Amplitude Solutions Research Articles

A semi-empirical model for vortex-induced vertical forces on a twin-box deck under turbulent wind flow

Long-span cable-supported bridges with twin-box decks have been constructed in recent years to enhance the stability of the bridges against flutter, but some of these bridges suffer large-amplitude vortex-induced vibration (VIV). Large-amplitude VIV can cause fatigue damage accumulation in structural components and reduce safety and comfort levels of road vehicles, trains, and pedestrians. Therefore, in the design of long-span bridges with twin-box decks, it is essential to know vortex-induced forces (VIF) and predict vortex-induced responses (VIR). Currently, there are several semi-empirical models for the estimation of VIF but most of them are established based on measured VIR for single-box decks and without considering turbulent wind effects.This paper presents a semi-empirical model for vertical VIF on a twin-box deck with and without turbulent wind flow. First, the two existing semi-empirical single-degree-of-freedom (SDOF) VIF models are discussed, and a refined SDOF model for VIF under smooth wind flow is proposed. The proposed SDOF VIF model includes all the non-conservative motion-induced force terms. An approximate analytical solution for the maximum amplitude of VIR is then deduced based on the proposed VIF model. Second, based on quasi-static assumption, the proposed VIF model is further extended to take account of turbulence wind effects by introducing a turbulence-induced positive damping term into the model. Finally, the validity of the proposed VIF model with and without turbulent wind effects is examined using a newly-developed wind tunnel test technique for an elastically-mounted twin-box section model. The comparative results show that the proposed VIF model can effectively predict the maximum VIR of a twin-box deck under different turbulent fields with different structural damping ratios. It is also found that the maximum VIR decreases nonlinearly with the increase of turbulence intensity.

Nonsynchronous Vibration of Planar Autobalancer/Rotor System With Asymmetric Bearing Support

Due to inherent nonlinearity of the autobalancer, the potential for other, undesirable, nonsynchronous limit-cycle vibration exists. In such undesirable situations, the balancer masses do not reach their desired synchronous balanced steady-state positions resulting in increased rotor vibration. Such behavior has been widely studied and is well understood for rotor systems on idealized bearings with symmetric supports. However, a comprehensive study into this nonlinear behavior of an imbalanced planar-rigid rotor/autobalancing device (ABD) system mounted on a general bearing holding asymmetric damping and stiffness forces including nonconservative effects cross-coupling ones has not been fully conducted. Therefore, this research primarily focuses on the unstable nonsynchronous limit-cycle behavior and the synchronous balancing condition of system under the influence of the general bearing support. Here, solutions for rotor limit-cycle amplitudes and the corresponding whirl speeds are obtained via a harmonic balance approach. Furthermore, the limit-cycle stability is assessed via perturbation and Floquet analysis, and all the possible responses including undesirable coexistence for the bearing parameters and operating speeds have been thoroughly studied. It is found that, due to asymmetric behavior of bearing support, the multiple limit cycles are encountered in the range of supercritical speeds and more complicate coexistences are invited into the ABD–rotor system compared to the case with idealized symmetric bearing supports. The findings in this paper yield important insights for researchers wishing to utilize automatic balancing devices in more practical rotor systems mounted on a asymmetric general bearing support.

Dynamical diagnostics of the SST annual cycle in the eastern equatorial Pacific: Part II analysis of CMIP5 simulations

In this study, a simple coupled framework established in Part І is utilized to investigate inter-model diversity in simulating the equatorial Pacific SST annual cycle (SSTAC). It demonstrates that the simulated amplitude and phase characteristics of SSTAC in models are controlled by two internal dynamical factors (the damping rate and phase speed) and two external forcing factors (the strength of the annual and semi-annual harmonic forcing). These four diagnostic factors are further condensed into a dynamical response factor and a forcing factor to derive theoretical solutions of amplitude and phase of SSTAC. The theoretical solutions are in remarkable agreement with observations and CMIP5 simulations. The great diversity in the simulated SSTACs is related to the spreads in these dynamic and forcing factors. Most models tend to simulate a weak SSTAC, due to their weak damping rate and annual harmonic forcing. The latter is due to bias in the meridional asymmetry of the annual mean state of the tropical Pacific, represented by the weak cross-equatorial winds in the cold tongue region.

