Non-linear oscillation of circular cylindrical shells

The method of multiple scales is used to analyze the non-linear forced response of circular cylindrical shells in the presence of a two-to-one internal (autoparametric) resonance to a harmonic excitation having the frequency Ω. If ω_r and a_r denote the frequency and amplitude of a flexural mode and ω_b and a_b denote the frequency and amplitude of the breathing mode, the steady-state response exhibits a saturation phenomenon when ω_b ≈ 2ω_r, if the excitation frequency Ω is near ω_b. As the amplitude ƒ of the excitation increases from zero, a_b increases linearly whereas a_r remains zero until a threshold is reached. This threshold is a function of the damping coefficients and ω_b−2ω_r. Beyond this threshold a_b remains constant (i.e. the breathing mode saturates) and the extra energy spills over into the flexural mode. In other words, although the breathing mode is directly excited by the load, it absorbs a small amount of the input energy (responds with a small amplitude) and passes the rest of the input energy into the flexural mode (responds with a large amplitude). For small damping coefficients and depending on the detunings of the internal resonance and the excitation, the response exhibits a Hopf bifurcation and consequently there are no steadystate periodic responses. Instead, the responses are amplitude- and phase-modulated motions. When Ω ≈ ω_r, there is no saturation phenomenon and at close to perfect resonance, the response exhibits a Hopf bifurcation, leading again to amplitude- and phase-modulated or chaotic motions.     

Nonlinear forced vibrations of thin structures with tuned eigenfrequencies: the cases of 1:2:4 and 1:2:2 internal resonances

This paper is devoted to the analysis of nonlinear forced vibrations of two particular three degrees-of-freedom (dofs) systems exhibiting second-order internal resonances resulting from a harmonic tuning of their natural frequencies. The first model considers three modes with eigenfrequencies ω ₁, ω ₂, and ω ₃ such that ω ₃?2ω ₂?4ω ₁, thus displaying a 1:2:4 internal resonance. The second system exhibits a 1:2:2 internal resonance, so that the frequency relationship reads ω ₃?ω ₂?2ω ₁. Multiple scales method is used to solve analytically the forced oscillations for the two models excited on each degree of freedom at primary resonance. A thorough analytical study is proposed, with a particular emphasis on the stability of the solutions. Parametric investigations allow to get a complete picture of the dynamics of the two systems. Results are systematically compared to the classical 1:2 resonance, in order to understand how the presence of a third oscillator modifies the nonlinear dynamics and favors the presence of unstable periodic orbits.     

Nonlinear resonances in infinitely long 1D continua on a tensionless substrate

In this work, we investigate the primary nonlinear resonance response of a one-dimensional continuous system, which can be regarded as a model for semi-infinite cables resting on an elastic substrate reacting in compression only, and subjected to a constant distributed load and to a small harmonic displacement applied to the finite boundary. By introducing a straightforward small amplitude expansion characterized by a smallness parameter ε and by performing a Fourier analysis, we first determine the frequencies of the oscillations of the system about the static solution at all orders. We find that, at each order, there exists a critical (cutoff) frequency, above which the solution behaves as a traveling wave toward infinity, while it decays exponentially below it. We then examine the resonance response of the system when an external harmonic excitation is applied at the finite boundary. To this aim, we scale the external excitation with the third power of ε and perform a Multiple-Time-Scale analysis, whose third-order consistency conditions give the differential equations which govern the behavior of the amplitude on the long time scale. In this way, we determine the third-order bending of the resonance curves, whose hardening or softening behavior depends upon the frequency of the chosen primary resonance.     

Non-linear dynamics of wind turbine wings

The paper deals with the formulation of non-linear vibrations of a wind turbine wing described in a wing fixed moving coordinate system. The considered structural model is a Bernoulli-Euler beam with due consideration to axial twist. The theory includes geometrical non-linearities induced by the rotation of the aerodynamic load and the curvature, as well as inertial induced non-linearities caused by the support point motion. The non-linear partial differential equations of motion in the moving frame of reference have been discretized, using the fixed base eigenmodes as a functional basis. Important non-linear couplings between the fundamental blade mode and edgewise modes have been identified based on a resonance excitation of the wing, caused by a harmonically varying support point motion with the circular frequency ω. Assuming that the fundamental blade and edgewise eigenfrequencies have the ratio of ω₂/ω₁?2, internal resonances between these modes have been studied. It is demonstrated that for ω/ω₁?0.66,1.33,1.66 and 2.33 coupled periodic motions exist brought forward by parametric excitation from the support point in addition to the resonances at ω/ω₁?1.0 and ω/ω₂?1.0 partly caused by the additive load term.     

