A Static and Nonlinear Dynamic Analysis of Resonant Microbeams

2007 ◽  
Author(s):  
Samir A. Emam ◽  
Mahmoud E. Khater ◽  
Emil H. Gad
Keyword(s):  
Periodic Orbits ◽  
Shooting Method ◽  
Nonlinear Dynamic Analysis ◽  
Beam Theory ◽  
Electrostatic Forces ◽  
Reduced Order ◽  
Resonant Sensors ◽  
Discretization Technique ◽  
The Stability ◽  
Euler Bernoulli Beam

An investigation into the response of microbeams to DC and AC electric actuation is presented. The beam is modeled according to the Euler-Bernoulli beam theory and small strains and moderate rotation approximations are assumed. The governing equation is a nonlinear integral-partial-differential equation in space and time. The model accounts for mid-plane stretching, applied axial load, DC electrostatic forces, and AC harmonic forces. A reduced-order model based on the Galerkin discretization technique is introduced to simulate the behavior of microswitches and resonant sensors. The static behavior of the microbeam under electrostatic forces is studied and compared to the results available in the literature. The dynamic behavior of resonant microbeams under AC harmonic forces is investigated. An analytical solution for the vibration modes and natural frequencies of the microbeam around its statically deflected position is obtained. A shooting method is used to numerically integrate the nonlinear discretized equations and obtain periodic orbits of the response. The stability of these periodic orbits is investigated using Floquet theory. The sensitivity of the device to small-amplitude excitations is also investigated.

scholarly journals Investigation of the Dynamics of a Clamped-Clamped Microbeam Near the Third Mode Using a Partial Electrode

2014 ◽  
Author(s):  
Karim M. Masri ◽  
Mohammad I. Younis
Keyword(s):  
Shooting Method ◽  
Electrode Configuration ◽  
Mass Detection ◽  
Order Model ◽  
Response Curves ◽  
Reduced Order ◽  
Resonant Sensors ◽  
The Third ◽  
Higher Order Modes ◽  
Multi Mode

We present an investigation of the dynamics of a clamped-clamped microbeam excited electrostatically near its third mode. To maximize the response at the third mode, a partial electrode configuration is utilized. A multi-mode Galerkin method is used to develop a reduced order model (ROM) of the beam. A shooting method to find the periodic motion is utilized to generate frequency response curves. The curves show hardenining behavior and dynamic pull-in. We show that the dynamic amplitude of the partial configuration is higher than that of a full electrode configuration. These results are promising for the use of higher-order modes for mass detection and for ultra sensitive resonant sensors.

Asymmetric Bifurcation of Initially Curved Nanobeam

2015 ◽  
Vol 82 (9) ◽  
Author(s):  
X. Chen ◽  
S. A. Meguid
Keyword(s):  
Symmetry Breaking ◽  
Degrees Of Freedom ◽  
Surface Elasticity ◽  
Beam Theory ◽  
Surface Effects ◽  
Beam Model ◽  
Bernoulli Beam ◽  
Reduced Order ◽  
Euler Bernoulli Beam ◽  
Bifurcation Behavior

In this paper, we investigate the asymmetric bifurcation behavior of an initially curved nanobeam accounting for Lorentz and electrostatic forces. The beam model was developed in the framework of Euler–Bernoulli beam theory, and the surface effects at the nanoscale were taken into account in the model by including the surface elasticity and the residual surface tension. Based on the Galerkin decomposition method, the model was simplified as two degrees of freedom reduced order model, from which the symmetry breaking criterion was derived. The results of our work reveal the significant surface effects on the symmetry breaking criterion for the considered nanobeam.

