Order Eigenmodes Research Articles

A semi-analytical approach for dynamic responses of monopile-supported offshore wind turbines subjected to accidental loads

Offshore wind energy is leading the way in sustainable energy generation. Offshore wind turbines (OWTs) need to be designed to withstand extreme or accidental loads in Ultimate Limit States (ULS) and Accidental Limit States (ALS), whereby high order eigenmodes may become significant. This paper introduces a semi-analytical approach for efficient and reliable analysis of the dynamic responses of monopile-supported OWTs subjected to accidental loads. A Rayleigh-Ritz solution is initially adopted to determine the high-order natural frequencies and eigenmodes of monopile OWTs, explicitly considering tapered towers and soil-pile interactions. Dynamic responses of OWTs are then calculated depending on the interaction schemes between accidental loads and the structures, e.g. soft contact loads (e.g. slamming, wind) and hard contact loads (e.g. ship collisions, ice impact). For soft contact loading conditions, the loads are considered independent of turbine responses, and the transient dynamic response is computed using the classical modal superposition method. While for hard contact loading conditions, the load actions depend on the contact stiffnesses and responses of the interacting bodies. A numerical contact algorithm is thus developed, and numerical iterations are performed to ensure convergence. The proposed approach is applied to the DTU 10 MW monopile-supported OWT subjected to extreme water slamming and ship collisions, respectively as example applications of soft and hard contact scenarios. The results are verified against nonlinear finite element analysis using USFOS and discussed with respect to the turbine natural frequencies and eigenmodes, contact forces and dynamic responses. A parametric analysis is conducted for ship-OWT collisions, exploring different impact scenarios by varying ship sizes, contact stiffnesses, and initial impact velocities. The proposed approach can serve as a promising tool for accidental load response analysis and the design of monopile OWTs.

Eigenmodes of fractal drums: A numerical student experiment

“Can one hear the shape of a drum?” was a question posed (and made famous) by mathematician Mark Kac in the mid-1960s. It addresses whether a deeper connection exists between the resonance modes (eigenmodes) of a drum and its shape. Here, we propose a numerical experiment, suitable for advanced undergraduate physics students, on the calculation of the eigenmodes of a square Koch fractal drum, for which experimental results do exist. This exercise is designed to develop the students' understanding of the vibrations of fractal drums, their eigenmodes, and potentially their integrated density of states. The students calculate the lowest order eigenmodes of the fractal drum, visualize these modes, and study their symmetry properties. As an extension, the students may investigate the integrated density of states of the fractal drum and compare their findings to the Weyl–Berry conjecture.

A Rayleigh-Ritz solution for high order natural frequencies and eigenmodes of monopile supported offshore wind turbines considering tapered towers and soil pile interactions

Offshore wind turbines may be exposed to extreme or accidental loads during the service life, such as extreme water slamming, ship collisions, earthquakes, extreme winds, ice loads, etc. Under the action of such loads, they will respond not only in the primary eigenmode, but also high order modes. This paper presents a Rayleigh-Ritz solution for high order natural frequencies and eigenmodes of monopile supported offshore turbines. The model considers explicitly tapered wind turbine towers and soil-pile interactions. Different soil conditions are considered using equivalent cross coupled springs at the mudline. The proposed Rayleigh-Ritz solution is applied to the DTU 10 MW wind turbine mounted on a monopile foundation. The predicted natural frequencies and eigenmodes are compared with nonlinear finite element simulations using USFOS. The results show excellent agreement for the first 3 eigenmodes. Even higher eigenmodes than the 3rd order are also calculated but are not presented. They can be readily obtained if needed.High order natural frequencies and eigenmodes using the proposed Rayleigh-Ritz solution can, together with the Duhamel's integral, be well utilized in the buckling and collapse analysis and design of monopile supported offshore wind turbines in Ultimate Limit States (ULS) and Accidental Limit States (ALS). In addition, the high accuracy of the predicted primary natural frequency makes it a useful design tool to help avoid undesired resonance from external excitation loads and to reduce resonance related fatigue effects in Fatigue Limit States (FLS).

