Long-wave asymptotic theories: The connection between functionally graded waveguides and periodic media

This article explores the deep connections that exist between the mathematical representations of dynamic phenomena in functionally graded waveguides and those in periodic media. These connections are at their most obvious for low-frequency and long-wave asymptotics where well established theories hold. However, there is also a complementary limit of high-frequency long-wave asymptotics corresponding to various features that arise near cut-off frequencies in waveguides, including trapped modes. Simultaneously, periodic media exhibit standing wave frequencies, and the long-wave asymptotics near these frequencies characterise localised defect modes along with other high-frequency phenomena. The physics associated with waveguides and periodic media are, at first sight, apparently quite different, however the final equations that distill the essential physics are virtually identical. The connection is illustrated by the comparative study of a periodic string and a functionally graded acoustic waveguide.     

Experimental study of surface waves with Faraday resonance excitation

The results of an investigation of standing two-dimensional gravity waves on the free surface of a homogeneous liquid, induced by the vertical oscillations of a rectangular vessel under Faraday resonance conditions, are presented. The frequency ranges of excitation are determined and resonance relationships for the second and third modes are obtained and analyzed. Nonlinearities of the waves generated, such as wave profile asymmetry and node oscillations, are evaluated. Wave breakdown and the onset of unstable oscillation modes are considered. Experimental results are compared with the theoretical data.The experimental studies [1–4], devoted to Faraday resonance, deal mainly with the conditions under which resonance arises and the frequency response.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 122–129, January–February, 1995.     

Aeolian tones of a plate cascade

A model for the aeroacoustic resonance effects (aeolian tones) excited around a plate cascade in a gas flow is suggested. Methods of calculating the frequencies of natural acoustic oscillations near the cascade are developed. The effect of the cascade geometry and the Mach number of the main flow on the frequencies, abundance, and modes of the natural oscillations is investigated. Anomalous acoustic oscillations near a cyclic plate cascade are shown to exist and are studied. It is shown that there always exist no less than two natural oscillation frequencies in the gas flow near any nontrivial cyclic plate cascade. It has been found that the natural oscillation frequencies can be combined in bundles such that in the case where the number of plates in a period is large the frequencies pertaining to each bundle occupy a certain interval with arbitrary density. The natural oscillations are classified with respect to the form of the eigenfunctions; the classification is based on the theory of representations of groups of locally plane symmetries of the cyclic plate cascade in the solution space. The correctness of the proposed model of the aeroacoustic resonance effects (aeolian tones) excited near a plate cascade in a gas flow is supported by a comparison with the available experimental and theoretical data. On the basis of the investigation performed, some previously unknown physical phenomena are predicted. Thus, the existence of frequency zones or main-flow Mach number ranges on which aeroacoustic resonance phenomena exist near a cyclic cascade with a large number of plates in a period is proved; it is shown that for certain frequencies of the natural oscillations near the cyclic plate cascade the resonance oscillations may be localized in the vicinity of the source; and the existence of narrow-band wave packets slowly propagating along the cascade is demonstrated. Novosibirsk, e-mail: sukhinin@hydro.nsc.ru. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 171–186, March–April, 2000.     

Heat and mass transfer of a fuel droplet evaporating in oscillatory flow

A numerical study of the heat and mass transfer from an evaporating fuel droplet in oscillatory flow was performed. The flow was assumed to be laminar and axisymmetric, and the droplet was assumed to maintain its spherical shape during its lifetime. Based on these assumptions, the conservation equations in a general curvilinear coordinate were solved numerically. The behaviors of droplet evaporation in the oscillatory flow were investigated by analyzing the effects of flow oscillation on the evaporation process of a n-heptane fuel droplet at high pressure.The response of the time history of the square of droplet diameter and space-averaged Nusselt numbers to the main flow oscillation were investigated in frequency band of 1–75 Hz with various oscillation amplitudes. Results showed that, depending on the frequency and amplitude of the oscillation, there are different modes of response of the evaporation process to the flow oscillation. One response mode is synchronous with the main flow oscillation, and thus the quasi-steady condition is attained. Another mode is asynchronous with the flow oscillation and is highly unsteady. As for the evaporation rate, however, in all conditions is more greatly enhanced in oscillatory flow than in quiescent air.To quantify the conditions of the transition from quasi-steady to unsteady, the response of the boundary layer around the droplet surface to the flow oscillation was investigated. The results led to including the oscillation Strouhal number as a criteria for the transition. The numerical results showed that at a low Strouhal number, a quasi-steady boundary layer is formed in response to the flow oscillation, whereas by increasing the oscillation Strouhal number, the phenomena become unsteady.     

