Frequency Dispersion Curves Research Articles

Characterization of elastic wave propagation in high-speed railway tracks considering temperature effects

This study establishes a three-dimensional infinite-length periodic ballasted track model, considering the effect of coupling between bending and torsion and the impact of sleeper width. The GPWE method is employed to investigate the energy bandgap structure of the track. The results show three vertical bending wave bandgaps below 1500 Hz: 0–109 Hz, 110–167 Hz, and 1081–1126 Hz. The node responses from a hammer impact experiment are used in the WSM to calculate the frequency dispersion curves, which confirms the validity of this model. The model is then utilized to analyze the influence of rail temperature forces and temperature-dependent fastener stiffness on the vibration transmission characteristics of the track within a temperature range of −35°C to 25°C. As temperature increases, the starting and cutoff frequency values and the width of bandgaps both decrease nonlinearly and later stabilize, with the width of the Bragg bandgap showing a significant reduction of 83%. The vibration attenuation rate in the bandgap gradually increases and finally stabilizes, especially above 15°C, while the vibration transfer characteristics are not significantly affected by temperature.

Theoretical investigation of collective motion in some liquid metals using pseudopotential approach

The collective motions are studied using a recently created pseudopotential and some elastic constants. This paper involves the calculations of the collective motions like phonon dispersion curves (PDCs) for longitudinal and transverse frequencies ([Formula: see text] and [Formula: see text]), longitudinal sound velocity ([Formula: see text]), transverse sound velocity ([Formula: see text]), isothermal bulk modulus (B), modulus of rigidity (G), Young’s modulus (Y), Poisson’s ratio [Formula: see text], Debye temperature ([Formula: see text]) of some monovalent (Na), divalent (Mg) and polyvalent liquid metals (Al, Pb, Bi). This investigation is based upon the theory of pseudopotential of second-order approach via the equations of Hubbard and Beeby (HB). [Formula: see text] The pair correlation function is calculated from the different mean spherical approximation (MSA), one-component plasma (OCP) and optimized random phase approximation (ORPA) structure factors employed in this work. Three different types of local field correction (screening) functions proposed by Hartree (H), Taylor (T) and Nagy (N) are utilized in this paper. The current findings are found to be in qualitative accord with experimental and theoretical data.

Characterization and application of submarine seismic ambient noise in the Bohai Sea and Yellow Sea

Submarine seismic ambient noise imaging combines current marine and on-land seismic detection technologies. Based on data from several broadband shallow-sea type ocean bottom seismometers (SOBSs) deployed in the Bohai Sea and north Yellow Sea, this paper analyzes the submarine seismic ambient noise characteristics. It explores the theory, technology, method and application of the submarine seismic ambient noise imaging using the single-point horizontal and vertical spectral ratio method (HVSR). The observations yield the following results: 1) Submarine seismic ambient noise has consistent and constant energy, making it an appropriate passive seismic source for submarine high-frequency surface wave investigation. 2) Using the HVSR approach, a single three-component OBS could differentiate between the basement and sediments. Array seismic observation could be utilized to extract the frequency dispersion curve and invert it to obtain the velocity structure for more accurate stratification. 3) The SOBS we use is suitable for submarine surface wave exploration. 4) Tomography results with greater resolution and deeper penetration could be obtained by combining active and passive sources in a simultaneous inversion of the HVSR and frequency dispersion curve. Seamless land-to-ocean seismic research can be accomplished with submarine seismic ambient noise imaging technologies.

Study on Corrosion Monitoring of Reinforced Concrete Based on Longitudinal Guided Ultrasonic Waves

The corrosion of reinforced concrete (RC) is one of the most serious durability problems in civil engineering structures, and the corrosion detection of internal reinforcements is an important basis for structural durability assessment. In this paper, the appropriate frequency required to cause excitation signals in the specimen is first analyzed by means of frequency dispersion curves. Subsequently, the effectiveness of five damage indexes (DIs) is discussed using random corrosion in finite elements. Finally, guided ultrasonic wave (GUW) tests are conducted on reinforcement and RC specimens at different corrosion degrees, and the test results are verified using a theoretical corrosion model. The results show that the larger the covered thickness is at the same frequency, the higher the modal order of the GUW in the frequency dispersion curve is, and the smaller the group velocity is. The SAD is the most sensitive to the corrosion state of the reinforcement compared with the other DIs, and it shows a linear increasing trend with the increase in the corrosion degree of the reinforcement. The SAD values of the RC specimens showed a three-stage change with the increase in the corrosion time, and the time until the appearance of corrosion cracks was increased with the increase in the covered thickness. It can be seen that increasing the covered thickness is an effective method to delay the time until the appearance of corrosion cracks in RC specimens.