Analysis of a photonic crystal fiber with ellipse air-holes for supporting 82 orbital angular momentum modes

We proposed a photonic crystal fiber (PCF) with ellipse air holes and high refractive index ring-core, which can stably support 82 orbital angular momentum states from the wavelength of 1.25–1.65 μm. The modes’ average confinement loss keeps as low as 10−9 dB m−1. In addition, the effective refractive index separation (Δn eff) between HE m +1,1 and EH m −1,1 (m = 2–21) is up to 10−3, which makes the mode degeneracy to LP mode can be almost neglected. The dispersion curves of low order eigenmodes are low and flat, especially at the wavelength of 1.4 μm the minimum absolute value of chromatic dispersion for HE1,1 mode is as low as 0.246 ps (nm·km)−1. Furthermore, we also analyze the effect of ellipticity change on the cladding air holes, which instructively show the favorable fault tolerance of confinement loss and dispersion. The proposed PCF structure will be a potential candidate for high-capacity optical fiber communications.

The Rail Surface Defects Recognition via Operating Service Rail Vehicle Vibrations

Rail surface defects will not only bring wheel rail noise during train operation, but also cause corresponding accidents. Most of the existing detection methods are manual detection, which is time-consuming, laborious, inefficient, and subjective. With the development of technology, automatic detection replaces manual detection, which reduces manual labor, improves efficiency, and objectively evaluates the surface state of rails, which is in line with the purpose of modern intelligent production. The automatic detection of a single sensor is usually not enough to complete the recognition, but multiple sensors need to be additionally installed and refitted on the service vehicle, which creates difficulty for on-site test conditions. Therefore, in order to overcome these shortages and to adapt to the actual vibration characteristics of service vehicles, a rail surface defect recognition method based on optimized VMD gray image coding and DCNN is proposed in this paper. Firstly, the optimization method of VMD mode number based on the maximum envelope kurtosis is proposed. The VMD after parameter optimization is used to decompose the four-channel axle box vibration signal, and the component with the largest correlation coefficient between each order eigenmode component and the original signal is extracted. Secondly, the filtered IMF components are arranged in sequence and encoded into grayscale images. Finally, the DCNN structure is designed, and the training set is input into the network for training, and the test set verifies the effectiveness of the network and realizes the recognition of rail surface defects. The test accuracy of railway data set measured on the serviced vehicle is 99.75%, and the results show that this method can accurately identify the category of rail surface defects. After adding Gaussian noise to the original signal, the test accuracy reaches 99.20%, which proves that the method has good generalization ability and anti-noise performance. Additionally, this method can ensure the safe operation of vehicles without adding new equipment, which reduces operation costs and improves the intelligent operation and maintenance of rails.

Features of resonant and non-resonant slab ion-temperature-gradient instabilities in weakly reversed magnetic shear configurations

Linear eigenmode analysis of slab ion-temperature-gradient (ITG) instabilities in weakly reversed magnetic shear configurations is performed based on a Gyro-Landau-Fluid model. Numerical results show that the linear features of the ITG instabilities are different in weakly reversed shear plasmas with double, single and non-rational surfaces, respectively. For the case with double rational surfaces separated far away, mode structures exhibit a global symmetry or antisymmetry versus the minimum of the safety factor q min with locally odd or even parity around each resonant surface. Linear properties are mainly determined by local magnetic shear as in normal magnetic shear case. With the distance between two surfaces decreasing, linear instability properties vary non-monotonically. Various high order eigenmodes are driven unstable with a moderate distance. The mode structures are eventually localized around q min surface in the single rational surface case, even nonresonant one. When there is no rational surface, the nonresonant modes are strongly driven unstable with growth rates even larger than the resonant modes. In addition, similar results in a cylinder plasma indicate the importance of non-resonant ITG instability in the short wavelength regime with weakly reversed magnetic shear.

Phase mismatch induced suppression of eigenmode resonance in terahertz metasurfaces.

