Showing papers on "Multiple-scale analysis published in 2011"

Electromechanical Modeling and Nonlinear Analysis of Axially Loaded Energy Harvesters

Abstract: To maximize the electromechanical transduction of vibratory energy harvesters, the resonance frequency of the harvesting device is usually tuned to the excitation frequency. To achieve this goal, some concepts call for utilizing an axial static preload to soften or stiffen the structure (Leland and Wright, 2006, "Resonance Tuning of Piezoelectric Vibration Energy Scavenging Generators Using Compressive Axial Preload, " Smart Mater. Struct., 15, pp. 1413-1420; Morris et al., 2008, "A Resonant Frequency Tunable, Extensional Mode Piezoelectric Vibration Harvesting Mechanism, " Smart Mater. Struct., 17, p. 065021). For the most part, however, models used to describe the effect of the axial preload on the harvester's response are linear lumped-parameter models that can hide some of the essential features of the dynamics and, sometimes, oppose the experimental trends. To resolve this issue, this study aims to develop a comprehensive understanding of energy harvesting using axially loaded beams. Specifically, using nonlinear Euler-Bernoulli beam theory, an electromechanical model of a clamped-clamped energy harvester subjected to transversal excitations and static axial loading is developed and discretized using a Galerkin expansion. Using the method of multiple scales, the general nonlinear physics of the system is investigated by obtaining analytical expressions for the steady-state response amplitude, the voltage drop across a resistive load, and the output power. These theoretical expressions are then validated against experimental data. It is demonstrated that in addition to the ability of tuning the harvester to the excitation frequency via axial load variations, the axial load aids in (i) increasing the electric damping in the system, thereby enhancing the energy transfer from the beam to the electric load, (ii) amplifying the effect of the external excitation on the structure, and (iii) enhancing the effective nonlinearity of the device. These factors combined can increase the steady-state response amplitude, output power, and bandwidth of the harvester.

Nonlinear Dynamics of Spring Softening and Hardening in Folded-MEMS Comb Drive Resonators

Abstract: This paper studies analytically and numerically the spring softening and hardening phenomena that occur in electrostatically actuated microelectromechanical systems comb drive resonators utilizing folded suspension beams. An analytical expression for the electrostatic force generated between the combs of the rotor and the stator is derived and takes into account both the transverse and longitudinal capacitances present. After formulating the problem, the resulting stiff differential equations are solved analytically using the method of multiple scales, and a closed-form solution is obtained. Furthermore, the nonlinear boundary value problem that describes the dynamics of inextensional spring beams is solved using straightforward perturbation to obtain the linear and nonlinear spring constants of the beam. The analytical solution is verified numerically using a Matlab/Simulink environment, and the results from both analyses exhibit excellent agreement. Stability analysis based on phase plane trajectory is also presented and fully explains previously reported empirical results that lacked sufficient theoretical description. Finally, the proposed solutions are, once again, verified with previously published measurement results. The closed-form solutions provided are easy to apply and enable predicting the actual behavior of resonators and gyroscopes with similar structures.

Nonlinear dynamic response of axially moving, stretched viscoelastic strings

Abstract: The dynamical response of axially moving, partially supported, stretched viscoelastic belts is investigated analytically in this paper. The Kelvin–Voigt viscoelastic material model is considered and material, not partial, time derivative is employed in the viscoelastic constitutive relation. The string is considered as a three part system: one part resting on a nonlinear foundation and two that are free to vibrate. The tension in the belt span is assumed to vary periodically over a mean value (as it occurs in real mechanisms), and the corresponding equation of motion is derived by applying Newton’s second law of motion for an infinitesimal element of the string. The method of multiple scales is applied to the governing equation of motion, and nonlinear natural frequencies and complex eigenfunctions of the system are obtained analytically. Regarding the resonance case, the limit-cycle of response is formulated analytically. Finally, the effects of system parameters such as axial speed, excitation characteristics, viscousity and foundation modulus on the dynamical response, natural frequencies and bifurcation points of system are presented.

