Laura Ruzziconi

An Experimental and Theoretical Investigation of Dynamic Pull-In in MEMS Resonators Actuated Electrostatically

We 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 dynamics of an electrically actuated imperfect microbeam resonator: experimental investigation and reduced-order modeling

We present a study of the dynamic behavior of a microelectromechanical systems (MEMS) device consisting of an imperfect clamped–clamped microbeam subjected to electrostatic and electrodynamic actuation. Our objective is to develop a theoretical analysis, which is able to describe and predict all the main relevant aspects of the experimental response. Extensive experimental investigation is conducted, where the main imperfections coming from microfabrication are detected, the first four experimental natural frequencies are identified and the nonlinear dynamics are explored at increasing values of electrodynamic excitation, in a neighborhood of the first symmetric resonance. Several backward and forward frequency sweeps are acquired. The nonlinear behavior is highlighted, which includes ranges of multistability, where the nonresonant and the resonant branch coexist, and intervals where superharmonic resonances are clearly visible. Numerical simulations are performed. Initially, two single mode reduced-order models are considered. One is generated via the Galerkin technique, and the other one via the combined use of the Ritz method and the Pade approximation. Both of them are able to provide a satisfactory agreement with the experimental data. This occurs not only at low values of electrodynamic excitation, but also at higher ones. Their computational efficiency is discussed in detail, since this is an essential aspect for systematic local and global simulations. Finally, the theoretical analysis is further improved and a two-degree-of-freedom reduced-order model is developed, which is also capable of capturing the measured second symmetric superharmonic resonance. Despite the apparent simplicity, it is shown that all the proposed reduced-order models are able to describe the experimental complex nonlinear dynamics of the device accurately and properly, which validates the proposed theoretical approach.

AN IMPERFECT MICROBEAM UNDER AN AXIAL LOAD AND ELECTRIC EXCITATION: NONLINEAR PHENOMENA AND DYNAMICAL INTEGRITY

This 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...

An Efficient Reduced-Order Model to Investigate the Behavior of an Imperfect Microbeam Under Axial Load and Electric Excitation

In 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 Pade 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.

Dynamical Integrity for Interpreting Experimental Data and Ensuring Safety in Electrostatic MEMS

A 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.

The dynamical integrity concept for interpreting/ predicting experimental behaviour: from macro- to nano-mechanics

The 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 ones 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)

NONLINEAR PHENOMENA IN THE SINGLE-MODE DYNAMICS OF A CABLE-SUPPORTED BEAM

In 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.

Theoretical Prediction of Experimental Jump and Pull-In Dynamics in a MEMS Sensor

The present research study deals with an electrically actuated MEMS device. An experimental investigation is performed, via frequency sweeps in a neighbourhood of the first natural frequency. Resonant behavior is explored, with special attention devoted to jump and pull-in dynamics. A theoretical single degree-of-freedom spring-mass model is derived. Classical numerical simulations are observed to properly predict the main nonlinear features. Nevertheless, some discrepancies arise, which are particularly visible in the resonant branch. They mainly concern the practical range of existence of each attractor and the final outcome after its disappearance. These differences are likely due to disturbances, which are unavoidable in practice, but have not been included in the model. To take disturbances into account, in addition to the classical local investigations, we consider the global dynamics and explore the robustness of the obtained results by performing a dynamical integrity analysis. Our aim is that of developing an applicable confident estimate of the system response. Integrity profiles and integrity charts are built to detect the parameter range where reliability is practically strong and where it becomes weak. Integrity curves exactly follow the experimental data. They inform about the practical range of actuality. We discuss the combined use of integrity charts in the engineering design. Although we refer to a particular case-study, the approach is very general.

Nonlinear dynamics of a MEMS resonator: Theoretical and experimental investigation

In this study we present a theoretical and experimental investigation of a microelectromechanical system (MEMS). The device is constituted of a clamped-clamped polysilicon microbeam electrostatically and electrodynamically actuated. The microbeam has a slightly curled up configuration, which is an imperfection commonly encountered as a consequence of the microfabrication process. Using a laser Doppler vibrometer, many experimental frequency sweeps are conducted in a neighborhood of the first symmetric natural frequency. To simulate the dynamics, we derive a single-mode reduced-order model. Extensive numerical investigations are performed, based on frequency response diagrams and behavior charts. The overall scenario of the response is explored, when both the frequency and the electrodynamic voltage are varying. This analysis is able to provide a very good matching with the experiments. Nevertheless, the theoretical predictions are not completely fulfilled in some aspects. In particular, the range of exist...

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.