Time reversal of flexural waves in a beam at audible frequency

There has been very limited work on the application of time reversal to the propagation of audible frequency waves in mechanical structures. The present work concentrates on the application of time reversal to the focusing of audible range, flexural waves in an infinite beam, and to the detection of local heterogeneity in such a beam. Practical applications of time reversal of flexural waves in structures include vibration energy focusing, detection of vibratory or acoustic sources, and detection of defects in mechanical structures. An analytical model of flexural wave propagation in the beam as well as sensing and emission using piezoelectric transducers is presented. Time reversal experiments are conducted and compared to the model results in either a homogeneous beam or a beam with point mass heterogeneities. In the various situations tested, it is shown that time reversal effectively compensates the spreading in time of the impulse due to the dispersive propagation of flexural waves. One interesting aspect of this property is the generation of large amplitude impulsive responses in the beam using remote actuators. Finally, the "Decomposition de l'Operateur de Retournement Temporel" approach is examined to detect and localize point mass scatterers in the beam.     

Interaction dynamics of elastic waves with a complex nonlinear scatterer through the use of a time reversal mirror

This Letter reports on work performed to locate and interrogate a nonlinear scatterer in a linearly elastic medium through the use of a time reversal mirror in combination with nonlinear dynamics. Time reversal provides the means to spatially and temporally localize elastic energy on a scattering feature while the nonlinear dynamics spectrum allows one to determine whether the scatterer is nonlinear (e.g., mechanical damage). Here elastic waves are measured in a solid and processed to extract the nonlinear elastic response. The processed elastic signals are then time reversed, rebroadcast, and found to focus on the nonlinear scatterer, thus defining a time-reversed nonlinear elastic wave spectroscopy process. Additionally, the focusing process illuminates the complexity of the nonlinear scatterer in both space and time, providing a means to image and investigate the origins and physical mechanisms of the nonlinear elastic response.     

Amplitude degradation of time-reversed pulses in nonlinear absorbing thermoviscous fluids

The linear wave equation in a lossless medium is time reversible, i.e., every solution p(x, t) has a temporal mirror solution p(x, -t). Analysis shows that time reversal also holds for the lossless nonlinear wave equation. In both cases, time-reversal invariance is violated when losses are present. For nonlinear propagation loses cannot normally be ignored; they are necessary to prevent the occurrence of multivalued waveforms. Further analysis of the nonlinear wave equation shows that amplification of a time-reversed pulse at the array elements also leads to a violation of time reversal even for lossless nonlinear acoustics. Numerical simulations are used to illustrate the effect of nonlinearity on the ability of a time-reversal system to effectively focus on a target in an absorbing fluid medium. We consider both the amplitude and arrival time of retrodirected pulses. The numerical results confirm that both shock generation (with the accompanying absorption) and amplification at the array, adversely affect the ability of a time-reversal system to form strong retrodirective sound fields.     

Electromagnetic retarded interaction and symmetry violation of time reversal in high order stimulated radiation and absorption processes of light

It has been proved that when the retarded effect (or multiple moment effect) of radiation fields is taken into account, the high order stimulated radiation and stimulated absorption probabilities of light are not the same so that time reversal symmetry would be violated, though the Hamiltonian of electromagnetic interaction is still unchanged under time reversal. The reason to cause time reversal symmetry violation is that certain filial or partial transition processes of bound atoms are forbidden or cannot be achieved due to the law of energy conservation and the special states of atoms themselves. These restrictions would cause the symmetry violation of time reversal of other filial or partial transition processes which can be actualized really. The symmetry violation is also relative to the asymmetry of initial states of bound atoms before and after time reversal. For the electromagnetic interaction between non-bound atoms and radiation field, there is no such kind of symmetry violation of time reversal. In this way, the current formula on the parameters of stimulated radiation and absorption of light with time reversal symmetry should be revised. A more reliable foundation can be established for the theories of laser and nonlinear optics in which non-equilibrium processes are involved.     

