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Nonlinear dynamics in microelectromechanical systems
No full textLa tesi affronta lo studio delle dinamiche nonlineari in alcuni dispositivi MEMS. Le tematiche di dinamica nonlineare attualmente affrontate in letteratura sono indispensabili per studiare la
loro risposta. L’accuratezza della modellazione dinamica nonlineare è importante per garantire l’affidabilità dei risultati e gli strumenti attuali di dinamica nonlineare riescono ad
interpretare scrupolosamente i dati sperimentali della risposta di questi dispositivi. La tesi considera due diversi casi-studio. Il primo caso-studio è un dispositivo MEMS con carico assiale, forma iniziale ad arco molto ribassato e attuazione elettrostatica ed elettrodinamica. È analizzato in un intorno della biforcazione da una singola ad una doppia buca di potenziale. Sia le configurazioni statiche nonlineari sia l’analisi dinamica lineare non possono essere risolte in forma chiusa e sono
approssimate con il metodo di Galerkin. Vengono usate per costruire un accurato modello ridotto delle dinamiche nonlineari ad un solo grado di libertà. In questo modello il termine del quinto ordine (che dipende dall’espansione in serie di Taylor nell’equazione del moto) è eliminato per avere una buona approssimazione delle buche di potenziale e del comportamento globale. Altri modelli ridotti sono considerati e paragonati. Si esegue l’analisi
dinamica nonlineare, con l’uso combinato di curve di risposta in frequenza, ritratti di fase attrattori-bacini e mappe di comportamento. In un intorno di ciascuna frequenza naturale, la risposta del dispositivo presenta le tipiche caratteristiche di un oscillatore softening. I casi di singola e doppia buca di potenziale vengono paragonati.
Il secondo caso-studio analizza i dati sperimentali di pull-in dinamico in risonanza primaria di
un dispositivo MEMS (un accelerometro capacitivo). Iniziando da questo caso particolare, si affronta la tematica dell’integrità dinamica in un sistema meccanico. Viene eseguito il suo
calcolo qualitativo, scegliendo gli strumenti più appropriati in base alle condizioni sperimentali considerate. Si evidenzia l’efficacia di questa analisi, mostrando l’accuratezza delle curve di percentuale costante di fattore di integrità nell’interpretare l’esistenza di
disturbi negli esperimenti e nella pratica. Inoltre, si mostra il loro utilizzo nella progettazione.This dissertation deals with the nonlinear dynamics in MEMS devices. The nonlinear
dynamic topics currently addressed in the literature are essential to investigate their response. The accuracy of the nonlinear dynamic modeling is important to guarantee the reliability of
the results and current nonlinear dynamic tools succeed in carefully interpreting the
experimental data of the response of these devices. The dissertation considers two different case-studies.
The first case-study is a MEMS device with axial load, very shallow arched initial shape and electrostatic and electrodynamic actuation. It is analyzed in the neighborhood of the bifurcation from a single potential well to a twin well. Both the nonlinear static configurations
and the linear dynamic analysis cannot be solved in closed form and they are approximated by the Galerkin technique. They are used to derive an accurate single degree of freedom reduced order model of the nonlinear dynamics. In this model the fifth order term (connected to the Taylor expansion in the equation of motion) is removed to obtain a good approximation of the
potential wells and of the global behavior. Other reduced order models are considered and
compared. The nonlinear dynamic analysis is performed, with the combined use of frequency
response curves, attractor-basins phase portraits and behavior charts. In a neighborhood of each natural frequency, the response of the device has the typical characteristics of a softening
oscillator. The cases of the single and the double potential well are compared.
