Frequency-dependent Model Research Articles

Modeling the effect of dispersion and attenuation for frequency-dependent amplitude variation with offset

As research in oil and gas exploration progresses, unconventional resources, such as shale gas, are increasingly becoming the focal point in the global pursuit of oil and gas resource. Shale gas reservoirs significantly differ from conventional sandstone reservoirs in aspects such as rock composition, pore type, occurrence mode, fluid, etc., thereby amplifying the challenges associated with geophysical modeling and the prediction of sweet spots. Since the formation and storage of shale gas are positively correlated with shale fracturing, a modeling approach based on Chapman theory is introduced to complete frequency-dependent petrophysical modeling. Additionally, the Frequency-dependent Amplitude Variation with Offset (FAVO) technique can estimate velocity dispersion by using the reflection coefficient information related to incidence angle and frequency. This method can more effectively identify fluids within shale reservoir. However, current FAVO forward modeling only considers the velocity dispersion and attenuation at the interface, neglecting the attenuation dispersion effects during interlayer propagation. To this end, we utilize Chapman-based petrophysical modeling as a foundation and conduct seismic forward modeling studies employing the compound matrix method. Through experimental analysis, we meticulously examine the attenuation dispersion effects at interfaces and within layers. Finally, we conduct FAVO simulations that vividly delineate the interplay between reservoir parameters and seismic responses.

A frequency-dependent model for bone remodeling using a micromorphic porous medium subjected to harmonic mechanical loading

In this paper, the bone tissue was modeled as a linear viscoelastic material saturated with interstitial fluid. We considered a specific case of harmonic loading and related the mechanical stimuli to the loading frequency. In this way, we could include the inertial effect in the model while not having to deal with the perturbation during each loading period. Two types of mechanical signals were considered: strain energy and dissipation energy. A parametric study revealed the dependency of the two signals on loading frequency and material property. The evolution of the apparent mass density supported the parametric study’s findings. Under the three different frequency loadings, the strain energy-stimulated samples experienced identical remodeling scenarios. The samples stimulated with dissipation energy, on the other hand, exhibited a strong frequency dependence. An additional study was performed to investigate the effect of long-term variations in the loading frequency on the remodeling process. This demonstrated the model’s capabilities in designing and evaluating load regimes for rehabilitation following a bone injury or bone reconstruction.

Electromechanical coupling optimization for a sandwich beam activation using piezoceramics

Sandwich beams composed of two stiff carbon fiber skins and a viscoelastic core are often used to passively dissipate energy through the activation of shearing in a specific frequency range. Semi-active or active energy dissipation can be made possible on those structures adding piezoelectric patches, coupling mechanical vibration energy and electrical energy. J.L. Guyader et al. [jsv, 1978] theorized a frequency dependent equivalent dynamic model of a sandwich structure, recently extended to the homogenization of anisotropic multi-layers by F.Marchetti et al. [jsv, 2020]. This frequency-dependent stiffness of the beam has to be taken into account in the piezoceramic thickness definition in order to optimize the global electromechanical coupling coefficient. An analytical model is derived from the works of F.Marchetti et al. and adapted to predict the first vibration mode of a finite cantilever beam instrumented with a piezoelectric patch. The electromechanical coupling evolution is then calculated with the proper boundary conditions and for different PZT ceramic thicknesses. This model is furtherly used to define the best transducer configuration for a given sandwich structure beam.

An analytical method for broadband acoustic analysis of 2D cavities containing or bounded by porous materials

Real-life vibro-acoustic systems often involve porous treatments, resulting in complex-valued and frequency-dependent models that are challenging to solve. Traditional prediction techniques like the finite element (FE) method requires huge computational cost, especially in the mid to high frequency ranges. This paper develops a novel spectral dynamic stiffness (SDS) formulation, using very few number of degrees of freedom but describing the broadband acoustic behaviour of acoustic cavities with porous materials highly accurately. The method employs frequency-dependent shape function that satisfies exactly the (damped) Helmholtz equation to describe the (equivalent) acoustic pressure field, and also features an innovative approach to use the fast-convergent Modified Fourier series to describe any arbitrary acoustic BCs. Finally, the SDS matrices for cavities containing or bounded by porous materials are formulated in an analytical manner. It is demonstrated that the method exhibits a much higher computational efficiency over the FE package COMSOL, at least 6 times faster than the COMSOL with a similar level of accuracy, and benchmark solutions are provided. This promising method can provide a powerful tool for systems with porous materials that have frequency-dependent characteristics, paving the way for efficient and accurate acoustic analysis in complex engineering applications.