Initial–boundary-value problems for one-dimensional compressible Navier–Stokes equations with degenerate transport coefficients

This paper is concerned with the construction of global, non-vacuum, strong, large amplitude solutions to initial–boundary-value problems for the one-dimensional compressible Navier–Stokes equations with degenerate transport coefficients. Our analysis derives the positive lower and upper bounds on the specific volume and the absolute temperature.

Stability of stationary solutions for inflow problem on the micropolar fluid model

In this paper, we study the asymptotic behavior of solutions to the initial boundary value problem for the micropolar fluid model in a half-line \(\mathbb {R}_{+}:=(0,\infty ).\) We prove that the corresponding stationary solutions of the small amplitude to the inflow problem for the micropolar fluid model are time asymptotically stable under small \(H^{1}\) perturbations in both the subsonic and degenerate cases. The microrotation velocity brings us some additional troubles compared with Navier–Stokes equations in the absence of the microrotation velocity. The proof of asymptotic stability is based on the basic energy method.

Energy partition, scale by scale, in magnetic Archimedes Coriolis weak wave turbulence.

Magnetic Archimedes Coriolis (MAC) waves are omnipresent in several geophysical and astrophysical flows such as the solar tachocline. In the present study, we use linear spectral theory (LST) and investigate the energy partition, scale by scale, in MAC weak wave turbulence for a Boussinesq fluid. At the scale k^{-1}, the maximal frequencies of magnetic (Alfvén) waves, gravity (Archimedes) waves, and inertial (Coriolis) waves are, respectively, V_{A}k,N, and f. By using the induction potential scalar, which is a Lagrangian invariant for a diffusionless Boussinesq fluid [Salhi et al., Phys. Rev. E 85, 026301 (2012)PLEEE81539-375510.1103/PhysRevE.85.026301], we derive a dispersion relation for the three-dimensional MAC waves, generalizing previous ones including that of f-plane MHD "shallow water" waves [Schecter et al., Astrophys. J. 551, L185 (2001)AJLEEY0004-637X10.1086/320027]. A solution for the Fourier amplitude of perturbation fields (velocity, magnetic field, and density) is derived analytically considering a diffusive fluid for which both the magnetic and thermal Prandtl numbers are one. The radial spectrum of kinetic, S_{κ}(k,t), magnetic, S_{m}(k,t), and potential, S_{p}(k,t), energies is determined considering initial isotropic conditions. For magnetic Coriolis (MC) weak wave turbulence, it is shown that, at large scales such that V_{A}k/f≪1, the Alfvén ratio S_{κ}(k,t)/S_{m}(k,t) behaves like k^{-2} if the rotation axis is aligned with the magnetic field, in agreement with previous direct numerical simulations [Favier etal., Geophys. Astrophys. Fluid Dyn. (2012)] and like k^{-1} if the rotation axis is perpendicular to the magnetic field. At small scales, such that V_{A}k/f≫1, there is an equipartition of energy between magnetic and kinetic components. For magnetic Archimedes weak wave turbulence, it is demonstrated that, at large scales, such that (V_{A}k/N≪1), there is an equipartition of energy between magnetic and potential components, while at small scales (V_{A}k/N≫1), the ratio S_{p}(k,t)/S_{κ}(k,t) behaves like k^{-1} and S_{κ}(k,t)/S_{m}(k,t)=1. Also, for MAC weak wave turbulence, it is shown that, at small scales (V_{A}k/sqrt[N^{2}+f^{2}]≫1), the ratio S_{p}(k,t)/S_{κ}(t) behaves like k^{-1} and S_{κ}(k,t)/S_{m}(k,t)=1.

Discrete breathers in a mass-in-mass chain with Hertzian local resonators.

We report on the existence of discrete breathers in a one-dimensional, mass-in-mass chain with linear intersite coupling and nonlinear, precompressed Hertzian local resonators, which is motivated by recent studies of the dynamics of microspheres adhered to elastic substrates. After predicting theoretically the existence of discrete breathers in the continuum and anticontinuum limits of intersite coupling, we use numerical continuation to compute a family of breathers interpolating between the two regimes in a finite chain, where the displacement profiles of the breathers are localized around one lattice site. We then analyze the frequency-amplitude dependence of the breathers by performing numerical continuation on a linear eigenmode (vanishing amplitude) solution of the system near the upper band gap edge. Finally, we use direct numerical integration of the equations of motion to demonstrate the formation and evolution of the identified localized modes in energy-conserving and dissipative scenarios, including within settings that may be relevant to future experimental studies.

