scholarly journals On the Nonlinear Dynamics of a Doubly Clamped Microbeam Near Primary Resonance

2017 ◽  
Vol 139 (4) ◽  
Author(s):  
Nizar R. Jaber ◽  
Karim M. Masri ◽  
Mohammad I. Younis
Keyword(s):  
Frequency Response ◽  
Primary Resonance ◽  
Single Frequency ◽  
Response Curves ◽  
Shooting Technique ◽  
Softening Behavior ◽  
Dc Bias ◽  
The Stability ◽  
Good Agreement ◽  
Structural Layer

This work aims to investigate theoretically and experimentally various nonlinear dynamic behaviors of a doubly clamped microbeam near its primary resonance. Mainly, we investigate the transition behavior from hardening, mixed, and then softening behavior. We show in a single frequency–response curve, under a constant voltage load, the transition from hardening to softening behavior demonstrating the dominance of the quadratic electrostatic nonlinearity over the cubic geometric nonlinearity of the beam as the motion amplitudes becomes large, which may lead eventually to dynamic pull-in. The microbeam is fabricated using polyimide as a structural layer coated with nickel from top and chromium and gold layers from the bottom. Frequency sweep tests are conducted for different values of direct current (DC) bias revealing hardening, mixed, and softening behavior of the microbeam. A multimode Galerkin model combined with a shooting technique are implemented to generate the frequency–response curves and to analyze the stability of the periodic motions using the Floquet theory. The simulated curves show a good agreement with the experimental data.

The Dynamics of a Doubly Clamped Microbeam Near the Primary Resonance: Experimental and Analytical Investigation

2015 ◽  
Author(s):  
Karim M. Masri ◽  
Nizar R. Jaber ◽  
Mohammad I. Younis
Keyword(s):  
Stable State ◽  
Primary Resonance ◽  
Noise Signal ◽  
Analytical Investigation ◽  
Unstable State ◽  
Response Curves ◽  
Softening Behavior ◽  
First Time ◽  
Multi Mode ◽  
Good Agreement

We present experimental and analytical investigation of the dynamics of a doubly clamped microbeam near its primary resonance. The microbeam is excited electrostatically by an electrode on the first half of the beam. These microbeams are fabricated using polyimide as structural layer coated with nickel from top and chromium and gold layers from bottom. A noise signal is applied to experimentally detect the natural frequencies. Then, frequency sweep tests are generated for various values of DC bias revealing hardening, transition, and softening behavior of the microbeam. We report for the first time the transition from lower stable state, to unstable state, and then to large stable state experimentally. A multi-mode Galerkin method is used to develop a reduced order model (ROM) of the beam. Shooting method is used to find the periodic motion and is utilized to generate frequency response curves. The curves show good agreement with the experimental results with hardening behavior at lower DC voltage then softening at higher voltage loads and dynamic pull-in.

Nonlinear Finite Element-Based Path Following of Periodic Solutions

2011 ◽  
Author(s):  
Andrea Arena ◽  
Giovanni Formica ◽  
Walter Lacarbonara ◽  
Harry Dankowicz
Keyword(s):  
Finite Element ◽  
Vector Field ◽  
Frequency Response ◽  
Periodic Solutions ◽  
Parametric Resonance ◽  
Primary Resonance ◽  
Path Following ◽  
State Variables ◽  
Response Curves ◽  
Space Dependence