On the Chaotic Vibrations of Electrostatically Actuated Arch Micro/Nano Resonators: A Parametric Study

2015 ◽  
Vol 25 (08) ◽  
pp. 1550106 ◽  
Author(s):  
Farid Tajaddodianfar ◽  
Mohammad Reza Hairi Yazdi ◽  
Hossein Nejat Pishkenari
Keyword(s):  
Nonlinear Dynamics ◽  
Nonlinear Partial Differential Equation ◽  
Beam Theory ◽  
Reduced Order ◽  
Electrostatically Actuated ◽  
Chaotic Vibrations ◽  
Numerical Tools ◽  
The Galerkin Method ◽  
Euler Bernoulli Beam ◽  
Insight Into

Motivated by specific applications, electrostatically actuated bistable arch shaped micro-nano resonators have attracted growing attention in the research community in recent years. Nevertheless, some issues relating to their nonlinear dynamics, including the possibility of chaos, are still not well known. In this paper, we investigate the chaotic vibrations of a bistable resonator comprised of a double clamped initially curved microbeam under combined harmonic AC and static DC distributed electrostatic actuation. A reduced order equation obtained by the application of the Galerkin method to the nonlinear partial differential equation of motion, given in the framework of Euler–Bernoulli beam theory, is used for the investigation in this paper. We numerically integrate the obtained equation to study the chaotic vibrations of the proposed system. Moreover, we investigate the effects of various parameters including the arch curvature, the actuation parameters and the quality factor of the resonator, which are effective in the formation of both static and dynamic behaviors of the system. Using appropriate numerical tools, including Poincaré maps, bifurcation diagrams, Fourier spectrum and Lyapunov exponents we scrutinize the effects of various parameters on the formation of chaotic regions in the parametric space of the resonator. Results of this work provide better insight into the problem of nonlinear dynamics of the investigated family of bistable micro/nano resonators, and facilitate the design of arch resonators for applications such as filters.

scholarly journals Existence and Stability of the Periodic Orbits Induced by Grazing Bifurcation in a Cantilever Beam System with Single Rigid Impacting Constraint

2021 ◽  
Author(s):  
Run Liu ◽  
Yuan Yue ◽  
Jianhua Xie
Keyword(s):  
Periodic Orbits ◽  
Cantilever Beam ◽  
Shooting Method ◽  
Jacobian Matrix ◽  
Mapping Method ◽  
Grazing Bifurcation ◽  
Beam System ◽  
Existence And Stability ◽  
The Stability ◽  
The Relationship

Abstract Grazing which can induce many nonclassical bifurcations, is a special dynamic phenomenon in some non-smooth dynamical systems such as vibro-impact systems with clearance. In this paper, the existence and stability of the periodic orbits induced by the grazing bifurcation in a cantilever beam system with impacts are uncovered. Firstly, the Poincaré mapping of the system is obtained by the discontinuous mapping method. Secondly, the periodic orbits are determined by means of shooting method, and Jacobian matrix in the case of non-impact is obtained subsequently. Thirdly, for various impacting patterns, a combination of inhomogeneous equations and inequations is obtained to determine the existence of period orbits after grazing. Furthermore, the stability criterion of the grazing-induced periodic orbits is given. Numerical results verify the effectiveness of theoretical analysis. What’s more, we also give a conjecture about the relationship between eigenvalues and the type of periodic orbits when eigenvalues are imaginary numbers.

COALESCENCE OF THE TWO SECONDARY RESPONSES IN COUPLED DUFFING EQUATIONS

2012 ◽  
Vol 22 (06) ◽  
pp. 1250149
Author(s):  
TING-YU LAI ◽  
PI-CHENG TUNG ◽  
YUNG-CHIA HSIAO
Keyword(s):  
Periodic Orbits ◽  
Harmonic Balance ◽  
Shooting Method ◽  
Harmonic Balance Method ◽  
The Novel ◽  
Duffing Equations ◽  
Duffing System ◽  
Parametric Continuation ◽  
Continuation Algorithm ◽  
The Stability

The novel coalescence of the secondary responses for the coupled Duffing equations are observed in this study. Two secondary responses that do not bifurcate from the primary responses merge into one due to saddle-node bifurcation generation within a specific parameter range. The frequency responses of the coupled Duffing equations are calculated using the harmonic balance method while the periodic orbits are detected by the shooting method. The stability of the periodic orbits is determined on utilizing Floquet theory. The parametric continuation algorithm is used to obtain the bifurcation points and bifurcation lines for a Duffing system with two varying parameters. The analytical results demonstrate the novel phenomenon that occurs in the Duffing equations.