A Closed Form Earthquake Response Analysis of Multistory Building on Compliant Soil*

A transfer matrix formulation is used to analyze structural response of an W-story building to earthquake excitation, taking in to account soil compliancy under the footing. The free-field ground acceleration is modeled as an evolutionary random process. Analysis is simplified by assuming that the superstructure is composed of N identically constructed story units and that soil behavior is characterized by a known impedance matrix or a compliance matrix. Closed form solutions are obtained for the frequency response functions of the system, permitting new insights to be gained into various parameters that affect the response. Ft is found that if the superstructure and the footing remain the same, the effect of soil-foundation interaction is determined by four frequency dependent functions: K_HH/K, K_HM/Kh K_MH/mhω², and K_MM/mh²ω² where the K's are the elements of the soil impedance matrix, m is the unit story mass of the superstructure, and h is the unit story height of the superstructure. A numerical example is given for an 8-story building, showing that soil compliancy reduces the levels of both displacement- and force-type response variables, as previously reported by other investigators. However, the additional degrees of freedom of footing rocking and translation do not necessarily add more peaks in the response     

Theoretical and Experimental Study on the Parametrically Excited Vibration of Mass-Loaded String

The parametrically excited lateral vibration of a mass-loaded string is investigated in this paper. Supposing that the mass at the lower end of the string is subjected to a vertical harmonic excitation and neglecting the higher-order vibration modes, the equation of motion for the mass-loaded string can be represented by a Mathieu's equation with cubic nonlinearity. Based on the stability criterion for Mathieu's equation, the critical conditions inducing parametric resonance are clarified. Theoretical analysis shows that when the natural frequency f _s of the string lateral vibration and the vertical excitation frequency f satisfy f _s= (n/2)f, n= 1, 2, 3, ..., parametric resonance occurs in the case of no damping. For a damped system, parametric resonance most likely occurs when f is close to 2f _s, and depends on the damping of the system and the vertical excitation. The critical excitation has been derived at different frequencies. If the natural frequency of the mass vertical vibration happens to be twice that of the string lateral vibration, the parametric resonance may occur due to a small disturbance. Numerical simulations show that the lateral vibration of the string does not increase infinitely at parametric resonance because the parametric excitation is self-tuned due to the coupling between the vertical and lateral vibrations. Finally, the theoretical results are supported by some experimental work.     

On non-linear vibration analyses of continuous systems with quadratic and cubic non-linearities

This paper examines the validity of non-linear vibration analyses of continuous systems with quadratic and cubic non-linearities. As an example, we treat a hinged-hinged Euler-Bernoulli beam resting on a non-linear elastic foundation with distributed quadratic and cubic non-linearities, and investigate the primary (Ω≈ω_n) and subharmonic (Ω≈2ω_n) resonances, in which Ω and ω_n are the driving and natural frequencies, respectively. The steady-state responses are found by using two different approaches. In the first approach, the method of multiple scales is applied directly to the governing equation that is a non-linear partial differential equation. In the second approach, we discretize the governing equation by using Galerkin's procedure, and then apply the shooting method to the obtained ordinary differential equations. In order to check the validity of the solutions obtained by the two approaches, they are compared with the solutions obtained numerically by the finite difference method.     

Global modes and superdirective acoustic radiation in low-speed axisymmetric jets

Global linear stability theory is used to study the resonances in a slowly diverging axisymmetric jet. The absolute frequency ω₀ is calculated as a function of slow axial position X , and analytic continuation into the complex X -plane allows a saddle point in ω₀ to be identified. A key element in the analysis is the approximation of ω₀(X) by rational functions which identifies a well-defined saddle point and leading-order global frequency. A preferred-mode Strouhal number, S_D=0.44 , is calculated which compares well with existing experimental values. The global frequency has negative imaginary part and the jet is interpreted as being marginally globally stable, so that forcing in the vicinity of the resonance frequency produces a large response above background, rather like that of a slightly damped linear oscillator. The axial shape of the global-mode amplitude is Gaussian and yields a superdirective acoustic field.     

