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Force inaccuracy considered in terms of displacement (limited range) for the various approximative force models. Note that as expected, the models produced by curve fitting over a specified domain exhibit the least error over the given domain.
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![Figure 6. The γ 3-λ 3 parameter space (From [16]). Those regions...](profile/Kimberly-Foster-7/publication/228683292/figure/fig2/AS:669997391151107@1536751234050/The-g-3-l-3-parameter-space-From-16-Those-regions-designated-I-and-VI-exhibit_Q320.jpg)
Electrostatically-actuated resonant microbeam devices have garnered significant attention due to their geometric simplicity and broad applicability. Recently, some of this focus has turned to comb-driven microresonators with purely-parametric excitation, as such systems not only exhibit the inherent benefits of MEMS devices, but also a general impr...
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... their name implies, these alternative models can be developed by fitting the spatially-dependent portion of the exact electrostatic force (in this case after modal truncation and projection) to a cubic function of displacement over a specified displacement operating regime, while enforcing a specified linear force constraint (as realized by manipulating coefficients contained in ν 1 , λ 1 , ν 3 and λ 3 ). The net result of such an approach, as detailed in figure 2, is an improvement in the approximate force model over the specified range. For example, the least-squares model fit over a displacement domain bounded by plus or minus one-half of the gap width, yields a maximum error of about 2.5%, approximately one- fourth of that produced by the truncated Taylor series model. ...
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Citations
... EMS with discontinuous parameters are frequently used in many engineering systems [3][4][5]. Many studies have shown that such systems may exhibit various types of behaviors [6][7][8], resulting in abrupt changes of the damping and stiffness coefficients. Yamapi et al. [9] demonstrated the complex behavior of non-smooth EMS due to the discontinuity in damping and elasticity. ...
This paper deals with the theoretical and experimental studies of non-smooth electromechanical system (EMS) actuated by Rayleigh-Duffing oscillator (RDO). The RDO exhibits sinusoidal and triangular signals depending on its nonlinear coefficient. The rate-equations of EMS under study are derived thanks to the Kirchhoff and Newton second laws. Based on the Routh-Hurwitz criterion, the stability of the two equilibrium points found are reported where one is unconditionally unstable and other one is stable or unstable based on systems parameters. The influence of the variation of the system parameters on the dynamics of the EMS is numerically studied by using the amplitude curves of the current and displacement of the EMS. Numerical results show that when the RDO excites the EMS with a sinusoidal signal, the movement of the EMS is either periodic oscillations or chaotic oscillations depending on the value of the elasticity coefficient. When the RDO excites the EMS with triangular signal, relaxation oscillations, periodic bursting oscillations or chaotic bursting oscillations are observed in the EMS depending on the value of the elasticity coefficient. The numerical analysis results are confirmed by the microcontroller implementation of EMS actuated by RDO.
... The study of oscillations of electrically actuated beams is often based on a quasilinear approach. In this case, the essentially nonlinear term is replaced by a few first terms of its Maclaurin series [7][8][9][10][11][12]. The harmonic balance method is used in Zhang and Meng [7], eigenmode expansion-in Refs. ...
... The harmonic balance method is used in Zhang and Meng [7], eigenmode expansion-in Refs. [9,[11][12][13]. Paper [14] is based on a linearization with respect to a static equilibrium position. ...
... The results given in Refs. [8,9,11,14] are based on incorrect original equations (for correct simplified boundary value problems describing nonlinear vibrations of beams, see Andrianov et al. [22]). ...
In this paper, oscillations of the electrically actuated microbeam are considered, described by strongly nonlinear differential equations. One of the essentially nonlinear effects is the pull‐in phenomenon, that is, the transition of the oscillatory regime to the attraction regime. This effect is considered in the paper on the basis of various approximate models. In particular, the error of replacing the original nonlinearity by its expansion in the Maclaurin series is estimated. It is shown that such an approximation can only be used for voltage values that are far from critical. Estimates are also given for initial displacements and velocities, which guarantee an oscillating character of solutions. The electrically activated oscillations of the microbeam are considered taking into account the geometric nonlinearity within the framework of the Kirchhoff model. The dependence of pull‐in value on geometric nonlinearity is investigated.
