Instability Boundary Research Articles

Nonlinear dynamics of lumped-parameter planetary gears with general mesh phasing

This work derives analytical solutions for resonances and parametric instabilities of all rotational, translational, and planet modes, regardless of the degeneracy/multiplicity of the natural frequencies, for planetary gears with time-varying mesh stiffness excitation and tooth separation nonlinearity. Degenerate modes are the focus, but the results apply to all modes. For a degenerate mode resonance, use of mesh phasing results and modal properties from the literature reduces a problem with multiple, unknown, coupled modal amplitudes to one with a single unknown modal amplitude. This reduction allows derivation of a closed-form amplitude–frequency relation that is algebraically prohibitive otherwise. The analytical solution reveals important features such as the peak resonant amplitude and tooth contact loss occurrence. An instability suppression rule governs the occurrence or suppression of a potential parametric instability between any two natural frequencies, regardless of their multiplicity. Closed-form instability boundaries that bound the range of mesh frequencies where an unsuppressed parametric instability occurs are derived with and without damping. Counter-intuitively, damping can destabilize a parametric instability between two different natural frequencies by enlarging the instability bandwidth. The critical damping at which a parametric instability is eliminated is determined analytically. For parametric instabilities of a common type, the amplitude of nonlinear response with tooth separation is derived in closed-form. Numerical integration of a planetary gear model verifies all the analytical results. These numerical results also show rich nonlinear behaviors near parametric instabilities such as jump phenomena, period-doubling/halving, and quasi-periodic response.

Theoretical analysis of Ledinegg instability and density wave oscillation using dimensionless numbers

• Dimensionless number criteria determining two phase flow stability is established. • Flow direction on stability is non-monotonic, the turning point expression is given. • Increasing flow rate and power proportionally can be good or bad for stability. • Contrary conclusions of flow direction and power level on stability are clarified. • The influences of tube length and diameter on stability are discussed and presented. Two-phase flow instability is influenced by many factors. The reported parameter effects on instability boundary are incomplete and sometimes conflict. Dimensionless numbers are more effective to reveal the mechanisms and describe the instability boundary. In this paper, dimensionless numbers are used to systematically analyze various parameters’ effects. One-dimensional governing equations describing the convective boiling process in a tube are simplified to a set of linear ordinary differential equations through integration and small perturbation methods. The judging inequalities expressed by six dimensionless numbers are obtained by the Routh stability criterion. The relationship between the judging inequalities and the instability types are identified theoretically for the first time. The boundary of density wave oscillation obtained from the fourth dimensionless number inequality agrees well with experimental data. The effects of Froude number, friction number, inlet temperature, inlet and outlet flow resistances, tube length and tube diameter are discussed in detail. Some contrary conclusions presented by previous investigators are clarified. The effect of varying Froude number (or flow orientation) on system stability is non-monotonic, and the turning point expression is given. The effects of geometric parameters are complex. The effects of tube length and tube diameter on the instability boundary are discussed systematically based on Froude number and friction number when operation parameters keep unchanged. Though many of the parameters’ influences are non-monotonic, some new universal laws or findings are extracted. For example, according to the conclusions of friction number effect, increasing the steam generator working power levels (heating power and mass flow rate increase proportionally) can be good or bad for stability according to different inlet flow resistances.

Analysis of the oscillation in the system of parallel narrow channels

In this paper, the RELAP5/MOD3.4 code is used to study the instability of the parallel narrow flow channel system in detail. First, the rationality of the RELAP5 code is verified with experimental data. By extracting the pressure drop in the channel under different node numbers, the importance of reasonable node model in pressure drop calculation using RELAP5 program is illustrated. Thus, the deep reason of the influence of the node number on the instability boundary is revealed. After that, the influences of various factors on the instability were investigated. The results show that: The more channels, the more unstable the system is, and the flow instability boundary tends to a fixed value; The fuel module separation has a significant effect on the instability in multi-channel system, and the single side heating channel has a stabilizing effect on the system; The flow instability of the system mainly depends on the hottest channel.