Justification of the Averaging Method for Multifrequency Systems with Pulse Action

We prove new estimates for the error of the averaging method for oscillation systems with slowly varying frequencies subjected to pulse action at fixed times. The main assumption is imposed not on all harmonics of the right-hand side of the system but only on resonance ones.__________Translated from Neliniini Kolyvannya, Vol. 8, No. 1, pp. 61–79, January–March, 2005.     

Nonlinear interactions in the planar dynamics of cable-stayed beam

An analytical model is proposed to study the nonlinear interactions between beam and cable dynamics in stayed-systems. The integro-differential problem, describing the in-plane motion of a simple cable-stayed beam, presents quadratic and cubic nonlinearities both in the cable equation and at the boundary conditions. Mainly studied are the effects of quadratic interactions, appearing at relatively low oscillation amplitude. To this end an analysis of the sensitivity of modal properties to parameter variations, in intervals of technical interest, has evidenced the occurrence of one-to-two and two-to-one internal resonances between global and local modes. The interactions between the resonant modes evidences two different sources of oscillation in cables, illustrated by simple 2dof discrete models.In the one-to-two global–local resonance, a novel mechanism is analyzed, by which cable undergoes large periodic and chaotic oscillations due to an energy transfer from the low-global to high-local frequencies.In two-to-one global–local resonance, the well-known parametric-induced cable oscillation in stayed-systems is correctly reinterpreted through the autoparametric resonance between a global and a local mode. Increasing the load the saturation of the global oscillations evidences the energy transfer from high-global to low-local frequencies, producing large cable oscillations. In both cases, the effects of detuning from internal and external resonance are presented.     

Optical properties of two-dimensional polymer photonic crystals after deformation-induced pattern transformations

The photonic band structure and optical transmittance of two-dimensional periodic elastomeric photonic crystals are studied computationally to understand the effects of large strains on optical properties of the structures. The large compressive deformation patterns of the two-dimensional periodic structure studied by Mullin and coworkers [Mullin, T., Deschanel, S., Bertoldi, K., Boyce, M.C., 2007. Pattern transformation triggered by deformation. Physical Review Letters 99(8), 084301] are first reproduced using hyperelastic material models for the elastomer SU-8. Finite element analysis is then used to solve Maxwell's equations to obtain light transmittance through both the undeformed and deformed structures; simultaneously the wave equation resulting from the appropriate two-dimensional form of Maxwell's equations is solved as an eigenvalue problem to obtain the band structure. The deformation-induced shift in transmission spectrum valleys for different bands is calculated, and the changes in the width of these reflectance peaks are also obtained. The band structure calculation shows that there are no complete photonic band gaps as expected for the low dielectric contrast system. However, the effect of the observed reversible, symmetry-breaking deformation pattern is to uncouple many of the photonic bands in all three high symmetry directions, i.e. Γ–X, X–M, and Γ–M. New non-degenerate deformation-induced optical modes appear in both the real space transmittance spectra and the band structure with lower reflectance values. Analyses of the deformation pattern, the optical mode shapes, and the photonic band structure reveal that localized regions of large rotation are responsible for the significant changes in optical transmittance. The results have practical importance for the design of strain-tunable optomechanical materials for sensing and actuation.     

Resonance in flow past oscillating cylinder under subcritical conditions

Flow around an oscillating cylinder in a subcritical region are numerically studied with a lattice Boltzmann method(LBM). The effects of the Reynolds number,oscillation amplitude and frequency on the vortex wake modes and hydrodynamics forces on the cylinder surface are systematically investigated. Special attention is paid to the phenomenon of resonance induced by the cylinder oscillation. The results demonstrate that vortex shedding can be excited extensively under subcritical conditions, and the response region of vibration frequency broadens with increasing Reynolds number and oscillation amplitude. Two distinct types of vortex shedding regimes are observed. The first type of vortex shedding regime(VSR I) is excited at low frequencies close to the intrinsic frequency of flow, and the second type of vortex shedding regime(VSR II)occurs at high frequencies with the Reynolds number close to the critical value. In the VSR I, a pair of alternately rotating vortices are shed in the wake per oscillation cycle,and lock-in/synchronization occurs, while in the VSR II, two alternately rotating vortices are shed for several oscillation cycles, and the vortex shedding frequency is close to that of a stationary cylinder under the critical condition. The excitation mechanisms of the two types of vortex shedding modes are analyzed separately.     

Combination Internal Resonances in Heated Annular Plates

Nonlinear modal interactions in the forced vibrations of a thermally loaded pre-buckled annular plate with clamped–clamped immovable boundary conditions are investigated. The mechanism responsible for the interaction is a combination internal resonance involving the natural frequencies of the three lowest axisymmetric modes. The in-plane thermal load acting on the plate is assumed to be axisymmetric and the plate is externally excited by a harmonic force. The nonlinear von Kármán plate equations along with the heat conduction equation are combined to model the behavior of the system. An analytical/numerical approach is used to examine the plate vibrations to a harmonic excitation near primary resonance of one of the modes.     