Prediction of frequency response of sub-frame bushing and study of high-order fractional derivative viscoelastic model

This paper presents experimental and dynamic modeling research on the rubber bushings of the rear sub-frame. The Particle Swarm Optimization algorithm was utilized to optimize a Backpropagation (BP) neural network, which was separately trained and tested across two frequency ranges: 1–40 Hz and 41–50 Hz, using wideband frequency sweep dynamic stiffness test data. The testing errors at amplitudes of 0.2 mm, 0.3 mm, and 0.5 mm were found to be 1.03%, 3.05%, and 1.96%, respectively. Subsequently, the trained neural network was employed to predict data within the frequency range of 51–70 Hz. To incorporate the predicted data into simulation software, a dynamic model of the rubber bushing was established, encompassing elastic, friction, and viscoelastic elements. Additionally, a novel model, integrating high-order fractional derivatives, was proposed based on the frequency-dependent model for the viscoelastic element. An enhanced Particle Swarm Optimization algorithm was introduced to identify the model's parameters using the predicted data. In comparison to the frequency-dependent model, the new model exhibited lower fitting errors at various amplitudes, with reductions of 3.84%, 3.61%, and 5.49%, respectively. This research establishes a solid foundation for subsequent vehicle dynamic modeling and simulation.

A hybrid finite-discrete element method for modelling cracking processes in sandy mudstone containing a single edge-flaw under cyclic dynamic loading

Rock mass deformation and failure are macroscopic manifestations of crack initiation, propagation, and coalescence. However, simulating the transition of rocks from continuous to discontinuous media under cyclic dynamic loading remains challenging. This study proposes a hybrid finite-discrete element method (HFDEM) to model crack propagation, incorporating a frequency-dependent cohesive-zone model. The mechanical properties of standard sandy mudstone under quasi-static and cyclic dynamic loading were simulated using HFDEM, and the method's reliability was verified through experimental comparison. The comparative analysis demonstrates that HFDEM successfully captures crack interaction mechanisms and accurately simulates the overall failure behavior of specimens. Additionally, the effects of pre-existing flaw inclination angle and dynamic loading frequency on rock failure mechanisms were investigated. The numerical results reveal that rock samples exhibit significantly higher compressive strength under dynamic loading compared to quasi-static loading, with compressive strength increasing with higher cyclic dynamic load frequencies. Furthermore, by analyzing the strength characteristics, crack propagation, and failure modes of the samples, insights into the failure mechanisms of rocks under different frequency loads were obtained. This study provides valuable insights into crack development and failure of rocks under seismic loads, offering guidance for engineering practices.