The Development of a Flight Test Platform to Study the Body Freedom Flutter of BWB Flying Wings

A flight test platform is designed to conduct an experimental study on the body freedom flutter of a BWB flying wing, and a flight test is performed by using the proposed platform. A finite element model of structural dynamics is built, and unsteady aerodynamics and aeroelastic characteristics of the flying wing are analyzed by the doublet lattice method and g-method, respectively. Based on the foregoing analyses, a low-cost and low-risk flying-wing test platform is designed and manufactured. Then, the ground vibration test is implemented, and according to its results, the structural dynamics model is updated. The flight test campaign shows that the body freedom flutter occurs at low flight speed, which is consistent with the updated analytical result. Finally, an active flutter suppression controller is designed using a genetic algorithm for the developed flying wing for future tests, considering the gains and sensor location as design parameters. The open- and closed-loop analyses in time- and frequency-domain analyses demonstrate that the designed controller can improve the instability boundary of the closed-loop system effectively.

Quantum properties near the instability boundary in optomechanical system

The properties of the system near the instability boundary are very sensitive to external disturbances, which is important for amplifying some physical effects or improving the sensing accuracy. In this paper, the quantum properties near the instability boundary in a simple optomechanical system have been studied by numerical simulation. Calculations show that the transitional region connecting the Gaussian states and the ring states when crossing the boundary is sometimes different from the region centered on the boundary line, but it is more essential. The change of the mechanical Wigner function in the transitional region directly reflects its bifurcation behavior in classical dynamics. Besides, quantum properties, such as mechanical second-order coherence function and optomechanical entanglement, can be used to judge the corresponding bifurcation types and estimate the parameter width and position of the transitional region. The non-Gaussian transitional states exhibit strong entanglement robustness, and the transitional region as a boundary ribbon can be expected to replace the original classical instability boundary line in future applications.

Numerical Simulation and Modeling Convention of Unsteady Fluidelastic Forces of Tube Arrays

Abstract This paper presents a numerical investigation on the unsteady fluidelastic forces of tube arrays. The key focus is on the consistency between the unsteady fluidelastic force model and the quasi-steady model for tube arrays at large reduced flow velocities, as well as comparing two well-known conventions for the unsteady model. Two-dimensional unsteady Reynolds-averaged Navier–Stokes (URANS) simulations are used to prove that the viscous damping coefficients of Tanaka's convention approach their quasi-steady values as the reduced flow velocity approaches infinity, whereas the hysteretic damping coefficients of Chen's modified convention always approach zero and hence result in low-resolution data plots as the reduced flow velocity becomes large. The nonconstant viscous damping coefficients of Tanaka's experimental data at high reduced flow velocities (which motivated the introduction of Chen's modified convention) might be induced by a systematic identification error in the phase of the fluidelastic force. A row of three flexible cylinders is used as a numerical example to analyze the effect of systematic phase error on the predicted stability boundary of the fluidelastic instability. Although identical fluidelastic forces are simulated by using the two conventions, Tanaka's convention is recommended due to its compatibility with the quasi-steady theory and optimal resolutions of data plots over any range of reduced flow velocities.

Experiment investigation on flow characteristics of open natural circulation system

Experimental research on flow characteristics of open natural circulation system was performed, to figure out the mechanism of the open natural circulation behaviors. The influence factors, such as the heating power, the inlet subcooled and the level of cooling tank on the flow characteristics of the system were examined. It was shown that within the scope of the experimental conditions, there are five flow types: single-phase stable flow, flash and geyser coexisting unstable flow, flash stable flow, flash unstable flow, and flash and boiling coexisting unstable flow. The geyser flow in flash and geyser coexisting unstable flow is different from classic geysers flow. The flow oscillation period and amplitude of the former are more regular, is a newly discovered flow pattern. By drawing the flow instability boundary diagram and sorting out the flow types, it is found that the two-phase unstable flow is mainly characterized by boiling and flash, which determine the behavior of open natural circulation respectively or jointly. Moreover, compared with full liquid level system, non-full liquid level system is more prone to boiling phenomenon, and the range of heat flux density and undercooling degree corresponding to unstable flow is larger.

