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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

Figures

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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.

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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].

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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].

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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.

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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.

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Fig. 8

Schematic safety chart for a system subjected to a harmonic excitation

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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]).

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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]).

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