Nonlinear Modal Interactions Research Articles

Vibration alleviation for wind turbine gearbox with flexible suspensions based on modal interaction

Wind turbine drivetrains play a fundamental role in converting wind power into electrical energy. The gearbox is one of the most important and expensive components in a wind turbine drivetrain. Since flexible suspensions mounted on the gearbox are mainly designed for isolating vibration transfer to other turbine components, the gearbox itself still suffers from complicated whole-body vibration. In view of this, a vibration absorption method based on modal interaction is put forward to alleviate the whole-body vibration of the wind turbine gearbox with flexible suspensions. A vibration absorber with adjustable control parameters is utilized to establish modal coupling with the wind turbine gearbox. Internal resonance is analyzed and used to construct a modal interaction mechanism between the vibration absorber mode and the controlled gearbox mode. With the help of modal interaction, the vibration energy of the controlled gearbox mode is successfully absorbed by the vibration absorber mode and effectively dissipated by the damping of the vibration absorber mode. Through numerical simulations and virtual prototyping simulations, its vibration alleviation performance is verified. Since the proposed method is designed in terms of the controlled mode of the wind turbine gearbox rather than external excitations, it is suitable for applications in a complicated working environment. Besides, the proposed method aims to transfer and dissipate vibration energy through nonlinear modal interaction rather than suppress vibration via external energy, and thus can effectively deal with strong vibration problems. More importantly, it can easily work together with the existing vibration isolation method and further alleviate the whole-body vibration of the gearbox. This research will contribute to improving the reliability and service life of wind turbine gearboxes.

Safety of a continuous spinning Shaft’s structure from nonlinear vibration with NIPPF

An ongoing spinning shaft’s nonlinear vibration is the subject of this paper's analysis and control. Investigations were conducted on the major resonances as well as the nonlinear modal interactions between the rotor and controller modes. The averaging procedure be analyzed to get the solution of the system’s equations with a Nonlinear Integration Positive Position Feedback controller (NIPPF). A good correlation is achieved between the approximate solutions and the numerical simulations when utilizing the Runge-Kutta method 4th-order (RK4). The linearized stability approach is applied in the autonomous system to provide stability close to fixed positions. Nonlinear systems' steady-state stability and amplitude were examined, both before and after control. At various values for something like the controller and system parameters, frequency response curves (FRCs) were assessed. The MATLB program was used to compare the analytical and numerical responses at time-history and FRCs to ensure their comparability. After conducting this investigation, we draw the conclusion that the NIPPF control technique offers the optimal model control. Finally, the settings reduce the vibration's intensity.

Mitigation of Structural Vibrations of MDOF Oscillators by Modal Coupling Due to Hysteretic Dampers

In civil engineering, structural elements characterized by hysteresis are often encountered, such as materials with limited elastic fields, microsliding friction and elastomeric absorbers. Hysteretic nonlinearities produce a wide variety of dynamical phenomena, such as significant modal coupling, bifurcations and superabundant modes. This paper investigates nonlinear modal interactions in the dynamic response of a two-degree-of-freedom system (2DOF) with hysteretic elements. These phenomena are notably important in internal resonance conditions, where modal interactions produce strong modifications in the response with possible beneficial effects. In specific conditions, the transfer of energy between the two modes leads to a notable reduction in the maximum response amplitude; the exploitation of this feature to achieve vibration mitigation of the forced response is the main goal of the paper. Two configurations are investigated: the hysteretic element at the top (vibration damper) and the hysteretic element at the base (isolator). In both cases, several internal resonance conditions occur since, by increasing the excitation intensity, the frequencies of the hysteretic system change, as well as their ratio. Qualitative similar results are obtained, characterized by a transfer of energy between the two modes. For both configurations, the usefulness of exploiting these nonlinear phenomena in vibration mitigation has been shown.