The second case-study analyzes the experimental dynamic pull-in data at primary resonance for a MEMS device (a capacitive accelerometer). Starting from this particular case, the issue of the dynamical integrity in a mechanical system is addressed. Its qualitative evaluation is
performed, choosing the most suitable tools according to the considered experimental
conditions. The effectiveness of this analysis is highlighted, showing the accuracy of the curves of constant percentage of integrity factor in interpreting the existence of disturbances
in experiments and practice. Also, their use in a design is proposed
Multiple internal resonance couplings and quasi-periodicity patterns in hybrid-shaped micromachined resonators
No full textMicro- and nano-electromechanical systems may experience internal resonances, which are inherently strengthened by the systems' nonlinearities. In the present paper, we investigate the dynamics of a hybrid resonator, combining straight and initially curved microbeam shapes. The curved part length is tailored to monitor the three lowest natural frequencies and induce simultaneous internal resonances between first and second modes and second and third modes. We examine the nonlinear interaction of dual hardening and softening bending of the fundamental frequency response curves. Due to the specific frequency ratios, different types of subcombination internal resonances emerge, with quasi-periodic energy transfer among the modes. The subcombination may result in frequencies closely-spaced, which leads to quasi-periodic beating among the frequencies involved and, due to the strong nonlinearities, to the emergence of intermodulation products. We analyze the different patterns underlying the quasi-periodicity, which are intrinsically related to the frequency ratios and deeply affected by the nonlinearities
NONLINEAR PHENOMENA IN THE SINGLE-MODE DYNAMICS OF A CABLE-SUPPORTED BEAM
No full textIn this paper we discuss the practical usefulness of nonlinear dynamical analysis for the design of a planar cable-supported beam: we refer to a feasible case, assuming the value of the parameters corresponding to a realistic pedestrian footbridge. We consider a one degree of freedom model, obtained by the classical Galerkin reduction technique: the ensuing ordinary differential equation has both quadratic and cubic terms, due to geometric nonlinearities. Extensive numerical simulations are performed: they point out that this model, in spite of its apparent simplicity, is able to highlight the complex dynamics of the cable-supported beam, describing several common and uncommon nonlinear phenomena. Each of them is interpreted in terms of oscillations of the considered mechanical system; we explain the relevance of all the obtained results in the design of the examined structure under steady loads as wind and pedestrians, but also under transient phenomena as earthquake and gust; the ensuing issues, the most dangerous ranges and also the sensibility to perturbations are discussed in detail. In particular we deal with the importance, for an engineering design, of a careful interpretation of: isola bifurcation, transition to chaos both by period doubling cascade and reverse boundary crisis, multistability with coexistence of chaotic and periodic attractors, fractal basins boundaries, erosion of immediate basins, interrupted sequence of period doubling bifurcations. Also the effects of secondary attractors are analyzed, and it is shown that in general they cannot be neglected even if their range of existence is very small. We underline that all these investigations are performed choosing the excitation frequency far from resonances in order to alert the designer that the system dynamics may be complex independently of the activation mechanism due to resonance
Global investigation of the nonlinear dynamics of carbon nanotubes
No full textUnderstanding the complex nonlinear dynamics of carbon nanotubes (CNTs) is essential to enable utilization of these structures in devices and practical applications. We present in this work an investigation of the global nonlinear dynamics of a slacked CNT when actuated by large electrostatic and electrodynamic excitations. The coexistence of several attractors is observed. The CNT is modeled as an Euler–Bernoulli beam. A reduced-order model based on the Galerkin method is developed and utilized to simulate the static and dynamic responses. Critical computational challenges are posed due to the complicated form of the electrostatic force, which describes the interaction between the upper electrode, consisting of the cylindrically shaped CNT, and the lower electrode. Toward this, we approximate the electrostatic force using the Padé expansion. We explore the dynamics near the primary and superharmonic resonances. The nanostructure exhibits several attractors with different characteristics. To achieve deep insight and describe the complexity and richness of the behavior, we analyze the nonlinear response from an attractor-basins point of view. The competition of attractors is highlighted. Compactness and/or fractality of their basins are discussed. Both the effects of varying the excitation frequency and amplitude are examined up to the dynamic pull-in instability
An Efficient Reduced-Order Model to Investigate the Behavior of an Imperfect Microbeam Under Axial Load and Electric Excitation
No full textIn this study an efficient reduced-order model for a MEMS device is developed and investigations of the nonlinear static and the dynamic behavior are performed. The device is constituted of an imperfect microbeam under an axial load and an electric excitation. The imperfections, typically due to microfabrication processes, are simulated assuming a shallow arched initial shape. The axial load is deliberately added with an elevated value. The structure has a bistable static configuration of double potential well with possibility of escape. We derive a single-mode reduced-order model via the Ritz technique and the Padé approximation. This model, while simple, is able to combine both a sufficient accuracy, which enables to detect the main qualitative features of the device response up to elevated values of electrodynamic excitation, and a remarkable computational efficiency, which is essential for systematic global nonlinear dynamic simulations. We illustrate the nonlinear phenomena arising in the device, such as the coexistence of various competing in-well and cross-well attractors, which leads to a considerable versatility of behavior. We discuss their physical meaning and their practical relevance for the engineering design of the microstructure, since this is an uncommon and very attractive aspect in applications
An Experimental and Theoretical Investigation of Dynamic Pull-In in MEMS Resonators Actuated Electrostatically