In this paper, we observe the distinguishable modulation of the different eigenmodes by lattice mode in terahertz U-shaped metasurfaces, and a remarkable lattice induced suppression of the high order eigenmode resonance is demonstrated. With the quantitative analysis of Q factor and loss of the resonances, we clarify that the peculiar phenomenon of suppression is originated from the phase mismatch of the metasurfaces via introducing the phase difference between the neighboring structures. These results provide new insights into the phase mismatch mediated transmission amplitude of eigenmode resonance in metasurfaces and open a new path to developing terahertz multifunctional devices.

Large scale coherent magnetohydrodynamic oscillations in a sunspot

Although theoretically predicted, the simultaneous excitation of several resonant modes in sunspots has not been observed. Like any harmonic oscillator, a solar magnetic flux tube can support a variety of resonances, which constitute the natural response of the system to external forcing. Apart from a few single low order eigenmodes in small scale magnetic structures, several simultaneous resonant modes were not found in extremely large sunspots. Here we report the detection of the largest-scale coherent oscillations observed in a sunspot, with a spectrum significantly different from the Sun’s global acoustic oscillations, incorporating a superposition of many resonant wave modes. Magnetohydrodynamic numerical modeling agrees with the observations. Our findings not only demonstrate the possible excitation of coherent oscillations over spatial scales as large as 30–40 Mm in extreme magnetic flux regions in the solar atmosphere, but also paves the way for their diagnostic applications in other astrophysical contexts.

A Rayleigh-Ritz Solution for High Order Natural Frequencies and Eigenmodes of Monopile Supported Offshore Wind Turbines Considering Tapered Towers and Soil Pile Interactions

A Rayleigh-Ritz Solution for High Order Natural Frequencies and Eigenmodes of Monopile Supported Offshore Wind Turbines Considering Tapered Towers and Soil Pile Interactions

Combustion dynamics of multi-element lean-premixed hydrogen-air flame ensemble

The crux of the technical issues facing the development of hydrogen gas turbine engines is the problem of how to enable low NOx, low dynamics combustor operations without detrimental flashback events in extremely fast lean-premixed hydrogen flames. While flashback mechanisms and nitrogen oxides emissions have been investigated extensively for high hydrogen content flames, the self-excited dynamics of lean-premixed pure hydrogen flame ensembles remain unclear. Here we use phase-resolved OH* chemiluminescence and OH PLIF imaging in a series of experiments in a multi-element injector configuration consisting of 293 small-scale injectors with an inner diameter of 3.0 mm. We collect a large experimental dataset to explore a variety of collective phenomena of the lean-premixed hydrogen-air flame ensemble, and perform systematic investigations of self-induced combustion instabilities. Our observations demonstrate that ultra-compact pure hydrogen flames generate high-amplitude pressure perturbations over a broad range of characteristic frequencies between 400 and 1800 Hz, corresponding to the third to tenth order eigenmodes. Low-frequency flame dynamics developed under relatively low equivalence ratio conditions involve a complex balance among several coexisting phenomena, including strong vortex interactions and periodic extinction-reignition processes, giving rise to large-scale asymmetric oscillations of the entire reaction zone. By contrast, the flame surface dynamics at an intermediate frequency of ~600 Hz exhibit prominent symmetric oscillations accompanied by merging and pinch-off of the constituent flames. Unexpectedly, high-frequency instabilities at approximately 1720 Hz (screech tones) are not influenced by such structurally complex flame dynamics, but by exceptionally simple, seemingly linear, flame surface motion without sudden flame area annihilation events like those observed for lower frequency cases. Despite the extremely low level of heat release rate fluctuations, on the order of less than 1%, the clustered premixed hydrogen flames are capable of producing disproportionately large pressure perturbations in excess of 12 kPa, originating from the synchronous phase dynamics of acoustic pressure and clustered flames’ heat release rate oscillations. These findings provide new insight into the driving mechanisms underlying high-frequency combustion dynamics of densely arranged pure hydrogen-air flames.

