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[47]  Chunbiao Gan,et al.  Studies on structural safety in stochastically excited Duffing oscillator with double potential wells , 2007 .

[48]  M. Younis,et al.  An Experimental and Theoretical Investigation of Dynamic Pull-In in MEMS Resonators Actuated Electrostatically , 2010, Journal of Microelectromechanical Systems.

[49]  J. R. de Souza Junior,et al.  An Investigation into Mechanisms of Loss of Safe Basins in a 2 D.O.F. Nonlinear Oscillator , 2002 .

[50]  Ekaterina Pavlovskaia,et al.  Non-linear dynamics of engineering systems , 2008 .

[51]  Giuseppe Rega,et al.  Bifurcation, response scenarios and dynamic integrity in a single-mode model of noncontact atomic force microscopy , 2013 .

[52]  J. M. T. Thompson,et al.  BASIN EROSION IN THE TWIN-WELL DUFFING OSCILLATOR: TWO DISTINCT BIFURCATION SCENARIOS , 1992 .

[53]  J. M. T. Thompson,et al.  Chaotic Phenomena Triggering the Escape from a Potential Well , 1991 .

[54]  Mohammad I. Younis,et al.  An Imperfect microbeam under an axial Load and Electric excitation: nonlinear Phenomena and Dynamical Integrity , 2013, Int. J. Bifurc. Chaos.

[55]  H. M. Chiu,et al.  Global analysis of a system with multiple responses including a strange attractor , 1987 .

[56]  Herbert A. Mang,et al.  Hilltop buckling as the A and O in sensitivity analysis of the initial postbuckling behavior of elastic structures , 2009 .

[57]  Giuseppe Rega,et al.  A DYNAMICAL SYSTEMS ANALYSIS OF THE OVERTURNING OF RIGID BLOCKS , 2004 .

[58]  W. T. Koiter.  Over de stabiliteit van het elastisch evenwicht , 1945 .

[59]  Ekaterina Pavlovskaia,et al.  Dynamic interactions between parametric pendulum and electro‐dynamical shaker , 2007 .

[60]  Frederico M. A. Silva,et al.  Nonlinear Dynamics, Safety, and Control of Structures Liable to Interactive Unstable Buckling , 2018, Global Nonlinear Dynamics for Engineering Design and System Safety.

[61]  Stefano Lenci,et al.  Controlling nonlinear dynamics in a two-well impact system. II. Attractors and bifurcation scenario under unsymmetric optimal excitation , 1998 .

[62]  Stefano Lenci,et al.  Identifying, evaluating, and controlling dynamical integrity measures in non-linear mechanical oscillators , 2005 .

[63]  Stefano Lenci,et al.  An efficient parallel implementation of cell mapping methods for MDOF systems , 2016, Nonlinear Dynamics.

[64]  J. M. T. Thompson,et al.  Ship stability criteria based on chaotic transients from incursive fractals , 1990, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[65]  Frederico M. A. Silva,et al.  The influence of uncertainties and random noise on the dynamic integrity analysis of a system liable to unstable buckling , 2015 .

[66]  M. Younis.  MEMS Linear and Nonlinear Statics and Dynamics , 2011 .

[67]  Stefano Lenci,et al.  Optimal numerical control of single-well to cross-well chaos transition in mechanical systems , 2003 .

[68]  Paulo B. Gonçalves,et al.  CHAOTIC BEHAVIOR RESULTING IN TRANSIENT AND STEADY STATE INSTABILITIES OF PRESSURE-LOADED SHALLOW SPHERICAL SHELLS , 2003 .

[69]  J. M. T. Thompson,et al.  Stochastic penetration of smooth and fractal basin boundaries under noise excitation , 1990 .

[70]  Stefano Lenci,et al.  Global optimal control and system-dependent solutions in the hardening Helmholtz–Duffing oscillator , 2004 .

[71]  STEFANO LENCI,et al.  Heteroclinic bifurcations and Optimal Control in the Nonlinear Rocking Dynamics of Generic and Slender Rigid Blocks , 2005, Int. J. Bifurc. Chaos.

[72]  Frederico M. A. Silva,et al.  Global stability analysis of parametrically excited cylindrical shells through the evolution of basin boundaries , 2007 .

[73]  Paulo B. Gonçalves,et al.  Influence of physical and geometrical system parameters uncertainties on the nonlinear oscillations of cylindrical shells , 2012 .

[74]  Giuseppe Rega,et al.  Asymptotic analysis of a noncontact AFM microcantilever sensor with external feedback control , 2015 .

[75]  Stefano Lenci,et al.  Experimental versus theoretical robustness of rotating solutions in a parametrically excited pendulum: A dynamical integrity perspective , 2011 .

[76]  Wanda Szemplinska-Stupnicka,et al.  Steady states in the twin-well potential oscillator: Computer simulations and approximate analytical studies. , 1993, Chaos.

[77]  Paulo B. Gonçalves,et al.  Nonlinear Oscillations and Stability of Parametrically Excited Cylindrical Shells , 2002 .

[78]  Mohammad I. Younis,et al.  An electrically actuated imperfect microbeam: Dynamical integrity for interpreting and predicting the device response , 2013 .

[79]  Andrew J. Dick,et al.  A parallelized multi-degrees-of-freedom cell mapping method , 2014 .

