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Microcantilever at reference position (a) and in a generic configuration (b); lines A and B represent the reference positions of the microcantilever and the sample surface, respectively
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Dynamical integrity of a noncontact AFM model with external feedback control is investigated to evaluate the effects of such local control procedure on the erosion of the basins of attraction of the system bounded solutions. Two-dimensional cross sections of the five-dimensional basins of attraction have been systematically constructed, and the rel...
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Scanning quantum dot microscopy is a recently developed high-resolution microscopy technique that is based on atomic force microscopy and is capable of imaging the electrostatic potential of nanostructures like molecules or single atoms. Recently, it could be shown that it not only yields qualitatively but also quantitatively cutting edge images ev...
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... The first group refers to the Helmholtz [13] and two-well Duffing [14,27,29] oscillators, the rocking rigid block [36], the parametrically excited pendulum [37], the asymmetrically constrained inverted pendulum [38][39][40], the shallow [24] von Mises truss [41], the Augusti model [17,42], the guyed pendulum [42,43], a coupled linear oscillator and nonlinear tuned mass damper [44], a nonlinear oscillator excited by nonideal energy source [45], and a truss model liable to snap-through and lateral instability [46]. The second group considers reduced models of structures in macro-mechanics (suspension bridge [47], shallow spherical shell [48], cylindrical shells under different excitations [49][50][51][52][53], general membrane and shell structures [54], composite thinwalled columns [55]), and micro-/nano-mechanics (beam-based devices and capacitive accelerometers for MEMS [56][57][58][59][60][61], micro-plate pressure sensor [62], carbon nanotube [28], noncontact [63][64][65], and tapping [66] mode atomic force microscopy). In some of the considered systems, the capability of DI analysis to account for the generally detrimental effect of system's imperfections on its overall safety has been shown, too. ...
... In fact, the topography of noncontact AFMs is crucially associated with the tip-sample distance, which can be sensibly modified by slight changes in the initial position and/or velocity. Consequently, the overall performance of the local feedback control has to be assessed, too, via the investigation of the system dynamical integrity, i.e. the detection of the basins of attraction of the main system responses, together with the analysis and quantification of how the relevant erosion processes do evolve [64]. ...
... The black circle represents the IF measure of the safe basin U = 0.598, responsible for unstabilization of the controlled solution, produces the downfall of the profiles from 50% to 0%, which reflects on the evident packing of the corresponding iso-GIM curves of Fig. 17b and underlines the unreliability of the feedback control when the system works around these high-amplitude values. Note that this occurs at a frequency value x ¼ 0:9 somehow shifted from the most pathological range of resonance where, at x ¼ 0:8, an even more dramatic shift of the feedback erosion profiles towards lower excitation amplitudes does occur, with respect to the uncontrolled ones [64,79]. Indeed, the overall worsening of system practical stability in Fig. 17b due to the presence of control is especially meaningful around the natural resonance frequency, with also a shift in the lowest peak of each curve, which moves from the nonlinear resonance frequency to the natural one. ...
In about the last two decades, global nonlinear dynamics has been evolving in a revolutionary way, with the development of sophisticated techniques employing concepts/tools of dynamical systems, bifurcation, and chaos theory, and applications to a wide variety of mechanical/structural systems. The relevant achievements entail a substantial change of perspective in dealing with vibration problems and are ready to meaningfully affect the analysis, control, and design of systems at different scales, in multiphysics contexts too. After properly framing the subject within some main stages of developments of nonlinear dynamics in solid/structural mechanics, as occurred over the last 40 years, the article focuses on highlighting the role played by global analysis in unveiling the nonlinear response and actual safety of engineering systems in different environments. Reduced order models of macro-/micro-structures are considered. Global dynamics of a laminated plate with von Kármán nonlinearities, shear deformability, and full thermomechanical coupling allows to highlight the meaningful effects entailed by the slow transient thermal dynamics on the fast steady mechanical responses. An atomic force microcantilever is referred to for highlighting the severe worsening of overall stability associated with the application of an external feedback control and the importance of global dynamics for conceiving and effectively implementing a control procedure aimed at enhancing engineering safety. The last part of the article dwells on the general role that a global dynamics perspective is expected to play in the safe design of real systems, in the near future, focusing on how properly exploiting concepts and tools of dynamical integrity to evaluate response robustness in the presence of unavoidable imperfections, and to improve the system’s actual load carrying capacity.