K2 Photometry on Oscillation Mode Variability: The New Pulsating Hot B Subdwarf Star EPIC 220422705

We present an analysis of oscillation mode variability in the hot B subdwarf star EPIC 220422705, a new pulsator discovered from ∼78 days of K2 photometry. The high-quality light curves provide a detection of 66 significant independent frequencies, from which we identified nine incomplete potential triplets and three quintuplets. Those g- and p-multiplets give rotation periods of ∼36 and 29 days in the core and at the surface, respectively, potentially suggesting a slightly differential rotation. We derived a period spacing of 268.5 s and 159.4 s for the sequence of dipole and quadrupole modes, respectively. We characterized the precise patterns of amplitude and frequency modulations (AM and FM) of 22 frequencies with high enough amplitude for our science. Many of them exhibit intrinsic and periodic patterns of AM and FM, with periods on a timescale of months as derived by the best fitting and Markov Chain Monte Carlo test. The nonlinear resonant mode interactions could be a natural interpretation for such AMs and FMs after other mechanisms are ruled out. Our results are the first step to building a bridge between mode variability from K2 photometry and the nonlinear perturbation theory of stellar oscillation.

Nonlinear Modal Interaction between Longitudinal and Bending Vibrations of a Beam Resonator under Periodic Thermal Loading

Nonlinear Modal Interaction between Longitudinal and Bending Vibrations of a Beam Resonator under Periodic Thermal Loading

Non-planar motions due to nonlinear interactions between unstable oscillatory modes in a cantilevered pipe conveying fluid

Cantilevered pipes conveying fluid with an end mass exhibit rich post-critical nonlinear behaviors and have attracted the interest of many researchers. In the present paper, we focus on non-planar motions due to nonlinear interactions between two unstable oscillatory modes. By applying the orthogonality condition between the eigenfunction and its adjoint function to governing equations, evolution equations governing mixed-mode non-planar motions are analytically derived. In a certain range of flow velocity, mixed-mode non-planar motion can be produced. In this region, second- and third-mode oscillations are in distinct planes, which intersect perpendicularly. In particular, three-dimensional non-contacting measurements were conducted to verify the theoretical predictions, and hidden nonlinear modal interactions were extracted from experimental data. We observed mixed-mode non-planar motion that qualitatively agrees with the theoretical predictions. Moreover, in further experiments with a long pipe for future study, a sequence of bifurcating non-planar motions was observed.

Fundamental study of oscillations in the Nigerian power system

The power systems in developing countries are usually stressed and operated near their stability limits. Consequently, accurate sources of oscillations and their controls can present a challenge. This paper reports a comprehensive study of the oscillations in the Nigerian 48-bus power system. The dominant modes, sensitive locations for faults, and the most responsible generators were identified by modal analysis. Uniquely, the potential for nonlinear modal interaction of these modes was carefully investigated. The low-frequency modes identified are 1 Hz, 1.14 Hz, and 1.37 Hz, and they are associated mainly with the generators at Kainji, Afam, and Delta power stations. The results indicate the existence of inter-area phenomena and nonlinear modal interactions among these modes. Also, the analysis revealed that the generator at the Kainji power station is most affected by the nonlinear interactions.

Numerical investigation of mode competition and cooperation on the combustion instability in a non-premixed combustor

The phenomena of combustion instabilities beyond limit cycles caused by the nonlinear interaction of various instability modes have recently received much attention. This paper numerically studied the mode competition and cooperation phenomena in the liquid rocket engines when combustion instabilities occur. The liquid rocket engine with the single-element structure is the focus of the present work, and the numerical model is developed employing the Large Eddy Simulation technology, with the complete conservation Navier-Stokes equations taken into account. The results first reveal the relationship between the acoustic mode and the intrinsic vorticity mode: the competition and cooperation between the two will result in the complex dynamic process, such as semi-stable oscillation and intermittency. What's more, a more detailed coupling process between intrinsic vortex shedding and acoustic oscillation is analyzed, indicating existence of four stability regimes in the liquid rocket engines and an important insight into combustion instability in non-premixed dump combustors is provided.

Controlling Nonlinear Interaction in a Many-Mode Laser by Tuning Disorder.

A many-mode laser with nonlinear modal interaction could serve as a model system to study many-body physics. However, precise and continuous tuning of the interaction strength over a wide range is challenging. Here, we present a unique method for controlling lasing mode structures by introducing random phase fluctuation to a nearly degenerate cavity. We show numerically and experimentally that as the characteristic scale of phase fluctuation decreases by two orders of magnitude, the transverse modes become fragmented and the reduction of their spatial overlap suppresses modal competition for gain, allowing more modes to lase. The tunability, flexibility, and robustness of our system provides a powerful platform for investigating many-body phenomena.

