A micromechanical mass sensing method based on amplitude tracking within an ultra-wide broadband resonance

We propose a mass sensing scheme in which amplitude shifts within a nonlinear ultra-wide broadband resonance serve as indicators for mass detection. To achieve the broad resonance bandwidth, we considered a nonlinear design of the resonator comprised of a doubly clamped beam with a concentrated mass at its center. A reduced-order model of the beam system was constructed in the form of a discrete spring-mass system that contains cubic stiffness due to axial stretching of the beam in addition to linear stiffness (Duffing equation). The cubic nonlinearity has a stiffening effect on the frequency response causing nonlinear bending of the frequency response toward higher frequencies. Interestingly, we found that the presence of the concentrated mass broadens the resonant bandwidth significantly, allowing for an ultra-wide operational range of frequencies and response amplitudes in the proposed mass sensing scheme. A secondary effect of the cubic nonlinearity is strong amplification of the third harmonic in the beam’s response. We computationally study the sensitivity of the first and third harmonic amplitudes to mass addition and find that both metrics are more sensitive than the linearized natural frequency and that in particular, the third harmonic amplitude is most sensitive. This type of open-loop mass sensing avoids complex feedback control and time-consuming frequency sweeping. Moreover, the mass resolution is within a functional range, and the design parameters of the resonator are reasonable from a manufacturing perspective.     

Input-shaping control of nonlinear MEMS

We develop a new technique for preshaping input commands to control microelectromechanical systems (MEMS). In general, MEMS are excited using an electrostatic field which is a nonlinear function of the states and the input voltage. Due to the nonlinearity, the frequency of the device response to a step input depends on the input magnitude. Therefore, traditional shaping techniques which are based on linear theory fail to provide good performance over the whole input range. The technique we propose combines the equations describing the static response of the device, an energy balance argument, and an approximate nonlinear analytical solution of the device response to preshape the voltage commands. As an example, we consider set-point stabilization of an electrostatically actuated torsional micromirror. The shaped commands are applied to drive the micromirror to a desired tilt angle with zero residual vibrations. Simulations show that fast mirror switching operation with almost zero overshoot can be realized using this technique. The proposed methodology accounts for the energy of the significant higher modes and can be used to shape input commands applied to other nonlinear micro- and macro-systems.     

Upper and lower bounds for the pull-in parameters of a micro- or nanocantilever on a flexible support

An analytical method is proposed to accurately estimate the pull-in parameters of a micro- or nanocantilever beam elastically constrained by a rotational spring at one end. The system is actuated by electrostatic force and subject to Casimir or van der Waals forces according to the beam size. The deflection of the beam is described by a fourth-order nonlinear boundary value problem, or equivalently in terms of a nonlinear integral equation. New a priori analytical estimates on the solution from both sides are first derived and then lower and upper bounds for the pull-in parameters are obtained, with no need of solving the nonlinear boundary value problem. The lower and upper bounds turn out to be very close each other and in excellent agreement with the numerical results provided by the shooting method. The approach also provides accurate predictions for the pull-in parameters of a freestanding nanoactuator.     

Test method for measuring strength of a curved sandwich beam

A fixture for testing curved sandwich beams in flexure was designed and evaluated. The test specimen is a continuous sandwich beam consisting of a central circular 90° region connected by two straight legs. The fixture was designed according to the four-point flexure principle to produce a pure bending moment in the curved region. The validity of the test fixture in producing the desired loading was examined by fitting a curved aluminum bar of similar bending stiffness as the sandwich beams considered. Strain gage readings were successfully compared to predictions from curved homogeneous beam theory. In addition, the deflection of the beam at the loading points was analyzed using straight and curved beam theory for the various sections of the beam, and predictions were compared to measured load-displacement response. Good agreement was achieved between experimental and analytical results lending confidence to the test principle. Curved sandwich beams consisting of glass/polyester face sheets over a PVC foam core were tested to failure and the loading response of the beams and their failure behavior are discussed. It was found that the beams failed at the upper face/core interface due to radial tension stress.     