... Now, we can employ the Galerkin approach to convert the partial differential equation into a nonlinear ordinary differential equation. This procedure entails separating the temporal and spatial components of d t ( x, t) as outlined in [41]. ...
Model predictive control (MPC) is a cutting-edge control technique, but its susceptibility to inaccuracies in the model remains a challenge for embedded systems. In this study, we propose a data-driven MPC framework to address this issue and achieve robust and adaptable performance. Our framework involves systematically identifying system dynamics and learning the MPC policy through function approximations. Specifically, we introduce a system identification method based on the Deep neural network (DNN) and integrate it with MPC. The function approximation capability of DNN enables the controller to learn the nonlinear dynamics of the system then the MPC policy is established based on the identified model. Also, through an added control term the robustness and convergence of the closed-loop system are guaranteed. Then the governing equation of a non-local strain gradient (NSG) nano-beam is presented. Finally, the proposed control scheme is used for vibration suppression in the NSG nano-beam. To validate the effectiveness of our approach, the controller is applied to the unknown system, meaning that solely during the training phase of the neural state-space-based model we relied on the data extracted from the time history of the beam’s deflection. The simulation results conclusively demonstrate the remarkable performance of our proposed approach in effectively suppressing vibrations.
... In this study, we use the Galerkin approach to transform the partial differential equation into a nonlinear ordinary differential equation. To do this, we decompose the temporal and spatial terms of ̅ , ̅ as follows [36]: ...
This research investigates the stabilization and control of an uncertain Euler–Bernoulli nano-beam with fixed ends. The governing partial differential equations of motion for the nano-beam are derived using Hamilton’s principle and the non-local strain gradient theory. The Galerkin method is then applied to transform the resulting dimensionless partial differential equation into a nonlinear ordinary differential equation. A novel fault-tolerant terminal sliding mode control technique is proposed to address the uncertainties inherent in micro/nano-systems and the potential for faults and failures in control actuators. The proposed controller includes a finite time estimator, the stability of which and the convergence of the error dynamics are established using the Lyapunov theorem. The significance of this study lies in its application to the field of micro/nano-mechanics, where the precise control and stabilization of small-scale systems is crucial for the development of advanced technologies such as nano-robotics and micro-electromechanical systems (MEMS). The proposed control technique addresses the inherent uncertainties and potential for faults in these systems, making it a valuable choice for practical applications. The simulation results are presented to demonstrate the effectiveness of the proposed control scheme and the high accuracy of the estimation algorithm.
... Perturbation methods such as the Method of Multiple Scales (MMS) [24][25][26][27][28][29] and averaging methods [30,31] have been widely applied to the investigation of the dynamic behaviour of parametrically excited systems. It has been shown that these methods can be useful techniques for predicting the response of such systems, in particular in a frequency range close to the principal parametric resonance [32]. ...
... Form ¼ 0:0569 (Fig. 6a), using Eqs. (30), (10), and (11), the undamped natural frequency of the system is obtained as f 0 ¼x 0 =2p ¼ 4:2000 HZ. The other system parameters are obtained asx 0 ¼ 3:1712, P ¼ À0:0445, g ¼ 27:5099 and a ¼ 6:6884 using Eqs. ...
The response of a parametrically excited cantilever beam (PECB) with a tip mass is investigated in this paper. The paper is mainly focused on accurate prediction of the response of the system, in particular, its hardening and softening characteristics when linear damping is considered. First, the method of varying amplitudes (MVA) and the method of multiple scales (MMS) are employed. It is shown that both Duffing nonlinearity and nonlinear inertia terms govern the hardening or softening behaviour of a PECB. MVA results show that for frequencies around the principal parametric resonance, the term containing a linear combination of nonlinear inertia and Duffing nonlinearity in the frequency response equation can tend to zero, resulting in an exponential growth of the vibrations, and results are validated by numerical results obtained from direct integration (DI) of the equation of motion, while the MMS fails to predict this critical frequency. A criterion for determining the hardening and softening characteristics of PECBs is developed and presented using the MVA. To verify the results, experimental measurements for a PECB with a tip mass are presented, showing good agreement with analytical and numerical results. Furthermore, it is demonstrated that the mass added at the cantilever tip can change the system characteristics, enhancing the softening behaviour of the PECB. It is shown that, within the frequency range considered, increasing the value of the tip mass decreases the amplitude response of the system and broadens the frequency range in which a stable response can exist.