Flow-excited membrane instability at moderate Reynolds numbers

In this paper, we study the fluid–structure interaction of a three-dimensional (3-D) flexible membrane immersed in an unsteady separated flow at moderate Reynolds numbers. We employ a body-conforming variational fluid–structure interaction solver based on the recently developed partitioned iterative scheme for the coupling of turbulent fluid flow with nonlinear structural dynamics. Of particular interest is to understand the flow-excited instability of a 3-D flexible membrane as a function of the non-dimensional mass ratio ( $m^{*}$ ), Reynolds number ( $Re$ ) and aeroelastic number ( $Ae$ ). For a wide range of parameters, we examine two distinct stability regimes of the fluid–membrane interaction: deformed steady state (DSS) and dynamic balance state (DBS). We propose stability phase diagrams to demarcate the DSS and DBS regimes for the parameter space of mass ratio versus Reynolds number ( $m^{*}$ - $Re$ ) and mass ratio versus aeroelastic number ( $m^{*}$ - $Ae$ ). With the aid of the global Fourier mode decomposition technique, the distinct dominant vibrational modes are identified from the intertwined membrane responses in the parameter space of $m^{*}$ - $Re$ and $m^{*}$ - $Ae$ . Compared to the deformed steady membrane, the flow-excited vibration produces relatively longer attached leading-edge vortices which improve the aerodynamic performance when the coupled system is near the flow-excited instability boundary. The optimal aerodynamic performance is achieved for lighter membranes with higher $Re$ and larger flexibility. Based on the global aeroelastic mode analysis, we observe a frequency lock-in phenomenon between the vortex-shedding frequency and the membrane vibration frequency causing self-sustained vibrations in the dynamic balance state. To characterize the origin of the frequency lock-in, we propose an approximate analytical formula for the nonlinear natural frequency by considering the added mass effect and employing a large deflection theory for a simply supported rectangular membrane. Through our systematic high-fidelity numerical investigation, we find that the onset of the membrane vibration and the mode transition has a direct dependence on the frequency lock-in between the natural frequency of the tensioned membrane and the vortex-shedding frequency or its harmonics. These findings on the fluid-elastic instability of membranes have implications for the design and development of control strategies for membrane wing-based unmanned systems and drones.

Hydrodynamic Model of Decaying Radial Oscillations in RU Cam

Calculations of stellar evolution up to the early white dwarf stage were carried out for stars with mass on the main sequence $M_0=0.82M_\odot$, $0.85M_\odot$, $0.9M_\odot$ and with initial abundances of helium and heavier elements $Y=0.25$ and $Z=10^{-3}$, respectively. For each value of $M_0$ the AGB and post--AGB evolutionary phases were computed with three values of the mass loss parameter in the Bl\"ocker formula: $\eta_B=0.02$, 0.05 and 0.1. The variable star RU~Cam with pulsation period $\Pi\approx 22$ day is shown to be in the post--AGB stage and the pulsation amplitude decrease in years 1962--1963 is due to movement of the star across the HR diagram beyond the pulsation instability region. Theoretical estimates of the mass and the luminosity of RU~Cam are $0.524M_\odot\le M\le 0.532M_\odot$ and $2.20\times 10^3L_\odot\le L\le 2.33\times 10^3L_\odot$, respectively. Hydrodynamic calculations of nonlinear stellar pulsations show that while the star approaches the instability boundary a significant reduction ($\approx 90\%$) in the pulsation amplitude occurs for nearly two years with subsequent slow decay of low--amplitude oscillations. Solution of the equations of hydrodynamics with time--dependent inner boundary conditions describing evolutionary changes in the radius and the luminosity at the bottom of the pulsating envelope allows us to conclude that decay of radial oscillations in RU~Cam is accompanied by the effect of oscillation hysteresis. In particular, the stage of large--amplitude limit cycle oscillations extends by $\approx 12$ years and the subsequent stage of decaying small--amplitude oscilations spreads beyond the formal boundary of pulsation instability.