Dynamic analysis of a novel wide-tunable microbeam resonator with a sliding free-of-charge electrode

In this paper, a MEMS-based resonator with a novel effective stiffness tunability is presented. The performance of the proposed resonator is based on the transversal vibration of the two porous cantilever microbeams with a rectangular microplate at the end of the structure. The microplate as a free-of-charge slider electrode is in contact with two other fixed substrate electrodes via the thin layer of dielectric material. Applying a constant DC voltage to the two fixed electrodes leads to the movement of free electrons in the slider and eventually to the formation of two series capacitors. As a result, the slider meets a nonlinear electrostatic force proportional to the square of the applied DC voltage. It will act as a nonlinear spring with a tunable stiffness during the oscillation of the resonator. The coupled nonlinear equations governing the longitudinal and transversal vibration of the resonator are extracted in the presence of the nonlinear voltage-sliding spring. Its steady-state solution is obtained based on a physically based learning method that makes it possible to obtain frequency response for the first harmony as well as for the higher harmonies and to predict primary and secondary resonances in different harmonies of the response. The effect of the applied tuning DC voltage, the geometrical parameters of the resonator, and the cantilever's porosity on the dynamic response of the resonator are investigated. It is shown that the tuning stiffness of this voltage-sliding spring provides a highly effective solution to realize an extreme tunable range. In the end, a modified tunable structure is introduced in which the folded beams are replaced with common ones. The modified resonator by making the nonlinear behavior of the resonator least can improve its performance significantly.

     

Symmetry breaking in an initially curved pre-stressed micro beam loaded by a distributed electrostatic force

The symmetric and asymmetric buckling of an initially curved micro beam subjected to an axial pre-stressing load and transversal distributed electrostatic force is studied. The analysis is based on a reduced order (RO) model resulting from the Galerkin decomposition with buckling modes of a straight beam used as the base functions. The criteria of symmetric limit point buckling and of non-symmetric bifurcation are derived in terms of the geometric parameters of the beam and the axial load. Two symmetry breaking conditions, defining the relations between the axial load and the geometric parameters of beams for which an asymmetric response bifurcates from the symmetric one, are obtained. The necessary criterion establishes the conditions for the appearance of bifurcation points on the unstable branch of the symmetric response curve; the sufficient criterion assures a realistic asymmetric buckling bifurcating from the stable branches of the symmetric response curve. A comparison between the RO model results and those obtained by direct numerical analysis shows good agreement between the two and indicates that the obtained criteria can be used to predict symmetric and non-symmetric buckling in electrostatically actuated curved pre-stressed micro beams. It is shown that while the symmetry breaking conditions are affected by the nonlinearity of the electrostatic force, its influence is less pronounced than in the case of the symmetric snap-through criterion. The nature of the latter and the relations between it and the symmetry breaking criteria are found to go through a prominent qualitative change as the initial distance between the beam and the electrode, characterizing the electrostatic force, changes.     

Three-to-One Internal Resonances in Hinged-Clamped Beams

The nonlinear planar response of a hinged-clamped beam to a primary excitation of either its first mode or its second mode is investigated. The analysis accounts for mid-plane stretching, a static axial load and a restraining spring at one end, and modal damping. For a range of axial loads, the second natural frequency is approximately three times the first natural frequency and hence the first and second modes may interact due to a three-to-one internal resonance. The method of multiple scales is used to attack directly the governing nonlinear partial-differential equation and derive two sets of four first-order nonlinear ordinary-differential equations describing the modulation of the amplitudes and phases of the first two modes in the case of primary resonance of either the first or the second mode. Periodic motions and periodically and chaotically modulated motions of the beam are determined by investigating the equilibrium and dynamic solutions of the modulation equations. For the case of primary resonance of the first mode, only two-mode solutions are possible, whereas for the case of primary resonance of the second mode, single- and two-mode solutions are possible. The two-mode equilibrium solutions of the modulation equations may undergo a supercritical or a subcritical Hopf bifurcation, depending on the magnitude of the axial load. A shooting technique is used to calculate limit cycles of the modulation equations and Floquet theory is used to ascertain their stability. The limit cycles correspond to periodically modulated motions of the beam. The limit cycles are found to undergo cyclic-fold bifurcations and period-doubling bifurcations, leading to chaos. The chaotic attractors may undergo boundary crises, resulting in the destruction of the chaotic attractors and their basins of attraction.     

