Imperfect Microbeam Research Articles

Parameter uncertainty and noise effects on the global dynamics of an electrically actuated microarch

This work aims to study the effect of uncertainties and noise on the nonlinear global dynamics of a micro-electro-mechanical arch obtained from an imperfect microbeam under an axial load and electric excitation. An adaptative phase-space discretization strategy based on an operator approach is proposed. The Ulam method, a classical discretization of flows in phase-space, is extended here to nondeterministic cases. A unified description is formulated based on the Perron–Frobenius, Koopman, and Foias linear operators. Also, a procedure to obtain global structures in the mean sense of systems with parametric uncertainties is presented. The stochastic basins of attraction and attractors’ distributions replace the usual basin and attractor concepts. For parameter uncertainty cases, the phase-space is augmented with the corresponding probability space. The microarch is assumed to be shallow and modeled using a nonlinear Bernoulli–Euler beam theory and is discretized by the Galerkin method using as interpolating function the linear vibration modes. Then, from the discretized multi degree of freedom model model, an accurate single degree of freedom reduced order model, based on theory of nonlinear normal modes, is derived. Several competing attractors are observed, leading to different (acceptable or unacceptable) behaviors. Extensive numerical simulations are performed to investigate the effect of noise and uncertainties on the coexisting basins of attraction, attractors’ distributions, and basins boundaries. The appearance and disappearance of attractors and stochastic bifurcation are observed, and the time-dependency of stochastic responses is demonstrated, with long-transients influencing global behavior. To consider uncertainties and noise in design, a dynamic integrity measure is proposed via curves of constant probability, which give quantitative information about the changes in structural safety. For each attractor, the basin robustness as a function of a stochastic parameter is investigated. The weighted basin area can quantify the integrity of nondeterministic cases, being also the most natural generalization of the global integrity measure. While referring to particular micro-electro-mechanical system, the relevance of the dynamical integrity analysis for stochastic systems to quantify tolerances and safety margins is underlined here.

Nonlinear forced vibration analysis of functionally graded non-uniform cylindrical microbeams applying the semi-analytical solution

The linear and nonlinear forced vibration response of axially functionally graded (AFG) cylindrical truncated conical and imperfect microbeam subjected to the dynamic harmonically load carried out in the presented research. Based on a couple of modified couple stress theory, the Euler-Bernoulli beam theory and von-Kármán theory, the linear and nonlinear governing equations and related boundary conditions for dynamic response of micro-size tubes are derived employing the Hamilton principle. We considered the uniform and nonuniform functions for the cross-section, in which the convex, linear and exponential functions are the nonuniform sections, and the porosity is regarded as an imperfection. The generalized differential quadrature method (GDQM) is used to prepare the initial conditions for homotopy perturbation (HP) techniques as the semi-analytical approach to calculate the linear and nonlinear results of dynamic responses. The obtained linear and nonlinear results of the free and forced vibration response show the negative and positive effects of some parameters such as the porosity parameter, the nonlinear amplitude, the small-scale parameter, AFG parameter, and different cross-section impact on the dynamic deflection and natural frequency of micro-scale tube and beams with both clamped and simply-supported boundary conditions.

Viscoelastic mechanics of Timoshenko functionally graded imperfect microbeams

A Timoshenko functionally graded (TFG) imperfect microscale beam is considered and the coupled viscoelastic mechanics is analysed in a nonlinear regime. Based on the Timoshenko beam theory, incorporating geometric imperfections, the Kelvin-Voigt method is used for internal viscosity, the rotary inertia is automatically generated due to the Timoshenko theory, and the Mari-Tanaka scheme is used for the mixture. After utilising the constitutive stress/strain relations, Hamilton’s principle is used giving three sets of coupled motion equations. A discretisation is performed using Galerkin’s method and numerical simulations, for mechanical behaviour analysis, are built upon a continuation method.

Thermo-hygro-mechanical bending and vibration of functionally graded material microbeams with microporosity defect

In this study, the thermo-hygro-mechanical bending and vibration of functionally graded material (FGM) microbeams with microporosity defect are investigated. The mathematical model of the system is developed by using the sinusoidal beam theory and the modified couple stress theory. Based on Hamilton’s principle, the governing equations and boundary conditions of imperfect FGM microbeams are derived. Then, Navier’s method is utilized to calculate the deflections and natural frequencies of the system. Analytical results are presented to illustrate the effects of the microporosity volume fraction, the microporosity distribution type, the power-law index, the temperature and the moisture on the static bending and free vibration of imperfect FGM microbeams.