Chirping ion cyclotron emission (ICE) on NSTX-U

We report here on the discovery on NSTX-U of a qualitatively new type of non-thermal emission from plasmas in the ion-cyclotron frequency range (ICE). This new type of ICE is weakly damped, γdamp/ω ICE ≈ 0.14%, low order eigenmodes, excited by a fast ion population radially located near an internal transport barrier. The ICE appears as repetitive bursts with durations of ≈1 ms, and the bursts exhibit frequency chirps upwards and/or downwards with maximum δf/f ≈ 1%. We refer to this chirping ICE as ch-ICE. The bursts are longer than the typical ICE bursts of ≈100 μs reported previously for NSTX and NSTX-U, for simplicity, NSTX(-U). The chirping resembles that seen in modeling of weakly unstable fast particle driven instabilities (cf, Berk et al 1997 Phys. Lett. A 234 213). These data also provide some preliminary indications of non-linear coupling between the ch-ICE harmonics. Despite extensive experimental and theoretical studies of ICE, theoretical understanding of ICE remains qualitative at best. The detailed experimental observations of ICE from NSTX(-U) find a wide variation in the ICE characteristics. The new observations of ch-ICE presented here, combined with the characteristics of ICE on NSTX(-U) previously reported (Fredrickson et al 2019 Phys. Plasmas 26 032111), could guide the development of more complete theoretical models of ICE.

Numerical simulation of the unsteady cavitating flow in a Francis turbine draft tube at Upper-Part-Load (UPL) conditions

At part-load conditions, Francis turbines experience the development of a precessing vortex rope in their draft tube (DT), rotating with a frequency between 0.2 and 0.5 times the runner frequency. It induces pressure pulsations at the precession frequency in the whole hydraulic system, undesirable vibrations and noise putting at risk the stability of the system and the lifetime of the machine components. In specific machines, a dramatic amplification of the noise and vibrations can be observed between 70% and 85% of the design conditions. In these cases, synchronous pressure pulsations with a high frequency, typically between 2 and 4 times the runner frequency, are observed, referred to as the so-called Upper-Part-Load (UPL) pulsations. These pulsations are induced by a self-excitation of the entire hydraulic system including the cavitation vortex rope at one of its high order eigenmodes. Besides, the cavitation vortex rope features an elliptical cross-section rotating around the vortex axis at a high frequency. In this manuscript, unsteady two-phase flow simulations using a Scaled-Adaptive-Simulation (SAS) turbulence model and ZGB cavitation model of a Francis turbine draft tube at 80% of the design condition are performed to clarify the mechanisms responsible for the formation of the elliptical shape of the vortex and the UPL pulsations. However, the numerical simulation with a constant inlet boundary condition cannot reproduce the UPL pulsation phenomenon since it is a self-excited oscillation at one eigenmode of the complete system, which is not considered. Therefore, the UPL pulsations component is extracted from the measured pressure fluctuations data by applying a band-pass filter and is then set as the inlet boundary condition in the numerical simulation. The flow field is therefore artificially excited which aims to confirm whether the same phenomenon can be observed and to compare with the flow field obtained with constant inlet boundary condition. The numerical simulations are validated by experimental results, and the wave propagation in the DT is clarified.

Effects of multiple families of nonlinear fibers on finite deformation near a crack tip in a neo-Hookean sheet

In this paper, we investigate the asymptotic crack tip fields in a neo-Hookean sheet reinforced by two families of nonlinear fibers, where the fibers are characterized by the standard reinforcing model. In the asymptotic analysis, the fibers with stronger stiffening compared to the matrix dominate the mechanical behavior at the crack tip, which simplifies the analysis. A hodograph transformation is used to solve for the leading and higher order eigenmodes for the nonlinear eigenvalue problem derived from the model. Asymptotic path-independent J-integrals are constructed to evaluate the leading order parameters separately. The asymptotic crack tip fields agree well with finite element results for different combinations of fiber orientation angles and modulus ratios. We find that the peak value of stress for the case with two families of fibers decreases significantly compared to the case for a single family of fibers.