... Most real systems are modeled by nonlinear differential equations, which play a crucial role in natural and physical simulations. In the last decades, significant progress has been made in using nonlinear equations to understand and accurately predict the stability of the motion of an oscillator [1][2][3][4][5][6]. ...
... They are developed based on micro-electronic technology. Consequently, they are used, for example, in the production of optical sensors, micro vibrators, and pressure sensors, and they also have many applications in engineering disciplines such as aerospace, optics, biomedicine, micro-switches, transistors, accelerometers, pressure sensors, micro-mirrors, micro-pumps, micro-grippers, bio-MEMS, and so on [1][2][3][4][5][6][7][8][9][10][11]. They are small integrated devices that combine electrical and mechanical components. ...
... Mathematics 2022, 10, x FOR PEER REVIEW 2 of 12 micro-mirrors, micro-pumps, micro-grippers, bio-MEMS, and so on [1][2][3][4][5][6][7][8][9][10][11]. They are small integrated devices that combine electrical and mechanical components. ...
In this paper, an analytical technique based on the global residue harmonic balance method (GRHBM) is applied in order to obtain higher-order approximate analytical solutions of an electrostatically actuated micro-beam. To illustrate the applicability and accuracy of the method, a high level of accuracy was established for the analytical solutions by comparing the results of the solutions with the numerical solution as well as the already published literature, such as the variational approach (VA), Hamiltonian approach (HA), energy balance method (EBM), and homotopy analysis method (HAM). It is shown that the GRHB method can be easily applied to nonlinear problems and provides solutions with a higher precision than existing methods. The obtained analytical expressions are employed to study the effects of axial force, initial gape, and electrostatic load on nonlinear frequency.
... Different types of the numerical methods used to calculate Static and Dynamic Pull-in Voltage and displacement are available in literature [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. The authors like Batra, Porfiri and Spinello [20], Rhoads, Shaw and Turner [21] have carried out wide review of the modeling of MEMS. ...
The Micro-electromechanical devices operate under the stable operating range. The demarcating Voltage and the displacement separating the stable and unstable operating range are called Pull-in parameters. The accurate determination of these parameters is one of the essential step in the MEMS design. In this paper the Static Analysis of electrostatically actuated micro-cantilever beam is carried out to obtain Pull-in Voltage and displacement considering fringing field effects using Galerkin Method. The fringing field effects are incorporated by considering three different fringing field Models and the results are compared.
... Unlike direct excitation applied to the resonators, parametric excitation as an alternative method is suggested to actuate M/NEMS resonators, which modulates the entire restoring force (including linear and nonlinear terms) as a function of time. Some studies have considered this technique [24,34,[267][268][269][270][271][272][273] to achieve large distinctive responses since many excitation methods including electrostatic actuation in M/NEMS field allow natural parametric excitation. The most basic model for this class of resonators is related to the linear Mathieu equation including a time-varying stiffness component. ...
Micro/nanoelectromechanical systems (M/NEMS), especially micro/nanomechanical resonators, provide an unprecedented platform for investigating a wide range of applications in both fundamental physical phenomena (such as quantum‐level problems) and engineering and applied sciences (like sensing physical quantities and signal processing). In sensing context, these tiny resonators are invaluable tools for detecting weak forces and single molecules. Measuring such small variations in sub‐nanometer amplitudes at high frequencies has created many practical challenges. Therefore, maximizing the responsivity to a specified input parameter and minimizing the responsivity to other inputs such as noise are considered as the optimal design for the excellent performance in nanoresonators. The present review aims to summarize some of the recent progress in this rapidly advancing field. Specifically, more attention is paid to the topics related to the dynamical behaviors of micro/nanoresonator and their practical applications. First, the theory behind micro/nanoresonators is described. Then a summary is presented of the basic concepts in resonant devices. In addition, several commonly dynamical techniques to enhance sensitivity are introduced. Finally, the devices are classified based on linear and nonlinear, single and array, symmetric and asymmetric, and frequency shift‐based and amplitude shift‐based resonators, along with the relative advantages and disadvantages of each one in engineering applications. Micro/nanomechanical resonators are invaluable tools for detecting weak forces and single molecules. This review deals with the linear and nonlinear dynamical behaviors of micro/nanoresonators and their practical applications. A comprehensive classification of the nanomechanical resonators based on the output signal, degrees of freedom, single and array resonators, and symmetric and asymmetric geometries is presented.
