Application of homotopy analysis method in studying dynamic pull-in instability of microsystems

In this study, homotopy analysis method is used to derive analytic solutions to predict dynamic pull-in instability of electrostatically-actuated microsystems. The model considers midplane stretching, initial stress, distributed electrostatic force and fringing fields effect. Influences of different parameters on dynamic pull-in instability are investigated. Results are in good agreement with numerical and experimental findings.     

An alternative reduced order model for electrically actuated micro-beams under mechanical shock

This study focuses on the effect of mechanical shock on dynamic pull-in instability of eclectically actuated micro-beams through an alternative reduced order model (ROM). The model's predictions for dynamic pull-in voltages are compared with available finite element (FE) results and six modes Galerkin approximations in the literature. It is shown that present results for high shock accelerations agree with FE predictions better than those obtained using six modes approximations. Furthermore, the present model can remove the limitation of previous methods in capturing dynamic pull-in instability for cases under enormous shock accelerations.     

Lyapunov exponents in discrete modelling of a cantilever beam impacting on a moving base

The dynamic behaviour of a cantilever beam of an unnegligible large mass and with a concentrated mass fixed at its end, which impacts on a movable base according to Hertz's damp law, is studied. A new finite element reference model of the system and its lower-dimensional substitutive models with one degree or two degrees of freedom are developed. The qualitative-type as well as quantitative-type applicability limits of these substitutive models are discussed - the latter ones are described in terms of the corresponding spectra of Lyapunov exponents.     

Lyapunov spectrum determination from the FEM simulation of a chaotic advecting flow

The problem of the determination of the Lyapunov spectrum in chaotic advection using approximated velocity fields resulting from a standard FEM method is investigated. A fourth order Runge–Kutta scheme for trajectory integration is combined with a third order Jacobian matrix method with QR ‐factorization. After checking the algorithm on the standard Lorenz and coupled quartic oscillator systems, the method is applied to a model 3‐D steady flow for which an analytical expression is known. Both linear and quadratic approximated velocity fields succeed in predicting the Lyapunov exponents as well as describing the chaotic or regular regions inside the flow with satisfactory accuracy. A more realistic flow is then studied in order to delineate the possible limitations of the approach. Copyright © 2005 John Wiley & Sons, Ltd.     

Non-linear dynamic instability of a double-sided nano-bridge considering centrifugal force and rarefied gas flow

Double-sided electromechanical nano-bridges can potentially be used as angular speed sensors and accelerometers in rotary systems such as turbine blades and vacuum pumps. In such applications, the influences of the centrifugal force and rarefied flow should be considered in the analysis. In the present study, the non-linear dynamic pull-in instability of a double-sided nano-bridge is investigated incorporating the effects of angular velocity and rarefied gas damping. The non-linear governing equation of the nanostructure is derived using Euler-beam model and Hamilton׳s principle including the dispersion forces. The strain gradient elasticity theory is used for modeling the size-dependent behavior of the system. The reduced order method is also implemented to discretize and solve the partial differential equation of motion. The influences of damping, centrifugal force, length scale parameters, van der Waals force and Casimir attraction on the dynamic pull-in voltage are studied. It is found that the dispersion and centrifugal forces decrease the pull-in voltage of a nano-bridge. Dynamic response of the nano-bridge is investigated by plotting time history and phase portrait of the system. The validity of the proposed method is confirmed by comparing the results from the present study with the experimental and numerical results reported in the literature.     

The analysis of the spectrum of Lyapunov exponents in a two-degree-of-freedom vibro-impact system

In the study of dynamical systems, the spectrum of Lyapunov exponents has been shown to be an efficient tool for analyzing periodic motions and chaos. So far, different calculating methods of Lyapunov exponents have been proposed. Recently, a new method using local mappings was given to compute the Lyapunov exponents in non-smooth dynamical systems. By the help of this method and the coordinates transformation proposed in this paper, we investigate a two-degree-of-freedom vibro-impact system with two components. For this concrete model, we construct the local mappings and the Poincaré mapping which are used to describe the algorithm for calculating the spectrum of Lyapunov exponents. The spectra of Lyapunov exponents for periodic motions and chaos are computed by the presented method. Moreover, the largest Lyapunov exponents are calculated in a large parameter range for the studied system. Numerical simulations show the success of the improved method in a kind of two-degree-of-freedom vibro-impact systems.     

