Asymptotic inverse scattering

Several inverse techniques are developed for determining the shape of an unknown scattering surface by analyzing backscattered acoustic or electromagnetic waves. These techniques are based on asymptotic high frequency representations of the fields and may be divided into three categories. The first one is the geometrical imaging method where the surface is reconstructed by means of a travel-time analysis which is here specified to the far field by utilizing Minkowski's support function. Furthermore, a geometrical method is developed for localizing edges from mid field data measured along a curve. The second category is called quasigeometrical imaging and uses geometric optics or higher order amplitudes for the reconstruction. It is shown that cross-polarized electromagnetic far field amplitude measurements permit one to deduce the complete quadratic approximation of the surface at the specular points from which the surface can be reconstructed pointwise. The third category may be subsumed under ‘asymptotic inverse scattering identities’. Here, asymptotic relations between scattered fields and distributions associated with the geometry of the scatterer are established. It is shown that the physical optics far field inverse scattering identity is only a leading order asymptotic relation but as such is also valid for non-convex scatterers. Furthermore, asymptotic inverse scattering identities are deduced which relate the singular function of a closed surface to the backscattered field data measured on a sphere enclosing the scatterer. This generalizes far field results of Cohen and Bleistein (Wave Motion 1 (1979), p. 153) to the mid field.     

The penetrable coated sphere embedded in a point source excitation field

A low frequency acoustic wave field emanates from a given point and fills up the whole space. A penetrable lossy sphere with a coeccentric spherical core, which is also penetrable and lossy but characterized by different physical parameters, disturbs the given point source field. We obtain zeroth- and first-order low frequency solutions of this scattering problem in the interior of the spherical core, within the spherical shell, and in the exterior medium of propagation. We also derive the leading nonvanishing terms of the normalized scattering amplitude, the scattering cross-section as well as the absorption cross-section. The special case of a penetrable sphere is recovered either by equating the physical parameters that characterize the media in the shell and in the exterior, or by reducing the radius of the core sphere to zero. By letting the compressional viscosity of the medium in the interior sphere, or in the shell, go to zero, we obtain corresponding results for the lossless case. The incident point source field is so modified as to be able to obtain the corresponding results for plane wave incidence in the limit as the source point approaches infinity. It is observed that a small scatterer interacts stronger with a point source generated field than with a plane wave. A detailed analysis of the influence that the geometrical and the physical parameters of the problem have on the scattering process is also included. An interesting conclusion is that if the point source is located at a distance more than five radii of the scatterer away from it, then no significant changes with the plane excitation case are observed.     

钛合金与钴铬钼切向微动磨损行为研究

本文中以Ti6Al4V与CoCrMo合金为研究对象,在不同介质中开展了球/面切向微动磨损的试验研究,结合多种微观分析手段,揭示了不同条件下钛合金与钴铬钼间切向微动运行特性和损伤机理.结果表明：钛合金球与钴铬钼平面间的切向微动主要运行于部分滑移区和混合区.随着微动振幅的增加,F_t-D曲线从直线型向椭圆型转变,其中振幅较大时微动末期F_t-D曲线呈平行四边型.在部分滑移区,摩擦系数较小且保持不变,磨痕表面磨损轻微.在混合区,摩擦系数的变化因振幅的不同各存在两种情况.在振幅较小的混合区,磨斑边缘有磨屑堆积,中心以黏着为主.在振幅较大的混合区,磨痕表面以磨粒磨损为主.     

A micromechanical mass sensing method based on amplitude tracking within an ultra-wide broadband resonance

We propose a mass sensing scheme in which amplitude shifts within a nonlinear ultra-wide broadband resonance serve as indicators for mass detection. To achieve the broad resonance bandwidth, we considered a nonlinear design of the resonator comprised of a doubly clamped beam with a concentrated mass at its center. A reduced-order model of the beam system was constructed in the form of a discrete spring-mass system that contains cubic stiffness due to axial stretching of the beam in addition to linear stiffness (Duffing equation). The cubic nonlinearity has a stiffening effect on the frequency response causing nonlinear bending of the frequency response toward higher frequencies. Interestingly, we found that the presence of the concentrated mass broadens the resonant bandwidth significantly, allowing for an ultra-wide operational range of frequencies and response amplitudes in the proposed mass sensing scheme. A secondary effect of the cubic nonlinearity is strong amplification of the third harmonic in the beam’s response. We computationally study the sensitivity of the first and third harmonic amplitudes to mass addition and find that both metrics are more sensitive than the linearized natural frequency and that in particular, the third harmonic amplitude is most sensitive. This type of open-loop mass sensing avoids complex feedback control and time-consuming frequency sweeping. Moreover, the mass resolution is within a functional range, and the design parameters of the resonator are reasonable from a manufacturing perspective.     

