Nonlinear Dynamics of an Imperfect Microbeam Under an Axial Load and Electric Excitation

2011 ◽  
Author(s):  
Laura Ruzziconi ◽  
Mohammad I. Younis ◽  
Stefano Lenci
Keyword(s):  
Nonlinear Dynamics ◽  
Microelectromechanical Systems ◽  
Complex Dynamics ◽  
Nonlinear Behavior ◽  
Single Mode ◽  
Phase Portraits ◽  
Nonlinear Phenomena ◽  
Theoretical Point ◽  
Initial Shape ◽  
Electric Excitation

This study is motivated by the growing attention, both from a practical and a theoretical point of view, toward the nonlinear behavior of microelectromechanical systems (MEMS). We analyze the nonlinear dynamics of an imperfect microbeam under an axial force and electric excitation. The imperfection of the microbeam, typically due to microfabrication processes, is simulated assuming the microbeam to be of a shallow arched initial shape. The device has a bistable static behavior. The aim is that of illustrating the nonlinear phenomena, which arise due to the coupling of mechanical and electrical nonlinearities, and discussing their usefulness for the engineering design of the microstructure. We derive a single-mode-reduced-order model by combining the classical Galerkin technique and the Pade´ approximation. Despite its apparent simplicity, this model is able to capture the main features of the complex dynamics of the device. Extensive numerical simulations are performed using frequency response diagrams, attractor-basins phase portraits, and frequency-dynamic voltage behavior charts. We investigate the overall scenario, up to the inevitable escape, obtaining the theoretical boundaries of appearance and disappearance of the main attractors. The main features of the nonlinear dynamics are discussed, stressing their existence and their practical relevance. We focus on the coexistence of robust attractors, which leads to a considerable versatility of behavior. This is a very attractive feature in MEMS applications. The ranges of coexistence are analyzed in detail, remarkably at high values of the dynamic excitation, where the penetration of the escape (dynamic pull-in) inside the double well may prevent the safe jump between the attractors.

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

2012 ◽  
Vol 8 (1) ◽  
Author(s):  
Laura Ruzziconi ◽  
Mohammad I. Younis ◽  
Stefano Lenci
Keyword(s):  
Axial Load ◽  
Single Mode ◽  
Reduced Order Model ◽  
Nonlinear Phenomena ◽  
Order Model ◽  
Dynamic Simulations ◽  
Initial Shape ◽  
Reduced Order ◽  
Electric Excitation ◽  
Nonlinear Static

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 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.

A review of the most important failure, reliability and nonlinearity aspects in the development of microelectromechanical systems (MEMS)

2017 ◽  
Vol 34 (1) ◽  
pp. 9-21 ◽  
Author(s):  
Peyman Rafiee ◽  
Golta Khatibi ◽  
Michael Zehetbauer
Keyword(s):  
Dynamic Response ◽  
Frequency Shift ◽  
Mode Coupling ◽  
Microelectromechanical Systems ◽  
Nonlinear Behavior ◽  
Nonlinear Phenomena ◽  
Mems Devices ◽  
Mechanical Loads ◽  
Content Type ◽  
Proper Operation

Purpose The purpose of this paper is to provide an overview of the major reliability issues of microelectromechanical systems (MEMS) under mechanical and environmental loading conditions. Furthermore, a comprehensive study on the nonlinear behavior of silicon MEMS devices is presented and different aspects of this phenomenon are discussed. Design/methodology/approach Regarding the reliability investigations, the most important failure aspects affecting the proper operation of the MEMS components with focus on those caused by environmental and mechanical loads are reviewed. These studies include failures due to fatigue loads, mechanical vibration, mechanical shock, humidity, temperature and particulate contamination. In addition, the influence of squeeze film air damping on the dynamic response of MEMS devices is briefly discussed. A further subject of this paper is discussion of studies on the nonlinearity of silicon MEMS. For this purpose, after a description of the basic principles of nonlinearity, the consequences of nonlinear phenomena such as frequency shift, hysteresis and harmonic generation and their effects on the device performance are reviewed. Special attention is paid to the mode coupling effect between the resonant modes as a result of energy transfer because of the nonlinearity of silicon. For a better understanding of these effects, the nonlinear behavior of silicon is demonstrated by using the example of Si cantilever beams. Findings It is shown that environmental and mechanical loads can influence on proper operation of the MEMS components and lead to early fracture. In addition, it is demonstrated that nonlinearity modifies dynamic response and leads to new phenomena such as frequency shift and mode coupling. Finally, some ideas are given as possible future areas of research works. Originality/value This is a review paper and aimed to review the latest manuscripts published in the field of reliability and nonlinearity of the MEMS structures.