[80]  A. Zubrzycki,et al.  The Global bifurcations that lead to Transient tumbling Chaos in a Parametrically Driven Pendulum , 2000, Int. J. Bifurc. Chaos.

[81]  C. Hsu,et al.  Cell-To-Cell Mapping A Method of Global Analysis for Nonlinear Systems , 1987 .

[82]  Giuseppe Rega,et al.  Local Versus Global Dynamics and Control of an AFM Model in a Safety Perspective , 2018, Global Nonlinear Dynamics for Engineering Design and System Safety.

[83]  A. M. A. Heijden.  W. T. Koiter-s Elastic Stability of Solids and Structures , 2012 .

[84]  Paulo B. Gonçalves,et al.  Low-Dimensional Galerkin Models for Nonlinear Vibration and Instability Analysis of Cylindrical Shells , 2005 .

[85]  Claude-Henri Lamarque,et al.  Bifurcation and Chaos in Nonsmooth Mechanical Systems , 2003 .

[86]  F. Moon.  Experiments on Chaotic Motions of a Forced Nonlinear Oscillator: Strange Attractors , 1980 .

[87]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[88]  Irving J. Oppenheim,et al.  The masonry arch as a four‐link mechanism under base motion , 1992 .

[89]  Stefano Lenci,et al.  Controlling practical stability and safety of mechanical systems by exploiting chaos properties. , 2012, Chaos.

[90]  A. M. Lyapunov.  The general problem of the stability of motion , 1992 .

[91]  J. Yorke,et al.  Crises, sudden changes in chaotic attractors, and transient chaos , 1983 .

[92]  Controlling Nonlinear Dynamics in a Two-Well Impact System. , 2011 .

[93]  Stefano Lenci,et al.  Optimal Control of Homoclinic Bifurcation: Theoretical Treatment and Practical Reduction of Safe Basin Erosion in the Helmholtz Oscillator , 2003 .

[94]  C. S. Hsu,et al.  Cell-to-Cell Mapping , 1987 .

[95]  S. Wiggins.  Introduction to Applied Nonlinear Dynamical Systems and Chaos , 1989 .

[96]  Stefano Lenci,et al.  A unified control framework of the non-regular dynamics of mechanical oscillators , 2004 .

[97]  B. Koch,et al.  Subharmonic and homoclinic bifurcations in a parametrically forced pendulum , 1985 .

[98]  Giuseppe Rega,et al.  Influence of a locally-tailored external feedback control on the overall dynamics of a non-contact AFM model , 2016 .

[99]  Mohammad I. Younis,et al.  An Efficient Reduced-Order Model to Investigate the Behavior of an Imperfect Microbeam Under Axial Load and Electric Excitation , 2013 .

[100]  Stefano Lenci,et al.  Optimal Control of Nonregular Dynamics in a Duffing Oscillator , 2003 .

[101]  Stefano Lenci,et al.  Competing Dynamic Solutions in a Parametrically Excited Pendulum: Attractor Robustness and Basin Integrity , 2007 .

[102]  Paul Kirkpatrick,et al.  Seismic measurements by the overthrow of columns , 1927 .

[103]  J. Michael T. Thompson,et al.  Dynamical Integrity: Three Decades of Progress from Macro to Nanomechanics , 2018, Global Nonlinear Dynamics for Engineering Design and System Safety.

[104]  Aik‐Siong Koh.  Rocking of rigid blocks on randomly shaking foundations , 1986 .

[105]  Mohammad I. Younis,et al.  Parameter identification of an electrically actuated imperfect microbeam , 2013 .

[106]  Giuseppe Rega,et al.  Global dynamics and integrity in noncontacting atomic force microscopy with feedback control , 2016 .

[107]  J. M. T. Thompson,et al.  Integrity measures quantifying the erosion of smooth and fractal basins of attraction , 1989 .

[108]  Y. Ueda,et al.  Basin boundary metamorphoses in the canonical escape equation , 1989 .

[109]  Raymond H. Plaut,et al.  Fractal behavior of an asymmetric rigid block overturning due to harmonic motion of a tilted foundation , 1996 .

[110]  Stefano Lenci,et al.  Load carrying capacity of systems within a global safety perspective. Part I. Robustness of stable equilibria under imperfections , 2011 .

[111]  John Perry.  NOTES ON THE ROCKING OF A COLUMN , 1881 .

[112]  René Thom,et al.  Structural stability and morphogenesis , 1977, Pattern Recognit..

[113]  Paulo B. Gonçalves,et al.  Influence of Modal Coupling on the Nonlinear Dynamics of Augusti’s Model , 2009 .

[114]  Stefano Lenci,et al.  Dynamical Integrity and Control of Nonlinear Mechanical Oscillators , 2008 .

[115]  Fumio Yamazaki,et al.  Response of rigid body assemblies to dynamic excitation , 1995 .

[116]  S. Bishop,et al.  Zones of chaotic behaviour in the parametrically excited pendulum , 1996 .

[117]  Matthew P. Cartmell,et al.  Rotating orbits of a parametrically-excited pendulum , 2005 .

[118]  P. Wahi,et al.  Initiation and directional control of period-1 rotation for a parametric pendulum , 2016, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.