Progress on a mathematical inversion technique for non destructive evaluation

In earlier work, computer output was presented for synthetic pulse-echo data which was processed according to a mathematical imaging technique. This technique was based on the physical optics farfield inverse scattering (acronym, POFFIS) formalism for scattering by volume defects. In this paper a number of theoretical advances in the POFFIS formalism are reported, with attendant revisions in the computer algorithm.Firstly, a revised POFFIS formalism was developed in which the surface of the scatterer is directly related to the scatteringi data. In this formalism, aperture limited scattering data yields an image of a corresponding aperture of the scattering surface of the defect. Secondly, this formalism will also yield an image of the scattering surface of a crack. Thirdly, for true amplitude data, the impedance or reflection coefficient may be read directly from the computer output. Related to this last result was the elimination of an “image fading” phenomenon at certain critical angles. Fourthly, the computer algorithm, which was originally designed to process data for a spherically symmetric “trailer hitch”, was modified (and tested) to process data when the range to the center of the coordinate system was different at each observational angle. Fifthly, the algorithm was modified (and tested) to process data when the average propagation speed varied with angle.Implementation on a real data set is discussed.     

Estimates for low-frequency elastic scattering by a rigid body

When a plane elastic wave is scattered by a rigid body the surface integral of the traction, projected along the direction of polarization of the incident wave, provides the leading low-frequency approximation for the scattering amplitudes. Two kinds of lower and upper bounds for the surface traction integral are given. One is based on the geometrical characteristics of the scatterer and is expressed in terms of corresponding values of the best fitting interior and exterior confocal triaxial ellipsoids. The case of best fitting interior and exterior spheres is examined as a special case. These bounds are sharp in the sense that they both become equalities when the scatterer degenerates to an ellipsoid. The other kind of lower and upper bounds involve the capacity of the scatterer. All estimates were obtained by using the generalized Dirichlet and Thomson Principles of Potential Theory in Elastostatics. Furthermore, all constants appearing in the bounds are given in terms of the ratio of the phase velocities for the transverse and the longitudinal wave. An upper bound for scattering by a cube at normal incidence is also included.This work was done while both authors were visiting the Department of Mathematics of The University of Tennessee at Knoxville. The second author wishes to acknowledge partial support from The University of Tennessee Science Alliance.     

Bifurcations in the Mean Angle of a Horizontally Shaken Pendulum: Analysis and Experiment

A pendulum excited by high-frequency horizontal displacement of its pivot point will vibrate with small amplitude about a mean position. The mean value is zero for small excitation amplitudes, but if the excitation is large enough the mean angle can take on non-zero values. This behavior is analyzed using the method of multiple time scales. The change in the mean angle is shown to be the result of a pitchfork bifurcation, or a saddle-node bifurcation if the system is imperfect. Analytical predictions of the mean angle as a function of frequency and amplitude are confirmed by physical experiment and numerical simulation.     

The singular function of a surface and physical optics inverse scattering

It is shown how to recover both the location and the reflection coefficient of a scatterer using only high frequency backscattered data. The result is obtained without use of the far field approximation although a separate identity is derived when this approximation is introduced. This latter result improves upon previously derived physical optics far field inverse scattering identities.     

Effect of flow regime change from subsonic to transonic on the air loads of an oscillating airfoil