轴向加速运动黏弹性梁受迫振动中的混沌动力学

研究了黏弹性轴向运动梁在外部激励和参数激励共同作用下横向振动的混沌非线性动力学行为. 引入有限支撑刚度, 并考虑黏弹性本构关系取物质导数, 同时计入由梁轴向加速度引起的沿径向变化的轴力, 建立轴向运动黏弹性梁横向非线性振动的偏微分-积分模型. 通过Galerkin截断方法研究了外部激励的频率和因速度简谐脉动引起的参数激励的频率在不可通约关系时轴向运动连续体的非线性动力学行为, 并对不同截断阶数的数值预测进行了对比. 基于对控制方程的Galerkin截断, 得到离散化的常微分方程组, 使用四阶Runge-Kutta方法求解. 基于此数值解, 运用非线性动力学时间序列分析方法, 通过Poincaré 映射, 观察到轴向运动梁随扰动速度幅值的倍周期分岔现象, 并比较了有无外部激励对倍周期分岔的影响. 分别在低速以及近临界高速运动状态下, 从相平面图、Poincaré 映射以及频谱分析的角度识别了系统中存在的准周期运动形态. 关键词： 轴向运动梁 非线性 混沌 分岔     

Eddy viscosity for time reversing waves in a dissipative environment

We present new results for the time reversal of weakly nonlinear pulses traveling in a random dissipative environment. Also we describe a new theory for calculating the eddy viscosity for weakly nonlinear waves propagating over a random surface. The turbulent viscosity is calculated from first principles, namely, without imposing any stress-strain hypothesis. A viscous shallow water model is considered and its effective viscosity characterized. We also show that weakly nonlinear waves can still be time reversed under weak dissipation. Incoherently scattered signals are recompressed, both for time reversal in transmission as well as in reflection. Under the weakly nonlinear, weakly dissipative regime, dissipation only affects the refocused pulse profile regarding its amplitude, but its shape is not corrupted. Numerical experiments are presented.     

TRC路径特征及其对时间反演信噪比的影响

时间反演具有时空聚焦特征,在许多方面有着潜在的应用。其中,基于时间反演腔的系统是一种主要的反演系统,可用于脉冲压缩、波束成形、微扰探测等。时间反演腔通常是一个电大的微波混沌腔,内部电磁波的传播具有明显的多径特征,即时间色散特征。因此,在时间反演过程中,反演腔可对反演信号进行相位补偿,重构出初始信号,从而在初始位置形成脉冲的时间压缩和空间聚焦。为了拓展时间反演腔的实际应用,本文基于多径信道模型研究腔体参数对反演性能的影响,重点分析路径的衰减特征、串扰特征和叠加特征对反演信噪比的影响,并总结给出影响反演信噪比的主要参数以及基本规律。     

Nonlinear dynamics of higher-dimensional system for an axially accelerating viscoelastic beam with in-plane and out-of-plane vibrations

In this paper, the bifurcations and chaotic motions of higher-dimensional nonlinear systems are investigated for the nonplanar nonlinear vibrations of an axially accelerating moving viscoelastic beam. The Kelvin viscoelastic model is chosen to describe the viscoelastic property of the beam material. Firstly, the nonlinear governing equations of nonplanar motion for an axially accelerating moving viscoelastic beam are established by using the generalized Hamilton’s principle for the first time. Then, based on the Galerkin’s discretization, the governing equations of nonplanar motion are simplified to a six-degree-of-freedom nonlinear system and a three-degree-of-freedom nonlinear system with parametric excitation, respectively. At last, numerical simulations, including the Poincare map, phase portrait and Lyapunov exponents are used to analyze the complex nonlinear dynamic behaviors of the axially accelerating moving viscoelastic beam. The bifurcation diagrams for the in-plane and out-of-plane displacements via the mean axial velocity, the amplitude of velocity fluctuation and the frequency of velocity fluctuation are respectively presented when other parameters are fixed. The Lyapunov exponents are calculated to identify the existence of the chaotic motions. From the numerical results, it is indicated that the periodic, quasi-periodic and chaotic motions occur for the nonplanar nonlinear vibrations of the axially accelerating moving viscoelastic beam. Observing the in-plane nonlinear vibrations of the axially accelerating moving viscoelastic beam from the numerical results, it is found that the nonlinear responses of the six-degree-of-freedom nonlinear system are much different from that of the three-degree-of-freedom nonlinear system when all parameters are same.     