Mechanism of Stability Enhancement with Shallow ‎Reversed Slot-Type Casing Treatment in a Transonic ‎Compressor

In this paper, the influence of a shallow reversed slot-type casing treatment on the performance of a tip-critical transonic compressor has been numerically investigated. Firstly, the complex flow fields in the rotor tip region are studied in details. It shows the severe blockage induced by suction surface boundary separation triggers compressor stall at 100% design speed, while the blockage due to tip leakage vortex dominates at 80% design speed. Secondly, the mechanism of stability extension is presented at different rotating speeds. The casing treatment alleviates greatly the tip blockage by manipulating the tip leakage flow, accompanied by the redistribution of aerodynamic loading and mass flux. As a result, the casing treatment is more efficient for the blockage induced by tip leakage vortex (at 80% design speed). Further analysis of the pressure field and passage shock distribution demonstrates that the passage shock intensity and its location will affect the effectiveness of casing treatment. Finally, the instability characteristics of compressor with casing treatment are revealed. The numerical results reflect when the mass flow approaching the instability boundary, the stator passage blockage presumably is dominant for triggering the compressor stall.

Electron Heat Flux in the Solar Wind: Generalized Approaches to Fluid Transport With a Variety of Skewed Velocity Distributions

AbstractIn the solar corona and solar wind, electron heat conduction is an important process that transports energy over large distances and helps determine the spatial variation of temperature. High‐density regions undergoing rapid particle‐particle collisions exhibit a heat flux described well by classical Spitzer‐Härm theory. However, much of the heliosphere is closer to a more collisionless state, and there is no standard description of heat conduction for fluid‐based (e.g., magnetohydrodynamic) models that applies generally. Some proposed models rely on electron velocity distributions that exhibit negative values of the phase‐space density. In this study, we explore how positive‐definite velocity distributions can be used in fluid‐based conservation equations for the electron heat flux along magnetic‐field lines in the corona and solar wind. We study both analytic forms of skewed distributions (e.g., skew‐normal distributions, two‐sided bi‐Maxwellians, and constant‐collision‐time electrostatic solutions) and empirical fits to measurements of core, halo, and strahl electrons in interplanetary space. We also present example solutions to a generalized conservation equation for the heat flux in the solar wind, with some limiting cases found to resemble known free‐streaming approximations. The resulting values of the electron heat flux vary as a function of radial distance and Knudsen number in ways that resemble observed data. We note that this model does not include the effects of kinetic instabilities (which may impose saturation limits when active), so for now its regime of applicability is limited to collisionless heat‐flux evolution away from the known instability boundaries in parameter space.

Theoretical analysis of a parametrically excited rotor system with electromechanically coupled boundary condition

In this paper, the dynamic behavior of a rotor system with electromechanically coupled boundary condition under periodic axial load is studied, where the boundary condition is derived from a ring-shaped piezoelectric damper which is developed for vibration control of rotor system. This damper is based on the piezoelectric shunt damping technique, which can produce much or little damping performance depends on the selection of shunt circuit parameters. By using assumed mode method and Lagrange equation, the equations of motion are derived. Actually these equations can also be derived from a general forced parametrically excited gyroscopic system. Thus, to analyze such system, a method of multiple scales is developed firstly, where a general procedure is proposed to establish solvability conditions. Subsequently, this procedure is applied to obtain the analytical instability boundaries and forced vibration responses. The analytical results show that the additional combination instability regions are created due to the introduction of shunt circuit. When the parametrically excited eccentric rotor is rotating, the parametric vibrations are superimposed on the unbalanced responses. As the resistance value of shunt circuit is increasing, both parametric vibrations and unbalance vibrations can be significantly suppressed. For the unbalanced responses, both the resonance and anti-resonance phenomena can be observed; whereas for the parametric vibrations, only the resonance phenomena exist. These phenomena indicate that we may achieve great vibration control performance for the rotor parametric vibrations by introducing the piezoelectric shunt damping, as long as the circuit parameters are well adjusted. To validate the obtained analytical expressions, the numerical methods are applied. Specifically, the discrete state transition matrix method (DSTM) is applied to validate the analytical instability boundaries and the Runge-Kutta method is conducted to verify the analytical frequency response functions.

Nearly invariant boundary entanglement in optomechanical systems* *Project supported by the National Natural Science Foundation of China (Grant Nos. 11574398, 11904402, 12074433, and 12004430) and the National Basic Research Program of China (Grant No. 2016YFA0301903).