Qualitative effect of internal resonance on the dynamics of two-dimensional resonator

Nonlinear modal interactions and associated internal resonance phenomena have recently been used to demonstrate improved oscillator performance and enhanced sensing capabilities. Here, we show tunable modal interaction in a molybdenum disulfide (MoS2) resonator. We achieve the tunability of coupling between these initially uncoupled modes by using electrostatic gate voltages. This tunable coupling enables us to make the modes commensurate and observe energy exchange between the modes. We attribute the strong energy exchange between the vibrational modes to 1:2 internal resonance. This interaction strongly impacts the dynamics of the modal response of such resonators. We observe peak splitting, a signature of energy exchange between the modes even when the modal response is in the linear regime. We model our device to explain the observed effect of excitation, detuning of modal frequencies, and intermodal coupling strength on the resonator dynamics. MoS2 resonators explored in this work are ideal for understanding the rich dynamics offered through the intermodal coupling.

Tidal Evolution and Diffusive Growth During High-eccentricity Planet Migration: Revisiting the Eccentricity Distribution of Hot Jupiters

Abstract High-eccentricity tidal migration is a potential formation channel for hot Jupiters. During this process, the planetary f-mode may experience a phase of diffusive growth, allowing its energy to quickly build up to large values. In Yu et al., we demonstrated that nonlinear mode interactions between a parent f-mode and daughter f- and p-modes expand the parameter space over which the diffusive growth of the parent is triggered. We extend that study by incorporating (1) the angular momentum transfer between the orbit and the mode, and consequently the evolution of the pericenter distance; (2) a prescription to regulate the nonlinear frequency shift at high parent mode energies; and (3) dissipation of the parent’s energy due to both turbulent convective damping of the daughter modes and strongly nonlinear wave-breaking events. The new ingredients allow us to follow the coupled evolution of the mode and orbit over ≳104 yr, covering the diffusive evolution from its onset to its termination. We find that the semimajor axis shrinks by a factor of nearly 10 over 104 yr, corresponding to a tidal quality factor  ∼ 10 . The f-mode’s diffusive growth terminates while the eccentricity is still high, at around e = 0.8–0.95. Using these results, we revisit the eccentricity distribution of proto-hot Jupiters. We estimate that less than 1 proto-HJ with eccentricity >0.9 should be expected in Kepler's data once the diffusive regime is accounted for, explaining the observed paucity of this population.

A data-driven model based on modal decomposition: application to the turbulent channel flow over an anisotropic porous wall

This article presents a data-driven model based on modal decomposition, applied to approximate the low-order statistics of the spatially averaged wall-shear stress in a turbulent channel flow over a porous wall with two anisotropic permeabilities, producing drag increase or reduction when compared with the case of an isotropic porous wall. The model is comparable to a neural network architecture using a linear map to a classification. To create this model, we use high-order dynamic mode decomposition (DMD) to identify the structures describing the main flow dynamics, and then test different linear combinations of these modes to estimate the time evolution of the stress at the porous interface. The coefficients of the model are obtained by training the model against the results of direct numerical simulations over different time intervals. Depending on the number and the way of combining the DMD modes, the reduced-order models presented can reconstruct the wall-shear stress with relative error smaller than 0.01 % and reproduce its statistical variations for at least 1500 time units with relative error in the standard deviation or the mean smaller than 5 %. The model has also been tested to approximate the statistics of the wall-shear stress over the whole wall, showing that the regeneration of the flow structures can be reproduced by the nonlinear interaction of modes. Finally, considering the DMD modes as communities in a neural network, we examine the influence of the mode-to-mode interaction on the nonlinear flow dynamics, which explains the performance of the different models.

Frequency comb generation in SNAP fiber resonator based on axial-azimuthal mode interactions.

We study nonlinear mode interaction in the cylindrical microresonator with nanoscale radius variation. We obtain a model which describes the interplay between azimuthal modes and investigate the nonlinear dynamics of axial-azimuthal modes. We also define regimes of generation depending on the pump parameters. Using a properly designed radius longitudinal profile, we obtain broadband generation of stable low-repetition-rate frequency combs based on axial-azimuthal modes.