Vibrations and Stability of an Axially Moving Rectangular Composite Plate

Abstract: The vibrations and stability are investigated for an axially moving rectangular antisymmetric cross-ply composite plate supported on simple supports. The partial differential equations governing the in-plane and out-of-plane displacements are derived by the balance of linear momentum. The natural frequencies for the in-plane and out-of-plane vibrations are calculated by both the Galerkin method and differential quadrature method. It can be found that natural frequencies of the in-plane vibrations are much higher than those in the out-of-plane case, which makes considering out-of-plane vibrations only is reasonable. The instability caused by divergence and flutter is discussed by studying the complex natural frequencies for constant axial moving velocity. For the axially accelerating composite plate, the principal parametric and combination resonances are investigated by the method of multiple scales. The instability regions are discussed in the excitation frequency and excitation amplitude plane. Finally, the axial velocity at which the instability region reaches minimum is detected.

Nonlinear vibrations of the blade with varying rotating speed

Abstract: Nonlinear behaviors of thin-walled Euler-Bernoulli beams with varying rotating speed which are attached to a rigid hub are investigated. Centrifugal force, aerodynamic load and the perturbed angular speed due to the inconstant air velocity are considered. The nonlinear factors are involved in displacement-strain relationships. The nonlinear governing partial differential equations of high-speed rotating thin-walled beam are established by using Hamiltonian Principle. Then, the ordinary differential equations of the rotating thin-walled beam are obtained by employing Galekin's approach during which Galekin's modes satisfy corresponding boundary conditions. The four-dimensional nonlinear averaged equations are obtained by applying the method of multiple scales. In this paper, the case of 1∶1 internal resonance is only considered. The results of the numerical simulation show that there exits complicated nonlinear behaviors in thin-walled Euler-Bernoulli beams with varying rotating speed.

Approximate and numerical analysis of nonlinear forced vibration of axially moving viscoelastic beams

Abstract: Steady-state periodical response is investigated for an axially moving viscoelastic beam with hybrid supports via approximate analysis with numerical confirmation. It is assumed that the excitation is spatially uniform and temporally harmonic. The transverse motion of axially moving beams is governed by a nonlinear partial-differential equation and a nonlinear integro-partial-differential equation. The material time derivative is used in the viscoelastic constitutive relation. The method of multiple scales is applied to the governing equations to investigate primary resonances under general boundary conditions. It is demonstrated that the mode uninvolved in the resonance has no effect on the steady-state response. Numerical examples are presented to demonstrate the effects of the boundary constraint stiffness on the amplitude and the stability of the steady-state response. The results derived for two governing equations are qualitatively the same, but quantitatively different. The differential quadrature schemes are developed to verify those results via the method of multiple scales.

Electromechanical Modeling and Normal Form Analysis of an Aeroelastic Micro-Power Generator

Abstract: Combining theories in continuous-systems vibrations, piezoelectricity, and fluid dynamics, we develop and experimentally validate an analytical electromechanical model to predict the response behavior of a self-excited micro-power generator. Similar to music- playing harmonica that create tones via oscillations of reeds when subjected to air blow, the proposed device uses flow-induced self-excited oscillations of a piezoelectric beam embedded within a cavity to generate electric power. To obtain the desired model, we adopt the non- linear EulerBernoulli beam's theory and linear constitutive relationships. We use Hamilton's principle in conjunction with electric circuits theory and the inextensibility condition to derive the partial differential equation that captures the transversal dynamics of the beam and the ordinary differential equation governing the dynamics of the harvesting circuit. Using the steady Bernoulli equation and the continuity equation, we further relate the exciting pressure at the surface of the beam to the beam's deflection, and the inflow rate of air. Subsequently, we employ a Galerkin's descritization to reduce the order of the model and show that a single- mode reduced-order model of the infinite-dimensional system is sufficient to predict the response behavior. Using the method of multiple scales, we develop an approximate analytical solution of the resulting reduced-order model near the stability boundary and study the normal form of the resulting bifurcation. We observe that a Hopf bifurcation of the super- critical nature is responsible for the onset of limit-cycle oscillations.

On the accuracy of the multiple scales method for non-linear vibrations of doubly curved shallow shells

Abstract: Non-linear free and forced vibrations of doubly curved isotropic shallow shells are investigated via multi-modal Galerkin discretization and the method of multiple scales. Donnell’s non-linear shallow shell theory is used and it is assumed that the shell is simply supported with movable edges. By deriving two different forms of the stress function, the equations of motion are reduced to a system of infinite non-linear ordinary differential equations with quadratic and cubic non-linearities. A quadratic relation between the excitation and the fundamental frequency is considered and it is shown that, although in case of hardening non-linearities the results resemble those found via numerical integration or continuation softwares, in case of softening non-linearity the solution breaks down as the amplitude becomes larger than the thickness. Results reveal that, expressing the relation between the excitation and fundamental frequency in this form, which was considered by many researchers as a useful tool in analyzing strong non-linear oscillators, yields in spurious results when the non-linearity becomes of softening type.