In this research, the effect of flow regime change from subsonic to transonic on the air loads of a pitching NACA0012 airfoil is investigated. To do this, the effect of change in flow regime on the lift and pitching moment coefficients hysteresis cycles is studied. The harmonic balance approach is utilized for numerical calculation due to its low computational time. Verifications are also made with previous works and good agreements are observed. The assessment of flow regime change on the aforementioned hysteresis cycles is accomplished in the Mach number range of M=0.65–0.755. The reduced frequency and pitch amplitude also vary from k=0.03 to 0.1 and α₀=1–2.51°, respectively. Results show that the effect of increase in Mach number is to increase and decrease the lift coefficient during downstroke and upstroke, respectively, whereas at low reduced frequencies, the effect of increase in Mach number may lead to a reverse manner when airfoil moves toward its extremum angle of attack. Results also reveal that as the pitch amplitude varies, the shape of lift coefficient hysteresis cycle depends more on the pitch amplitude than on the appearance of shock. It is shown that as the Mach number increases, the incidence angles correspond to the extremum pitching moment, and depending on the reduced frequency, lie between zero and extremum angle of attack. These incidence angles shift toward the extremum angle of attack as the reduced frequency decreases. Results also show that the increase in pitch amplitude at low Mach number, in such a way that leads to the formation of shock around the extremum angle of attack, causes the extremum pitching moment to appear around these angles and at high Mach number, depending on the reduced frequency, the extremum pitching moment incidence angles would be between zero and extremum incidence angle.     

An Approximate Method for Oscillation Analysis of a Wheel Pair and a Bogie

An approximate method is considered for determining the amplitude of self-oscillations of a freight bogie. Apart from frequency, the determining parameters of the self-oscillations are the amplitudes and the phase shifts of the peak and zero values of the variables relative to the peak and zero values of one of the phase variables.     

Degenerate weakly non-linear elastic plane waves

Weakly non-linear plane waves are considered in hyperelastic crystals. Evolution equations are derived at a quadratically non-linear level for the amplitudes of quasi-longitudinal and quasi-transverse waves propagating in arbitrary anisotropic media. The form of the equations obtained depends upon the direction of propagation relative to the crystal axes. A single equation is found for all propagation directions for quasi-longitudinal waves, but a pair of coupled equations occurs for quasi-transverse waves propagating along directions of degeneracy, or acoustic axes. The coupled equations involve four material parameters but they simplify if the wave propagates along an axis of material symmetry. Thus, only two parameters arise for propagation along an axis of twofold symmetry, and one for a threefold axis. The transverse wave equations decouple if the axis is fourfold or higher. In the absence of a symmetry axis it is possible that the evolution equations of the quasi-transverse waves decouple if the third-order elastic moduli satisfy a certain identity. The theoretical results are illustrated with explicit examples.     

Flow-induced oscillation of two circular cylinders in tandem arrangement at low Re

Results are presented for flow-induced vibrations of a pair of equal-sized circular cylinders of low nondimensional mass (m^*=10) in a tandem arrangement. The cylinders are free to oscillate both in streamwise and transverse directions. The Reynolds number, based on the free-stream speed and the diameter of the cylinders, D is 100 and the centre-to-centre distance between the cylinders is 5.5D. The computations are carried out for reduced velocities in the range 2≤U^*≤15. The structural damping is set to zero for enabling maximum amplitudes of oscillation. A stabilized finite element method is utilized to carry out the computations in two dimensions. Even though the response of the upstream cylinder is found to be qualitatively similar to that of an isolated cylinder, the presence of a downstream cylinder is found to have significant effect on the behaviour of the upstream cylinder. The downstream cylinder undergoes very large amplitude of oscillations in both transverse and streamwise directions. The maximum amplitude of transverse response of the downstream cylinder is quite similar to that of a single cylinder at higher Re beyond the laminar regime. Lock-in and hysteresis are observed for both upstream and downstream cylinders. The downstream cylinder undergoes large amplitude oscillations even beyond the lock-in state. The phase between transverse oscillations and lift force suffers a 180 jump for both the cylinders almost in the middle of the synchronization regime. The phase between the transverse response of the two cylinders is also studied. Complex flow patterns are observed in the wake of the freely vibrating cylinders. Based on the phase difference and the flow patterns, the entire flow range is divided into five sub-regions.     