Normal oscillations of a rotating liquid drop

The problem of determining the frequencies and profiles of surface waves of a liquid drop rotating under conditions of weightlessness is considered. For an ideal liquid, Ritz's method can be applied to a certain quadratic functional to calculate the oscillation frequencies on a computer. For a low-viscosity liquid, using the boundary-layer method, formulas are obtained for the decay rate and the corrections to the oscillation frequencies. The results of the calculations are presented in the form of graph and a table.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 78–87, July–August, 1979.I should like to thank N. D. Kopachevskii for supervision and assistance in the work.     

Study of the shock motion in a hypersonic shock system/turbulent boundary layer interaction

Wall pressure fluctuations and surface heat transfer signals have been measured in the hypersonic turbulent boundary layer over a number of compression-corner models. The distributions of the separation shock oscillation frequencies and periods have been calculated using a conditional sampling algorithm. In all cases the oscillation frequency distributions are of broad band, but the most probable frequencies are low. The VITA method is used for deducing large scale disturbances at the wall in the incoming boundary layer and the separated flow region. The results at present showed the existence of coherent structures in the two regions. The zero-cross frequencies of the large scale structures in the two regions are of the same order as that of the separation shock oscillation. The average amplitude of the large scale structures in the separated region is much higher than that in the incoming boundary layer. The length scale of the separation shock motion region is found to increase with the disturbance strength. The results show that the shock oscillation is of inherent nature in the shock wave/turbulent boundary layer interaction with separation. The shock oscillation is considered to be the consequence of the coherent structures in the separated region.This work was supported by the Chinese National Science Foundation. Thanks for Prof. Z. B. Lin and Miss X. Y. Feng for their helps. The authors wish to express thanks to Professor W. Merzkirch who has helped us to check the paper again and again.     

Wave transmission characteristics in periodic media of finite length: multilayers and fiber arrays

Wave transmission characteristics in elastic media that have periodic microstructure over a finite spatial length are examined theoretically as well as numerically. Two classes of such media are demonstrated, namely, one-dimensional multilayered media with finite-length periodicity and two-dimensional composite media with square arrays of aligned fibers within a finite length. From these one-dimensional and two-dimensional analyses, the influence of the finite-length periodicity on the wave transmission characteristics is discussed. In these media, there are frequency bands (stop bands) where the energy transmission coefficient appears to vanish or takes very low values, while in pass bands it oscillates with the frequency due to the finite-length periodicity. It is theoretically demonstrated in the one-dimensional case of multilayers how the frequency intervals of the oscillation in the transmission spectrum depend on the repeating number of the periodic cells as well as other acoustic and geometrical parameters. The results of the two-dimensional fiber arrays, which are obtained numerically by solving the equations of the SH wave multiple scattering, are shown to fit well in the one-dimensional framework of multilayered structures up to a certain frequency encompassing the first stop band. This similarity between two classes of problems is demonstrated by appropriately identifying the one-dimensional reduced transfer matrix for a single cell that is representative of the periodic fiber array.     

Mathematical simulation of unsteady flow past a wing with jets

The various approximate approaches to the investigation of the unsteady aerodynamic characteristics of an airfoil with jet flap [1–3] are applicable only for an airfoil, low jet intensity, and low oscillation frequencies. In the present paper, the method of discrete vortices [4] is generalized to the case of unsteady flow past a wing with jets and arbitrary shape in plan. The problem is solved in the linear formulation; the conditions used are standard: no flow through the wing and jet, finite velocities at the trailing edges where there is no jet, and also a dynamical condition on the jet. The wing and jet are assumed to be thin and the medium inviscid and incompressible.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 139–144, May–June, 1982.     

Fundamental and subharmonic resonances in a system with a ‘1-1’ internal resonance

The fundamental and subharmonic resonances of a two degree-of-freedom oscillator with cubic stiffness nonlinearities and linear viscous damping are examined using a multiple-seales averaging analysis. The system is in a 1–1 internal resonance, i.e., it has two equal linearized eigenfrequencies, and it possesses nonlinear normal modes. For weak coupling stiffnesses the internal resonance gives rise to a Hamiltonian Pitchfork bifurcation of normal modes which in turn affects the topology of the fundamental and subharmonic resonance curves. It is shown that the number of resonance branches differs before and after the mode bifurcation, and that jump phenomena are possible between forced modes. Some of the steady state solutions were found to be very sensitive to damping: a whole branch of fundamental resonances was eliminated even for small amounts of viscous damping, and subharmonic steady state solutions were shifted by damping to higher frequencies. The analytical results are verified by a numerical integration of the equations of motion, and a discussion of the effects of the mode bifurcation on the dynamics of the system is given.     