Three-to-One Internal Resonances in Parametrically Excited Hinged-Clamped Beams

The nonlinear planar response of a hinged-clamped beam to a principal parametric resonance of either its first or second mode or a combination parametric resonance of the additive type of its first two modes is investigated. The analysis accounts for mid-plane stretching, a static axial load, a restraining spring at one end, and modal damping. The natural frequency of the second mode is approximately three times the natural frequency of the first mode for a range of static axial loads, resulting in a three-to-one internal resonance. The method of multiple scales is used to attack directly the governing nonlinear integral-partial-differential equation and associated boundary conditions and derive three sets of four first-order nonlinear ordinary-differential equations describing the modulation of the amplitudes and phases of the first two modes in the cases of (a) principal parametric resonance of either the first or the second mode, and (b) a combination parametric resonance of the additive type of these modes. Periodic motions and periodically and chaotically modulated motions of the beam are determined by investigating the equilibrium and dynamic solutions of the modulation equations. For the case of principal parametric resonance of the first mode or combination parametric resonance of the additive type, trivial and two-mode solutions are possible, whereas for the case of parametric resonance of the second mode, trivial, single, and two-mode solutions are possible. The trivial and two-mode equilibrium solutions of the modulation equations may undergo either a supercritical or a subcritical Hopf bifurcation, depending on the magnitude of the axial load. For some excitation parameters, we found complex responses including period-doubling bifurcations and blue-sky catastrophes.     

Dynamics of a mass–spring–beam with 0:1:1 internal resonance using the analytical and continuation method

The nonlinear harmonic response of an autoparametric system comprised of a linear oscillator with a vertically attached flexural beam is investigated and the capability of the beam as a vibration absorber is assessed. A weak torsional spring is used for constraining the rotation of the beam giving rise to an almost non-flexural rotational mode with a low frequency. The system parameters are also tuned to enforce the zero-to-one-to-one internal resonance condition. The Lagrange’s formulation accompanied by the assumed-mode method is used to derive the discretized equations based on the order three nonlinear Euler–Bernoulli beam theory. An analytical solution is developed based on the method of multiple scales where the generalized coordinate corresponding to the non-flexural rotational mode is approximated by higher order perturbation expansion than the other coordinates, due to much larger contribution of the non-flexural rotation to the response. Comprehensive response and bifurcation analysis are performed using analytical and direct numerical solutions. The results are obtained for vertically-aligned and also initially inclined beams and various interesting behaviors are recognized for different non-dimensional system parameters. Different types of bifurcations such as the Pitch-fork, Hopf, Period-doubling and symmetry breaking bifurcations are observed in the solution of slow-flow equations and some of them are found to be beneficial for vibration absorption in a limited range of excitation amplitudes and frequencies.     

Nonlinear coupling of transverse modes of a fixed–fixed microbeam under direct and parametric excitation

Tuning of linear frequency and nonlinear frequency response of microelectromechanical systems is important in order to obtain high operating bandwidth. Linear frequency tuning can be achieved through various mechanisms such as heating and softening due to DC voltage. Nonlinear frequency response is influenced by nonlinear stiffness, quality factor and forcing. In this paper, we present the influence of nonlinear coupling in tuning the nonlinear frequency response of two transverse modes of a fixed–fixed microbeam under the influence of direct and parametric forces near and below the coupling regions. To do the analysis, we use nonlinear equation governing the motion along in-plane and out-of-plane directions. For a given DC and AC forcing, we obtain static and dynamic equations using the Galerkin’s method based on first-mode approximation under the two different resonant conditions. First, we consider one-to-one internal resonance condition in which the linear frequencies of two transverse modes show coupling. Second, we consider the case in which the linear frequencies of two transverse modes are uncoupled. To obtain the nonlinear frequency response under both the conditions, we solve the dynamic equation with the method of multiple scale (MMS). After validating the results obtained using MMS with the numerical simulation of modal equation, we discuss the influence of linear and nonlinear coupling on the frequency response of the in-plane and out-of-plane motion of fixed–fixed beam. We also analyzed the influence of quality factor on the frequency response of the beams near the coupling region. We found that the nonlinear response shows single curve near the coupling region with wider width for low value of quality factor, and it shows two different curves when the quality factor is high. Consequently, we can effectively tune the quality factor and forcing to obtain different types of coupled response of two modes of a fixed–fixed microbeam.