Wideband and High-Gain D-Band Antennas for Next-Generation Short-Distance Wireless Communication Chips

This article presents two kinds of low-cost antennas for D-band short-distance wireless communication applications. Both the antennas are excited by a substrate integrated waveguide (SIW) feed line based on the printed circuit board (PCB) process. The first type is a stainless-steel-based horn antenna, fabricated by electric-discharging machining (EDM) process and selective laser melting (SLM) 3-D printing, respectively, for reducing cost. To integrate the horn antennas with the SIW feed line, a wideband transition structure from SIW to the rectangular waveguide is designed by suppressing higher order eigenmodes. The experimental data show that the integrated horn antenna can achieve 18.9% (129.5-156.5 GHz) -10 dB impedance bandwidth and a peak gain of 15.46 dBi. The second type of antenna is a single-layer rectangular patch antenna fed by SIW. By locally widening the SIW feed line to generate an equivalent shunt capacitance instead of using complex resonators, a wide impedance bandwidth can be realized. The measured -10 dB impedance bandwidth and the peak gain of the patch antenna are 20.8% (125.3-154.4 GHz) and 7.26 dBi, respectively. The two proposed antennas can provide comparable performance for different application requirements under the premise of a low fabrication cost.

Simulation of defocusing effect based on two-step ABCD algorithm while a modal decomposition

Depicting the multimode laser beam by modal decomposition can potentially assess light field variations in the fiber, during propagation. The practical engineering conditions in the lab however could not realize ideal levels, hence further research on factors influencing this method, such as defocus, is especially necessitated. The grid spacing in observation plane by Fast Fourier Transform is fixed and unchangeable within diffraction imaging, hence possibly yielding erroneous data during obtaining light field intensities. Our research resolves these issues via a Two-step ABCD algorithm, applied in the modal decomposition to characterize various guided modes at the output of multimode fibers. A direct benefit is that the image plane size can be altered, further refining laser facula clarity. Furthermore, the quantitative expressions that analyze defocus factors impacting modal decomposition are acquired. The conclusions thereby prove the modal decomposition algorithm can keep effectiveness in the range of −0.25% to 0.25% of relative defocus for low order eigenmodes, having no suitable limited band for high order eigenmodes, with reference value in engineering applications.

Structurally-integrated resonators for broadband panel vibration suppression

There is limited understanding on interaction mechanisms that govern vibration attenuation when a small coverage of lightweight resonators are applied to practical engineering structures. To shed light on the unknowns, this research investigates lightweight elastomeric half cylindrical resonators attached to an aluminum panel using a small mass ratio around 3% and only around 1.7% coverage over the panel area. Finite element modeling of the system dynamics is complemented by corresponding experimental undertaking. The eigenfrequencies and eigenmodes of the resonators are scrutinized for the respective contributions provided towards broadband panel vibration suppression. The first order eigenmodes of the resonators are found to exert great influence on the starting mode for greater vibration attenuation, which may be tuned by the Young’s modulus of the resonators. The concept of displacement polarization is established to probe how the resonator eigenmodes quantitatively contribute to attenuate the forced panel vibration. This study reveals how flexural panel vibration may be attenuated by transferring the vibration from the panel to resonators by virtue of modal interaction, and prepares a generalized analytical technique that may be used by other researchers studying multi-modal interactions between host structures and applied resonators. These findings may guide the future development of lightweight resonators with a small coverage area for vibration suppression in engineering applications.

Orbital angular momentum generation in a dual-ring fiber based on the phase-shifted coupling mechanism and the interference of supermodes.