Explosive attractor solutions to a universal cubic delay equation

New explosive attractor solutions have been found in a universal cubic delay equation that has been studied in both the plasma and the fluid mechanics literature. Through computational simulations and analytic approximations, it is found that the oscillatory component of the explosive mode amplitude solutions are described through multi-frequency Fourier expansions with respect to a pseudo-time variable. The spectral dependence of these solutions as a function of a system parameter, ϕ, is studied. The mode amplitude that is described in the explosive regime has two main features: a well-known envelope (t0−t)−5/2, with t0 the blow-up time of the amplitude, and a spectrum of discrete oscillations with ever-increasing frequencies, which may give experimental information about the properties of a system's equilibrium.

Localized discrete breather modes in neuronal microtubules

We made an attempt to provide a realistic picture of the localization of energy in microtubules (MTs), and we intend to model the nonlinear dynamics of MTs using the “double-well” $$\phi ^4$$ form of the potential describing the dipole–dipole interactions. We investigate the modulational instability (MI) of the nonlinear plane wave solutions by considering both the wave vector (q) of the basic states and the wave vector (Q) of the perturbations as free parameters. A set of explicit criteria of MI is derived, and under the plane-wave perturbation, the constant amplitude solution becomes unstable and localized discrete breathers (DBs) solutions appear. We show numerically that MI is also an indicator of the presence of discrete breathers. We suggest that an electric field favourably leads the DB excitations towards the properly aligned end triggering a dissembly of the protofilament due to the energy release. These DBs could catalyse MT-associated proteins attachment/detachment and promote or inhibit the kinesin walk. We establish that the electromechanical vibrations in MTs can generate an electromagnetic field in the form of an electric pulse (breathers) which propagates along MT serving as signalling pathway in neuronal cells. The DBs in MT can be viewed as a bit of information whose propagation can be controlled by an electric filed. They might perform the role of elementary logic gates, thus implementing a subneuronal mode of computation. The generated DBs present us with novel possibilities for the direct interaction between the local electromagnetic field and the cytoskeletal structures in neurons. Thus, we emphasize that the effect of discreteness and electric field plays a significant role in MTs.

Resonance Principle for the Design of Flapping Wing Micro Air Vehicles

Achieving resonance in flapping wings has been recognized as one of the most important principles to enhance power efficiency, lift generation, and flight control performance of high-frequency flapping wing micro air vehicles (MAVs). Most work on the development of such vehicles have attempted to achieve wing flapping resonance. However, the theoretical understanding of its effects on the response and energetics of flapping motion has lagged behind, leading to suboptimal design decisions and misinterpretations of experimental results. In this work, we systematically model the dynamics of flapping wing as a forced nonlinear resonant system, using both nonlinear perturbation method and linear approximation approach. We derived an analytic solution for steady-state flapping amplitude, energetics, and characteristic frequencies including natural frequency, damped natural frequency, and peak frequency. Our results showed that both aerodynamic lift and power efficiency are maximized by driving the wing at natural frequency, instead of other frequencies. Interestingly, the flapping velocity is maximized at natural frequency as well, which can lead to an easy experimental approach to identify natural frequency and validate the resonance design. Our models and analysis were validated with both simulations and experiments on ten different wings mounted a direct-motor-drive flapping wing MAV. The result can serve as a systematic design principle and guidance in the interpretations of empirical results.

Sustainable theory of a logistic model - Fisher information approach

Information theory provides a useful tool to understand the evolution of complex nonlinear systems and their sustainability. In particular, Fisher information has been evoked as a useful measure of sustainability and the variability of dynamical systems including self-organising systems. By utilising Fisher information, we investigate the sustainability of the logistic model for different perturbations in the positive and/or negative feedback. Specifically, we consider different oscillatory modulations in the parameters for positive and negative feedback and investigate their effect on the evolution of the system and Probability Density Functions (PDFs). Depending on the relative time scale of the perturbation to the response time of the system (the linear growth rate), we demonstrate the maintenance of the initial condition for a long time, manifested by a broad bimodal PDF. We present the analysis of Fisher information in different cases and elucidate its implications for the sustainability of population dynamics. We also show that a purely oscillatory growth rate can lead to a finite amplitude solution while self-organisation of these systems can break down with an exponentially growing solution due to the periodic fluctuations in negative feedback.