A two-dimensional pseudospectral model for time reversal and nonlinear elastic wave spectroscopy

One way to characterize metallic materials in the presence of defects like dislocation networks is to measure their large dynamic nonlinear elastic response. In this numerical study, a new method combining the nonlinear elastic wave spectroscopy (NEWS) method with a time reversal (TR) process is proposed. This method, called NEWS-TR, uses nonlinear analysis as a pretreatment of time reversal and then consists of retrofocusing only nonlinear components on the defect position. A two-dimensional pseudospectral time domain algorithm is developed here to validate the NEWS-TR method as a potential technique for damage location. Hysteretic nonlinear behavior of the materials being studied is introduced using the Preisach-Mayergoyz model. Moreover, in order to extend this solver in two dimensions, the Kelvin notation is used to modify the elastic coefficient tensor. Simulations performed on a metallic sample show the feasibility and value of the NEWS-TR methodology for microdamage imaging. Retrofocusing quality depends on different parameters such as the filtering method used to keep only nonlinear components and the nonlinear effect measured. In harmonic generation, pulse inversion filtering seems to be a more appropriate filtering method than classical harmonic filtering for most defect positions, mainly because of its ability to filter all fundamental components.     

Beam focusing by metallic nano-slit array containing nonlinear material

A two-dimensional nonlinear finite difference time domain (NFDTD) method is performed to investigate the beam focusing phenomenon in metallic nano-slit array containing nonlinear material. Illuminated by an incident TM-polarized Gauss beam, the arrayed nano-slits transport electro-magnetic energy in the form of surface plasmons (SPs) and provide desired phase retardations for beam focusing, owing to the nonlinear response. The position of focus can be actively controlled by parameters of incident beam, such as the intensity and the beam waist. The physical origin of the focusing effect is discussed by considering SPs mode and nonlinear optical theory. PACS 73.20.Mf; 78.66.Bz; 41.85.Lc     

Detection of fatigue-induced micro-cracks in a pipe by using time-reversed nonlinear guided waves: a three-dimensional model study

Localization of fatigue-related micro-cracks in pipelines is of increasing importance in industrial applications. A three-dimensional (3D) fatigue-crack imaging technique combining nonlinear guided waves with time reversal is proposed in this paper for potential applications in pipeline inspections. By using this method, the non-classical nonlinear guided waves generated from micro-cracks with hysteretic behavior are recorded, and the third harmonic waves are used to reconstruct the fatigue-crack images in a pipe by using a time reversal (TR) process. The feasibility of this method is examined by the imaging simulations for a steel pipe with varied defect areas. A finite-difference time-domain (FDTD) code is programmed to solve the wave equations under cylindrical coordinates, and simulate the experimental process of wave propagation. The results show that: (1) the proposed technique has excellent spatial retrofocusing capability; (2) the accuracy of defect localization and sizing depends on the crack orientation and the adopted guided wave mode; and (3) different displacement/stress components have varied sensitivities to the crack orientation.     