一种基于超声共振谱的低Q值材料共振频率提取方法

超声共振谱技术通过测量样本在超声激励下产生的固有共振频率来计算弹性参数,而共振频率的提取是整个测量过程的关键.低$Q$值(品质因数)材料由于其衰减特性,导致共振谱平缓并无法直观地从谱图上观察得到共振频率,为从中提取更为有效的共振频率, 本文提出了一种新的共振频率提取方法.采用经验模态分解法将材料频率响应自适应分解为有限个具有特殊振荡特性的固有模态函数分量,根据材料的超声共振谱先验信息选择具有共振频率特性的固有模态函数分量,并从中提取共振频率. 以短切纤维环氧树脂材料(仿骨材料, $Q \approx$25)为例, 通过实验与传统线性预测方法进行对比,计算弹性系数和工程模量. 实验结果表明新方法的计算效率高,对弱激发模态更为敏感,共振频率的匹配数量(26)多于传统方法(21)且满足5倍于弹性系数的估计要求,优化后的弹性模量更接近标准值.新方法可从低$Q$值材料平缓的频谱中提取数量足够且有效的共振频率,不仅有效提升了力学参数估计的可靠性,而且拓展了超声共振谱技术的应用范围.      

Parametric pulse excitation in distributed mechanical systems with nonstationary boundaries

In a distributed system whose parameters vary with time the natural oscillation modes are interconnected and so it is possible to get parametric excitation of several synchronized harmonic modes simultaneously. If the natural oscillation spectrum of such a system consists of almost equally spaced lines, then a periodic change of the parameters with time can lead to the excitation of pulse-type oscillations [1]. This phenomenon can occur both in systems whose size varies with time and in systems whose boundary properties are nonstationary. The present paper is devoted to a study of the instability in these systems.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 145–151, July–August, 1976.     

Axisymmetric waves in a fluid-filled,hollow, elastic cylinder surrounded by a fluid

The laws of propagation of axisymmetric normal modes in a hollow cylinder filled with and surrounded by fluid media are investigated. Dispersion curves are plotted, exhibiting functional relations between the complex propagation constant and the dimensionless frequency. Distinctive attributes of the dispersion curves and the energy characteristics of the investigated waveguide structure are analyzed.Institute of Hydromechanics, National Academy of Sciences of Ukraine, Kiev. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 30, No. 9, pp. 15–23, September, 1994.     

Effects of the local resonance in bending on the longitudinal vibrations of reticulated beams

This work investigates the dynamic behaviour of reticulated beams obtained by repeating a unit cell made up of interconnected beams or plates forming an unbraced frame. As beams are much stiffer in tension–compression than in bending, the longitudinal modes of such structures (governed by tension–compression at the macroscopic scale) can appear in the same frequency range as the bending modes of the elements. The condition of scale separation being respected for compression, the homogenization method of periodic discrete media is used to rigorously derive the macroscopic behaviour at the leading order. In the absence of bending resonance, the longitudinal vibrations of the structure are described at the macroscopic scale by the usual equation for beams in tension–compression. When there is resonance, the form of the equation is unchanged but the real mass of the structure is replaced by an effective mass which depends on the frequency. This induces an abnormal response in the neighbourhood of the natural frequencies of the resonating elements. This paper focuses on the consequences on the modal properties and the transfer function of the reticulated structure. The same macroscopic mode shape can be associated with several natural frequencies of the structure (but the deformation of the elements at the local scale is different). Moreover the vibrations are not transmitted when the effective mass is negative. These phenomena are first evidenced theoretically and then illustrated with numerical simulations.     

Gas bubble oscillations in a fluid in the presence of 2:1 resonance between the radial and deformation oscillation frequencies

Small nonlinear oscillations of an ellipsoidal bubble in a fluid in the presence of 2:1 frequency resonance between the radial and ellipsoidal modes are considered. The equations of motion are reduced to Hamiltonian form. The quadratic and cubic terms are taken into account in the expansion of the Hamiltonian. The Hamilton function is transformed to the normal form using the invariant normalization method in the first approximation. This makes it possible to construct an analogy between the system considered and the well-known problem of a pendulous spring. The radial and ellipsoidal bubble oscillation modes correspond to the vertical and horizontal coordinates of a material point, respectively. In the absence of resonance the solution of the nonlinear equations differs from the solution of the linear equations by only a small (quadratic in the amplitude) change in the oscillation frequency. In the resonance case the radial and ellipsoidal oscillation modes periodically change places and the energy of one mode is converted into that of the other. The interest in the system in resonance is associated with precisely this fact. The question of the dissipation effect in real media is considered. The decay rate depends significantly on the physical properties of the material and, in certain special cases, can be small enough for the energy transfer effect to manifest itself.