     

Shearing effects on the breathing mechanism of a cracked beam section in bi-axial flexure

The main purpose of this paper is to complete the works presented by Andrieux and Varé (2002) and El Arem et al. (2003) by taking into account the effects of shearing in the constitutive equations of a beam cracked section in bi-axial flexure. The paper describes the derivation of a lumped cracked beam model from the three-dimensional formulation of the general problem of elasticity with unilateral contact conditions on the crack lips. Properties of the potential energy and convex analysis are used to reduce the three-dimensional computations needed for the model identification, and to derive the final form of the elastic energy that determines the nonlinear constitutive equations of the cracked transverse section. We aim to establish a relation of behavior between the applied forces and the resulting displacements field vectors, which is compatible with the beams theory in order to allow the model exploitation for shafts dynamics analysis. The approach has been applied to the case of a cracked beam with a single crack covering the half of its circular cross section.     

Piecewise-linear restoring force surfaces for semi-nonparametric identification of nonlinear systems

A method for identifying a piecewise-linear approximation to the nonlinear forces acting on a system is presented and demonstrated using response data from a micro-cantilever beam. It is based on the Restoring Force Surface (RFS) method by Masri and Caughey, which is very attractive when initially testing a nonlinear system because it does not require the user to postulate a form for the nonlinearity a priori. The piecewise-linear fitting method presented here assures that a continuous piecewise-linear surface is identified, is effective even when the data does not cover the phase plane uniformly, and is more computationally efficient than classical polynomial based methods. A strategy for applying the method in polar form to sinusoidally excited response data is also presented. The method is demonstrated on simulated response data from a cantilever beam with a nonlinear electrostatic force, which highlights some of the differences between the local, piecewise-linear model presented here and polynomial-based models. The proposed methods are then applied to identify the force-state relationship for a micro-cantilever beam, whose response to single frequency excitation, measured with a Laser Doppler Vibrometer, contains a multitude of harmonics. The measurements suggest that an oscillatory nonlinear force acts on the cantilever when its tip velocity is near maximum during each cycle.     

Dynamic properties of interactional solutions for the (4 + 1)-dimensional Fokas equation

The internal resonances between the longitudinal and transversal oscillations of a forced Timoshenko beam with an axial end spring are studied in depth. In the linear regime, the loci of occurrence of 1 : ir, \(ir \in \mathbb {N}\), internal resonances in the parameters space are identified. Then, by means of the multiple time scales method, the 1 : 2 case is investigated in the nonlinear regime, and the frequency response functions and backbone curves are obtained analytically, and investigated thoroughly. They are also compared with finite element numerical simulations, to prove their reliability. Attention is paid to the system response obtained by varying the stiffness of the end spring, and it is shown that the nonlinear behaviour instantaneously jumps from hardening to softening by crossing the exact internal resonance value, in contrast to the singular (i.e. tending to infinity) behaviour of the nonlinear correction coefficient previously observed (without properly taking the internal resonance into account).