Nationwide frequency-dependent seismic site amplification models for Iceland

Seismic wave amplification due to localized site conditions or site-effects, is an important aspect of regional seismic hazard assessment. In the absence of systematic studies of frequency-dependent site-effects during strong Icelandic earthquakes, various local site proxies of large-scale studies in other seismic regions have been used and/or proposed for application in Iceland. Recently, however, earthquake site-effects were rigorously quantified for 34 strong-motion stations in Southwest Iceland for the first time and correlated to distinct Icelandic geological units of hard rock, rock, lava rock, and sedimentary soil. These units are prevalent throughout Iceland and in this study therefore, we present 1) nationwide maps of proxies (e.g. slope, Vs30, geological units) that may contribute to a better estimation of site effects and associated, 2) frequency-dependent site-amplification maps of Iceland relative to the median ground-motion prediction of Rahpeyma et al. (2023) [55]. We particularly focus on the two transform zones of the country, on the basis of digital elevation models and detailed geological maps. Specifically, the frequency-dependent site factors for each geological unit are presented at 1, 2, 5, 7, 10–30 Hz, and PGA. Finally, for comparison, we generate site amplification maps based on recent large-scale models developed in other seismic regions and/or applied in other studies (e.g., ESRM20) as well as various site proxies they are based on (e.g., geology- and slope-based inferred VS30, geomorphological sedimentary thickness). We compare site-proxy maps and amplification maps from both Icelandic and large-scale, non-Icelandic, models. Specifically, neither spatial patterns nor amplification levels in either proxy or amplification maps from large-scale non-Icelandic studies resemble those observed from local quantitative strong-motion research as presented in this study. We attribute the discrepancy primarily to the young geology of Iceland and its formation history. Additionally, this study compares model performance across frequencies by assessing the bias of model predictions against empirical site amplifications in the South Iceland Seismic Zone, accounting for site-to-site variability of residuals indicating the superior performance of the local amplification model. The results presented in this study now allow a more informed estimation of earthquake ground motion amplitudes on various geological units that are expected to result in more reliable and information-based seismic hazard estimates in Iceland.

Experimental and theoretical investigation on magnetorheological elastomers containing carbonyl iron particles coated with silane coupling agent

Magnetorheological (MR) elastomer composites, comprising soft silicone rubber, various additives, and different weight fractions of carbonyl iron particles (CIPs) coated with silane coupling agent, are produced via a novel manufacturing process in an anisotropic state. This study encompasses both experimental and modeling investigations into the dynamic viscoelastic properties of magnetorheological elastomer (MREs) in shear mode under varying magnetic fields, displacement amplitudes, and frequencies. Two MRE vibration mitigation devices are fabricated to experimentally assess the shear storage modulus and the loss factor of MREs. The experimental findings reveal a pronounced MR effect in the MRE devices, where both the shear storage modulus and the loss factor increase with rising magnetic fields, frequencies, and particle weight fractions, yet decrease with higher displacement amplitudes. A modified fractional-derivative equivalent parametric model, grounded in a magnetic field- and frequency-dependent shear modulus model along with internal variable theory, is proposed to describe the effects of these key influencing factors on the MREs’ dynamic viscoelastic properties. Comparative analysis of experimental and numerical data demonstrates that this refined mathematical model can accurately represent the dynamic viscoelastic properties of MREs.

3D attenuation tomography of the Uttarakhand, NW Himalaya: Linkage to fluid or partial melt zones - Seismic hazard

This work proposes the shear-wave attenuation tomography of the Garhwal and Kumaun regions, Uttarakhand Himalaya, to investigate the crustal state. Based on attenuation characteristics, the high attenuated layer identified starts at ∼10 km in the Garhwal region and ∼5 km in the Kumaun region. The obtained model revealed that quality factor (Q) values vary from 16 to 664 and 49 to 866 at 1 Hz frequency for the Kumaun and Garhwal regions, respectively. The high attenuation rate (1/Q) in the Kumaun region compared to the Garhwal region may be due to the high attenuated layer at a shallower depth in the Kumaun region. The comparatively low attenuation rate of the Garhwal characterizes it as a region with high seismic hazard potential. Pioneeringly, layered frequency-dependent attenuation models are proposed up to a depth of 30 km for six different layers with 5 km thickness each, which will provide new insight into seismic hazard evaluation.

Modelling dielectric behaviour and anomalous relaxation in chromonic liquid crystals

ABSTRACT We discuss the feasibility of applying a frequency-dependent diffusive model to study the dielectric response of complex fluids. The model considers the effect of free and bound particles by solving a non-exponential relaxation behaviour and anomalous diffusion phenomena into a coupled system of differential equations and boundary conditions. To illustrate the potentialities of the model in the data interpretation, we used the experimental results that provide the frequency spectra of a system formed by an aqueous solution of sunset yellow, a chromonic liquid crystal. Our model yields a considerably better adjustment of the experimental data compared to the simple Havriliak-Negami model and provides the relaxation frequency due to the director mode associated with the stack fluctuations about their short axis. Furthermore, we obtained parameters associated with the ion motion nature of the sample, including diffusion coefficient, Debye length and the ionic diffusive regime, which present a similar subdiffusive behaviour found in nematic thermotropic liquid crystals.