Identification of pre-tensioning of rebars based on the acoustoelastic effect

Rebar as the core vertical prestressing stressing member of the bridge structure. The stress loss can not be effectively controlled during the whole process of anchorage, coupled with the influence of environmental corrosion. The result is that the vertical prestressing force cannot meet the design requirements, seriously threatening the durability and safety of the structure. This paper combines ultrasonic guided wave technology with acoustic elasticity theory to detect the effective prestressing of rebar. Guided wave propagation characteristics and acoustic elasticity effect under different tensile forces after bare rebar and outsourcing mortar are investigated. Tension tests and guided wave propagation tests were carried out on the specimens to analyze the variation rule between the first wave speed of guided waves and the tension force. The results show that the first wave speed of the guided wave decreases with the increase of the excitation frequency, which is consistent with the theoretical frequency dispersion curve. With the gradual increase of the excitation power, the amplitude of the guided waveform received by the rebar increases gradually only in the time and frequency domains. The law between first wave speed and tension after the outsourcing mortar is consistent with that of bare rebar. It is feasible to establish an identification method of prestressing rebar based on acoustoelastic effect.

FREE VIBRATION MODELING IN A FUNCTIONALLY GRADED HOLLOW CYLINDER USING THE LEGENDRE POLYNOMIAL APPROACH

Introduction: The building industry is under increasing pressure to maximize performance while reducing the costs and the environmental impact. To solve this problem, a new type of materials, i.e., functionally graded materials (FGMs), are proposed. These materials have the advantage of being able to withstand harsh environments without losing their properties. Purpose of the study: The paper aims to further extend the understanding of the propagation modes and characteristics of guided waves in FGM cylinders with infinite lengths. In the course of the study, we analyzed a cylindrical shell composed of three annular layers, each separated by a gradient layer across the wall thickness. A modeling tool based on the Legendre orthogonal polynomial method is proposed in the paper. Methods: The method applied results in an eigenvalue/eigenvector problem. The boundary conditions are integrated into the constitutive equations of guided wave propagation. The phase velocity and normalized frequency dispersion curves are calculated. Besides, the displacement distributions and stress field profiles for a functionally graded cylinder with various graded indices in both modes (axisymmetric and symmetric) are calculated and discussed. The results show a constant fluctuation of effective FGM material. Results: It was found that the phase velocity curves of the same mode decrease as the exponents of the power law increase. In addition, the boundary conditions have a greater impact on the normal stresses. The accuracy and effectiveness of the improved orthogonal polynomial method are demonstrated through a comparison of the exact solution obtained by an analytical-numerical method and our numerical results.

Experimental study on various factors contributing to poor microtremor data processing results using ESAC method

In practical applications, microtremor data often encounter contamination from various sources of noise, leading to suboptimal processing outcomes. This study aimed to provide a comprehensive analysis of the impact of noise on the microtremor method. The focus was on presenting explicit and quantitative evidence to demonstrate the extent to which these noises influence the accuracy and reliability of the method. Initially, a ten-station triangular array was employed to gather a sequence of high-precision microtremor data. Next, various influencing factors, such as high-amplitude random noise, were generated and incorporated into the meticulously collected real data in order to simulate the actual perturbation process. Based on both the raw and the artificially contaminated data, a series of dispersion curves of the fundamental-mode Rayleigh waves were obtained by applying the extended spatial autocorrelation method (ESAC). Based on the obtained results, several valuable conclusions have been derived that can effectively inform the procedures for acquiring and processing microtremor data. If the amplitude and duration of sudden onset vibrations are considerable, more data should be collected to reduce the Negative impact. If non-random noise exhibits similar frequency characteristics as microtremor data, and the duration and energy of the noise are relatively small, it can be ignored; otherwise, more data should be collected. The dispersion curve, determined by the principle of the ESAC method, remains unaffected by any changes in the amplitude of one or more traces. The likelihood of frequency dispersion curve distortion increases as the impact of entire data anomalies on the coherence curve becomes greater. In general, when the collection positions of abnormal data are more widely spaced, the impact on the frequency dispersion curve tends to decrease. It is important to note that, in a nested-triangular array configuration, the central geophone plays a crucial role in ensuring accurate and dependable outcomes.