     

Nonlinear ferro-electro-elastic beam theory

Motivated by the uniqueness and potential of the nonlinear range of piezoelectric and ferroelectric smart materials and structures, a static physically nonlinear ferro-electro-elastic beam theory which takes the effect of domain switching into account is developed. The kinematic assumptions adopt the geometrically linear Bernoulli–Euler form for the mechanical components and a first-order theory for the electrical potential, and lay the basis for further augmentation to higher order theories. The beam theory includes the field equations that correspond to the static case, the boundary conditions and the constitutive equations of ferro-electro-elasticity. The general 3-D constitutive equations are reduced to comply with the beam theory and formulated as ordinary differential equations by means of a set of generalized electro-mechanical stiffnesses. A micromechanical constitutive model that accounts for the loading history and for the domain switching phenomenon is adopted and an iterative solution procedure that incorporates the micromechanical approach is suggested. A numerical example that demonstrates the impact of the domain switching on the nonlinear electromechanical static response of a ferro-electro-elastic beam is presented and discussed. The quantitative assessment of this behavior takes a step towards new structural applications that cope with or even take advantage of the nonlinear ferro-electro-elastic range.     

SI-Traceable Spring Constant Calibration of Microfabricated Cantilevers for Small Force Measurement

A variety of methods exist to measure the stiffness of microfabricated cantilever beams such as those used as mechanical sensors in atomic force microscopy (AFM). In order for AFM to be used as a quantitative small force measurement tool, these methods must be validated within the International System of Units (SI). To this end, two different contact techniques were used to calibrate the spring constant of a cantilever beam. First, a dynamic indentation-based method was used to measure the spring constant of a rectangular cantilever beam. These results were then compared against an SI-traceable spring constant measurement from an electrostatic force balance (EFB). The measurements agree within experimental uncertainty and within 2% for spring constants greater than 2 N/m. The use of this cantilever beam as a transfer artifact for in situ AFM cantilever calibration was then evaluated in comparison to the thermal calibration method. Excellent agreement is seen between these techniques, establishing the consistency of the thermal and dynamic indentation methods with SI-traceable contact cantilever calibration for the rectangular cantilever geometry tested. Disclaimer: This article is authored by employees of the U.S. federal government, and is not subject to copyright. Commercial equipment and materials are identified in order to adequately specify certain procedures. In no case does such identification imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose.     

Symmetry breaking in an initially curved micro beam loaded by a distributed electrostatic force

The asymmetric buckling of a shallow initially curved stress-free micro beam subjected to distributed nonlinear deflection-dependent electrostatic force is studied. In order to highlight the symmetry breaking phenomenon and the approach to its analysis, the subsidiary simplified problem of a curved beam attached to a linearly elastic foundation, and subjected to uniformly distributed “mechanical” load, which is independent of deflections, is addressed first. The analysis is based on a two degrees of freedom reduced order (RO) model resulting from the Galerkin decomposition with linear undamped eigenmodes of a straight beam used as the base functions. Simple approximate expressions are derived defining the geometric parameters of beams for which an asymmetric response bifurcates from the symmetric one. The necessary criterion establishes the conditions for the appearance of bifurcation points on the unstable branch of the symmetric limit point buckling curve; the sufficient criterion assures a realistic asymmetric buckling bifurcating from the stable branches of the curve. It is shown that while the symmetry breaking conditions are affected by the nonlinearity of the electrostatic force, its influence is less pronounced than in the case of the symmetric snap-through criterion. A comparison between the RO model results and those obtained by direct numerical analysis shows good agreement between the two and indicates that the obtained criteria can be used to predict non-symmetric buckling in electrostatically actuated bistable micro beams.     

Bistability of curved microbeams actuated by fringing electrostatic fields

In this work we investigate the feasibility of two-directional switching of an initially curved or pre-buckled electrostatically actuated microbeams using a single electrode fabricated from the same structural layer. The distributed electrostatic force, which is engendered by the asymmetry of the fringing fields in the deformed state, acts in the direction opposite to the deflection of the beam and can be effectively viewed as a reaction of a nonlinear elastic foundation with stiffness parameterized by the voltage. The reduced order model was built using the Galerkin decomposition with linear undamped modes of a straight beam as base functions and verified using the results of the numerical solution of the differential equation. The electrostatic force was approximated by means of fitting the results of three-dimensional numerical solution of the electrostatic problem. Static stability analysis reveals that the presence of the restoring electrostatic force may result in the suppression of the snap-through instability as well as in the appearance of additional stable configurations associated with higher buckling modes of the beam that are not observed in “mechanically” loaded structures. We show that two-directional switching of a pre-buckled beam between two stable configurations cannot be achieved using quasistatic loading. Furthermore, we show that switching is both associated with the dynamic snap-through mechanism and possible within certain interval of actuation voltages. Using a single-degree-of-freedom (lumped) model, estimation voltage boundaries are obtained. Theoretical results illustrate the feasibility of the suggested operational principle as an efficient mechanism in the arena of non-volatile mechanical memory devices.     