Frequency-dependent elastic properties of fracture-induced VTI rocks in a fluid-saturated porous and microcracked background

Fractures are widely distributed underground. The stiffness matrix of fractured rocks has been extensively investigated in a fluid-saturated porous background medium. However, the existing stiffness models only incorporate the attenuation mechanism of wave-induced fluid flow (WIFF). For macroscopic fractures, the elastic scattering (ES) of fractures cannot be ignored. To alleviate this issue, a frequency-dependent stiffness matrix model is developed, including the mesoscopic WIFF between fractures and background (FB-WIFF), the microscopic squirt flow, and the macroscopic ES from the fractures. By combining the far-field scattered wavefields of normal incident P and SV waves with the linear slip theory, the dynamic full-stiffness matrices for fracture-induced effective vertically transversely isotropic rocks in a fluid-saturated porous and microcracked background are derived. Then, the P, SV, and SH-wave velocities and attenuation can be obtained through the Kelvin-Christoffel equation. The results indicate that the FB-WIFF mechanism significantly affects the velocities and attenuation of the P and SV waves but has almost no effect on the SH wave, whereas the squirt flow and ES mechanisms affect the velocities and attenuation of all the P, SV, and SH waves. For validation, the model is compared with existing models and previous experimental ultrasonic data.

Determination of elastic loss of piezoelectric materials by impedance curve fitting using intelligent algorithms

Understanding the loss parameters of piezoelectric materials is crucial for designing effective piezoelectric sensors. Traditional elastic loss parameter measurement techniques mainly rely on three methods: 3 dB bandwidth, impedance fitting, and ultrasonic attenuation. However, the elastic losses obtained through these methods are constant and frequency-independent, which does not align with the actual vibration characteristics of piezoelectric materials. Therefore, there is a need for a fast, accurate, and frequency-dependent method to obtain the elastic loss of piezoelectric materials. This paper introduces an approach that utilizes intelligent algorithms for fitting impedance curve to calculate elastic loss parameters. A frequency-dependent second-order energy loss model for piezoelectric materials is established. Then, a genetic algorithm is introduced to obtain the optimal elastic loss parameters. The results demonstrate a high consistency between theoretical and experimental impedances, with an error less than 5%. The elastic loss parameters obtained through intelligent algorithm-based impedance curve fitting match well with stress experiment results, with an error less than 6%. This method provides a rapid, accurate, and cost-effective way to obtain frequency-dependent second-order elastic loss parameters for piezoelectric materials.

From frequency-dependent models to frequency-independent enriched continua for mechanical metamaterials

Mechanical metamaterials have recently gathered increasing attention for their uncommon mechanical responses enabling unprecedented applications for elastic wave control. Many research efforts are driven towards the conception of always new metamaterials’ unit cells that, due to local resonance or Bragg-Scattering phenomena, may produce unorthodox macroscopic responses such as band-gaps, cloaking, focusing, channeling, negative refraction, etc. To model the mechanical response of large samples made up of these base unit cells, so-called homogenization or upscaling techniques come into play trying to establish an equivalent continuum model describing these macroscopic metamaterials’ characteristics. A rather common approach is to assume a priori that the target continuum model is a classical linear Cauchy continuum featuring the macroscopic displacement as the only kinematical field. This implies that the parameters of such continuum models (density and/or elasticity tensors) must be considered to be frequency-dependent to capture the complex response of the considered mechanical systems in the frequency domain. These frequency-dependent models can be useful to describe some of the aforementioned macroscopic metamaterials’ properties, yet, they suffer some drawbacks such as featuring negative masses and/or elastic coefficients in some frequency ranges which are close to resonance frequencies of the underlying microstructure. This implies that the considered Cauchy continuum is not positive-definite for all the considered frequencies. In this paper, we present a procedure, based on the definition of extra kinematical variables (with respect to displacement alone) and through the use of the inverse Fourier transform in time, to convert a frequency-dependent model into an enriched continuum model of the micromorphic type. All the parameters of the associated enriched model are constant (i.e., frequency-independent) and the model itself remains positive-definite for all the considered frequency ranges. The response of the frequency-dependent model and the associated micromorphic model coincide in the frequency domain, in particular when looking at the dispersion curves. Moreover, the micromorphic (frequency-independent) model results to be well defined both in time- and in the frequency-domain, while the Cauchy (frequency-dependent) model can only exist in the frequency domain This paper aims to build a bridge between the upscaling techniques usually found in the literature and our persuasion that macroscopic continua of the micromorphic type should be used to model metamaterials’ response at the macroscopic scale.