... In this regard, various experimental [12,34], analytical [11,55,60,59], [61,14,42,47], and numerical [8,2,4,6,7,55,64,88] investigations have been conducted on the pull-in instability behavior of micro/nano-beams subjected to electrostatic loadings. Numerical methods including finite difference method (FDM) [51,55,62], finite element method (FEM) [33,45], differential quadrature method (DQM) [33,74], [80], [4], and reduced order model (ROD) [14,44,54,82] can predict the pull-in voltage with a very high degree of accuracy; however, these methods are computationally intensive. ...
Because of fasttechnological development, electrostatic nanoactuator devices like nanosensors, nanoswitches, and nanoresonators are highly considered by scientific community. Thus, this article presents a new solution technique in solving highly nonlinear integro-differential equation governing electrically actuated nanobeams made of functionally graded material. The modified couple stress theory and Gurtin–Murdoch surface elasticity theory are coupled together to capture the size effects of the nanoscale thin beam in the context of Euler–Bernoulli beam theory. For accurate modelling, all the material properties of the bulk and surface continuums of the FG nanoactuator are varied continuously in thickness direction according to power law. The nonlinearity arising from the electrostatic actuation, fringing field, mid-plane stretching effect, axial residual stress, Casimir dispersion, and van der Waals forces are considered in mathematical formulation. The nonlinear nonclassical equilibrium equation of FG nanobeam-based actuators and associated boundary conditions are exactly derived using Hamilton principle. The new solution methodology is combined from three phases. The first phase applies Galerkin method to get an integro-algebraic equation. The second one employs particle swarm optimization method to approximate the integral terms (i.e. electrostatic force, fringing field, and intermolecular forces) to non-integral cubic algebraic equation. Then, solved the system easily in last phase. The resulting algebraic model provides means for obtaining critical deflection, pull-in voltage, detachment length, minimum gap, and freestanding effects. A reasonable agreement is found between the results obtained from the present method and those in the available literature. A parametric study is performed to investigate the effects of the gradient index, material length scale parameter, surface energy, intermolecular forces, initial gap, and beam length on the pull-in response and freestanding phenomena of fully clamped and cantilever FG nanoactuators.
... A shortcoming of this approach is that it restricts the applicability of resonators, since many resonators for mass sensing usually operate in loss media such as air or water [4] , and raises costs in the packaging phase. The other approach exploits the nonlinear excitation methods, such as the electrostatic actuation, parametric excitation [5] and secondary resonances [6] , to enhance the oscillation ampli- * Corresponding author. ...
Noise-induced motions are a significant source of uncertainty in the response of electrostatic MEMS where electrical and mechanical sources contribute to noise and can result in sudden and dramatic loss of stability. In this manuscript, we present an analysis for the stabilization of an electrostatically actuated MEMS resonator in the presence of noise processes using a stochastic optimal control scheme. A stochastic lumped-mass model that accounts for the uncertainty in mass, mechanical restoring force, bias voltage, and AC voltage amplitude of the resonator is presented. The Ito^ equations, describing the averaged modulations of the resonator's total energy and phase difference, are obtained via the stochastic averaging of energy envelope. The stochastic optimal control law aiming at minimization the pull-in probability of the resonator is determined via the stochastic dynamic programming equations associated with the Ito^ equations. We show that the resulting stochastic optimal feedback control, with a careful selection of its control gain, can be used to suppress the noise-activated pull-in instability of the disturbed resonator. The impact of the time delay inherent in the feedback control input on the control effectiveness is investigated. In addition, the control scheme also shows superior performance in counteracting the deterministic pull-in instability of the resonator operating in the dynamic pull-in frequency band. Good agreement between the theoretical results and numerical simulation is demonstrated.