Time-delayed control of a continuous flexible rotor via the saturation phenomenon

This paper aims to analyze and control nonlinear vibrations of a continuous spinning shaft, including a nonlinear saturation-based controller with time delays near the major critical speed and the presence of 2:1 internal resonance. The nonlinear modal interactions between the rotor and controller modes, as well as the primary and internal resonances, cause an energy bridge between the rotor and controller, so energy is transferred from the rotor to the controller, and a saturation phenomenon occurs. The nonlinear delay equations of motion are touched on analytically using the perturbation technique. Using the NSC gain coefficients, the saturation region and the frequency resonance bandwidth of the rotor are controlled. It is shown that, for negative values of the detuning parameter, as the internal detuning parameter increases, lower energies from the rotor are transferred to the controller, and the controller will be activated sooner. However, for positive values of the detuning parameter, by increasing the internal detuning parameter, the controller amplitude is increased, and the NSC will be later activated. Moreover, the stable zone of the rotor and NSC is studied.

Non-linear modal interactions during rub-impact of a rotating flexible shaft

An understating on interaction between a rotating flexible member and a stationary element has a wide range of industrial applications. The focus of this study was to understand the modal interactions using a conceptual model. The conceptual model of the system consists of multiple disc rotors with a localised conduit. A theoretical model of the system was developed using finite element analysis. An eigenvalue analysis was carried out on the system and a Campbell diagram of the system was developed. The Campbell diagram provided information about the rotor speed wherein synchronization with the whirling modes occur. Further, a reduced order non-linear contact model was developed. A numerical bifurcation analysis with rotor speed as parameter was carried out on the reduced model. Analysis was carried out using randomized initial condition for each rotation speed. During the bifurcation study, certain rotation speed produced a high whirling response at speeds different from the critical speed. The increase in the response was associated with the internal resonance of whirling modes and rotor speed. The participating modes were identified using the Campbell diagram and full spectrum. The results from the theoretical model were validated using an experimental test rig. The identification of these internal resonance rotor speed can be useful in identifying the rotor speed exclusion zone and thereby improves the rotor life.

Nonlinear modal interaction between longitudinal and bending vibrations of a beam resonator under periodic thermal loading

The paper investigates the nonlinear modal interaction of longitudinal and bending vibrations of a beam resonator under periodic thermal loading. The mode of parametric oscillations is investigated under conditions of internal multiple resonance between some flexural and longitudinal forms of free oscillations of the resonator. The possibility of generation in the system of the longitudinal-bending mode was found, the frequency of the slow envelope of which essentially depends on the parameter of the internal frequency detuning, which is directly related to the magnitude of external disturbances subject to high-precision measurement.

Effect of geometric imperfections and circumferential symmetry on the internal resonances of cylindrical shells

In the present work the resonant response of an imperfect cylindrical shell is investigated using Donnell’s nonlinear shallow shell theory. For this, a reduced order multi degree of freedom model is obtained by discretizing the partial differential equation of motion using the Galerkin procedure and a consistent modal expansion which captures the nonlinear modal interactions and couplings between two asymmetric vibration modes with near commensurable natural frequencies in a 1:2 ratio. As a result of the circumferential symmetry each mode exhibits a 1:1 internal resonance, leading to a possible 1:1:2:2 multiple internal resonances. These modes are coupled through quadratic and cubic nonlinearities arising from the shell curvature and nonlinear strain–displacement relation. The existence and stability of solutions and their bifurcations are investigated using numerical continuation methods for bifurcation analysis and their stability are studied using Floquet theory. It is known that geometric imperfections have a strong influence on the response of thin shell structures. Here, a detailed parametric analysis shows the influence of different forms of geometric imperfections on the shell natural frequencies and bifurcations in the main resonance region. Several branches of solutions due to multiple bifurcations are detected leading to dynamic jumps under increasing and decreasing frequency sweep. Steady-state harmonic and quasi-periodic responses resulting from Neimark–Sacker bifurcations are detected. The reduced order model demonstrates the influence of the geometric imperfection shape and magnitude on the bifurcation scenario and the energy transfer among the four interacting modes.