Effect of a dc magnetic field on the magnetization relaxation of uniaxial single-domain ferromagnetic particles driven by a strong ac magnetic field

The nonlinear ac stationary response of the magnetization of noninteracting uniaxial single-domain ferromagnetic particles acted on by superimposed dc and ac magnetic fields applied along the anisotropy axis is evaluated from the Fokker-Planck equation, expressed as an infinite hierarchy of recurrence equations for Fourier components of the relaxation functions governing longitudinal relaxation of the magnetization. The exact solution of this hierarchy comprises a matrix continued fraction, allowing one to evaluate the ac nonlinear response and reversal time of the magnetization. For weak ac fields, the results agree with perturbation theory. It is shown that the dc bias field changes substantially the magnetization dynamics leading to new nonlinear effects. In particular, it is demonstrated that for a nonzero bias field as the magnitude of the ac field increases the reversal time first increases and having attained its maximum at some critical value of the ac field, decreases exponentially.     

时间反转聚焦在组织中形成损伤的数值建模

为考察基于时间反转方法的高强度聚焦超声治疗在预设目标点处的组织损伤情况,使用三维有限差分算法求解Westervelt方程,建立非线性声波传播数值模型,采用97阵元相控阵结合虚拟源的时间反转方法进行超声聚焦,分析其形成的声场和热场,并考察目标点偏离轴线时的组织损伤形成规律。结果表明随着目标点偏轴距离的增大,声压旁瓣开始增多。旁瓣的温升较低,不足以形成组织损伤。时间反转方法可用于多点聚焦,在一定的范围内,形成多点目标损伤而不产生额外的周围组织损伤。同时多点聚焦可以形成一个较大的损伤区域,减少超声治疗时间。     

Fighting against diffraction: apodization and near field diffraction structures

Diffraction is a natural phenomenon, which occurs when waves propagate or encounter an obstacle. Diffraction is also a fundamental aspect of modern optics: all imaging systems are diffraction systems. However, like a coin has two sides, diffraction also leads to some unfavorable effects, such as an increase in the size of a beam during propagation, and a limited minimal beam size after focusing. To overcome these disadvantages set by diffraction, many techniques have been developed by various groups, including apodization techniques to reduce the divergence of a laser beam and increase the resolution, and time reversal, STED microscopy, super lenses and optical antennas to obtain resolution down to nano‐scale. This review concentrates on the diffraction of electromagnetic waves, and the ways to overcome beam divergence and the diffraction limit.     

Strongly nonlinear beats in the dynamics of an elastic system with a strong local stiffness nonlinearity: Analysis and identification

We consider a linear cantilever beam attached to ground through a strongly nonlinear stiffness at its free boundary, and study its dynamics computationally by the assumed-modes method. The nonlinear stiffness of this system has no linear component, so it is essentially nonlinear and nonlinearizable. We find that the strong nonlinearity mostly affects the lower-frequency bending modes and gives rise to strongly nonlinear beat phenomena. Analysis of these beats proves that they are caused by internal resonance interactions of nonlinear normal modes (NNMs) of the system. These internal resonances are not of the classical type since they occur between bending modes whose linearized natural frequencies are not necessarily related by rational ratios; rather, they are due to the strong energy-dependence of the frequency of oscillation of the corresponding NNMs of the beam (arising from the strong local stiffness nonlinearity) and occur at energy ranges where the frequencies of these NNMs are rationally related. Nonlinear effects start at a different energy level for each mode. Lower modes are influenced at lower energies due to larger modal displacements than higher modes and thus, at certain energy levels, the NNMs become rationally related, which results in internal resonance. The internal resonances of NNMs are studied using a reduced order model of the beam system. Then, a nonlinear system identification method is developed, capable of identifying this type of strongly nonlinear modal interactions. It is based on an adaptive step-by-step application of empirical mode decomposition (EMD) to the measured time series, which makes it valid for multi-frequency beating signals. Our work extends an earlier nonlinear system identification approach developed for nearly mono-frequency (monochromatic) signals. The extended system identification method is applied to the identification of the strongly nonlinear dynamics of the considered cantilever beam with the local strong nonlinear stiffness at its free end.     