Turbulence in a skewed three‐dimensional wall‐bounded shear flow: effect of mean vorticity on structure modification

Mean‐flow three‐dimensionalities affect both the turbulence level and the coherent flow structures in wall‐bounded shear flows. A tailor‐made flow configuration was designed to enable a thorough investigation of moderately and severely skewed channel flows. A unidirectional shear‐driven plane Couette flow was skewed by means of an imposed spanwise pressure gradient. Three different cases with 8°, 34°and 52°skewing were simulated numerically and the results compared with data from a purely two‐dimensional plane Couette flow. The resulting three‐dimensional flow field became statistically stationary and homogeneous in the streamwise and spanwise directions while the mean velocity vector V and the mean vorticity vector Ω remained parallel with the walls. Mean flow profiles were presented together with all components of the Reynolds stress tensor. The mean shear rate in the core region gradually increased with increasing skewing whereas the velocity fluctuations were enhanced in the spanwise direction and reduced in the streamwise direction. The Reynolds shear stress is known to be closely related to the coherent flow structures in the near‐wall region. The instantaneous and ensemble‐averaged flow structures were turned by the skewed mean flow. We demonstrated for the medium‐skewed case that the coherent structures should be examined in a coordinate system aligned with V to enable a sound interpretation of 3D effects. The conventional symmetry between Case 1 and Case 2 vortices was broken and Case 1 vortices turned out to be stronger than Case 2. This observation is in conflict with the common understanding on the basis of the spanwise (secondary) mean shear rate. A refined model was proposed to interpret the structure modifications in three‐dimensional wall‐flows. What matters is the orientation of the mean vorticity vector Ω relative to the vortex vorticity vector ω _v, that is, the sign of Ω · ω _v. In the present situation, Ω · ω _v > 0 for the Case 1 vortices causing a strengthening relative to the Case 2 vortices. Copyright © 2011 John Wiley & Sons, Ltd.     

Dynamic analysis of piezoelectric energy harvester under combination parametric and internal resonance: a theoretical and experimental study

In this work, theoretical and experimental analysis of a piezoelectric energy harvester with parametric base excitation is presented under combination parametric resonance condition. The harvester consists of a cantilever beam with a piezoelectric patch and an attached mass, which is positioned in such a way that the system exhibits 1:3 internal resonance. The generalized Galerkin’s method up to two modes is used to obtain the temporal form of the nonlinear electromechanical governing equation of motion. The method of multiple scales is used to reduce the equations of motion into a set of first-order differential equations. The fixed-point response and the stability of the system under combination parametric resonance are studied. The multi-branched non-trivial response exhibits bifurcations such as turning point and Hopf bifurcations. Experiments are performed under various resonance conditions. This study on the parametric excitation along with combination and internal resonances will help to harvest energy for a wider frequency range from ambient vibrations.

     

VIBRATION OF CAVITATING ELASTIC WING IN A PERIODICALLY PERTURBED FLOW: EXCITATION OF SUBHARMONICS

The vibration of an elastic wing with an attached cavity in periodically perturbed flows is analyzed. Because the cavity thickness and length L also are perturbed, an excitation with a fixed frequency ω leads to a parametric vibration of the wing, and the amplitudes and spectra of its vibration have nonlinear dependencies on the amplitude of the perturbation. Numerical analysis was carried out for a two-dimensional flow of ideal fluid. Wing vibration was described by means of the beam equation. As a result, two frequency bands of a significant vibration increase were found. A high-frequency band is associated mainly with an elastic resonance of the wing, and a cavity can add a certain damping. A low-frequency band is associated with cavity-volume oscillations. The governing parameter for the low-frequency vibration is the cavity length-based Strouhal number St_C=ωL/U, where U is the free-stream speed. The most significant vibration in the low-frequency band corresponds to approximately constant values of Sh_Cand has the most extensive subharmonics.     

Linear viscoelasticity of thermoassociative chitosan-g-poly(N-isopropylacrylamide) copolymer

Chitosan-g-poly(N-isopropylacrylamide) (chitosan-g-PNIPAM) was synthesized and characterized rheologically in aqueous solutions. The copolymer solution exhibits a thermoassociative behavior in which its elastic response dramatically increases when temperature is above the critical temperature or the association temperature, T _assoc. The copolymer at low concentration shows typical solution property. When the temperature is increased up to the critical temperature, the copolymer exhibits a gel-like characteristic due to the formation of physical cross-links between chitosan backbones through the self-aggregation of PNIPAM side chains. At high concentration, the system exhibits a weak elastic response due to the entanglement of the copolymer at 25°C. As temperature is raised above T _assoc, the system shows a strong elastic behavior due to the formation of additional physical cross-links via the aggregation of PNIPAM side chains. Chitosan-g-PNIPAM offers an attractive associating behavior in aqueous solution at temperature close to the body temperature, thus providing potential applications in pharmaceutical and medical industries.     