Viscoelastic dynamics of axially FG microbeams

A viscoelastic axially functionally graded (FG) imperfect microbeam is considered and the complex nonlinear dynamics is analysed for the first time. The distribution of the material properties from beam's left end to the right one is of an exponential form. The small nature of the system is incorporated using the couple stress theory (CST) in a modified form. The viscosity between microbeam's infinitesimal elements is incorporated through the Kelvin-Voigt scheme. The elastic part of the energy is formulated via the elastic component of the stress and its couple; the viscous parts are incorporated via a negative work. The energy of kinetic type is also formulated. Without ignoring the axial displacement/inertia, Hamilton's principle of energy is used to formulate the viscoelastic mechanical model of the microsystem. A high-dimensional truncation/discretisation is conducted and numerical simulations are performed based upon it. The complex nonlinear dynamics of the axially FG imperfect beam is analysed in the presence/absence of viscosity.

Size-dependent post-buckling behaviors of geometrically imperfect microbeams

In this paper, the size-dependent buckling and post-buckling analysis of the imperfect microbeams is presented. By accounting the mid-plane stretching, the nonlinear integral partial-differential equation governing the transverse motion of the imperfect microbeam is derived employing Hamilton’s principle. The modified couple stress theory is employed to capture the size effects of microbeams. The exact analytical solutions of the critical buckling load and the static response are acquired for the buckling problem of imperfect microbeams with different boundary conditions. Moreover, the dynamic behavior around the static configurations in the preand postbuckling domain is also investigated. It is found that the buckling of imperfect microbeam occurs through a saddlenode bifurcation, and the responses in postbuckling domain are asymmetrical. The numerical results are depicted to illustrate the influence of the dimensionless scale parameter and the imperfection amplitude on the circuital buckling load, static response and the dynamic properties of the imperfect microbeams.

Experimental and mathematical analysis of a piezoelectrically actuated multilayered imperfect microbeam subjected to applied electric potential

In this paper we propose to fabricate a microcantilever beam actuator using thin layer PZT material. The multilayered microbeam is initially curved due to fabrication process. The device is also modeled using a the Euler-Bernoulli beam theory and takes into account the multilayer structure of the device. The inclusion of these imperfection into the theoretical model generates geometric nonlinearities. Static and dynamic experimental analysis is conducted to examine the physiomechanical behavior of the considered microstructure. To this end, The model is first compared and validated with experimental findings for an applied constant voltage. Very good agreement depicts the accuracy of the proposed mathematical model. When time varying voltage is applied, the nonlinear differential equation is solved using a perturbation technique. Again good agreement with experiments are found. Finally, a parametric study is carried out to explore the influences of different parameters on the frequency response of the piezoelectrically actuated imperfect microbeams.

Viscoelastic resonant responses of shear deformable imperfect microbeams

A viscoelastic model for the nonlinear analysis of the coupled transverse, longitudinal, and rotational oscillations of an imperfect shear deformable microbeam is developed, for the first time, based on the modified couple stress theory. An energy dissipation mechanism is developed via use of the Kelvin–Voigt internal energy dissipation mechanism. For the stress and deviatoric part of the symmetric couple stress tensors, the viscous components along with the corresponding work terms are obtained. The size-dependent elastic energy along with the kinetic energy of the viscoelastic microsystem is formulated in terms of the displacement field together with system geometric and physical parameters. The internal energy dissipation is developed via the work done by the viscous components of the stress and the deviatoric part of the symmetric couple stress tensors by means of the Kelvin–Voigt mechanism. These work and energy terms are inserted into Hamilton’s principle together with the work due to an external force in order to obtain three viscoelastically coupled equations governing the transverse, longitudinal, and rotational motions with cubic and quadratic nonlinear terms. A high-dimensional Galerkin approximation method is applied for all the three equations, yielding three sets of second-order coupled ordinary differential equations with cubic and quadratic nonlinearities. Upon application of a transformation, a continuation technique along with the backward differentiation formula (BDF) is employed in order to obtain the time-variant response of the system subject to a harmonic load. Special attention is paid to the effect of the Kelvin–Voigt type viscoelasticity on the system response in the presence of the length-scale parameter.

Resonance Responses of Geometrically Imperfect Functionally Graded Extensible Microbeams

This paper aims at analyzing the size-dependent nonlinear dynamical behavior of a geometrically imperfect microbeam made of a functionally graded (FG) material, taking into account the longitudinal, transverse, and rotational motions. The size-dependent property is modeled by means of the modified couple stress theory, the shear deformation and rotary inertia are modeled using the Timoshenko beam theory, and the graded material property in the beam thickness direction is modeled via the Mori–Tanaka homogenization technique. The kinetic and size-dependent potential energies of the system are developed as functions of the longitudinal, transverse, and rotational motions. On the basis of an energy method, the continuous models of the system motion are obtained. Upon application of a weighted-residual method, the reduced-order model is obtained. A continuation method along with an eigenvalue extraction technique is utilized for the nonlinear and linear analyses, respectively. A special attention is paid on the effects of the material gradient index, the imperfection amplitude, and the length-scale parameter on the system dynamical response.