In order to understand our previous numerical finding that steady-state entanglement along the instability boundary remains unchanged in a three-mode optomechanical system [Phys. Rev. A 101 023838 (2020)], we investigate in detail the boundary entanglement in a simpler two-mode optomechanical system. Studies show that both the mechanism to generate entanglement and the parameter dependence of boundary entanglement are quite similar in these two models. Therefore, the two-mode system has captured the main features in the three-mode system. With the help of analytical calculations and discussing in a much bigger parameter interval, we find that the unchanging behavior previously discovered is actually an extremely slow changing behavior of the boundary entanglement function, and most importantly, this nearly invariant boundary entanglement is a general phenomenon via parametric down conversion process in the weak dissipation regime. This is by itself interesting as threshold quantum signatures in optomechanical phonon lasers, or may have potential value in related applications based on boundary quantum properties.

Unveiling the structure and dynamics of peeling mode in quiescent high-confinement tokamak plasmas

Quiescent high-confinement mode plasmas with edge-harmonic oscillations do not exhibit the explosive instabilities associated with edge-localized modes. Instead, an additional means of enhanced transport is considered to maintain the plasma edge under conditions just below the boundary of the peeling mode instability. Although the potential of the peeling mode has been widely recognized in plasma physics, no direct evidence for this mode has been revealed previously because decisive diagnostics were lacking. Herein, we report evidence of the structure and dynamical steady state of peeling mode in quiescent high-confinement mode. Edge-harmonic oscillations are dominated by fundamental mode at both the low- and high-field sides. Edge perturbations are confirmed to have kink parity and exhibit the frozen-in-condition predicted by linear stability analysis. The envelope signal of the fundamental mode exhibits repeated cycles of growth and damping in association with minor changes in the edge gradient. Results from this study are quantitatively consistent with limit-cycle-oscillation model.

Validation of variable helix milling instability islands

During machining, it is well-known that unstable self-excited vibrations known as regenerative chatter can limit productivity. There has been a great deal of research that has sought to understand regenerative chatter, and to avoid it through modifications to the machining process. One promising approach is the use of variable helix tools. Here, the time delay between successive tooth passes is intentionally modified, in order to improve the boundary of instability. Previous research has predicted that such tools can offer significant performance improvements whereby islands of instability occur in the stability lobe diagram. By avoiding these islands, it is possible to avoid regenerative chatter, at depths of cut that are orders of magnitude higher than for traditional tools. However, to the authors’ knowledge, these predictions have not been experimentally validated, and there is limited understanding of the parameters that can give rise to these improvements. The present study seeks to address this shortfall. A recent approach to analysing regenerative chatter stability is modified, and its numerical convergence is shown to outperform alternative methods. It is then shown that islands of instability only emerge at relatively high levels of structural damping, and that they are particularly susceptible to model convergence effects. The model predictions are validated against detailed experimental data that uses a specially designed configuration to minimise experimental error. To the authors’ knowledge, this provides the first experimentally validated study of unstable islands in variable helix milling, whilst also demonstrating the importance of structural damping and numerical convergence on the prediction accuracy.

Auto-parametric sloshing in a rectangular aqueduct excited by wind vortex-induced vibration: experimental investigation and stability analysis

The vortex-induced vibration of the aqueduct girder may excite the parametric sloshing of the contained water which will have substantial impacts on the normal running and structural safety of the aqueduct bridge. This paper first conducts a wind-tunnel experiment to investigate the auto-parametric sloshing caused by wind vortex-induced vibration for an elastically-hung rectangular aqueduct with contained water. When the vortex excitation frequency is close to the natural frequency of the structure and approximately equal to twice the natural frequency of the fluid the vortex-induced resonance of the structure and the strong auto-parametric sloshing (internal resonance) of water successively occur in the test. The auto-parametric sloshing for the non-internal resonance case is also experimentally investigated. The contained water can be simplified as a rigid mass for the stability analysis of sloshing. The instability boundaries of sloshing and the corresponding wind speed regions are theoretically determined and compared with the experimental results

Study on flow instability in natural circulation loop with parallel channels under moving condition