On primary resonances of weakly nonlinear delay systems with cubic nonlinearities

Abstract: We implement the method of multiple scales to investigate primary resonances of a weakly nonlinear second-order delay system with cubic nonlinearities. In contrast to previous studies where the implementation is confined to the assumption of linear delay terms with small coefficients (Hu et al. in Nonlinear Dyn. 15:311, 1998; Ji and Leung in Nonlinear Dyn. 253:985, 2002), in this effort, we propose a modified approach which alleviates that assumption and permits treating a problem with arbitrarily large gains. The modified approach lumps the delay state into unknown linear damping and stiffness terms that are functions of the gain and delay. These unknown functions are determined by enforcing the linear part of the steady-state solution acquired via the method of multiple scales to match that obtained directly by solving the forced linear problem. We examine the validity of the modified procedure by comparing its results to solutions obtained via a harmonic balance approach. Several examples are discussed demonstrating the ability of the proposed methodology to predict the amplitude, softening-hardening characteristics, and stability of the resulting steady-state responses. Analytical results also reveal that the system can exhibit responses with different nonlinear characteristics near its multiple delay frequencies.

On nonlinear response near-half natural frequency of electrostatically actuated microresonators

Abstract: This paper deals with the nonlinear response of electrostatically actuated cantilever beam microresonators near-half natural frequency. A first-order fringe correction of the electrostatic force, viscous damping, and Casimir effect are included in the model. Both forces, electrostatic and Casimir, are nonlinear. The dynamics of the resonator is investigated using the method of multiple scales (MMS) in a direct approach of the problem. The reduced order model (ROM) method, based on Galerkin procedure, is used as well. Steady-state motions are found. Numerical simulations are conducted for uniform microresonators. The influences of damping, actuation, and fringe effect on the resonator response are found.

The method of multiple scales for forced oscillators with some real-power nonlinearities in the stiffness and damping force

Abstract: The steady-state response of forced damped nonlinear oscillators is considered, the restoring force of which has a non-negative real power-form nonlinear term and the linear term of which can be negative, zero or positive. The damping term is also assumed in a power form, thus covering polynomial and non-polynomial damping. The method of multiple scales with a new expansion parameter is presented in order to cover the cases when the nonlinearity is not necessarily small. Amplitude-frequency equations and approximate solutions for the steady-state response at the frequency of excitation are obtained and compared with numerical results, showing good agreement.

Stability Analysis of a Nonlinear Jointed Beam under Distributed Follower Force

Abstract: This paper deals with the stability of a beam subjected to thrust. The thrust acts upon the structure as a follower non-conservative force, thus the structure can lose its stability by flutter or divergence depending on the system parameters. The model consists of two beams interconnected by a nonlinear joint. The joint is a combination of linear and nonlinear springs and a damper. Follower force is assumed to be linearly distributed along the length of beam, so the governing equation has variable coefficients, so that only an approximate solution is possible. We divided the beam into a number of segments so that force distributions could be approximated as constants and then we used the method of multiple scales to obtain the analytical solution of the system. The flutter and divergence and post-critical behavior are then obtained.

Non-Linear Transverse Vibrations and 3:1 Internal Resonances of a Tensioned Beam on Multiple Supports

Abstract: In this study, nonlinear transverse vibrations of a tensioned Euler-Bernoulli beam resting on multiple supports are investigated. The immovable end conditions due to simple supports cause stretching of neutral axis and introduce cubic nonlinearity to the equations of motion. Forcing and damping effects are included in the analysis. The general arbitrary number of support case is investigated and 3, 4, and 5 support cases analyzed in detail. A perturbation technique (the method of multiple scales) is applied to the equations of motion to obtain approximate analytical solutions. 3:1 internal resonance case is also considered. Natural frequencies and mode shapes for the linear problem are found for the tensioned beam. Nonlinear frequencies are calculated; amplitude and phase modulation figures are presented for different forcing and damping cases. Frequency-response and force-response curves are drawn. Different internal resonance cases between modes are investigated.