Investigating the stabilizing effect of stochastic excitation on a chaotic dynamical system

The response of an interactive Mathieu–Duffing system in R ⁴, subjected to a harmonic excitation is investigated. For a deterministic circular frequency, chaotic behavior is observed. Subsequently it is shown that when the excitation becomes stochastic, chaos is subsided and trajectories tend to a diffused attracting set. The stabilizing effect of stochastic excitation is verified by finding the largest Lyapunov exponent for the two cases.     

Tuning the primary resonances of a micro resonator,using piezoelectric actuation

Microbeam dynamics is important in MEMS filters and resonators. In this research, the effect of piezoelectric actuation on the resonance frequencies of a piezoelectrically actuated capacitive clamped-clamped microbeam is studied. The microbeam is sandwiched with piezoelectric layers throughout its entire length. The lower piezoelectric layer is exposed to a combination of a DC and a harmonic excitation voltage. The DC electrostatic voltage is applied to prevent the doubling of the excitation frequency. The traditional resonators are tuned using DC electrostatic actuation, which tunes the resonance frequency only in backward direction on the frequency domain. The proposed model enables tuning the resonance frequencies in both forward and backward directions. For small amplitudes of harmonic excitation and high enough quality factor, the frequency response curves obtained by the shooting method are validated with those of the multiple time scales technique. Unlike the perturbation technique, which imposes limitation on both the amplitude of the harmonic excitation and the quality factor to be applicable, the shooting method can be applied to capture the periodic attractors regardless of how big the amplitude of harmonic excitation and the quality factor are.     

Mechanical behaviors of electrostatic microresonators with initial offset imperfection: qualitative analysis via time-varying capacitors

This paper investigates analytically and numerically the effect of initial offset imperfection on the mechanical behaviors of microbeam-based resonators. Symmetry breaking of DC actuation, due to different initial offset distances of microbeam to lower and upper electrodes, is concerned. For qualitative analysis, time-varying capacitors are introduced and a lumped parameter model, considering nonlinear electrostatic force and midplane stretching of microbeam, is adopted to examine the system statics and dynamics. The Method of Multiple Scales (MMS) is applied to determine the primary resonance solution under small vibration assumption. Meanwhile, the Finite Difference Method (FDM) combined with Floquet theory is utilized to generate frequency response curves for medium- and large-amplitude vibration simulations. Static bifurcation, phase portrait and Hamiltonian function are firstly investigated to examine the system inherent behaviors. Besides, basins of attraction are briefly depicted to grasp the effects of initial offset and AC excitation on the system global dynamics. Then, variation of equivalent natural frequency versus DC voltage is analyzed. Results show that initial offset may induce complex frequency rebound phenomenon as well as a separate frequency branch under secondary pull-in condition. In what follows, emergences of softening, linear and hardening vibration are classified through discussing a key parameter obtained from the frequency response equation. New linear behavior induced by initial offset imperfection is found, which exhibits much higher sensitivity to DC voltage. Medium- and large-amplitude in-well motions are also investigated, indicating the existence of alternations of softening and hardening behaviors. Finally, lumped parameters are deduced via Galerkin procedure, and case studies are provided to illustrate the effectiveness of the whole analysis.     

Flutter suppression of a plate-like wing via parametric excitation

The present work is motivated by the well known stabilizing effect of parametric excitation of some dynamical systems such as the inverted pendulum. The possibility of suppressing wing flutter via parametric excitation along the plane of highest rigidity in the neighborhood of combination resonance is explored. The nonlinear equations of motion in the presence of incompressible fluid flow are derived using Hamilton's principle and Theodorsen's theory for modeling aerodynamic forces. In the presence of air flow, the bending and torsion modes possess nearly the same frequency. Under parametric excitation and in the absence of air flow, each mode oscillates at its own natural frequency. In the neighborhood of combination resonance, the nonlinear response is determined using the multiple scales method at the critical flutter speed and at slightly higher airflow speed. The domains of attraction and bifurcation diagrams are obtained to reveal the conditions under which the parametric excitation can provide stabilizing effect. The basins of attraction for different values of excitation amplitude reveal the stabilizing effect that takes place above a critical excitation level. Below that level, the response experiences limit cycle oscillations, cascade of period doubling, and chaos. For flow speed slightly higher than the critical flutter speed, the response experiences a train of spikes, known as ‘firing,’ a term that is borrowed from neuroscience, followed by ‘refractory’ or recovery effect, up to an excitation level above which the wing is stabilized. The results of the multiple scales method are verified using numerical simulation of the original nonlinear differential equations.     