Effects of the strain gradients on the band structures of the elastic waves propagating in 1D phononic crystals: An analytical approach

In this paper, an analytical method is proposed for the analysis of the effects of the strain gradients on the frequency band structures and band-gaps of the elastic waves propagating in one-dimensional (1D) nano-sized phononic crystals (PCs). The considered 1D nano-sized PCs are made of periodic unit-cells containing a metal section and an epoxy section. The strain gradients are taken into account to obtain the governing equations by using the strain energy, the kinetic energy and the Hamilton's principle. An analytical solution based on the transfer matrix method is derived to compute the frequency dispersion curves or band structures of the elastic waves. The continuity conditions on the interface between the neighboring sections and the periodic boundary conditions for the unit-cell are satisfied using the proposed analytical method. Two band-gap characteristics, namely the starting frequency and the width of each band-gap, are defined and investigated. The effects of the small-scale parameters in the strain-gradient theory on the band structures and the band-gap characteristics are analyzed and discussed in detail. It is demonstrated that the strain gradients may significantly influence the frequency band structures and the band-gap characteristics of the elastic waves propagating in 1D nano-sized PCs.

Acoustic Characteristics Analysis of Double-Layer Liquid-Filled Pipes Based on Acoustic–Solid Coupling Theory

Based on the theory of acoustic–solid coupling, the phase velocity-thickness product of a double-layer liquid-filled pipeline is analyzed, and the dispersion relationship between angular frequency and wavenumber–thickness product is analyzed, providing a theoretical basis for ultrasonic guided wave detection. The wave number analytical expression of the double-layer liquid-filled pipeline is constructed, and the dispersion relationship of the double-layer liquid-filled pipeline under different frequency–thickness products and wavenumber–thickness products is calculated through parameter scanning. The dispersion curves of the double-layer liquid-filled pipeline are numerically analyzed in the domains of pressure acoustics, solid mechanics, and acoustic–solid coupling. The numerically simulated dispersion curves show high consistency with the analytically calculated dispersion curves. The analysis of the phase velocity frequency–thickness product indicates that the axial mode dispersion curves of the pipe wall decrease with the increase in frequency–thickness product in the coupling domain, and then tend to be flat and intersect with the radial mode dispersion curves in the coupling domain; these intersection points cannot be used for ultrasonic guided wave detection. The T(0,1) mode dispersion curve in the coupling domain of the pressure acoustics domain remains smooth from low frequency to high frequency. It is found that the dispersion curves of the phase velocity frequency–thickness product, angular frequency wavenumber–thickness product, and the acoustic pressure distribution map of the double-layer liquid-filled pipeline based on acoustic–solid coupling can provide theoretical support for ultrasonic guided wave detection of pipelines.

Study on bandgap characteristics and vibration attenuation mechanism of double-oscillator power-exponent prism phononic crystal plate

This study introduces a local resonance mechanism to a periodic acoustic black hole (ABH) structure to achieve vibration control of plate structures and proposes a double-oscillator power-exponent prism phononic crystal. Results show that the periodic power-exponent prism can generate a high-frequency bandgap, the interior oscillator can generate a low-frequency bandgap, and the top oscillator can separate the frequency dispersion curve at around 700 Hz to form a bandgap with a width of 189 Hz. The double-oscillator power-exponent prism phononic crystal, composed of two types of oscillators and a power-exponent prism, can simultaneously have high-, middle-, and low-frequency bandgaps. Simulations and experiments show that it has a good attenuation effect on flexural vibration in the bandgap frequency band. The present results can provide a useful reference for bandgap design based on the combination of multiple mechanisms.