A computational framework is proposed to path follow the periodic solutions of nonlinear spatially continuous systems and more general coupled multiphysics problems represented by systems of partial differential equations with time-dependent excitations. The set of PDEs is cast in first order differential form (in time) u˙ = f(u,s,t;c) where u(s,t) is the vector collecting all state variables including the velocities/time rates, s is a space coordinate (here, one-dimensional systems are considered without lack of generality for the space dependence) and t denotes time. The vector field f depends, in general, not only on the classical state variables (such as positions and velocities) but also on the space gradients of the leading unknowns. The space gradients are introduced as part of the state variables. This is justified by the mathematical and computational requirements on the continuity in space up to the proper differential order of the space gradients associated with the unknown position vector field. The path following procedure employs, for the computation of the periodic solutions, only the evaluation of the vector field f. This part of the path following procedure within the proposed combined scheme was formerly implemented by Dankowicz and coworkers in a MATLAB software package called COCO. The here proposed procedure seeks to discretize the space dependence of the variables using finite elements based on Lagrangian polynomials which leads to a discrete form of the vector field f. A concurrent bifurcation analysis is carried out by calculating the eigenvalues of the monodromy matrix. A hinged-hinged nonlinear beam subject to a primary-resonance harmonic transverse load or to a parametric-resonance horizontal end displacement is considered as a case study. Some primary-resonance frequency-response curves are calculated along with their stability to assess the convergence of the discretization scheme. The frequency-response curves are shown to be in close agreement with those calculated by direct integration of the PDEs through the FE software called COMSOL Multiphysics. Besides primary-resonance direct forcing conditions, also parametric forcing causing the principal parametric resonance of the lowest two bending modes is considered through construction of the associated transition curves. The proposed approach integrates algorithms from the finite element and bifurcation domains thus enabling an accurate and effective unfolding of the bifurcation and post-bifurcation scenarios of nonautonomous PDEs with the underlying structures.

The Dynamics of MEMS Arches of Non-Ideal Boundary Conditions

2011 ◽  
Author(s):  
Sami A. Alkharabsheh ◽  
Mohammad I. Younis
Keyword(s):  
Boundary Conditions ◽  
Natural Frequencies ◽  
Electrostatic Force ◽  
Dynamical Behavior ◽  
Experimental Investigations ◽  
Harmonic Load ◽  
Ideal Boundary ◽  
Shooting Technique ◽  
Softening Behavior ◽  
The Stability

We present an investigation into the dynamics of MEMS arches when actuated electrically including the effect of their flexible (non-ideal) supports. First, the eigenvalue problem of a nonlinear Euler-Bernoulli shallow arch with torsional and transversal springs at the boundaries is solved analytically. Several results are shown to demonstrate the possibility of tuning the theoretically obtained natural frequencies of an arch to match the experimentally measured. Then, simulation results are shown for the forced vibration response of an arch when excited by a DC electrostatic force superimposed to an AC harmonic load. Shooting technique is utilized to find periodic motions. The stability of the captured periodic motion is examined using the Floquet theory. The results show several jumps in the response during snap-through motion and pull-in. Theoretical and experimental investigations are conducted on a microfabricated curved beam actuated electrically. Results show softening behavior and superharmonic resonances. It is demonstrated that non-ideal boundary conditions can have significant effect on the qualitative dynamical behavior of the MEMS arch, including its natural frequencies, snap-through behavior, and dynamic pull-in.

Nonlinear Two-to-One Internal Resonance for Broadband Energy Harvesting

2015 ◽  
Author(s):  
D. X. Cao ◽  
S. Leadenham ◽  
A. Erturk
Keyword(s):  
Energy Harvesting ◽  
Frequency Response ◽  
Internal Resonance ◽  
Numerical Solutions ◽  
Primary Resonance ◽  
Frame Structure ◽  
Response Curves ◽  
Current Effort ◽  
Bandwidth Enhancement ◽  
Piezoelectric Coupling