Generation of Gravity Waves by Singular Potential Vorticity Disturbances in Shear Flows

Abstract The linear mechanism of generation of gravity waves by potential vorticity (PV) disturbances in flows with constant horizontal and vertical shears is studied. The case of the initial singular distribution of PV, in which the PV is localized in one coordinate and is periodic with respect to other coordinates, is considered. In a stratified rotating medium, such a distribution induces a vortex wave (continuous mode), the propagation of which is accompanied by the emission of gravity waves. To find the emission characteristics, a linearized system of dynamical equations is reduced to wave equations with sources that are proportional to the initial distributions of PV. The asymptotic solutions of the equations are constructed for small Rossby numbers (horizontal shear) and large Richardson numbers (vertical shear). When passing through the inertial levels symmetrically located with respect to a vortex source, the behavior of the solutions for wave amplitudes radically changes. Directly in the vicinity of the source, the solutions are of monotonic character, corresponding to a quasigeostrophic vortex wave. At long distances from the source, the solutions oscillate. The horizontal momentum flux and the Eliassen–Palm flux are estimated using asymptotic solutions. It is found that, within the indicated range of both Rossby and Richardson numbers, these fluxes are exponentially small: that is, the emission of waves is weak.

Real-time measurements of spontaneous breathers and rogue wave events in optical fibre modulation instability

Modulation instability is a fundamental process of nonlinear science, leading to the unstable breakup of a constant amplitude solution of a physical system. There has been particular interest in studying modulation instability in the cubic nonlinear Schrödinger equation, a generic model for a host of nonlinear systems including superfluids, fibre optics, plasmas and Bose–Einstein condensates. Modulation instability is also a significant area of study in the context of understanding the emergence of high amplitude events that satisfy rogue wave statistical criteria. Here, exploiting advances in ultrafast optical metrology, we perform real-time measurements in an optical fibre system of the unstable breakup of a continuous wave field, simultaneously characterizing emergent modulation instability breather pulses and their associated statistics. Our results allow quantitative comparison between experiment, modelling and theory, and are expected to open new perspectives on studies of instability dynamics in physics.

Evaluation of nonlinear effects at the focus of a hemispherical acoustic source

Several theoretical models exist for simulating nonlinear generation of harmonics for weakly or moderately focused waves. Under these conditions, the acoustic field can be considered as a directed beam. In the case of hemispherical sources, this assumption is no longer valid, so development of different models is needed. Here, a theoretical approach is presented for evaluating the amplitude level of the second harmonic generated in hemispherically focused fields. The approach is based on the successive approximation method and includes several steps. First, the linear acoustic problem is solved for a hemispherical source using a spherical harmonics series expansion. Next, the volume sources for the second harmonic generation are calculated using the solution to the linear problem. Finally, these sources are used for obtaining an analytic solution for the second harmonic amplitude at the focus. The developed method was applied to study the acoustic field of the ExAblate 3000 clinical transducer that has a hemispherical shape of 30 cm diameter, operating frequency of 670 kHz, and acoustic power of several hundred watts. Calculations predict that nonlinear effects are weak even for the free-field focusing in water at acoustic power levels up to 1 kW. [Work supported by RSF 14-15-00665.]

Force transmissibility of a two-stage vibration isolation system with quasi-zero stiffness

A quasi-zero stiffness (QZS) vibration isolator outperforms other passive control strategies in vibration attenuation especially in a low-frequency band, but it also has an intrinsic limitation of low roll-off rate in the effective frequency range of vibration isolation. To overcome this limitation, a two-stage QZS vibration isolation system (VIS) is proposed, in which the QZS feature is realized by combining a vertical liner spring with two parallel cam–roller–spring mechanisms. Considering a possible disengagement between the cam and the roller under large amplitude vibration, a piecewise nonlinear dynamical model is developed and approximately solved by the averaging method. The analytical solutions for amplitude–frequency relationship and force transmissibility are derived. The results reveal that the two-stage QZS VIS has both advantages of low-frequency vibration isolation and high roll-off rate. It is also found that the second resonance can be eliminated when heavy damping is present in the upper stage, and hence, a broader effective frequency range of isolation can be achieved. High intermediate mass and soft vertical springs in the lower stage are also found to result in high-quality isolation performance.