From ambient vibration data analysis to 1D ground-motion prediction of the Mj 5.9 and the Mj 6.5 Kumamoto earthquakes in the Kumamoto alluvial plain, Japan

We present horizontal ground motion predictions at a soft site in the Kumamoto alluvial plain for the Mj 5.9 and Mj 6.5 Kumamoto earthquakes of April 2016, in the framework of an international blind prediction exercise. Such predictions were obtained by leveraging all available information which included: (i) analysis of earthquake ground motions; (ii) processing of ambient vibration data (AMV); and (iii) 1D ground response analysis. Spectral analysis of earthquake ground-motion data were used to obtain empirical estimates of the prediction site amplification function, with evidence of an amplification peak at about 1.2 Hz. Horizontal-to-vertical spectral ratio analysis of AMV confirmed this resonance frequency and pointed out also a low-frequency resonance around 0.3 Hz at the prediction site. AMV were then processed by cross-correlation, modified spatial autocorrelation and high-resolution beamforming methods to retrieve the 1D shear-wave velocity (Vs) structure at the prediction site by joint inversion of surface-wave dispersion and ellipticity curves. The use of low frequency dispersion curve and ellipticity data allowed to retrieve a reference Vs profile down to few thousand meters depth which was then used to perform 1D equivalent-linear simulations of the M 5.9 event, and both equivalent-linear and nonlinear simulations of the M 6.5 event at the target site. Adopting quantitative goodness-of-fit metrics based on time–frequency representation of the signals, we obtained fair-to-good agreement between 1D predictions and observations for the Mj 6.5 earthquake and a poor agreement for the Mj 5.9 earthquake. In terms of acceleration response spectra, while ground-motion overpredictions were obtained for the Mj 5.9 event, simulated ground motions for the Mj 6.5 earthquake severely underestimate the observations, especially those obtained by the nonlinear approach.Graphical

Influence of Aspect Ratio on Wave Propagation in Granular Crystals Consisting of Ellipse-Shaped Particles

The behavior of waves in granular materials is very complex and closely related to the macro-micro properties such as particle shape, packing, and particle size distribution. In this study, the influence of the aspect ratio (AR) and confining stress on wave propagation in 2D granular crystals consisting of ellipse-shaped particles is investigated based on discrete element method. The energy attenuation, wave velocity, wavefront shape, and frequency dispersion are mainly focused. The results show that the energy attenuation exhibits an increasing trend with bigger confining stress and smaller AR in the direction of wave propagation. The wave velocities increase with increasing AR, and the relation between compressive wave velocities and the confining stress and AR is obtained. Wavefront has the elliptical shape which has the same AR as elliptical particles of specimen, and its analytical expression is also obtained. The dispersion is affected by AR, which is mainly reflected in the modified particle spacing in sine fitted frequency dispersion curve. Larger confining stress allows a larger maximum frequency, and produces stronger dispersion. The above conclusion can provide a reference for wave-guided granular crystals design.

Study on the Stage Characteristics of Noise Dispersion in the Deformation Evolution of Red Sandstone

The uniaxial compression experiment of red sandstone is observed simultaneously by using acoustic emission and digital speckle correlation methods. The deformation evolution of red sandstone is divided into microfracture random expansion stage, deformation localization stage, subinstability stage, and instability failure stage. Green’s function and dispersion curve of each stage are obtained from the noise data picked up by acoustic emission equipment, and the dispersion characteristics of each evolution stage are analyzed. The results show the following: (1) In the stage of random propagation of microcracks, the noise in the low frequency range passes through at a higher phase velocity, the phase velocity changes periodically, the correlation coefficient is high at the initial time, and the variation trend of frequency dispersion curve is relatively consistent. (2) In the deformation localization stage, the frequency range without zero phase velocity moves to the high frequency range, and the phase velocity changes periodically. (3) In the subinstability stage, dense phase velocity zeros appear on the dispersion curve image, the dispersion curve tends to be disordered, various indicators change obviously, and the correlation coefficient decreases rapidly. (4) In the unstable failure stage, the fracture evolution is completed, and the variation trend of each index of the frequency dispersion curve is consistent. (5) The size and difference of the sensitive kernel function of the two layers are related to the evolution region of the fracture. The sensitivity kernel value of the medium layer where the fracture evolves is high, and the longer the evolution time, the greater the difference.