TTS in LAOS: validation of time-temperature superposition under large amplitude oscillatory shear

The validation of time-temperature superposition of non-linear parameters obtained from large amplitude oscillatory shear is investigated for a model viscoelastic fluid. Oscillatory time sweeps were performed on a 11?wt.% solution of high molecular weight polyisobutylene in pristane as a function of temperature and frequency and for a broad range of strain amplitudes varying from the linear to the highly non-linear regime. Lissajous curves show that this reference material displays strong non-linear behaviour when the strain amplitude is exceeding a critical value. Elastic and viscous Chebyshev coefficients and alternative non-linear parameters were obtained based on the framework of Ewoldt et al. (J Rheol 52(6):1427?C1458, 2008) as a function of temperature, frequency and strain amplitude. For each strain amplitude, temperature shift factors a _T (T) were calculated for the first order elastic and viscous Chebyshev coefficients simultaneously, so that master curves at a certain reference temperature T _ref were obtained. It is shown that the expected independency of these shift factors on strain amplitude holds even in the non-linear regime. The shift factors a _T (T) can be used to also superpose the higher order elastic and viscous Chebyshev coefficients and the alternative moduli and viscosities onto master curves. It was shown that the Rutgers-Delaware rule also holds for a viscoelastic solution at large strain amplitudes.     

Nonlinear Isolator Dynamics at Finite Deformations: An Effective Hyperelastic,Fractional Derivative,Generalized Friction Model

In presenting a nonlinear dynamic model of a rubber vibrationisolator, the quasistatic and dynamic motion influences on theforce response are investigated within the time and frequencydomain. It is found that the dynamic stiffness at the frequency ofa harmonic displacement excitation, superimposed upon the longterm isolator response, is strongly dependent on staticprecompression, dynamic amplitude and frequency. The problems ofsimultaneously modelling the elastic, viscoelastic and frictionforces are removed by additively splitting them, modelling theelastic force response by a nonlinear, shape factor basedapproach, displaying results that agree with those of aneo-Hookean hyperelastic isolator at a long term precompression.The viscoelastic force is modeled by a fractional derivativeelement, while the friction force governs from a generalizedfriction element displaying a smoothed Coulomb force. A harmonicdisplacement excitation is shown to result in a force responsecontaining the excitation frequency and its every otherhigher-order harmonic, while using a linearized elastic forceresponse model, whereas all higher-order harmonics are present forthe fully nonlinear case. It is furthermore found that the dynamicstiffness magnitude increases with static precompression andfrequency, while decreasing with dynamic excitationamplitude – eventually increasing at the highest amplitudes due tononlinear elastic effects – with its loss angle displaying amaximum at an intermediate amplitude. Finally, the dynamicstiffness at a static precompression, using a linearized elasticforce response model, is shown to agree with the fully nonlinearmodel except at the highest dynamic amplitudes.     

Comparative Analysis of the Profile Evolutions of an Elastic Harmonic Wave Caused by the Second and Third Harmonics

The influence of the second and third harmonics on the evolution of a harmonic longitudinal wave propagating through a nonlinearly elastic material has been simulated for real composite materials (the most typical plots are presented for a granulated composite with copper granules and molybdenum matrix). The frequency and initial amplitude are varied beginning from conditionally small values (at which visible distortions appear after a great number of oscillations) to extremely large values (at which the profile becomes distorted already after the second or third oscillation). Four and three different stages of profile evolution due to the influence of the second and third harmonics, respectively, are observed. It is found out that the effect of the initial amplitude on the evolution process is weaker for the second harmonic, and the effect of the frequency on the evolution process is weaker for the third harmonics. It is also revealed that the ranges of frequencies and initial amplitudes within which the evolution caused by different harmonics is very intensive are different—the effect of the third harmonic is stronger at larger values of both parameters. The effects of both harmonics are tantamount within the boundary ranges where the second harmonic is already predominant and the third harmonic is at the early stage of development     

Transonic limit cycle oscillation behavior of an aeroelastic airfoil with free-play

The limit cycle oscillation (LCO) behaviors of an aeroelastic airfoil with free-play for different Mach numbers are studied. Euler equations are adopted to obtain the unsteady aerodynamic forces. Aerodynamic and structural describing functions are employed to deal with aerodynamic and structural nonlinearities, respectively. Then the flutter speed and flutter frequency are obtained by V-g method. The LCO solutions for the aeroelastic airfoil obtained by using dynamically linear aerodynamics agree well with those obtained directly by using nonlinear aerodynamics. Subsequently, the dynamically linear aerodynamics is assumed, and results show that the LCOs behave variously in different Mach number ranges. A subcritical bifurcation, consisting of both stable and unstable branches, is firstly observed in subsonic and high subsonic regime. Then in a narrow Mach number range, the unstable LCOs with small amplitudes turn to be stable ones dominated by the single degree of freedom flutter. Meanwhile, these LCOs can persist down to very low flutter speeds. When the Mach number is increased further, the stable branch turns back to be unstable. To address the reason of the stability variation for different Mach numbers at small amplitude LCOs, we find that the Mach number freeze phenomenon provides a physics-based explanation and the phase reversal of the aerodynamic forces will trigger the single degree of freedom flutter in the narrow Mach number range between the low and high Mach numbers of the chimney region. The high Mach number can be predicted by the freeze Mach number, and the low one can be estimated by the Mach number at which the aerodynamic center of the airfoil lies near its elastic axis. Influence of angle of attack and viscous effects on the LCO behavior is also discussed.     