The matric algorithm of Lyapunov exponent for the experimental data obtained in dynamic analysis

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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.     

Practical Evaluation of Invariant Measures for the Chaotic Response of a Two-Frequency Excited Mechanical Oscillator

This paper presents results which characterize the chaotic response of alow-dimensional mechanical oscillator. An experimental system based on acart rolling on a two-well potential surface has been shown to closelyapproximate a modified form of Duffing's equation. Two-frequency forcingis applied, providing a useful means of varying the dimension of theresponse. Computation of correlation dimension and Lyapunov spectra areperformed on both experimental and numerical data in order to assess theutility of these measures in a practical setting. A specific focus isthe distinction between subharmonic and quasi-periodic forcing, sincethis has a subtle, and interesting, effect on the subsequent dynamics.The results tend to highlight the statistical nature of the measures andthe caution that should be used in their interpretation.     

Lyapunov functions for a non-linear model of the X-ray bursting of the microquasar GRS 1915+105

This paper introduces a biparametric family of Lyapunov functions for a non-linear mathematical model based on the FitzHugh-Nagumo equations able to reproduce some main features of the X-ray bursting behaviour exhibited by the microquasar GRS 1915+105. These functions are useful to investigate the properties of equilibrium points and allow us to demonstrate a theorem on the global stability. The transition between bursting and stable behaviour is also analyzed.     

An eigenanalysis-based bifurcation indicator proposed in the framework of a reduced-order modeling technique for non-linear structural analysis

The Koiter–Newton method is a reduced order modeling technique which allows us to trace efficiently the entire equilibrium path of a non-linear structural analysis. In the framework of buckling the method is capable to handle snap-back and snap-through phenomena but may fail to predict reliably bifurcation branches along the equilibrium path. In this contribution we extend the original Koiter–Newton approach with a reliable and accurate bifurcation indicator which is based on an eigenanalysis of the reduced order tangent stiffness matrix. The proposed indicator has a negligible numerical effort since all computations refer to the reduced order model which is typically of very small dimension. The extension allows the identification of bifurcation points and a tracing of corresponding bifurcation branches in each sector of the equilibrium path. The performance of the method in terms of reliability, accuracy and computational effort is demonstrated with several examples.     

On the convergence and causality of a frequency domain method for dynamic structural analysis

Venanico-Filho et al. developed an elegant matrix formulation for dynamic analysis by frequency domain (FD), but the convergence, causality and extended period need further refining. In the present paper, it was argued that: (1) under reasonable assumptions (approximating the frequency response function by the discrete Fourier transform of the discretized unitary impulse response function), the matrix formulation by FD is equivalent to a circular convolution; (2) to avoid the wraparound interference, the excitation vector and impulse response must be padded with enough zeros; (3) provided that the zero padding requirement satisfied, the convergence and accuracy of direct time domain analysis, which is equivalent to that by FD, are guaranteed by the numerical integration scheme; (4) the imaginary part of the computational response approaching zero is due to the continuity of the impulse response functions. The English text was polished by Yunming Chen     

Symmetric mode instability of the slipflow model for flows in a microchannel with a vanishing Reynolds number

The slipflow model is usually used to study microflows when the Knudsen number lies between 0.01 and 0.1. The instability due to microscale effect seems to have never been studied before. In this paper we present preliminary results for the instability (not physical instability) of this model when applied to microchannel flow with a vanishing Reynolds number. The present paper is restricted to symmetrical mode. Both first-order and second-order slip boundary conditions will be considered. The project supported by the National Natural Science Foundation of China (10025210) and Tsinghua Project for Basic Research (2002)     

An investigation on the lift force of a wing pitching in dynamic stall for a comfort control vessel

On the basis of a comfort control system for ocean vessels, the control forces and moments in the form of lift forces from active wings are of important interest. In an ocean vessel comfort control system, active wings or fins are commonly used and constantly adjust their angles of attack to produce optimal sea-keeping conditions. The unsteady nature of the flow field around a wing, and the behaviour of the generated lift force must be understood in order to optimize the comfort control system. This paper presents experimental data on the flow past a pitching wing, paying particular attention to the lagging effects between the fluid dynamic lift force and the motion of the wing at large angles of attack as a function of peak angle of attack and reduced frequency of oscillation. The range of motion investigated has been chosen according to the applicability of a comfort control wing surface. Numerical data is also included to aid explanation on some of the witnessed phenomena.     