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

2009 ◽  
Vol 19 (03) ◽  
pp. 923-945 ◽  
Author(s):  
STEFANO LENCI ◽  
LAURA RUZZICONI
Keyword(s):  
Complex Dynamics ◽  
Excitation Frequency ◽  
Period Doubling ◽  
Single Mode ◽  
Activation Mechanism ◽  
Dynamical Analysis ◽  
Nonlinear Phenomena ◽  
Nonlinear Dynamical ◽  
Transient Phenomena ◽  
Bifurcation Transition

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.

An Electrically Actuated Microbeam-Based MEMS Device: Experimental and Theoretical Investigation

2017 ◽  
Author(s):  
Laura Ruzziconi ◽  
Nizar Jaber ◽  
Lakshmoji Kosuru ◽  
Mohammed L. Bellaredj ◽  
Stefano Lenci ◽  
...  
Keyword(s):  
Microelectromechanical Systems ◽  
Nonlinear Behavior ◽  
Single Mode ◽  
Parametric Identification ◽  
Unknown Parameters ◽  
Single Degree Of Freedom ◽  
Reduced Order ◽  
Electrically Actuated ◽  
Nonlinear Features ◽  
Frequency Sweeps

The present paper deals with the dynamic behavior of a microelectromechanical systems (MEMS). The device consists of a clamped-clamped microbeam electrostatically and electrodynamically actuated. Our objective is to develop a theoretical analysis, which is able to describe and predict all the main relevant aspects of the experimental response. In the first part of the paper an extensive experimental investigation is conducted. The microbeam is perfectly straight. The first three experimental natural frequencies are identified and the nonlinear dynamics are explored at increasing values of electrodynamic excitation. Several backward and forward frequency sweeps are acquired. The nonlinear behavior is highlighted. The experimental data show the coexistence of the nonresonant and the resonant branch, which perform a bending toward higher frequencies values before undergoing jump or pull-in dynamics. This kind of bending is not particularly common in MEMS. In the second part of the paper, a theoretical single degree-of-freedom model is derived. The unknown parameters are extracted and settled via parametric identification. A single mode reduced-order model is considered, which is obtained via the Galerkin technique. To enhance the computational efficiency, the contribution of the electric force term is computed in advance and stored in a table. Extensive numerical simulations are performed at increasing values of electrodynamic excitation. They are observed to properly predict all the main nonlinear features arising in the device response. This occurs not only at low values of electrodynamic excitation, but also at higher ones.

Nonlinear Dynamic Response of an Electrically Actuated Imperfect Microbeam Resonator

2013 ◽  
Author(s):  
Laura Ruzziconi ◽  
Ahmad M. Bataineh ◽  
Mohammad I. Younis ◽  
Weili Cui ◽  
Stefano Lenci
Keyword(s):  
Nonlinear Dynamics ◽  
Theoretical Analysis ◽  
Ritz Method ◽  
Nonlinear Behavior ◽  
Single Mode ◽  
Reduced Order Models ◽  
Nonlinear Dynamic Response ◽  
Reduced Order ◽  
Combined Use ◽  
Experimental Complex

We present a study of the dynamic behavior of a MEMS device constituted 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 and the nonlinear dynamics are explored at increasing values of electrodynamic excitation, in a neighborhood of the first symmetric resonance. The nonlinear behavior is highlighted, which includes ranges of multistability, where the non-resonant 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 Padé 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 capable also to capture 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

2013 ◽  
Vol 23 (02) ◽  
pp. 1350026 ◽  
Author(s):  
LAURA RUZZICONI ◽  
STEFANO LENCI ◽  
MOHAMMAD I. YOUNIS
Keyword(s):  
Axial Load ◽  
Single Mode ◽  
Quantitative Information ◽  
Structural Safety ◽  
Nonlinear Phenomena ◽  
Practical Case ◽  
Electric Excitation ◽  
Dynamic Voltage ◽  
Final Response ◽  
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 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.

NONLINEAR DYNAMICS OF CELL CYCLES WITH STOCHASTIC MATHEMATICAL MODELS

2009 ◽  
Vol 17 (03) ◽  
pp. 425-460 ◽  
Author(s):  
RODERICK V. N. MELNIK ◽  
XILIN WEI ◽  
GABRIEL MORENO–HAGELSIEB
Keyword(s):  
Nonlinear Dynamics ◽  
Cell Cycle ◽  
Regulatory Networks ◽  
Deterministic Model ◽  
Phase Portraits ◽  
Nonlinear Phenomena ◽  
Cellular Dynamics ◽  
Cell Cycles ◽  
Special Events ◽  
Wide Range