An investigation of the parametric excitation of an eccentrically tensioned bar

Results of an investigation of the parametric excitation of an eccentrically tensioned bar are presented. Two distinct types of resonant behavior are observed. For one, the excitation-response frequency ratio is one and, for the other, the ratio is two. In germs of oscillation amplitude and size of the region of parameter space in which resonance occurs, the latter response exhibits the greater potential for producing damage. The transition from small to large amplitude oscillations occurs abruptly and coincides with the occurrence of a snap-through behavior.     

The dynamic behavior of MEMS arch resonators actuated electrically

In this paper, we investigate the dynamic behavior of clamped-clamped micromachined arches when actuated by a small DC electrostatic load superimposed to an AC harmonic load. A Galerkin-based reduced-order model is derived and utilized to simulate the static behavior and the eigenvalue problem under the DC load actuation. The natural frequencies and mode shapes of the arch are calculated for various values of DC voltages and initial rises. In addition, the dynamic behavior of the arch under the actuation of a DC load superimposed to an AC harmonic load is investigated. A perturbation method, the method of multiple scales, is used to obtain analytically the forced vibration response of the arch due to DC and small AC loads. Results of the perturbation method are compared with those obtained by numerically integrating the reduced-order model equations. The non-linear resonance frequency and the effective non-linearity of the arch are calculated as a function of the initial rise and the DC and AC loads. The results show locally softening-type behavior for the resonance frequency for all DC and AC loads as well as the initial rise of the arch.     

FLUID FORCES AND DYNAMICS OF A HYDROELASTIC STRUCTURE WITH VERY LOW MASS AND DAMPING

In this paper we present some central new results from a study of the dynamics and fluid forcing on an elastically mounted rigid cylinder, constrained to oscillate transversely to a free stream. With very low damping, and with a low specific mass that is around 1% of the value used in the classic study of Feng (1968), we show that the cylinder excitation regime extends over a large range of normalized velocity (around four times that found by Feng), with a large amplitude which is around twice that of Feng. Four distinct regions of response are identified, namely the initial excitation region, the “upper branch” (of very high amplitude response), the “lower branch” (of moderate amplitude response), and the desynchronization region. There are distinct differences in the character of mode transitions, as follows. As normalized velocity is increased, there is a hysteretic jump from an initial excitation regime to the upper branch, whereas the jump from the upper to the lower branch involves an intermittent switching, which is illustrated by plotting the instantaneous phase between lift force and displacement using the Hilbert transform. Contrary to classical “lock-in”, whereby the oscillation frequency matches the structural natural frequency, we find that the oscillation frequency increases markedly above the natural frequency, through the excitation regime. Finally, we present the first lift force measurements for such a freely vibrating cylinder experiment, yielding a maximum lift coefficient of around 4·5, whereas a maximum drag coefficient of 6·0 is also measured. The lift is comparable, but somewhat higher, than the forces measured (C_L∼2·0) in the equivalent free-vibration experiments of Hoveret al.(1997), involving force-feedback and on-line computer-simulation of the modelled structure. Both the lift and drag maxima exhibit at least a five-fold increase over the stationary cylinder case. Perhaps the largest effect is found for the fluctuating drag, which is found to be upto 100 times that measured for a static cylinder.     

Influence of Horizontal Excitations on Dynamic Stability of a Slender Beam Under Vertical Excitation

Experimental studies have been conducted to clarify the influence of horizontal harmonic excitations on the dynamic stability of a slender cantilever beam under vertical harmonic excitation. Three kinds of aluminum test beams with rectangular cross section have been used. The test beam being clamped at one end and free at the other end, was vertically stood, and was harmonically excited to both vertical and horizontal directions simultaneously. The direction of the horizontal excitation was taken parallel to one of the beam side faces, i.e. two directions were considered as X and Y directions which have the largest and smallest flexural rigidity, respectively. By varying the horizontal excitation amplitude, keeping the amplitude of excitation in the vertical direction, the influence of the horizontal excitation has been investigated on the principal instability regions in which unstable vibration of the fundamental vibration mode occurs. The excitation frequency in the vertical excitation was taken around twice the fundamental natural frequency 2f _{Y 1} in smallest rigidity direction, while that in the horizontal direction was taken around both the fundamental natural frequency f _{Y 1} and twice of it 2f _{Y 1}. Obtained experimental results present useful fundamental data for aseismatic design of structures under earthquake containing both vertical and horizontal excitation components.