A Novel Approach for Harmonic Assessment of Power Systems With Large Penetration of IBRs – A U.K. Case Study

This work provides a high-level assessment of the harmonic performance of the future UK power system with high penetration of Renewable Energy Sources (RESs). A few variations to the standard harmonic analysis are proposed to allow for a more accurate representation of Inverter-Based Resources (IBRs) in frequency-domain studies. In the first part of the paper, two reduced network models (for 2020 and 2050, respectively) are developed, including both transmission and distribution system representation. A frequency-dependent impedance model for individual and aggregated IBRs is adopted, and a novel distribution load model with harmonic injection is proposed. Additionally, the variability of harmonic phase angles is modelled taking into account the physical behavior of the IBRs. Probabilistic Harmonic Load Flows (PHLFs) are carried out on the reduced networks by varying the harmonic injection magnitudes and phase angles of each harmonic source. This work shows that, while overall harmonic levels are dependent on the changes in demand, the inverter impedance has significant impact on local resonances, and accurate models for IBRs are required.

The Normal Casimir Force for Lateral Moving Planes with Isotropic Conductivities

We consider the two planes at zero temperature with isotropic conductivity that are in relative lateral motion with velocity v and interplane distance a. Two models of conductivity are taken into account—the constant and frequency-dependent Drude models. The normal (perpendicular to planes) Casimir force is analyzed in detail for two systems—(i) two planes with identical conductivity and (ii) one plane that is a perfect metal. The velocity correction to the Casimir energy, ΔvE∝v2, for small enough velocities is used for all considered cases. In the case of constant conductivity, η, the energy correction is ΔvE∝η/a3v/η2 for v≪η≪1.

Cross-correlation of the thermal Sunyaev-Zel’dovich and CMB lensing signals in Planck PR4 data with robust CIB decontamination

We use the full-mission PR4 data to measure the CMB lensing convergence (κ)–thermal Sunyaev-Zel’dovich (tSZ, y) cross-correlation signal, Cℓyκ. This is only the second measurement to date of this signal, following Hill and Spergel []. We perform the measurement using foreground-cleaned tSZ maps built from the PR4 frequency maps via a tailored needlet internal linear combination (NILC) code in our companion paper [F. McCarthy and J. C. Hill, companion paper, .], in combination with the PR4 κ maps and various systematic-mitigated PR3 κ maps. A serious systematic is the residual cosmic infrared background (CIB) signal in the tSZ map, as the high CIB—κ cross-correlation can significantly bias the inferred tSZ—κ cross-correlation. We mitigate this contamination by deprojecting the CIB in our NILC algorithm, using a moment deprojection approach to avoid leakage due to incorrect modeling of the CIB frequency dependence. We validate this method on mm-wave sky simulations. We fit a theoretical halo model to our measurement, finding a best-fit amplitude of A=0.82±0.21 (for the highest signal-to-noise PR4 κ map) or A=0.56±0.24 (for a PR3 κ map built from a tSZ-deprojected CMB map), indicating that the data are consistent with our fiducial model within ≈1−2σ. Although our error bars are similar to those of the previous measurement [J. C. Hill and D. N. Spergel, ], our method is significantly more robust to CIB contamination. Our moment-deprojection approach lays the foundation for future measurements of this signal with higher signal-to-noise κ and y maps from ground-based telescopes, which will precisely probe the astrophysics of the intracluster medium of galaxy groups and clusters in the intermediate-mass (M∼1013–1014h−1M⊙), high-z (z≲1.5, c.f. z≲0.8 for the tSZ auto-power signal) regime, as well as CIB-decontaminated measurements of tSZ cross-correlations with other large-scale structure probes. Published by the American Physical Society 2024