Topological spin Meissner effect in spinor exciton-polariton condensate: Constant amplitude solutions, half-vortices, and symmetry breaking

We generalize the spin Meissner effect for exciton-polariton condensate confined in annular geometries to the case of non-trivial topology of the condensate wavefunction. In contrast to the conventional spin Meissner state, topological spin Meissner states can in principle be observed at arbitrary high magnetic field not limited by the critical magnetic field value for the condensate in a simply-connected geometry. One special example of the topological Meissner states are half-vortices. We show that in the absence of magnetic field half-vortices in a ring exist in a form of superposition of elementary half-vortex states which resolves recent experimental results where such puzzling superposition was observed. Furthermore, we show that if a pure half-vortex state is to be observed, a non-zero magnetic field of a specific magnitude needs to be applied. Studying exciton-polariton in a ring in presence of TE-TM splitting, we observe spin Meissner states which break rotational symmetry of the system by developing inhomogeneous density distributions. We classify various states arising in presence of non-zero TE-TM splitting based on what states they can be continued from by increasing the TE-TM splitting parameter from zero. With further increasing TE-TM splitting, states with broken symmetry may transform into stable half-dark solitons and therefore may serve as a useful tool to generate various non-trivial states of a spinor condensate.

Global well-posedness in energy space of small amplitude solutions for klein-gordon-zakharov equation in three space dimension

The Cauchy problem of the Klein-Gordon-Zakharov equation in three dimensional space {utt−Δu+u=−nu, (x,t)∈ℝ3×ℝ+,ntt−Δu=Δ|u|2, (x,t)∈ℝ3×ℝ+,u(x,0)=u0(x), ∂tu(x,0)=u1(x), n(x,0)=n0(x), ∂tn(x,0)=n1(x),is considered. It is shown that it is globally well-posed in energy space H1×L2×L2×H−1 if small initial data (u0(x),u1(x),n0(x),n1(x))∈(H1×L2×L2×H−1). It answers an open problem: Is it globally well-posed in energy space H1×L2×L2×H−1 for 3D Klein-Gordon-Zakharov equation with small initial data [1,2]? The method in this article combines the linear property of the equation (dispersive property) with nonlinear property of the equation (energy inequalities). We mainly extend the spaces Fs and Ns in one dimension [3] to higher dimension.

A note on decay rates of solutions to a system of cubic nonlinear Schrödinger equations in one space dimension

We consider the initial value problem for a system of cubic nonlinear Schr\"odinger equations with different masses in one space dimension. Under a suitable structural condition on the nonlinearity, we will show that the small amplitude solution exists globally and decays of the rate $O(t^{-1/2}(\log t)^{-1/2})$ in $L^\infty$ as $t$ tends to infinity, if the system satisfies certain mass relations.

Sitnikov problem in the square configuration: elliptic case

This paper is extension to the classical Sitnikov problem, when the four primaries of equal masses lie at the vertices of a square for all time and moving in elliptic orbits around their center of mass of the system, the distances between the primaries vary with time but always in such a way that their mutual distances remain in the same ratio. First we have established averaged equation of motion of the Sitnikov five-body problem in the light of Jalali and Pourtakdoust (Celest. Mech. Dyn. Astron. 68:151–162, 1997), by applying the Van der Pol transformation and averaging technique of Guckenheimer and Holmes (Nonlinear oscillations, dynamical system bifurcations of vector fields, Springer, Berlin, 1983). Next the Hamiltonian equation of motion has been solved with the help of action angle variables \(I\) and \(\phi\). Finally the periodicity and stability of the Sitnikov five-body problem have been examined with the help of Poincare surfaces of section (PSS). It is shown that chaotic region emerging from the destroyed islands, can easily be seen by increasing the eccentricity of the primaries to \(e = 0.21\). It is valid for bounded small amplitude solutions \(z_{\mathrm{max}} ( z_{\mathrm{max}} = 0.65 )\) and \(0 \le e < 0.3\).

Controlled thiol-ene polymer microsphere production using a low-frequency acoustic excitation coaxial flow method

A novel technique for the production of thiol-ene microspheres using acoustic resonance and coaxial flow is reported. The method utilizes low-frequency acoustically driven mechanical perturbations to disrupt the flow of a thiol-ene liquid jet, resulting in small thiol-ene droplets that are photochemically polymerized to yield thiol-ene microspheres. Tuning of the frequency, amplitude, and monomer solution viscosity are critical parameters impacting the diameter of the microspheres produced. Characterization by optical microscopy, scanning electron microscopy, and dynamic light scattering reveal microspheres of diameters <10 μm, with narrow particle distributions.