基于浸入边界算法的振动钝体绕流模拟研究

传统CFD方法在振动钝体绕流计算中常借助动网格技术,网格再生任务繁重。针对于此,本文利用可在静止网格中计算动边界绕流问题的浸入边界算法(IBM),编写数值模拟程序,分别对竖向强迫正弦振动方柱(Re=UD/v=103、振幅恒定、振动频率变化)以及桥梁断面(Re=UB/v=7.5×103、振幅、振动频率均变化)展开气动特性和流场特征结构分析。初步研究结果表明,振幅恒定为方柱高度的14%时,其涡脱锁定区长度为0.06~0.2,锁定区后端(Stc0.2)振动方柱涡脱频率回归静止涡脱频率;不同振幅下的桥梁断面阻力系数均在静止涡脱频率处产生峰值,桥梁断面升力系数则在此处均出现归零效应,且振幅越大,归零效应愈明显。     

On three-dimensional acoustic-gravity waves in model non-isothermal atmospheres

The propagation of acoustic-gravity waves is studied in a family of model non-isothermal atmospheres, including temperature profiles with any initial and asymptotic temperature and adjustable maximum temperature gradient. The equation for the vertical velocity of linear acoustic-gravity waves is solved exactly in terms of hypergeometric functions, the wave field being described to all orders in the scattering parameter kL, at all frequencies and distances, including wavelengths comparable to the scale of temperature change and atmospheric layers with a large temperature gradient. It is found that since the temperature is bounded, in the asymptotic regime waves grow in amplitude exponentially and phase increases linearly with altitude. The growth in amplitude is larger than exponential and the phase increases faster than linearly for atmospheres whose temperature increases with altitude, the effect being more marked for high frequency waves in regions of large temperature gradients. The accumulation of these effects leads to a wave field which is equivalent to the isothermal case at asymptotic temperature modified by a constant amplitude factor and phase shift which account for the history of propagation of the wave through the temperature gradients.     

Limit-cycle oscillations in unsteady flows dominated by intermittent leading-edge vortex shedding

High-frequency limit-cycle oscillations of an airfoil at low Reynolds number are studied numerically. This regime is characterized by large apparent-mass effects and intermittent shedding of leading-edge vortices. Under these conditions, leading-edge vortex shedding has been shown to result in favorable consequences such as high lift and efficiencies in propulsion/power extraction, thus motivating this study. The aerodynamic model used in the aeroelastic framework is a potential-flow-based discrete-vortex method, augmented with intermittent leading-edge vortex shedding based on a leading-edge suction parameter reaching a critical value. This model has been validated extensively in the regime under consideration and is computationally cheap in comparison with Navier–Stokes solvers. The structural model used has degrees of freedom in pitch and plunge, and allows for large amplitudes and cubic stiffening. The aeroelastic framework developed in this paper is employed to undertake parametric studies which evaluate the impact of different types of nonlinearity. Structural configurations with pitch-to-plunge frequency ratios close to unity are considered, where the flutter speeds are lowest (ideal for power generation) and reduced frequencies are highest. The range of reduced frequencies studied is two to three times higher than most airfoil studies, a virtually unexplored regime. Aerodynamic nonlinearity resulting from intermittent leading-edge vortex shedding always causes a supercritical Hopf bifurcation, where limit-cycle oscillations occur at freestream velocities greater than the linear flutter speed. The variations in amplitude and frequency of limit-cycle oscillations as functions of aerodynamic and structural parameters are presented through the parametric studies. The excellent accuracy/cost balance offered by the methodology presented in this paper suggests that it could be successfully employed to investigate optimum setups for power harvesting in the low-Reynolds-number regime.