Viscoelastically coupled size-dependent behaviour of imperfect extensible microbeams

This paper aims at investigating the size-dependent nonlinear behaviour of a viscoelastic imperfect extensible microbeam taking into account both transverse and longitudinal displacements and inertia. The size-dependent potential energy is formulated in the framework of the modified couple stress theory. The works due to the viscous parts of the stress tensor and the deviatoric part of the couple stress tensor are obtained in terms of system parameters. The kinetic energy as well as the work of external dynamic loading is obtained as functions of the displacement field. Hamilton’s principle is employed in order to balance the work and energy terms which results in the coupled nonlinear equations of motion for the longitudinal and transverse directions. A high-dimensional weighted-residual technique is employed so as to discretise the coupled equations of longitudinal and transverse motions and hence the continuous system with infinite number of degrees of freedom is truncated into a reduced-order model with sufficient degrees of freedom for accurate results capable of capturing almost all modal interactions. This high-dimensional nonlinear coupled reduced-order model is solved for the fundamental coupled nonlinear resonant response via use of a continuation method as well as direct time integration for response characterisation with special consideration to the coupled effect of the viscousity, initial imperfection, and length-scale parameter on the system response in the longitudinal and transverse directions. It is shown that the deviation between the response of viscoelastic and elastic systems is substantial for fairly large excitation forces.

Size-dependent parametric dynamics of imperfect microbeams

The nonlinear parametric dynamics of a geometrically imperfect microbeam subject to a time-dependent axial load is investigated in this paper. Based on the Euler–Bernoulli beam theory and the modified couple stress theory, continuous models for kinetic and potential energies are developed and balanced via use of Hamilton's principle. A model reduction procedure is carried out by applying the Galerkin scheme coupled with an assumed-mode technique, yielding a high-dimensional second-order reduced-order model. A linear analysis is performed upon the linear part of the reduced-order model in order to obtain the linear size-dependent natural frequencies. A nonlinear analysis is performed on the reduced-order model using the pseudo-arclength continuation method and a direct time-integration technique, yielding generalised coordinates, and hence the system parametric response. It is shown that, the steady-state frequency-response curves possess a trivial solution, both stable and unstable, throughout the solution space, separated by period-doubling bifurcation points, from which non-trivial solution branches bifurcate.

Thermo-mechanical dynamics of perfect and imperfect Timoshenko microbeams

This paper investigates the three-dimensional motion characteristics of perfect and imperfect Timoshenko microbeams under mechanical and thermal forces; the mechanical properties of the microbeam are considered temperature-dependent. The centerline of the microbeam is considered to be extensible and the equations of motion for the longitudinal, transverse, and rotational motions are derived by means of the extended Hamilton’s principle and the modified couple stress theory. These three coupled nonlinear partial differential equations are discretized by means of Galerkin’s technique, yielding a set of second-order nonlinear ordinary differential equations. These equations are solved by means of the pseudo-arclength continuation technique and via an eigenvalue analysis, for the nonlinear and linear analyses, respectively. The geometrically perfect microbeam remains stable at its original static equilibrium position up to the temperature when it loses stability by divergence via a supercritical pitchfork bifurcation; the post-buckling state is obtained and resonant response over it is analysed. For the initially imperfect microbeam, as the temperature is increased, the initial curvature amplitude increases and no instabilities occur; the resonant response of the system over the new deflected configuration is examined numerically. The effect of different parameters on the nonlinear behaviour of the system is studied.

Coupled longitudinal-transverse behaviour of a geometrically imperfect microbeam

Based on the modified couple stress theory, the coupled longitudinal-transverse nonlinear behaviour of an imperfect microbeam is investigated numerically. The equations governing the longitudinal and transverse motions are obtained using Hamilton’s principle for the system with an initial geometric imperfection. The Galerkin scheme is employed to discretize the two partial differential equations of motion, yielding a set of second-order nonlinear ordinary differential equations with coupled terms. This set is cast into new set of first-order nonlinear ordinary differential equations and solved by means of the pseudo-arclength continuation technique. The nonlinear resonant response of the system along with bifurcations are presented via frequency–response curves. Moreover, the effect of different system parameter on the frequency–response curves is highlighted.