Nonlinear study of the dynamic behavior of a string-beam coupled system under combined excitations

Abstract: In this paper, the nonlinear dynamic behavior of a string-beam coupled system subjected to external, parametric and tuned excitations is presented. The governing equations of motion are obtained for the nonlinear transverse vibrations of the string-beam coupled system which are described by a set of ordinary differential equations with two degrees of freedom. The case of 1:1 internal resonance between the modes of the beam and string, and the primary and combined resonance for the beam is considered. The method of multiple scales is utilized to analyze the nonlinear responses of the string-beam coupled system and obtain approximate solutions up to and including the second-order approximations. All resonance cases are extracted and investigated. Stability of the system is studied using frequency response equations and the phase-plane method. Numerical solutions are carried out and the results are presented graphically and discussed. The effects of the different parameters on both response and stability of the system are investigated. The reported results are compared to the available published work.

An analytical study of the non-linear vibrations of cylindrical shells

Abstract: The primary resonance response of simply supported circular cylindrical shells is investigated using the perturbation method. Donnell's non-linear shallow-shell theory is used to derive the governing partial differential equations of motion. The Galerkin technique is then employed to transform the equations of motion into a set of temporal ordinary differential equations. Considering only the primary resonance case, the method of multiple scales is used to study the periodic solutions and their stability. The necessary and sufficient conditions for appearance of the so-called companion mode are also discussed. To this end, a range of the possible multi-mode solution is obtained for response and excitation amplitudes and also excitation frequency as a function of damping, geometry and material properties of the shell. Parametric studies are performed to illustrate the effect of different values of thickness, length and material composition on the possibility of the companion mode participation in primary resonance response.

Dynamics of Axially Accelerating Beams With an Intermediate Support

Abstract: The transverse vibrations of an axially accelerating Euler―Bernoulli beam resting on simple supports are investigated The supports are at the ends, and there is a support in between The axial velocity is a sinusoidal function of time varying about a constant mean speed Since the supports are immovable, the beam neutral axis is stretched during the motion, and hence, nonlinear terms are introduced to the equations of motion Approximate analytical solutions are obtained using the method of multiple scales Natural frequencies are obtained for different locations of the support other than end supports The effect of nonlinear terms on natural frequency is calculated for different parameters Principal parametric resonance occurs when the velocity fluctuation frequency is equal to approximately twice of natural frequency By performing stability analysis of solutions, approximate stable and unstable regions were identified Effects of axial velocity and location of intermediate support on the stability regions have been investigated

In-Plane Nonlinear Dynamics of Wind Turbine Blades

Abstract: The partial differential equation that governs the in-plane motion of a wind turbine blade subject to gravitational loading and which accommodates for aerodynamic loading is developed using the extended Hamilton principle. This partial differential equation includes nonlinear terms due to nonlinear curvature and nonlinear foreshortening, as well as parametric and direct excitation at the frequency of rotation. The equation is reduced using an assumed cantilevered beam mode to produce a single second-order ordinary differential equation (ODE) as an approximation for the case of constant rotation rate. Embedded in this ODE are terms of a nonlinear forced Mathieu equation. The forced Mathieu equation is analyzed for resonances by using the method of multiple scales. Superharmonic and subharmonic resonances occur. The effect of various parameters on the response of the system is demonstrated using the amplitude-frequency curve. A superharmonic resonance persists for the linear system as well.Copyright © 2011 by ASME

Multi-pulse orbits dynamics of composite laminated piezoelectric rectangular plate

Abstract: The multi-pulse orbits and chaotic dynamics of a simply supported laminated composite piezoelectric rectangular plate under combined parametric excitation and transverse excitation are studied in detail. It is assumed that different layers are perfectly bonded to each other with piezoelectric actuator patches embedded. The nonlinear equations of motion for the laminated composite piezoelectric rectangular plate are derived from von Karman-type equation and third-order shear deformation plate theory of Reddy. The two-degree-of-freedom dimensionless equations of motion are obtained by using the Galerkin approach to the partial differential governing equation of motion for the laminated composite piezoelectric rectangular plate. The four-dimensional averaged equation in the case of primary parametric resonance and 1:3 internal resonances is obtained by using the method of multiple scales. From the averaged equation, the theory of normal form is used to find the explicit formulas of normal form. Based on the normal form obtained, the energy phase method is utilized to analyze the multi-pulse global bifurcations and chaotic dynamics for the laminated composite piezoelectric rectangular plate. The analysis of the global dynamics indicates that there exist multi-pulse jumping orbits in the perturbed phase space of the averaged equation. Based on the averaged equation obtained, the chaotic motions and the Shilnikov type multi-pulse orbits of the laminated composite piezoelectric rectangular plate are also found by numerical simulation. The results obtained above mean the existence of the chaos in the Smale horseshoe sense for the simply supported laminated composite piezoelectric rectangular plate.