Dynamic Analysis of Three Parallel Beams With Electrostatic Force Interaction

2011 ◽  
Author(s):  
Pezhman A. Hassanpour ◽  
Patricia M. Nieva ◽  
Amir Khajepour
Keyword(s):  
Time Domain ◽  
Stability Condition ◽  
Initial Conditions ◽  
Electrostatic Force ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Stability Margin ◽  
Modal Damping ◽  
The Stability ◽  
Euler Bernoulli Beam

In this paper, the dynamics of a micro-machined structure with three parallel cantilevers is investigated. The cantilevers are electrically charged and apply electrostatic force to each other. The governing equations of motion are derived using Euler-Bernoulli beam theory and considering structural modal damping. The stability condition of the beams for various electric charges is also studied. In addition, the equations of motion are integrated to obtain the response of the beams in time-domain for a range of initial conditions. This response is used to study the behavior of the beams at the stability margin. The end application of the structure under investigation is in the device characterization. The dynamic stability condition and time-domain responses are used to investigate the reliability of the characterization. Once translated back to physical quantities, these results can be used for improving the measurements.

scholarly journals Stability Analysis of a Periodic Fluid-Conveying Heterogeneous Nanotube System

2020 ◽  
Vol 33 (6) ◽  
pp. 756-769
Author(s):  
Jiayin Dai ◽  
Yongshou Liu ◽  
Guojun Tong
Keyword(s):  
Dynamic Stiffness ◽  
Beam Theory ◽  
Longitudinal Direction ◽  
Length Ratio ◽  
Governing Equations ◽  
Periodic Distribution ◽  
The Stability ◽  
Material Length ◽  
Euler Bernoulli Beam ◽  
Selection Of

AbstractIn this paper, the stability of a periodic heterogeneous nanotube conveying fluid is investigated. The governing equations of the nanotube system are derived based on the nonlocal Euler–Bernoulli beam theory. The dynamic stiffness method is employed to analyze the natural frequencies and critical flow velocities of the heteronanotube. The results and discussions are presented from three aspects which reveal the influences of period number, material length ratio and boundary conditions. In particular, we make comparisons between the heterogeneous nanotubes with periodic structure and the homogeneous ones with the same integral values of material properties along the longitudinal direction to isolate the influences of periodic distribution. According to the simulation results, we can conclude that with a proper selection of period number in terms of length ratio, the stability of the constructed nanotube can be improved.

Static Response of Microbeams due to Capillary and Electrostatic Forces

2015 ◽  
Author(s):  
Ahmad M. Bataineh ◽  
Hassen M. Ouakad ◽  
Mohammad I. Younis
Keyword(s):  
Electrostatic Force ◽  
Capillary Forces ◽  
Electrostatic Forces ◽  
Restoring Force ◽  
Nonlinear Beam ◽  
Reduced Order ◽  
Electrode Size ◽  
Beam Equations ◽  
Beam Parameters ◽  
The Stability