... Currently, there are no detailed works on global dynamics of TM-AFM, which (i) evaluate the escape boundaries, (ii) estimate dynamical integrity, and (iii) perform detailed analysis of bifurcations. Existing literature has focused on the dynamical integrity and bifurcation scenarios of non-contact AFM [19,21]. But basins of attraction and erosion process of basin portraits in TM-AFM are lacking in the literature. ...
In this work, we perform a comprehensive analysis of the robustness of attractors in tapping mode atomic force microscopy. The numerical model is based on cantilever dynamics driven in the Lennard–Jones potential. Pseudo-arc-length continuation and basins of attraction are utilized to obtain the frequency response and dynamical integrity of the attractors. The global bifurcation and response scenario maps for the system are developed by incorporating several local bifurcation loci in the excitation parameter space. Moreover, the map delineates various escape thresholds for different attractors present in the system. Our work unveils the properties of the cantilever oscillation in proximity to the sample surface, which is governed by the so-called in-contact attractor. The robustness of this attractor against operating parameters is quantified by means of integrity profiles. Our work provides a unique view into global dynamics in tapping mode atomic force microscopy and helps establishing an extended topological view of the system.
... Extensive simulations were performed, and the stability regions were detected to evaluate both the effectiveness of the control actuation and the possible criticalities on the overall response. Developing systematic dynamical integrity simulations (Settimi and Rega 2016b), the considered control technique was demonstrated to work well for the "local" purpose for which it is specifically designed. Nevertheless, some meaningful drawbacks arise. ...
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.
... Numerical investigations involving computational approaches for continuation have proven to be effective for these purposes. Nu-45 merous studies have indeed exploited numerical tools to obtain bifurcation diagrams and stability charts, which allowed to achieve a comprehensive description of the dynamical behavior of different mechanical devices in terms of the main system parameters [9][10][11][12][13]. ...
https://authors.elsevier.com/a/1X3Px4jvQ~EMU
... Local bifurcation analyses allow us to verify the occurrence of the modified response patterns. However, a comprehensive and reliable understanding of the actual scenario of dynamic response can only be obtained by complementing local analyses with global ones [31][32][33], conducted via proper 2D cross sections of the four-dimensional (4D) basins of attraction. This is needed, in particular, if we are aiming to highlight the possibly non-trivial steady effects entailed by the considered thermomechanical coupling with respect to the global scenario obtainable with the uncoupled model, in which thermal phenomena are taken into account as solely steady mean excitations entering the equation of motion, upon independently solving the underlying, but uncoupled, thermal equations. ...
The nonlinear response of a reduced model of an orthotropic single-layered plate with thermomechanical coupling is investigated in the presence of thermal excitations, in addition to mechanical ones. Different issues are addressed via accurate and extended local and global analyses. (i) Assessing the possible occurrence, disappearance or modification of mechanical buckling as a result of thermal aspects; (ii) exploiting global dynamics to unveil the effects of coupling; (iii) highlighting the crucial role played by the slow thermal transient evolution in modifying the fast steady mechanical response; (iv) framing the influence of coupling and underlining the need to use a thermomechanical model to grasp the actual plate dynamics; and (v) getting hints of technical interest as to the outcome robustness with respect to variations in the external/internal thermal parameters.
... The global dynamics of this feedback controlled model has been already investigated 6 by systematically construct- ing 2D cross-sections of the five-dimensional basins of attraction, and by obtaining the relevant erosion profiles for increasing excitation amplitude. The erosion process has been quantified by means of two different integrity mea- sures 7 , the Global Integrity Measure (GIM) and the Integrity Factor (IF), respectively accounting for and getting rid of the basins fractal parts. ...
The role of a global dynamics analysis to assess a system robustness and actual safety in operating conditions is investigated by studying the effect of different local and global control techniques on the nonlinear behavior of a noncontact AFM via dynamical integrity concepts and tools.
... Projections of a 4D phase space, describing the oscillations and stability of the same mechanical model have been also proposed 13 : with the use of basins of attraction the authors highlight the instability phenomena that may arise under loading conditions such as a parametric excitation of flexural modes, and the escape phenomenon from the pre-buckling potential well. By using 2D cross sections of the 5D basins of attraction, erosion profiles and integrity measures for the parametrically excited noncontacting atomic force microscopy problem are obtained in 14 . In 15 the sizes variation of basins of attraction is analysed in a periodically forced pendulum with oscillating support in the case of time-varying dissipation. ...