No full textWe present experimental and theoretical investigations of dynamic pull-in of electrostatically actuated resonators. Several experimental data are presented, showing regimes of ac forcing amplitude versus ac frequency, where a resonator is forced to pull in if operated within these regimes. Results are shown for primary and secondary resonance excitations. The influences of the initial conditions of the system, the ac excitation amplitude, the ac frequency, the excitation type, and the sweeping type are investigated. A shooting technique to find periodic motions and a basin-of-attraction analysis are used to predict the limits of the pull-in bands. When compared with the experimental data, the results have shown that the pull-in limits coincide with 30%-40% erosion lines of the safe basin in the case of primary resonance and 5%-15% erosion lines of the safe basin in the case of subharmonic resonance. Bifurcation diagrams have been constructed, which designers can use to establish factors of safety to reliably operate microelectromechanical-systems resonators away from pull-in bands and the danger of pull-in, depending on the expected disturbances and noise in the systems
Nonlinear oscillations, transition to chaos and escape in the Duffing system with non-classical damping
Full text linkThe nonlinear dynamics of a single-degree-of-freedom oscillator with an external excitation and complex non-viscous damping is examined. The complex nature of the damper introduces a hidden variable to the set of equations of motion. We examine nonlinear oscillations, bifurcations and the escape from the potential well in that system. The shape of the resonance curve is obtained by the multiple time scales method and it is confirmed numerically. By treating the excitation and damping effects as perturbations we found the heteroclinic orbits connecting the saddle points of the Hamiltonian and estimate the range of system parameters leading to a chaotic behaviour by means of the Melnikov method. This result is also confirmed by numerical simulations. The mechanism of escape from the potential well is analyzed by means of behaviour charts and basins of attraction
The dynamical integrity concept for interpreting/ predicting experimental behaviour: from macro- to nano-mechanics
No full textThe dynamical integrity, a new concept proposed by J.M.T. Thompson, and developed by the authors, is used to interpret experimental results. After reviewing the main issues involved in this analysis, including the proposal of a new integrity measure able to capture in an easy way the safe part of basins, attention is dedicated to two experiments, a rotating pendulum and a micro-electro-mechanical system, where the theoretical predictions are not fulfilled. These mechanical systems, the former at the macro-scale and the latter at the micro-scale, permit a comparative analysis of different mechanical and dynamical behaviours. The fact that in both cases the dynamical integrity permits one to justify the difference between experimental and theoretical results, which is the main achievement of this paper, shows the effectiveness of this new approach and suggests its use in practical situations. The men of experiment are like the ant, they only collect and use; the reasoners resemble spiders, who make cobwebs out of their own substance. But the bee takes the middle course: it gathers its material from the flowers of the garden and field, but transforms and digests it by a power of its own. Not unlike this is the true business of philosophy (science); for it neither relies solely or chiefly on the powers of the mind, nor does it take the matter which it gathers from natural history and mechanical experiments and lay up in the memory whole, as it finds it, but lays it up in the understanding altered and digested. Therefore, from a closer and purer league between these two faculties, the experimental and the rational (such as has never been made), much may be hoped. (Francis Bacon 1561-1626) But are we sure of our observational facts? Scientific men are rather fond of saying pontifically that one ought to be quite sure of one's observational facts before embarking on theory. Fortunately those who give this advice do not practice what they preach. Observation and theory get on best when they are mixed together, both helping one another in the pursuit of truth. It is a good rule not to put overmuch confidence in a theory until it has been confirmed by observation. I hope I shall not shock the experimental physicists too much if I add that it is also a good rule not to put overmuch confidence in the observational results that are put forward until they have been confirmed by theory. (Arthur Stanley Eddington 1882-1944)
AN IMPERFECT MICROBEAM UNDER AN AXIAL LOAD AND ELECTRIC EXCITATION: NONLINEAR PHENOMENA AND DYNAMICAL INTEGRITY
No full textThis work deals with the nonlinear dynamics of a microelectromechanical system constituted by an imperfect microbeam under an axial load and an electric excitation. The device is characterized by a bistable static configuration. We analyze the single-mode dynamics and describe the overall scenario of the response, up to the inevitable escape, when both the frequency and the electrodynamic voltage are considered as driving parameters. We observe the presence of several competing attractors leading to a considerable versatility of behavior, which may have many feasible applications. Extensive numerical simulations are performed. The frequency-dynamic voltage behavior chart is obtained, which detects the theoretical boundaries of appearance and disappearance of the main attractors. Taking into account the erosion of the double well, we investigate the final response when each attractor vanishes. All these results represent the limit when disturbances are absent, which never occurs in practice. To extend them to the practical case where disturbances exist, we develop a dynamical integrity analysis. This is performed via curves of constant percentage of local integrity measure, which give quantitative information about the changes in the structural safety. For each attractor, we examine both the practical disappearance, by analyzing the robustness of its basin along the range of existence, and the practical final response, by detecting where safe jump to another attractor may be ensured and where instead dynamic pull-in may arise. These curves may be used to establish safety factors in order to operate the device according to the desired outcome, depending on the expected disturbances
Dynamical Integrity for Interpreting Experimental Data and Ensuring Safety in Electrostatic MEMS
No full textA dynamical integrity analysis is performed for an electrostatic micro-electro-mechanical system (MEMS) device. The analysis starts from the experimental data of dynamic pull-in due to a frequency-sweeping process in a capacitive accelerometer. The loss of dynamical integrity is investigated by curves of constant percentage of integrity factor. We found that these curves follow exactly the experimental data and succeed in interpreting the existence of disturbances. On the other hand, instead, the theoretical curves of disappearance of the attractors represent the limit when disturbances are absent, which never occurs in practice. Also, the obtained behavior chart can serve as a design guideline in order to ensure safety of the device
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