Global Nonlinear Dynamics for Engineering Design and System Safety

During the explosion of interest in applied nonlinear dynamics and chaos in the 1980s, a key concept was quickly seen to be the concept of dynamical integrity which was required to cope with, for example, the erosion of basins of attraction by fractal incursions. Articles by Thompson and Soliman laid down the fundamental ideas, but major contributions by Rega and Lenci established this integrity as a central issue in the design of (for example) structures in their inevitable dynamic environment. They extended and developed the various integrity measures and pioneered ways of controlling the basin erosion phenomenon, applying the ideas to a wide range of mechanical problems. The present paper offers a review of this progress, highlighting key conceptual ideas and some of the more interesting applications.

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.

The present chapter highlights the importance of the dynamical integrity theory for micro and nanoapplications. Three case-studies of devices at different scales are presented (a capacitive accelerometer, a microbeam-based micro-electro-mechanical system, and a single-walled slacked carbon nanotube) and different issues commonly addressed in the engineering design are examined via dynamical integrity concepts. The iso-integrity curves are observed to follow exactly the experimental data. They are able to detect the parameter range where each attractor can be reliably observed in practice and where, instead, becomes vulnerable. Also, they may be used to simulate and predict the expected dynamics under different (smaller or larger) experimental disturbances. While referring to particular case-studies, we show the relevance of the dynamical integrity analysis for the engineering design of a mechanical system, in order to operate it in safe conditions, according to the desired outcome and depending on the expected disturbances.

The nonlinear dynamics of two archetypal structural systems exhibiting interactive modal post-buckling behavior is addressed, the discrete Augusti’s model and a reduced-order model of the axially loaded cylindrical shell. The uncoupled models exhibit a stable post-buckling response. However, the modal interaction leads to unstable post-buckling paths that entail a complex dynamic behavior and imperfection sensitivity, with a marked influence on the dynamic integrity and safety. Perfect and imperfect Augusti’s models are investigated in terms of static buckling, linear vibrations, nonlinear normal modes, local and global nonlinear response to harmonic excitation, dynamic integrity, control of global bifurcations aimed at increasing the load carrying capacity. Then, as an example of a continuous system exhibiting strong modal coupling and interaction, a two-degree-of-freedom model of the thin-walled cylindrical shell is investigated in terms of global behavior and dynamic integrity. The influence of uncertainties on the nonlinear response and dynamic integrity is also shortly addressed. The chapter shows how a judicious use of the tools of nonlinear dynamics sheds light on the actual safety of structural systems liable to unstable buckling under static and dynamic loads.

The role of local and global dynamics to assess a system robustness and actual safety in operating conditions is investigated, by also studying the effect of different local and global control techniques on the nonlinear behavior of a noncontact AFM. First, the nonlinear dynamical behavior of a single-mode model of noncontact AFM is analyzed in terms of stability of the main periodic solutions, as well as attractors robustness and basins integrity. To the same AFM model, an external feedback control is inserted during its nonlinear continuum formulation, with the aim to keep the system response to an operationally suitable one. The dynamical analysis of the controlled system is developed to investigate and verify the effects of control into the system overall behavior, which could be unexpectedly influenced by the local nature of the control technique. A different control technique is finally applied to the AFM model, acting on global bifurcation events to obtain an enlargement of the systems safe region in parameters space. The analytical procedure, based on Melnikov method, is applied to the homoclinic bifurcation involving the system hilltop saddle, and its practical effects as regards possibly increasing the system overall robustness are numerically investigated by means of a dynamical integrity analysis. Then, a fully numerical procedure is implemented to possibly control global bifurcations involving generic saddles. The method proves to succeed in delaying the drop down of the erosion profile, thus increasing the overall robustness of the system during operating conditions.

This chapter discusses recent applications and algorithm developments of the cell mapping methods, which were created by C. S. Hsu in 1980s for global analysis of nonlinear dynamical systems. Such systems can have multiple steady-state responses including equilibrium states, periodic motions, chaotic attractors as well as domains of attraction of these steady-state responses. Without the cell mapping methods, these dynamical responses would have been far more difficult to obtain. Since the creation of them, the cell mapping methods have enjoyed attention from the research communities. New extensions of the methods and new algorithms including parallel computing have been developed in the past few decades. The cell mapping methods have also been applied to global analysis and control design of deterministic, stochastic and fuzzy dynamical systems. Representative examples of new applications are presented in this chapter.