Nonlinear dynamic response of an edge-cracked functionally graded Timoshenko beam under parametric excitation

This paper investigates the nonlinear flexural dynamic behavior of a clamped Timoshenko beam made of functionally graded materials (FGMs) with an open edge crack under an axial parametric excitation which is a combination of a static compressive force and a harmonic excitation force. Theoretical formulations are based on Timoshenko shear deformable beam theory, von Karman type geometric nonlinearity, and rotational spring model. Hamilton’s principle is used to derive the nonlinear partial differential equations which are transformed into nonlinear ordinary differential equation by using the Least Squares method and Galerkin technique. The nonlinear natural frequencies, steady state response, and excitation frequency-amplitude response curves are obtained by employing the Runge–Kutta method and multiple scale method, respectively. A parametric study is conducted to study the effects of material property distribution, crack depth, crack location, excitation frequency, and slenderness ratio on the nonlinear dynamic characteristics of parametrically excited, cracked FGM Timoshenko beams.     

Investigation of size-dependent quasistatic response of electrically actuated nonlinear viscoelastic microcantilevers and microbridges

This study investigates the size-dependent quasistatic response of a nonlinear viscoelastic microelectromechanical system (MEMS) under an electric actuation. To have this problem in view, the deformable electrode of the MEMS is modelled using cantilever and doubly-clamped viscoelastic microbeams. The modified couple stress theory in conjunction with Bernoulli–Euler beam theory are used for mathematical modeling of the size-dependent instability of microsystems in the framework of linear viscoelastic theory. Simultaneous effect of electrostatic actuation including fringing field, residual stress, mid-plane stretching and Casimir and van der Waals intermolecular forces are considered in the theoretical model. A single element of the standard linear solid element is used to simulate the viscoelastic behavior. Based on the extended Hamilton’s variational principle, the nonlinear governing integro-differential equation and boundary conditions are derived. Thereafter, a new generalized differential-integral quadrature solution for the nonlinear quasistatic response of electrically actuated viscoelastic micro/nanobeams under two different boundary conditions; doubly-clamped microbridge and clamped-free microcantilever. The developed model is verified and a good agreement is obtained. Finally, a comprehensive study is conducted to investigate the effects of various parameters such as material relaxation time, durable modulus, material length scale parameter, Casimir force, van der Waals force, initial gap and beam length on the pull-in response of viscoelastic microbridges and microcantilevers in the framework of viscoelasticity.     

Primary and secondary resonance analyses of clamped–clamped micro-beams

Nonlinear harmonic vibration of a micro-electro-mechanical beam is investigated, and the micro-actuator, which is considered in this study, is a special kind of electrostatic symmetric actuators. A fully clamped micro-beam with a uniform thickness is modeled as an electrostatic micro-actuator with two symmetric potential walls. The nonlinear forced vibration of the micro-beam is analyzed, and the non-dimensional governing equation of motion, using the Galerkin method, is developed. Higher-order nonlinear terms in the equation of motion are taken into account for the first time, and the perturbation method is utilized regarding these terms and hence, all the resonant cases have been considered. The multiple scales method is employed to solve the nonlinear equations, and therefore, the problem does not deal with the large deformations. The primary and secondary resonance conditions are determined, and the corresponding secular terms in each case are recognized. Harmonic responses are obtained for different cases of resonance, and eventually, the stable and unstable portions of the responses are identified. A parametric sensitivity study is carried out to examine the effects of different parameters on the amplitude–frequency characteristic equations.