The application of active filters supported by Pulse Width Modulated Inverters in the harmonic simulation of the high power electric traction

The paper shows the present results of a long term research work. The authors are working on the modelling of active filtering of harmonics caused by the railways. To simulate the harmonic penetration and filtering effects of the power electric traction the combination of frequency and time dependent model should be used. This novel method, called double domain simulation is improving the accuracy of the frequency domain simulation. To calculate the sophisticated model of the electric locomotive and active harmonic filter as a non-linear load a time dependent model must be used. The traction supply system together with the equivalent supply network impedance could be calculated in frequency domain. An iteration algorithm is developed converting the variables in every iteration step between the time and frequency domain.

Effect on power quality of the high power electric traction (Double domain computer simulation vs. site measurements).

The paper shows the present results of a long term research work. The authors are working on the modelling of penetration of harmonics caused by the railways. In a former paper [1] a detailed model of the traction system was discussed in frequency domain. However this paper presents the combination of frequency and time dependent model. This novel method, called double domain method is used to improve the accuracy of the frequency domain simulation. To calculate the sophisticated model of the electric locomotive as a non-linear load a time dependent model must be used. The traction supply system together with the equivalent supply network impedance could be calculated in frequency domain. An iteration algorithm is developed to converting the variables in every iteration step between the time and frequency domain.

Co-evaluation of power system frequency performance and operational reliability considering the frequency regulation capability of wind power

With the increasing proportion of renewable energy, the power system inertia decreases, and the operation uncertainty rises. It brings concerns about the system frequency and operational reliability. However, the impacts of the power system frequency performance on the reliability parameters of generation units have not been fully investigated. This paper studies the frequency performance and the operational reliability co-evaluation for power systems considering wind turbines. Firstly, a power system frequency regulation model is established considering the regulation capability of wind turbines. Then, the cluster of equivalent wind turbines is incorporated into the frequency regulation architecture of thermal power units, which accelerates the analysis of frequency performance. Then, the frequency performance of the power system with the participation of wind turbines under the operation uncertainty and the unit random faults is quantitatively analyzed. A frequency-dependent generator reliability parameter model is derived. Next, a multi-time scale co-evaluation framework is proposed to realize the co-evaluation of frequency performance and operational reliability. Case studies are carried out on the modified IEEE RTS-79 system and a provincial power system. Results show that compared with the existing research, the proposed method can obtain the frequency performance and reliability results efficiently.

Stress and frequency dependence of wave velocities in saturated rocks based on acoustoelasticity with squirt-flow dissipation

SUMMARY We perform seismic and ultrasonic measurements in carbonate and shaley sandstone samples as a function of differential pressure. The velocities show a strong frequency and pressure dependence. The dispersion disappears with increasing pressure and the squirt flow in turn inhibits the pressure dependence. To model these effects, we combine the Gurevich's squirt-flow model with the Mori–Tanaka scheme and the David Zimmerman model, and extend it with third-order elastic constants, to obtain a frequency-dependent acoustoelasticity model. Comparisons between measurements from this study and literature and modelling results show that the P-wave velocity increases non-linearly first and then nearly linearly, dominated by crack closure and acoustoelasticity, respectively. The pressure dependence of wave velocities is reduced by liquid substitution and further by the squirt-flow mechanism. The effects of fluid properties and crack closure on P-wave velocity decrease with differential pressure. The results will feed a new model and help better understanding the wave propagation in pre-stressed rocks at different scales.