Symmetric-Galerkin boundary element analysis of the dynamic T-stress for the interaction of a crack with an auxetic inclusion

In this work, the symmetric-Galerkin boundary element method (SGBEM) for 2-D elastodynamics in the Laplace-space frequency domain (Laplace domain) is employed to study the dynamic stress intensity factors (DSIFs) and the dynamic T-stress (DTS) during the interaction between a crack and an auxetic inclusion under impact loading conditions. It is found that, while the auxeticity has virtually no effect on the DSIFs, its influence on the DTS is noticeable. This finding is particularly important as it implies the imperative need of fracture criteria based on both the DSIFs and DTS for predicting crack growth in composite materials with auxetic phases.     

A constrained theory of a Cosserat point for the numerical solution of dynamic problems of non-linear elastic rods with rigid cross-sections

A constrained theory of a Cosserat point has been developed for the numerical solution of non-linear elastic rods. The cross-sections of the rod element are constrained to remain rigid but tangential shear deformations and axial extension are admitted. As opposed to the more general theory with deformable cross-sections, the kinetic coupling equations in the numerical formulation of the constrained theory are expressed in terms of the simple physical quantities of force and mechanical moment applied to the common ends of neighboring elements. Also, in contrast with standard finite element methods, the Cosserat element uses a direct approach to the development of constitutive equations. Specifically the kinetic quantities are determined by algebraic expressions which are obtained by derivatives of a strain energy function. Most importantly, no integration is needed over the element region. A number of example problems have been considered which indicate that the constrained Cosserat element can be used to model large deformation dynamic response of non-linear elastic rods.     

Numerical investigations of the vortex interactions for a flow over a pitching foil at different stages

The fluid–structure interaction is investigated numerically for a two-dimensional flow (Re=2.5·10⁶) over a sinusoid-pitching foil by the SST (Shear Stress Transport) k–ω model. Although discrepancies in the downstroke phase, which are also documented in other numerical studies, are observed by comparing with experimental results, our current numerical results are sufficient to predict the mean features and qualitative tendencies of the dynamic stall phenomenon. These discrepancies are evaluated carefully from the numerical and experimental viewpoints.In this study, we have utilized Λ, which is the normalized second invariant of the velocity gradient tensor, to present the evolution of the Leading Edge Vortex (LEV) and Trailing Edge Vortex (TEV). The convective, pressure, and diffusion terms during the dynamic stall process are discussed based on the transport equation of Λ. It is found that the pressure term dominates the rate of the change of the rotation strength inside the LEV. This trend can hardly be observed directly by using the vorticity transport equation due to the zero baroclinic term for the incompressible flow.The mechanisms to delay the stall are categorized based on the formation of the LEV. At the first stage before the formation of the LEV in the upper surface, the pitching foil provides extra momentum into the fluid flows to resist the flow separation, and hence the stall is delayed. At the second stage, a low-pressure area travels with the evolution of the LEV such that the lift still can be maintained. Three short periods at the second stage corresponds to different flow patterns during the dynamic stall, and these short periods can be distinguished according to the trend of the pressure variation inside the LEV. The lift stall occurs when a reverse flow from the lower surface is triggered during the shedding of the LEV. For a reduced frequency k_f=0.15, the formation of the TEV happens right after the lift stall, and the lift can drop dramatically. With a faster reduced frequency k_f=0.25, the shedding of the LEV is postponed into the downstroke, and the interaction between the LEV and TEV becomes weaker correspondingly. Thus, the lift drops more gently after the stall. In order to acquire more reliable numerical results within the downstroke phase, the Large Eddy Simulation (LES), which is capable of better predictions for the laminar-to-turbulent transition and flow reattachment process, will be considered as the future work.