The transformation of waste vibration energy into low-power electricity has been heavily researched to enable self-sustained wireless electronic components. Monostable and bistable nonlinear oscillators have been explored by several researchers in an effort to enhance the frequency bandwidth of operation. Linear two degree of freedom (2-DOF) configurations as well as combination of a nonlinear single-DOF harvester with a linear oscillator to constitute a nonlinear 2-DOF harvester have also been explored to develop broadband energy harvesters. In the present work, the concept of nonlinear internal resonance in a continuous frame structure is explored for broadband energy harvesting. The L-shaped beam-mass structure with quadratic nonlinearity was formerly studied in the nonlinear dynamics literature to demonstrate modal energy exchange and the saturation phenomenon when carefully tuned for two-to-one internal resonance. In the current effort, piezoelectric coupling is introduced, and electromechanical equations of the L-shaped energy harvester are employed to explore the primary resonance behaviors around the first and the second linear natural frequencies for bandwidth enhancement. Simulations using approximate analytical frequency response equations as well as time-domain numerical solutions reveal that 2-DOF configuration with quadratic and two-to-one internal resonance could extend the bandwidth enhancement capability. Both electrical power and shunted vibration frequency response curves of steady-state solutions are explored in detail. Effects of various electromechanical system parameters, such as piezoelectric coupling and load resistance, on the overall dynamics of the internal resonance energy harvesting system are reported.

Vibration Control by Time Delayed Linear and Non-Linear Acceleration Feedback

2014 ◽  
Vol 592-594 ◽  
pp. 2107-2111 ◽  
Author(s):  
Ranjan Kumar Mitra ◽  
Atul Krishna Banik ◽  
Shymal Chatterjee
Keyword(s):  
Frequency Response ◽  
Duffing Oscillator ◽  
Primary Stability ◽  
Linear Acceleration ◽  
Response Curves ◽  
Gradual Reduction ◽  
Acceleration Feedback ◽  
Good Agreement ◽  
Peak Value ◽  
Appreciable Reduction

The forced Duffing oscillator is investigated by time-delayed linear and nonlinear acceleration feedbacks. The method of Slowly Varying Parameter (SVP) can efficiently develop frequency response curves with stability of solutions. Appreciable reduction in peak value of response and gradual reduction in the skew-ness in frequency response curve is observed with the introduction of gain and delay. For some values of control parameters the entrainment phenomena are also observed. Appropriate choice of gain and delay parameters are made from the primary stability zone of linear stability analysis graph. The results obtained by this method are compared with numerical integration (NI) method and they are in good agreement.

Transverse Vibrations of a Centrally Clamped Rotating Circular Disk

1999 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Adel Jilani ◽  
Piergiuseppe Manzione
Keyword(s):  
Multiple Scales ◽  
Primary Resonance ◽  
Circular Disk ◽  
Qualitative Behavior ◽  
Equilibrium Solutions ◽  
Transverse Loads ◽  
Transverse Vibrations ◽  
Critical Rotational Speed ◽  
The Stability ◽  
Good Agreement

Abstract The transverse vibrations of a circular disk of uniform thickness rotating about its axis with constant angular velocity are analyzed. The results specialized to the linear case of disks clamped at the center and free at the periphery are in good agreement with those reported in the literature. The natural frequencies of spinning hard and floppy disks are obtained for various nodal diameters and nodal circles. Primary resonance is shown to occur at the critical rotational speed at which, in the linear analysis, the spinning disk is unable to support arbitrary spatially fixed transverse loads. Using the method of multiple scales, we determine a set of four nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of two interacting modes. The symmetry of the system and the loading conditions are reflected in the symmetry of the modulation equations. They are reduced to an equivalent set of two first-order equations whose equilibrium solutions are determined analytically. The stability characteristics of these solutions is studied; the qualitative behavior of the response is independent of the mode being considered.

Subharmonic Excitation for Electrically Actuated Microbeams

2005 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Mohammad I. Younis
Keyword(s):  
Frequency Response ◽  
Rf Mems ◽  
Electrostatic Force ◽  
Primary Resonance ◽  
Resonance Excitation ◽  
Global Dynamics ◽  
Response Curves ◽  
Electrically Actuated ◽  
Subharmonic Excitation ◽  
Order Of Magnitude

We present analysis of the global dynamics of electrically actuated microbeams under subharmonic excitation. The microbeams are excited by a DC electrostatic force and an AC harmonic force with a frequency tuned near twice their fundamental natural frequencies. We show that the dynamic pull-in instability can occur in this case for an electric load much lower than that predicted with static analysis and the same order-of-magnitude as that predicted in the case of primary-resonance excitation. We show that, once the subharmonic resonance is activated, all frequency-response curves reach pull-in, regardless of the magnitude of the AC forcing. Our results show a limited influence of the quality factor on the frequency response. This result and the fact that the frequency-response curves have very steep passband-to-stopband transitions make the combination of a DC voltage and a subhormonic of order one-half a promising candidate for designing improved high-sensitive RF MEMS filters.