Tunable unidirectional sound propagation through a sonic-crystal-based acoustic diode

Nonreciprocal wave propagation typically requires strong nonlinear materials to break time reversal symmetry. Here, we utilized a sonic-crystal-based acoustic diode that had broken spatial inversion symmetry and experimentally realized sound unidirectional transmission in this acoustic diode. These novel phenomena are attributed to different mode transitions as well as their associated different energy conversion efficiencies among different diffraction orders at two sides of the diode. This nonreciprocal sound transmission could be systematically controlled by simply mechanically rotating the square rods of the sonic crystal. Different from nonreciprocity due to the nonlinear acoustic effect and broken time reversal symmetry, this new model leads to a one-way effect with higher efficiency, broader bandwidth, and much less power consumption, showing promising applications in various sound devices.     

克尔介质中皮秒亮孤子多光子J-C模型粒子数反转特性

利用全量子理论研究了克尔介质中皮秒孤子光场与二能级原子多光子相互作用系统中粒子数反转随时间的变化特性,用MATLAB软件编程模拟并讨论了初始平均光子数、克尔介质与场模非线性相互作用强弱、孤子光场与原子耦合强度及相互作用过程中跃迁光子数目对粒子数反转的影响.数值计算结果表明：初始平均光子数越大、克尔介质与场模的非线性相互作用越强、孤子光场与原子耦合强度越小或相互作用过程中跃迁光子数目越多,粒子数反转崩坍与恢复的振荡幅度越小、平均值越大.     

Nonlinear Plasma Dynanics in the Plasma Wakefield Accelerator

Excitation of nonlinear plasma oscillations by an ultrarelativistic electron beam is considered in this paper. It is shown, by analytical solutions of the fully relativistic nonlinear fluid equations in one dimension, that under certain conditions on the relative densities of the electron beam and the plasma, extremely large longitudinal electric fields can be generated in the wake of the beam. This scheme can be considered as a nonlinear regime of the plasma wakefield accelerator (PWFA), and is seen to have the advantage that the transformer ratio, the ratio of the maximum amplitude of the accelerating field behind the driving beam over the maximum amplitude of the decelerating field inside of the beam, can be made arbitrarily large, dependent only on the length of the driving beam. The effects of beam loading on the efficiency of this scheme are considered, and are shown to be equivalent to those predicted in the linear regime.     

On the possibility of suppression of self-focusing of a femtosecond laser pulse during its propagation in a cubic nonlinear medium

Based on a computer simulation, the self-focusing of an axially symmetric beam in a cubic nonlinear medium under the anomalous dispersion conditions is studied with the account for the time dispersion of nonlinear response, which manifests for femtosecond pulses. It is shown that, at a certain value of the parameter of linear modulation of the pulse (or of its chirp), the dispersion of nonlinear response can lead either to the suppression of formation of a nonlinear focus and to the possibility of formation of optical shock waves in time or even to a change in the regime of the beam self-action owing to the action of the local response, i.e., to the change from the self-focusing of the beam to the regime of its defocusing.     

Numerical approaches to simulating nonlinear ultrasound fields generated by diagnostic-type transducers

Two theoretical approaches for simulating nonlinear focused ultrasound fields generated by a diagnostic convex array are compared. The first model is based on the three-dimensional Westervelt equation and describes the full structure of the array field with high accuracy. However, it requires great computational resources and is technically difficult. The second model is based on an axially symmetric form of the parabolic KZK equation for estimating the strength of nonlinear effects in the focal region of a beam, which reduces the computational time by a factor of several hundreds. To establish the boundary conditions to the KZK model, the radius and the focal length of a circular piston source are defined such that the simulated field on the beam axis in the linear case fits the real structure of the field in the focal region. It is shown that the parabolic model can be used to accurately describe the spatial and temporal structure of the field generated by a diagnostic transducer in the focal region of the beam along its axis and in the plane of the beam’s electronic focusing.