Based on the phased-shifted interference between supermodes, a novel method that can directly convert LP01 mode to orbital angular momentum (OAM) mode in a dual-ring microstructure optical fiber is proposed. In this fiber, the resonance between even and odd HE11 modes in inner ring and higher order mode in outer ring will form two pairs of supermodes, and the intensities and phases of the complete superposition mode fields for the involved supermodes created by the resonance at different wavelengths and propagating lengths are investigated and exhibited in this paper. We demonstrate that OAM mode can be generated from π/2-phase-shifted linear combinations of supermodes, and the phase difference of the even and odd higher order eigenmodes can accumulate to π/2 during the coupling process, which is defined as "phase-shifted" conversion. We build a complete theoretical model and systematically analyze the phase-shifted coupling mechanism, and the design principle and optimization method of this fiber are also illustrated in detail. The proposed microstructure fiber is compact, and the OAM mode conversion method is simple and flexible, which could provide a new approach to generate OAM states.

Towards asteroseismology of core-collapse supernovae with gravitational wave observations – II. Inclusion of space–time perturbations

Improvements in ground-based, advanced gravitational wave (GW) detectors may allow in the near future to observe the GW signal of a nearby core-collapse supernova. For the most common type of progenitors, likely with slowly rotating cores, the dominant GW emission mechanisms are the post-bounce oscillations of the proto-neutron star (PNS) before the explosion. We present a new procedure to compute the eigenmodes of the system formed by the PNS and the stalled accretion shock in general relativity including spacetime perturbations. The new method improves on previous results by accounting for perturbations of both the lapse function and the conformal factor. We apply our analysis to two numerical core-collapse simulations and show that our improved method is able to obtain eigenfrequencies that accurately match the features observed in the GW signal and to predict the qualitative behaviour of quasi-radial oscillations. Our analysis is possible thanks to a newly developed algorithm to classify the computed eigenmodes in different classes (f-, p-, and g-modes), improving previous results which suffered from misclassification issues. We find that most of the GW energy is stored in the lowest order eigenmodes, in particular in the $^2g_1$ mode and in the $^2f$ mode. Our results also suggest that a low-frequency component of the GW signal attributed in previous works to the characteristic frequency of the Standing Accretion Shock Instability should be identified as the fundamental quadrupolar f-mode. We also develop a formalism to estimate the contribution of quasi-radial ($l=0$) modes to the quadrupolar component of the GW emission in the case of a deformed background, with application to rapidly rotating cores. This work provides further support for asteroseismology of core-collapse supernovae and the inference of PNS properties based on GW observations.

Exploration of Fermi-Pasta-Ulam Behavior in a Magnetic System.

We study nonlinear spin motion in one-dimensional magnetic chains. We find significant differences from the classic Fermi-Pasta-Ulam (FPU) problem examining nonlinear elastic motion in a chain. We find that FPU behavior, the transfer of energy among low order eigenmodes, does not occur in magnetic systems with only exchange and external fields, but does exist if a uniaxial anisotropy is also present. The FPU behavior may be altered or turned off through the magnitude and orientation of an external magnetic field. A realistic micromagnetic model shows such behavior could be measurable.

Investigation on Simulation Methods of Initial Geometric Imperfection Distribution in Elasto-plastic Stability Analysis of Single-layer Reticulated Shells

Based on the theory of probability and mathematical statistics and on the results of nearly 2000 cases of elasto-plastic stability analysis for single-layer reticulated shells, a new simulation method of initial geometric imperfection distribution, N-order eigenmode imperfection method, is developed. The elasto-plastic stability analyses of four types of commonly used single-layer reticulated domes were conducted with the random imperfection mode method, the eigenmode imperfection method and the N-order eigenmode imperfection method, respectively, by means of ANSYS. The results show that, though the random imperfection mode method can simulate the initial geometric imperfection distribution realistically, the calculation cost of this method is too large to popularize in design practice. Adopting the first order eigenmode to simulate the distribution of initial geometric imperfection may fail to derive the most unfavorable buckling load. The reliability of buckling load obtained from the eigenmode imperfection method cannot be guaranteed effectively. The N-order eigenmode imperfection method can evaluate the influence of initial geometric imperfection on structural stability reasonably, and capture the buckling load satisfying the requirement of the ‘3σ’ principle with less calculation. This method is a convenient and efficient way for designers to estimate stability performance of single-layer reticulated shells accurately and safely.