Micro-sensors or micro-switches usually operate under the effect of electrostatic force and could face some environmental effects like humidity, which may lead to condensation underneath the beams and create strong capillary forces. Those tiny structures are principally made of microbeams that can undergo instabilities under the effect of those created huge capillary forces. In fact, during the fabrication of microbeams, there is an important step to separate the beam from its substrate (wet etching). After this step, the microstructure is dried, which may causes the onset of some droplets of water trapped underneath the beam that could bring about a huge capillary force pulling it toward its substrate. If this force is bigger than the microbeam’s restoring force, it will become stuck to the substrate. This paper investigates the instability scenarios of both clamped-clamped (straight and curved) and cantilever (straight and curled) microbeams under the effect of capillary and/or electrostatic forces. The reduced order modeling (ROM) based on the Galerkin procedure is used to solve the nonlinear beam equations. The non-ideal boundaries are modeled by adding springs. The volume of the fluid between the beam and the substrate underneath it is varied and the relation between the volume of the water and the stability of the beam is shown. An analysis for the factors of which should be taken in to consideration in the fabrication processes to overcome the instability due to huge capillary forces is done. Also the size of the electrode for the electrostatic force is varied to show the effect on the micro-switch stability. A variation of the pull-in voltage with some specific beam parameters and with more than one case of electrode size is shown. It is found that capillary forces have a pronounced effect on the stability of microbeams. It is also found that the pull-in length decreases as the electrode size increases. It is also shown that the pull-in voltage decreases as the amount of fluid underneath the beam increases.

scholarly journals Analysis on Free Vibration and Stability of Rotating Composite Milling Bar with Large Aspect Ratio

2020 ◽  
Vol 10 (10) ◽  
pp. 3557
Author(s):  
Jingmin Ma ◽  
Jianfeng Xu ◽  
Yongsheng Ren
Keyword(s):  
Aspect Ratio ◽  
Free Vibration ◽  
Beam Theory ◽  
Inertial Effect ◽  
Large Aspect Ratio ◽  
Internal Damping ◽  
Gyroscopic Effect ◽  
The Stability ◽  
The Time Domain ◽  
Euler Bernoulli Beam

The free vibration behavior and stability of composite milling bar with large aspect ratio are analyzed. Specifically, the free vibration equations are derived based on Euler Bernoulli beam theory and Hamilton’s principle and solved by Galerkin method. In addition, to investigate the stability of the cutting system with a rotating composite milling bar, this study develops an analytical model for regeneration and cutting force fluttering, in which the internal damping, external damping, gyroscopic effect, and inertial effect are considered. The subsequent stability lobes are obtained by the time domain method. Then, the stability of composite and conventional metal milling bar is compared. The effects of internal damping, external viscous damping, ply angle, gyroscopic effect, and inertial effect on cutting stability are analyzed.

Reduced Order Model of Coaxial Vibrations of Electrostatically Actuated Double Wall Carbon Nanotubes: Frequency Response of Primary Resonance

2018 ◽  
Author(s):  
Ezequiel Juarez ◽  
Dumitru I. Caruntu
Keyword(s):  
Carbon Nanotubes ◽  
Frequency Response ◽  
Multiple Scales ◽  
Primary Resonance ◽  
Approximate Solutions ◽  
Reduced Order ◽  
Electrostatically Actuated ◽  
The Stability ◽  
Euler Bernoulli Beam ◽  
Steady State Solutions

In this paper, the Reduced Order Method (ROM) and the Method of Multiple Scales (MMS) are used to investigate the influences of dimensionless damping and voltage parameters on the amplitude-frequency response of an electrostatically actuated double-walled carbon nanotube (DWCNT). The forces responsible for the nonlinearities in the vibrational behavior are intertube van der Waals and electrostatic forces. Soft AC excitation and small viscous damping forces are assumed. Herein, the coaxial case is investigated. In this mode of vibration, the outer and inner carbon nanotubes move synchronously (in-phase) with the same maximum tip deflection. The DWCNT structure is modelled as a cantilever beam with Euler-Bernoulli beam assumptions since the DWCNT is characterized with high length-diameter ratio. The results shown assume steady-state solutions in the first-order MMS solution. The analytical approximate solutions provided by MMS are validated numerically by two-term (2T) Time Reponses and AUTO-07P. The two methods in this paper are found to be in excellent agreement at lower amplitudes. Additionally, the two methods are assessed for their advantages and limitations. The importance of the results in this paper are the effect of damping and voltage on the stability of the DWCNT vibration.

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