Asteroseismology of a High-amplitude δ Scuti Star GSC 4552-1498: Mode Identification and Model Fitting

Abstract In this paper, the pulsation behavior of high-amplitude δ Scuti star GSC 4552-1498 was analyzed. Using the high-precision photometric data from the Transiting Exoplanet Survey Satellite, two new independent frequencies F1 = 22.6424(1) day−1 and F2 = 28.6803(5) day−1 were identified for this source, along with the fundamental one F = 17.9176(7) day−1, which was previously known. In addition, the classical O − C analysis was conducted to give a new ephemeris formula of BJDmax = T 0 + P × E = 2453321.534716(4) + 0.055811(0) × E. The O − C diagram reveals a continuous period increase, but the rate of (1/P)(dP/dt) = 1.11(3) × 10−7 yr−1 seems much larger (about hundreds) than predicted by evolution theories, which is long been noticed but not well understood, possibly related to nonlinear mode interaction. Based on frequency parameters (i.e., F, F1, and F2), a series of theoretical models were conducted by employing the stellar evolution code. It turns out that F1 should be a non-radial mode and F2 is the second overtone radial mode. Due to the mass–metallicity degeneracy, the stellar parameter of the star can however not be determined conclusively. We suggest high-resolution spectral observation is highly desired in the future to further constrain models. We note GSC 4552-1498 is located on the main sequence in the H-R diagram.

Investigation of low-power comb generation in silicon microresonators from dual pumps

Here, the influence of pump powers for silicon microresonator-based Kerr comb generation (KCG) was investigated theoretically and experimentally in two types of dual-pumped KCG. One employs dual pumps at different free spectral range (FSR) spacings of the resonator for broadband comb spectrum and the other to produce comb lines at one FSR spacing for low pump powers. The KCG is modeled by including nonlinear absorption and mode interaction into the Lugiato–Lefever equation. Both the theoretical and the experimental results indicate that the KCG with pump powers inside the ring of tens of milliwatts would be limited by high nonlinear absorption even with a 25 V reverse bias to deplete free carriers. The most efficient comb spectrum occurs with recirculating pump power in the ring cavity of 13 mW (0.3 mW pump power in the bus waveguide) at 25 V reverse bias. Generally, for silicon KCG at the telecom wavelengths, the use of low pump powers and reverse bias is essential. At 25 V reverse bias, increasing the stored pump power in the ring to 160 mW would suppress the KCG and produce a similar number of comb lines as 1 mW recirculating pump power. This work advances our understandings of silicon KCG in the telecom band.

Tides in the High-eccentricity Migration of Hot Jupiters: Triggering Diffusive Growth by Nonlinear Mode Interactions

Abstract High-eccentricity migration is a possible formation channel for hot Jupiters. However, in order for it to be consistent with the observed population of planets, tides must circularize the orbits in an efficient manner. A potential mechanism for such rapid circularization is the diffusive growth of the tidally driven planetary f-mode. Such growth occurs if the f-mode phase at pericenter varies chaotically from one pericenter passage to the next. Previous studies focused on the variation of the orbital period due to tidal back-reaction on the orbit as the source of chaos. Here we show that nonlinear mode interactions can also be an important source. Specifically, we show that nonlinear interactions between a parent f-mode and daughter f-/p-modes induce an energy-dependent shift in the oscillation frequency of the parent. This frequency shift varies randomly from orbit to orbit because the parents’ energy varies. As a result, the parents’ phase at pericenter varies randomly, which we find can trigger it to grow diffusively. We show that the phase shift induced by nonlinear mode interactions in fact dominates the shift induced by tidal back-reaction and lowers the one-kick energy threshold for diffusive growth by about a factor of 5 compared to the prediction from the linear theory. For a given pericenter distance, this reduces the semimajor axis needed to trigger diffusive growth from ≳2 au to ≃1 au. Nonlinear interactions could thus enhance the formation rate of hot Jupiters through the high-eccentricity migration channel and potentially mitigate the discrepancy between the observed and predicted occurrence rates for close-in gas giants as compared to those farther from the star.