Nonlinear vibrations and frequency response analysis of a cantilever beam under periodically varying magnetic field

Abstract: In this paper, nonlinear vibration of a cantilever beam with tip mass subjected to periodically varying axial load and magnetic field has been studied. The temporal equation of motion of the system containing linear and nonlinear parametric excitation terms along with nonlinear damping, geometric and inertial types of nonlinear terms has been derived and solved using method of multiple scales. The stability and bifurcation analysis for three different resonance conditions were investigated. The numerical results demonstrate that while in simple resonance case with increase in magnetic field strength, the system becomes unstable, in principal parametric or simultaneous resonance cases, the vibration can be reduced significantly by increasing the magnetic field strength. The present work will be very useful for feed forward vibration control of magnetoelastic beams which are used nowadays in many industrial applications.

Nonlinear Electrohydrodynamic Stability of Two Superposed Streaming Finite Dielectric Fluids in Porous Medium with Interfacial Surface Charges

Abstract: A weakly nonlinear stability analysis of wave propagation in two superposed dielectric fluids streaming through porous media in the presence of vertical electric field producing surface charges is investigated in three dimensions. The method of multiple scales is used to obtain a dispersion relation for the linear problem and a nonlinear Klein–Gordon equation with complex coefficients describing the behavior of the perturbed system at the critical point of the neutral curve. In the linear case, we found that the system is always unstable for all physical quantities (including the dimension l), even in the presence of electric fields and porous medium, in the nonlinear case, novel stability conditions are obtained, and the effects of various parameters on the stability of the system are discussed numerically in detail.

Non-linear response of a magneto-elastic translating beam with prismatic joint for higher resonance conditions

Abstract: The non-linear response of a magneto-elastic translating beam having prismatic joint for higher resonance conditions is studied. A periodically varying transverse magnetic field is applied to the system. Two frequencies of prismatic motion and oscillating transverse magnetic field are implemented to the system. The method of multiple scales as one of the perturbation techniques is used to derive two first order ordinary differential equations that govern the time variation of the amplitude and phase of the response. Then a stability analysis is conducted for subharmonic resonance and simultaneous resonance conditions. A parametric study is performed to investigate the effect of magnetic field strength, amplitude of prismatic motion, damping and payload mass on the frequency response curves for both the resonance conditions. The catastrophic failure of the system may occur due to the presence of saddle-node and pitchfork bifurcations. The results obtained by method of multiple scales are compared with those obtained by numerically integrating the reduced equations and are found to be in good agreement. The developed results can be applied to control the vibration of a beam with prismatic joint subjected to magnetic field for third order subharmonic resonance and simultaneous resonance conditions.

Analytical approximations for the periodic motion of the Duffing system with delayed feedback

Abstract: In this study, the homotopy analysis method is developed to give periodic solutions of delayed differential equations that describe time-delayed position feedback on the Duffing system. With this technique, some approximate analytical solutions of high accuracy for some possible solutions are captured, which agree well with the numerical solutions in the whole time domain. Two examples of dynamic systems are considered, focusing on the periodic motions near a Hopf bifurcation of an equilibrium point. It is found that the current technique leads to higher accurate prediction on the local dynamics of time-delayed systems near a Hopf bifurcation than the energy analysis method or the traditional method of multiple scales.

A homogenization analysis of the field theoretic approach to the quasi-continuum method

Abstract: Using the orbital-free density functional theory as a model theory, we present an analysis of the field theoretic approach to quasi-continuum method. In particular, by perturbation method and multiple scale analysis, we provide a formal justification for the validity of numerical coarse-graining of various fields in the quasi-continuum reduction of field theories by taking the homogenization limit. Further, we derive the homogenized equations that govern the behavior of electronic fields in regions of smooth deformations. Using Fourier analysis, we determine the far-field solutions for these fields in the presence of local defects, and subsequently estimate cell-size effects in computed defect energies.