Parametric Excitation of a Microbeam-String With Asymmetric Electrodes: Multimode Dynamics and the Effect of Nonlinear Damping

2017 ◽  
Vol 139 (4) ◽  
Author(s):  
Karin Mora ◽  
Oded Gottlieb
Keyword(s):  
Evolution Equations ◽  
Multiple Scales ◽  
Orbital Stability ◽  
Nonlinear Damping ◽  
Nonlinear Beam ◽  
Reduced Order ◽  
Nonlinear Viscoelastic ◽  
Current Voltage ◽  
Spatio Temporal ◽  
Asymmetric Electrodes

The dynamic motion of a parametrically excited microbeam-string affected by nonlinear damping is considered asymptotically and numerically. It is assumed that the geometrically nonlinear beam-string, subject to only modulated alternating current voltage, is closer to one of the electrodes, thus resulting in an asymmetric dual gap configuration. A consequence of these novel assumptions is a combined parametric and hard excitation in the derived continuum-based model that incorporates both linear viscous and nonlinear viscoelastic damping terms. To understand how these assumptions influence the beam's performance, the conditions that lead to both principal parametric resonance and a three-to-one internal resonance are investigated. Such conditions are derived analytically from a reduced-order nonlinear model for the first three modes of the microbeam-string using the asymptotic multiple-scales method which requires reconstitution of the slow-scale evolution equations to deduce an approximate spatio-temporal solution. The response is investigated analytically and numerically and reveals a bifurcation structure that includes coexisting in-phase and out-of-phase solutions, Hopf bifurcations, and conditions for the loss of orbital stability culminating with nonstationary quasi-periodic solutions and chaotic strange attractors.

Nonlinear Reduced-Order Models for Aerodynamic Lift of Oscillating Airfoils

2016 ◽  
Author(s):  
Muhammad Saif Ullah Khalid ◽  
Imran Akhtar
Keyword(s):  
Numerical Simulations ◽  
Strouhal Number ◽  
Lift Force ◽  
Multiple Scales ◽  
Primary Resonance ◽  
Nonlinear Damping ◽  
Van Der Pol Oscillator ◽  
Reduced Order ◽  
Naca 0012 ◽  
Unsteady Lift

For the present study, setting Strouhal Number as the control parameter, we perform numerical simulations for the flow over oscillating NACA-0012 airfoil at Reynolds Number 103. Temporal profiles of the unsteady lift, and their respective spectra clearly indicate the solution to be a period-1 attractor for low Strouhal numbers. This study reveals that aerodynamic forces produced by the oscillating airfoils are independent of the initial kinematic conditions that proves existence of the limit cycle. Frequencies present in the oscillating lift force are composed of the fundamental harmonics, and its odd harmonics. Using these numerical simulations, we observe the shedding frequencies nearly equal to the excitation frequencies in all the cases. Hence, considering it as a primary resonance case, we model the unsteady lift force through a modified van der Pol oscillator. Using the method of multiple scales and spectral analysis of the steady-state CFD solutions, we estimate the frequencies and the damping terms in the reduced-order model. We prove the applicability of this model to all the planar motions of airfoil; heaving, pitching and flapping. With the increasing Strouhal number, the nonlinear damping terms for all types of motion approach similar magnitudes. Another important aspect in one of the currently-proposed model is capturing the time-averaged value of the aero-dynamic lift force.

Nonlinear Reduced-Order Models for Aerodynamic Lift of Oscillating Airfoils2

2017 ◽  
Vol 12 (5) ◽  
Author(s):  
Muhammad Saif Ullah Khalid ◽  
Imran Akhtar
Keyword(s):  
Numerical Simulations ◽  
Strouhal Number ◽  
Lift Force ◽  
Multiple Scales ◽  
Primary Resonance ◽  
Nonlinear Damping ◽  
Van Der Pol Oscillator ◽  
Reduced Order ◽  
Naca 0012 ◽  
Aerodynamic Lift

For the present study, setting Strouhal number as the control parameter, we perform numerical simulations for the flow over oscillating NACA-0012 airfoil at a Reynolds number of 1000. This study reveals that aerodynamic forces produced by oscillating airfoils are independent of the initial kinematic conditions suggesting the existence of limit cycle. Frequencies present in the oscillating lift force are composed of the fundamental harmonics and its odd harmonics. Using these numerical simulations, we analyze the shedding frequencies close to the excitation frequencies. Hence, considering it as a primary resonance case, we model the unsteady lift force with a modified van der Pol oscillator. Using the method of multiple scales and spectral analysis of the steady-state computational fluid dynamics (CFD) solutions, we estimate the frequencies and the damping terms in the reduced-order model (ROM). We show the applicability of this model to all planar motions of the airfoil; heaving, pitching, and flapping. With increasing the Strouhal number, the nonlinear damping terms for all types of motion approach similar magnitudes. Another important aspect in one of the proposed model is its ability to capture the time-averaged value of the aerodynamic lift force. We also notice that increase in the magnitude of the lift force is due to the effect of destabilizing linear damping parameter.