Nonlinear liquid sloshing in a square tank subjected to obliquely horizontal excitation

2012 ◽  
Vol 700 ◽  
pp. 304-328 ◽  
Author(s):  
Takashi Ikeda ◽  
Raouf A. Ibrahim ◽  
Yuji Harata ◽  
Tasuku Kuriyama
Keyword(s):  
Theoretical Analysis ◽  
Frequency Response ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Excitation Frequency ◽  
Frequency Range ◽  
Response Curves ◽  
Chaotic Motions ◽  
Nonlinear Responses ◽  
Good Agreement

AbstractNonlinear responses of surface waves in rigid square and nearly square tanks partially filled with liquid subjected to obliquely horizontal, sinusoidal excitation are investigated theoretically and experimentally. Two predominant modes of sloshing are significantly coupled nonlinearly because their natural frequencies are nearly identical resulting in 1:1 internal resonance. Therefore, if only one of these modes is directly excited, the other mode is indirectly excited due to the nonlinear coupling. In the nonlinear theoretical analysis, the modal equations of motion are derived for the two predominant sloshing modes as well as five higher sloshing modes. The linear viscous terms are incorporated in order to consider the damping effect of sloshing. The expressions for the frequency response curves are determined using van der Pol’s method. The influences of the excitation direction and the aspect ratio of the tank cross-section on the frequency response curves are numerically examined. Planar and swirl motions of sloshing, and Hopf bifurcations followed by amplitude modulated motions including chaotic motions, are predicted when the excitation frequency is close to one of the natural frequencies of the two predominant sloshing modes. Lyapunov exponents are calculated and reveal the excitation frequency range over which liquid chaotic motions occur. In addition, bifurcation sets are shown to clarify the influences of the parameters on the change in the structural stability. The theoretically predicted results are in good agreement with the measured data, thus the theoretical analysis was experimentally validated.

Dynamics of Variable-Geometry Electrostatic Microactuators

2006 ◽  
Author(s):  
F. Najar ◽  
S. Choura ◽  
E. M. Abdel-Rahman ◽  
S. El-Borgi ◽  
A. H. Nayfeh
Keyword(s):  
Frequency Response ◽  
Cross Section ◽  
Limit Cycles ◽  
Floquet Theory ◽  
Differential Quadrature Method ◽  
Variable Geometry ◽  
Response Curves ◽  
Differential Integral Equation ◽  
Nonlinear Algebraic System ◽  
The Stability

This paper investigates the dynamic behavior of a microbeam-based electrostatic microactuator. The cross-section of the microbeam under consideration varies along its length. A mathematical model, accounting for the system nonlinearities due to mid-plane stretching and electrostatic forcing, is adopted and used to examine the microbeam dynamics. The Differential Quadrature Method (DQM) and Finite Difference Method (FDM) are used to discretize the partial-differential-integral equation representing the microbeam dynamics. The resulting nonlinear algebraic system is solved for the limit cycles of various microstructure geometries under combined DC-AC loads and the stability of these limit cycles is examined using Floquet theory. Results are presented to show the effect of variations in the geometry on the frequency-response curves of the microactuator. We examine the effect of varying the gap size and the microbeam thickness and width on the frequency-response curves for hardening-type and softening-type behaviors. We found that it is possible to tune the geometry of the microactuator to eliminate dynamic pull-in.

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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