Leak Source Beam-forming Location of Spacecraft in Orbit Based on Dispersion Characteristics of Lamb Wave

In recent years, the number of space debris is increasing rapidly, threatening the safety of spacecraft. More and more spacecraft are at risk of colliding with space debris, which may cause air leakages on the sealed container of spacecraft. This is a huge danger for spacecraft and astronauts. In this paper, a leak source location method based on beam-forming and dispersion characteristics of Lamb wave is proposed. The ‘L’ shaped sensor array with 8 acoustic emission sensors is used to collect the Lamb waves. Due to the dispersion characteristics, a 10 kHz narrow band pass filter is used firstly to process the signals. And then with a priori knowledge of guided-mode frequency dispersion curves of Lamb waves, the corresponding propagation speeds of waves are determined to obtain the time delay matrixes. By the delay-and-sum beam-forming algorithm, the leak source can be directed. The experiment result shows that the accuracy rate of the leak location is more than 90% evaluated by location error less than 5°.

Application of the Ritz–Rayleigh method for Lamb waves in extremely anisotropic media

We propose a numerical approach for the calculation of frequency–dispersion curves in a flat anisotropic waveguide based on the Ritz–Rayleigh method, offering several significant benefits over commonly used analytical and numerical models. The problem is linearized using a tailored functional basis based on Legendre polynomials, utilizing all symmetries provided by the problem to reduce the computational demands, which brings significant benefits for an inverse procedure, and thus for material characterization. The approach is completely general in terms of elastic properties and direction of propagation. As an exemplary calculation, frequency–dispersion curves of phase and group velocity are calculated for a (001)-oriented free-standing film of the Ni–Mn–Ga alloy exhibiting a strong cubic anisotropy. In the case of propagation along the [100] direction, the dispersion curves oscillate strongly, creating pairs of symmetric and antisymmetric modes. Taking advantage of the generality of the approach, the curves are also presented for the case of extreme anisotropy. Sets of curves for directions [110] and [1 0.9 0] (i.e., 3∘ off [110]), where the pseudo-surface wave exists, are also calculated. High-frequency asymptotes are shown to correspond with surface and bulk modes propagating along the plate.

Study on Mode Repulsion of Ultrasonic-Guided Waves in Pipe Bends

Abstract The wave motion of guided waves in pipe bends is still veiled in some mystery, which hinders the application of guided-wave techniques in the inspection of pipelines with bends. Mode repulsion, which exists in the wavenumber versus frequency dispersion curves of guided waves in pipe bends, is an intriguing phenomenon deserving in depth study. The governing equation of wave motion in pipe bends, deduced by the semi-analytical finite element (SAFE) method, can be regarded as an eigenvalue problem. The eigenvalue derivatives, with respect to the wavenumber, are investigated to determine whether mode repulsion will occur or not. A term in the second derivative of the eigenvalue is identified to determine the mode repulsions. With respect to symmetry, it is found that mode repulsion only occurs between modes of one and the same type, such as symmetric or antisymmetric modes, and does not occur between modes of different type, like between symmetric and antisymmetric modes. A specific case of mode repulsion in a small-bore thin-walled pipe in the low-frequency range, where relatively fewer modes exist, is further studied, and the interactions between these modes are clarified. The evolutions of mode shapes before and after mode repulsion are further illustrated.

Particle filtering with adaptive resampling scheme for modal frequency identification and dispersion curves estimation in ocean acoustics

The goal of this work is to accurately estimate the modal frequencies and dispersion curves from a measured ocean acoustics signal. A particle filtering approach, a class of sequential Monte Carlo methods, is developed for modal frequency identification and dispersion curves estimation from a time-frequency representation of ocean acoustics signal. The adaptive resampling algorithm for enhancing the quality of a set of particles after likelihood calculation is implemented to improve the accuracy of the modal estimates as well as the dispersion curves of the signal. Results demonstrate the advantages in implementing the adaptive resampling into the conventional sequential importance sampling particle filter (SIS-PF) instead of using the sequential importance resampling (SIR) scheme. The noise robustness of the proposed method is demonstrated through examples where the realizations of different Signal-to-Noise Ratio (SNR) levels were used to test the performance of the adaptive resampling method. The results display the evidences that the adaptive resampling particle filter (AR-PF) is superior to the SIR-PF. Via root mean square error (RMSE), the AR-PF delivers smaller errors than those obtained by the SIR-PF for all SNR levels, emphasizing the benefit in incorporating the adaptive resampling into the PF for modal frequency identification and dispersion curves estimation of ocean acoustics signal.