The Static and Dynamic Behavior of MEMS Arches Under Electrostatic Actuation

2009 ◽  
Author(s):  
Hassen M. Ouakad ◽  
Mohammad I. Younis ◽  
Fadi M. Alsaleem ◽  
Ronald Miles ◽  
Weili Cui
Keyword(s):  
Multiple Scales ◽  
Perturbation Technique ◽  
Dynamical Behavior ◽  
Primary Resonance ◽  
Reduced Order Model ◽  
Harmonic Load ◽  
Order Model ◽  
Reduced Order ◽  
Electrostatically Actuated ◽  
Model Equations

In this paper, we investigate theoretically and experimentally the static and dynamic behaviors of electrostatically actuated clamped-clamped micromachined arches when excited by a DC load superimposed to an AC harmonic load. A Galerkin based reduced-order model is used to discretize the distributed-parameter model of the considered shallow arch. The natural frequencies of the arch are calculated for various values of DC voltages and initial rises of the arch. The forced vibration response of the arch to a combined DC and AC harmonic load is determined when excited near its fundamental natural frequency. For small DC and AC loads, a perturbation technique (the method of multiple scales) is also used. For large DC and AC, the reduced-order model equations are integrated numerically with time to get the arch dynamic response. The results show various nonlinear scenarios of transitions to snap-through and dynamic pull-in. The effect of rise is shown to have significant effect on the dynamical behavior of the MEMS arch. Experimental work is conducted to test polysilicon curved microbeam when excited by DC and AC loads. Experimental results on primary resonance and dynamic pull-in are shown and compared with the theoretical results.

Reduced-Order Nonlinear Damping Model: Formulation and Application to Postflutter Aeroelastic Behavior

AIAA Journal ◽  
2021 ◽  
pp. 1-11
Author(s):  
X. Q. Wang ◽  
Pengchao Song ◽  
Marc P. Mignolet ◽  
P. C. Chen
Keyword(s):  
Nonlinear Damping ◽  
Model Formulation ◽  
Reduced Order ◽  
Damping Model

Full discretisation of second-order nonlinear evolution equations: strong convergence and applications

2011 ◽  
Vol 11 (4) ◽  
pp. 441-459 ◽  
Author(s):  
Etienne Emmrich ◽  
David Šiška
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Strong Convergence ◽  
Evolution Equations ◽  
Martensitic Transformations ◽  
Nonlinear Damping ◽  
Nonlinear Evolution Equations ◽  
Discrete Approximations ◽  
Partial Differential ◽  
Vibrating Membrane

Abstract Recent results on convergence of fully discrete approximations combining the Galerkin method with the explicit-implicit Euler scheme are extended to strong convergence under additional monotonicity assumptions. It is shown that these abstract results, formulated in the setting of evolution equations, apply, for example, to the partial differential equation for vibrating membrane with nonlinear damping and to another partial differential equation that is similar to one of the equations used to describe martensitic transformations in shape-memory alloys. Numerical experiments are performed for the vibrating membrane equation with nonlinear damping which support the convergence results.

scholarly journals A Class of Shock Wave Solution to Singularly Perturbed Nonlinear Time-Delay Evolution Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Yi-Hu Feng ◽  
Lei Hou
Keyword(s):  
Shock Wave ◽  
Time Delay ◽  
Wave Solution ◽  
Evolution Equations ◽  
Multiple Scales ◽  
Expansion Method ◽  
Initial Boundary ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problem ◽  
Shock Wave Solution

Nonlinear singularly perturbed problem for time-delay evolution equation with two parameters is studied. Using the variables of the multiple scales method, homogeneous equilibrium method, and approximation expansion method from the singularly perturbed theories, the structure of the solution to the time-delay problem with two small parameters is discussed. Under suitable conditions, first, the outer solution to the time-delay initial boundary value problem is given. Second, the multiple scales variables are introduced to obtain the shock wave solution and boundary layer corrective terms for the solution. Then, the stretched variable is applied to get the initial layer correction terms. Finally, using the singularly perturbed theory and the fixed point theorem from functional analysis, the uniform validity of asymptotic expansion solution to the problem is proved. In addition, the proposed method possesses the advantages of being very convenient to use.

Effects of Nonlinear Damping Washers on the Automatic Ball Balancer for Optical Disk Drives

2005 ◽  
Author(s):  
Cheng-Kuo Sung ◽  
Paul C. P. Chao ◽  
Ben-Cheng Yo
Keyword(s):  
Nonlinear Dynamics ◽  
Steady State ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Dynamic Performance ◽  
Nonlinear Damping ◽  
Disk Drives ◽  
Optical Disc ◽  
Scaled Model ◽  
Jump Phenomena

This study is devoted to explore the effect of nonlinear dynamics of damping washers on the dynamic performance of automatic ball balancer (ABB) system installed in optical disc drives. The ABB is generally used on rotational system to reduce vibration. Researches have been conducted to study the performance of the ABB by investigating the nonlinear dynamics of the system; however, the model adopted often consider the damping washer in a typical ABB suspension system as a linear one, which does not reflect the fact that the practical washers are inevitably exhibit nontrivial nonlinear dynamics at some range of operation, deviating the ABB performance away from the expecteds. In this study, a complete dynamic model of the ABB including a detailed nonlinear model of the damping washers based on experimental data for practical wahers is established. The method of multiple scales is then applied to formulate a scaled model to find all possible steady-state ball positions and analyze stabilities. It is found that with reasonable level of nonlinearity, the balancing balls of the ABB are still reside at the desired positions at steady state, rendering expected vibration reduction; however, jump phenomena also occurs as the spindle operated through natural frequency of the suspension, causing unwanted system vibrations. Numerical simulations and experiments are conducted to verify the theoretical findings. The obtained results are used to predict the level of residual vibration, with which the guidelines on choices of the nonlinear damping washers are distilled to achieve desired performance.

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