Numerical integrations represent a time-consuming element in the long-term dynamics analysis of mechanical systems. This limits the resolution of the computations and the dimension of the system to be investigated numerically. In fact, even pushing memory resources to their thresholds, only few tools can deal with higher-dimensional systems. This work illustrates, in a preliminary manner, the results that can be obtained reducing the aforementioned constraints thanks to the implementation of algorithms based on a parallel computing approach. In particular, by focusing on basins of attraction, four applications are discussed. i) The full domain of attraction for a four-dimensional (4D) system describing a linear oscillator coupled with a nonlinear absorber is calculated. ii) The variation of a safe basin with respect to the system dimension is then analyzed. It is highlighted how 4D and 3D analyses provide more confident results with respect to 2D analyses. iii) The parametric variation of a 2D system with a reduced step is performed by building a 3D representation which allows to highlight a smooth transition between the states. iv) A convergence study of a basin of attraction resolution is carried out. The integrity factor is used as a comparison measure.
... In this respect, previous studies by the authors have analyzed the local nonlinear dynamics of a reduced model of noncontact AFM with an external feedback control [Settimi et al., 2015;Settimi & Rega, 2016b] and have pointed out a generalized detrimental effect of the control technique on the dynamical stability of the system, especially around the resonance frequencies which are the most critical regions to be taken under control to avoid the dangerous phenomenon of jump-to-contact. Moreover, the investigation of the global dynamics of the controlled system [Settimi & Rega, 2016a] has confirmed the criticality and dangerousness of the resonance regions, highlighting that here the system safety is strongly reduced even for very low values of the excitation amplitude due to the penetration of tongues of unbounded response inside the potential well. Nevertheless, from a practical perspective, an adequate level of safety (i.e. an acceptable residual dynamical integrity [Rega & Lenci, 2015]) has to be guaranteed to ensure a proper AFM operation, since it is undisputed that the system security depends as much on the stability of its responses as on the uncontrolled basins of attraction surrounding them. ...
A control technique exploiting the global dynamical features is applied to a reduced order model of noncontact AFM, aiming to obtain an enlargement of the system’s safe region in parameters space. The method consists of optimally modifying the shape of the system excitation by adding controlling superharmonics, to delay the occurrence of the global events (i.e. homo/heteroclinic bifurcations of some saddle) which trigger the erosion of the basins of attraction leading to loss in safety. The system’s main saddles and the bifurcations involving the relevant manifolds are detected through accurate numerical investigations, and their topological characterization allows the determination of the global event responsible for the sharp reduction in the system dynamical integrity. Since an analytical treatment is impossible in applying the control, a fully numerical procedure is implemented. Besides being effective in detecting the value of the optimal superharmonic to be added for shifting the global bifurcation to a higher value of forcing amplitude, the method also proves to succeed in delaying the drop down of the erosion profile, thus increasing the overall robustness of the system during operating conditions.
The role of “Dynamics and Stability” in the history of AIMETA Conferences is discussed. It is emphasized that these subjects, initially away from the interests of the scientific community, have assumed increasing importance over time, culminating in the foundation of the AIMETA Group of Dynamics and Stability (GADeS), which today collects scientific contributions from many members belonging to different scientific sectors. The first timid steps taken by “Dynamics and Stability” in AIMETA are recalled in a historical key, and the causes that first slowed down and then determined their growth are conjectured. With reference to Dynamics and to its classic distinction between linear and nonlinear behaviour, the perturbation method is seen as a key to interpreting nonlinear phenomena. With reference to Stability and Bifurcation, the existence of non-communicating worlds, namely static and dynamic bifurcations, is noted in the world scientific panorama. Once again, the perturbation method can be recognized as the tool that acts as a bridge between the two worlds. Finally, the main topics that have been debated in the GADeS meetings and mini-symposia are briefly reviewed. This paper, in addition to representing a historical synthesis and a cross-section of contemporary research in “Dynamics and Stability” within AIMETA, offers critical considerations on the fragmentation of knowledge, and encourages the development of a unifying vision of the two disciplines.KeywordsDynamicsStabilityStatic and dynamic bifurcationsCenter manifoldPerturbation methodStructural mechanics
In this work, the nonlinear dynamics and control of an atomic force microscopy (AFM) in fractional-order are investigated. Numerical simulations show the existence of chaotic behavior for some regions in the parameter space, whose behavior is characterized using the power spectral density and the 0-1 test. To bring the system from a chaotic state to a periodic one, the nonlinear saturation control (NLSC) and the time-delayed feedback control (TDFC) techniques for the fractional-order systems are applied with and without accounting for the fractional-order. Numerical results show the influence of fractional-order derivative on the dynamics of the AFM system. Due to that, some phenomena arise, which are confirmed through detailed numerical investigations by the 0-1 test. The NLSC and TDFC techniques showed to be efficient in controlling the chaotic behavior of the AFM in fractional-order.