Micromagnetic Study of Reducing Forbidden Bandgaps and its Width in a Triangular Antidot Array Waveguide With Different Orientations

Dipole-exchange spin waves (SWs) propagating in a 1-D magnonic waveguide embedded with different orientations of triangular antidots are investigated through the micromagnetic simulation. The dipole-exchange SW propagation is affected due to the effect of changing the triangular antidots orientation. This leads to reduction in the number of forbidden bands and their gap width in the magnonic waveguide. The SW characteristics are interpreted by plotting frequency dispersion curves and analyzing the power and phase distribution profile (PPDP) and static demagnetization field profile to substantiate the occurrence of bandgaps. The number of forbidden bands and their gap width get reduced due to the formation of symmetric and antisymmetric modes, and it will be further clarified by the PPDPs. For θ = 0°, four wide bandgaps are recorded; I and II bandgaps are due to symmetric modes, whereas the remaining gaps are due to antisymmetric modes. From θ = 0° to 90°, the number of forbidden bands is reduced from 4 to 2 because of the inhomogeneous distribution of the internal field introduced by the presence of triangular array antidots in the transverse direction, and it is confined via the static demagnetization field. The way to develop magnonic antidot-based waveguide for SW filter applications is paved by opening, closing, and tuning of the gap width. This can be effectively brought about by changing the orientation of triangular antidots.

Dispersion curves for a natural fibre composite panel: Experimental and numerical investigation

The design of orthotropic panels, in which some mechanical properties are associated to high degrees of uncertainty, such as for the natural fibres, is very arduous and often techniques that give local information are needed. In this paper a unidirectional flax–polyethylene panel is experimentally and numerically investigated. Experimentally, for a given distance source–receiver, the group velocities are obtained by processing the guided waves time-transient signals via a time-frequency transform leading to an estimation of the dispersion curves for different fibre directions and frequencies. Test results are then compared and used to validate numerical ones, obtained by means of both Finite Element and Wave Finite Element models.

Analytical solutions of thermo-piezoelectric interactions in a solid fiber of polygonal cross-sections immersed in fluid

PurposeThe purpose of this paper is to study the analytical solutions of transversely isotropic thermo-piezoelectric interactions in a polygonal cross-sectional fiber immersed in fluid using the Fourier expansion collocation method.Design/methodology/approachA mathematical model is developed for the analytical study on a transversely isotropic thermo-piezoelectric polygonal cross-sectional fiber immersed in fluid using a linear form of three-dimensional piezothermoelasticity theories. After developing the formal solution of the mathematical model consisting of partial differential equations, the frequency equations have been analyzed numerically by using the Fourier expansion collocation method (FECM) at the irregular boundary surfaces of the polygonal cross-sectional fiber. The roots of the frequency equation are obtained by using the secant method, applicable for complex roots.FindingsFrom the literature survey, it is evident that the analytical formulation of thermo-piezoelectric interactions in a polygonal cross-sectional fiber contact with fluid is not discussed by any researchers. Also, in this study, a polygonal cross-section is used instead of the traditional circular cross-sections. So, the analytical solutions of transversely isotropic thermo-piezoelectric interactions in a polygonal cross-sectional fiber immersed in fluid are studied using the FECM. The dispersion curves for non-dimensional frequency, phase velocity and attenuation coefficient are presented graphically for lead zirconate titanate (PZT-5A) material. The present analytical method obtained by the FECM is compared with the finite element method which shows a good agreement with present study.Originality/valueThis paper contributes the analytical model to find the solution of transversely isotropic thermo-piezoelectric interactions in a polygonal cross-sectional fiber immersed in fluid. The dispersion curves of the non-dimensional frequency, phase velocity and attenuation coefficient are more prominent in flexural modes. Also, the surrounding fluid on the various considered wave characteristics is more significant and dispersive in the hexagonal cross-sections. The aspect ratio (a/b) of polygonal cross-sections is critical to industry or other fields which require more flexibility in design of materials with arbitrary cross-sections.