Cell cycles are fundamental components of all living organisms and their systematic studies extend our knowledge about the interconnection between regulatory, metabolic, and signaling networks, and therefore open new opportunities for our ultimate efficient control of cellular processes for disease treatments, as well as for a wide variety of biomedical and biotechnological applications. In the study of cell cycles, nonlinear phenomena play a paramount role, in particular in those cases where the cellular dynamics is in the focus of attention. Quantification of this dynamics is a challenging task due to a wide range of parameters that require estimations and the presence of many stochastic effects. Based on the originally deterministic model, in this paper we develop a hierarchy of models that allow us to describe the nonlinear dynamics accounting for special events of cell cycles. First, we develop a model that takes into account fluctuations of relative concentrations of proteins during special events of cell cycles. Such fluctuations are induced by varying rates of relative concentrations of proteins and/or by relative concentrations of proteins themselves. As such fluctuations may be responsible for qualitative changes in the cell, we develop a new model that accounts for the effect of cellular dynamics on the cell cycle. Finally, we analyze numerically nonlinear effects in the cell cycle by constructing phase portraits based on the newly developed model and carry out a parametric sensitivity analysis in order to identify parameters for an efficient cell cycle control. The results of computational experiments demonstrate that the metabolic events in gene regulatory networks can qualitatively influence the dynamics of the cell cycle.

Nonlinear Dynamics and Application of the Four Dimensional Autonomous Hyper-Chaotic System

2014 ◽  
Author(s):  
Ge Kai ◽  
Wei Zhang
Keyword(s):  
Nonlinear Dynamics ◽  
Numerical Simulation ◽  
Differential Equations ◽  
Chaotic System ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Phase Portraits ◽  
State Variables ◽  
Chaotic Motions ◽  
Finance System

In this paper, we establish a dynamic model of the hyper-chaotic finance system which is composed of four sub-blocks: production, money, stock and labor force. We use four first-order differential equations to describe the time variations of four state variables which are the interest rate, the investment demand, the price exponent and the average profit margin. The hyper-chaotic finance system has simplified the system of four dimensional autonomous differential equations. According to four dimensional differential equations, numerical simulations are carried out to find the nonlinear dynamics characteristic of the system. From numerical simulation, we obtain the three dimensional phase portraits that show the nonlinear response of the hyper-chaotic finance system. From the results of numerical simulation, it is found that there exist periodic motions and chaotic motions under specific conditions. In addition, it is observed that the parameter of the saving has significant influence on the nonlinear dynamical behavior of the four dimensional autonomous hyper-chaotic system.

Slow–Fast Behaviors and Their Mechanism in a Periodically Excited Dynamical System with Double Hopf Bifurcations

2021 ◽  
Vol 31 (08) ◽  
pp. 2130022
Author(s):  
Miaorong Zhang ◽  
Xiaofang Zhang ◽  
Qinsheng Bi
Keyword(s):  
Dynamical System ◽  
Hopf Bifurcation ◽  
Frequency Domain ◽  
Autonomous System ◽  
Single Mode ◽  
Phase Portraits ◽  
Hopf Bifurcations ◽  
Double Hopf Bifurcation ◽  
Exciting Frequency ◽  
Slow Passage

This paper focuses on the influence of two scales in the frequency domain on the behaviors of a typical dynamical system with a double Hopf bifurcation. By introducing an external periodic excitation to the normal form of the vector field with double Hopf bifurcation at the origin and taking the exciting frequency far less than the natural frequency, a theoretical model with two scales in the frequency domain is established. Regarding the whole exciting term as a slow-varying parameter leads to a generalized autonomous system, in which the equilibrium branches and their bifurcations with the variation of the slow-varying parameter can be derived. With the increase of the exciting amplitude, different types of bifurcations may be involved in the generalized autonomous system, resulting in several qualitatively different forms of bursting attractors, the mechanism of which is presented by overlapping the transformed phase portraits and the bifurcations of the equilibrium branches. It is found that the single mode 2D torus may evolve to the bursting attractors with mixed modes, in which the trajectory alternates between the single mode oscillations and the mixed mode oscillations. Furthermore, the transitions between the quiescent states and the spiking states may not occur exactly at the bifurcation points because of the slow passage effect, while Hopf bifurcations may cause different forms of repetitive spiking oscillations.

Nonlinear Coupled Transverse and Axial Vibration of Variable Cross-Section Beam Flexures Interconnecting Rigid Body

2016 ◽  
Author(s):  
Hasan Malaeke ◽  
Hamid Moeenfard ◽  
Amir H. Ghasemi
Keyword(s):  
Differential Equations ◽  
Rigid Body ◽  
Cross Section ◽  
Multiple Scales ◽  
Nonlinear Behavior ◽  
Single Mode ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Closed Form Solutions ◽  
Dynamic Performances

The objective of this paper is to analytically study the nonlinear behavior of variable cross-section beam flexures interconnecting an eccentric rigid body. Hamilton’s principle is utilized to obtain the partial differential equations governing the nonlinear vibration of the system as well as the corresponding boundary conditions. Using a single mode approximation, the governing equations are reduced to a set of two nonlinear ordinary differential equations in terms of end displacement components of the beam which are coupled due to the presence of the transverse eccentricity. The method of multiple scales are employed to obtain parametric closed-form solutions. The obtained analytical results are compared with the numerical ones and excellent agreement is observed. These analytical expressions provide design insights for modeling and optimization of more complex flexure mechanisms for improved dynamic performances.

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