Parameter identification of an electrically actuated imperfect microbeam

In this study we consider a microelectromechanical system (MEMS) and focus on extracting analytically the model parameters that describe its non-linear dynamic features accurately. The device consists of a clamped-clamped polysilicon microbeam electrostatically and electrodynamically actuated. The microbeam has imperfections in the geometry, which are related to the microfabrication process, resulting in many unknown and uncertain parameters of the device. The objective of the present paper is to introduce a simple but appropriate model which, despite the inevitable approximations, is able to describe and predict the most relevant aspects of the experimental response in a neighborhood of the first symmetric resonance. The modeling includes the main imperfections in the microstructure. The unknown parameters are settled via parametric identification. The approach is developed in the frequency domain and is based on matching both the frequency values and, remarkably, the frequency response curves, which are considered as the most salient features of the device response. Non-linearities and imperfections considerably complicate the identification process. Via the combined use of linear analysis and non-linear dynamic simulations, a single first symmetric mode reduced-order model is derived. Extensive numerical simulations are performed at increasing values of electrodynamic excitation. Comparison with experimental data shows a satisfactory concurrence of results not only at low electrodynamic voltage, but also at higher ones. This validates the proposed theoretical approach. We highlight its applicability, both in similar case-studies and, more in general, in 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 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 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.

Nonlinear dynamics of a geometrically imperfect microbeam based on the modified couple stress theory

The present study investigates the nonlinear dynamics of a geometrically imperfect microbeam numerically on the basis of the modified couple stress theory. Hamilton’s principle is used to obtain the nonlinear partial differential equation of motion for an initially curved beam. The equation of motion is discretized and reduced to a set of nonlinear ordinary differential equations by means of the Galerkin scheme. This set of equations is solved numerically by means of the pseudo-arclength continuation technique which allows the continuation of both stable and unstable solution branches as well as determination of different types of bifurcation. An eigenvalue analysis is also conducted to obtain the linear natural frequencies of the system. The frequency-response curves are constructed for the system with different initial imperfections. Moreover, the frequency-response curves of the system are plotted together as a specific system parameter is varied, in order to highlight the effect of each parameter on the resonant dynamics of the system.

An electrically actuated imperfect microbeam: Dynamical integrity for interpreting and predicting the device response

In this study we deal with a microelectromechanical system (MEMS) and develop a dynamical integrity analysis to interpret and predict the experimental response. The device consists of a clamped-clamped polysilicon microbeam, which is electrostatically and electrodynamically actuated. It has non-negligible imperfections, which are a typical consequence of the microfabrication process. A single-mode reduced-order model is derived and extensive numerical simulations are performed in a neighborhood of the first symmetric natural frequency, via frequency response diagrams and behavior chart. The typical softening behavior is observed and the overall scenario is explored, when both the frequency and the electrodynamic voltage are varied. We show that simulations based on direct numerical integration of the equation of motion in time yield satisfactory agreement with the experimental data. Nevertheless, these theoretical predictions are not completely fulfilled in some aspects. In particular, the range of existence of each attractor is smaller in practice than in the simulations. This is because these theoretical curves represent the ideal limit case where disturbances are absent, which never occurs under realistic conditions. A reliable prediction of the actual (and not only theoretical) range of existence of each attractor is essential in applications. To overcome this discrepancy and extend the results to the practical case where disturbances exist, a dynamical integrity analysis is developed. After introducing dynamical integrity concepts, integrity profiles and integrity charts are drawn. They are able to describe if each attractor is robust enough to tolerate the disturbances. Moreover, they detect the parameter range where each branch can be reliably observed in practice and where, instead, becomes vulnerable, i.e. they provide valuable information to operate the device in safe conditions according to the desired outcome and depending on the expected disturbances.

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 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.

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 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.

Theoretical and experimental investigation of the nonlinear response of an electrically actuated imperfect microbeam

In this study a theoretical and experimental investigation of the nonlinear response of an electrically actuated microbeam is performed. A mechanical model is proposed, which accounts for two common imperfections of microbeams, due to microfabrications, which are the compliant support conditions and the initially deformed profile. A computationally efficient single-mode reduced-order model is derived by combining the Ritz technique and the Pade approximation. Numerical simulations of the harmonic response of the device near primary resonance are shown illustrating nonlinear phenomena arising in the device response. Experimental investigation is conducted on a polysilicon imperfect microbeam confirming the simulation results. The concurrence between the theoretical results and the experimental data reveals that this model, while simple, is capable of properly capturing the response both at low and, especially, at higher electrodynamic voltages.