In this study, the mathematical models suitable for natural circulation, parallel channels, and general forms of additional force under moving conditions were established. An analysis code was developed using SIMPLE algorithm. Based on the validated code, the natural circulation and parallel flow instability under typical moving conditions such as inclination, rolling, heaving, and coupled motion were discussed, and the corresponding flow instability boundaries were obtained. The results show that under the moving conditions, the natural circulation flow change is manifested as the superposition of the two-phase instability fluctuation and the motion-induced fluctuation. The heaving condition has the most drastic effect on flow rate with fluctuation amplitude exceeding 30%, but the average flow rate remains unchanged. The boundary of natural circulation flow instability will be slightly extended affected by motions and the variation of the boundary quality is less than 0.03. The inclination and rolling tend to destabilize the parallel channels system and reduce the boundary quality by more than 0.1, whereas the effect of heaving on the parallel instability is not significant. For the parallel flow instability, the adverse effect is mainly attributed to the decrease of the mean flow rate at the entrance of the parallel channels caused by the motions, while the flow re-distribution between parallel channels and natural circulation flow fluctuation induced by the moving conditions do not contribute much.

Searching for the shortest path to voltage instability boundary: From Euclidean space to algebraic manifold

Voltage instability is one of the main causes of power system blackouts. This paper focuses on finding the shortest path to the boundary of the singularity-induced voltage instability problem. Instead of using the Euclidean distance, we propose to use the arc length of the path on the network constraint manifold. This formulation is further converted into an optimal control framework to solve for the shortest path on the manifold. We rigorously show that the global solution of the proposed problem formulation always ends on the correct singular boundary. However, the traditional Euclidean distance formulation does not achieve this crucial topological property, thus, can lead to a wrong voltage collapse direction and/or a very conservative estimation of the stability margin. Numerical simulations are firstly performed on a low-dimensional example to fully visualize the algebraic manifold, the singular submanifold, and the optima for both problem formulations. Then, a larger 39-bus example is investigated in three different cases for both formulations. The results validate our theoretical statements that our proposed formulation always identifies the shortest path towards the correct voltage instability boundary. A broad range of potential applications using the proposed method are further discussed.

Higher-order stability analysis of a rotating BDFG tapered beam with time-varying velocity

In this paper, dynamic stability analysis is carried out for a rotating tapered cantilever beam made of bi-directional functionally graded (BDFG) materials with time-dependent rotating velocity. Rayleigh-Ritz method is employed to obtain the eigenfrequencies and mode shapes of the beam. Hamilton’s principle and the Galerkin method are adopted to establish the equation of motion with periodic coefficients. Dynamic instability problem due to the periodic rotating velocity is solved by Bolotin’s method with higher-order approximation. Both the sub-harmonic and harmonic parametric instability regions are investigated. Floquet theory is applied to verify the dynamic instability boundaries. The results indicate that the typical first-order approximate Bolotin’s method cannot meet the accuracy requirements of dynamic instability analysis for rotating BDFG tapered beam, and a second-order approximation is needed to improve computation accuracy. The effects of mean rotating velocity, hub radius, dynamic amplitude factor, material FG indexes and taper ratio on the natural frequencies and dynamic instability characteristics of the rotating BDFG tapered beam are discussed under different parameters.

Avoidance of vertical displacement events in DIII-D using a neural network growth rate estimator

Robust disruption avoidance techniques are critical for the development of reliable fusion reactor devices. A viable reactor will require non-disruptive, long pulse operation where simply shutting down a discharge is undesirable. To achieve such performance, the plasma must be controlled to continuously avoid hazardous regimes instead of asynchronously aborting. A recent experiment on DIII-D demonstrated for the first time real-time control of proximity to a disruptive instability boundary. In particular, the vertical growth rate, an eigenvalue that characterizes the degree of instability of the plasma's vertical position, was regulated so as not to exceed DIII-D's vertical controllability limit. The open-loop growth rate was estimated in real time on the DIII-D plasma control system using a neural network model trained with tens of thousands of DIII-D shots. The model was trained to replicate the results of RZRIG [1], a rigid displacement code for calculating the growth rate. Once trained, producing an estimate using the neural network is multiple orders of magnitude faster than RZRIG, thereby making the calculation suitable for real-time execution. The control system regulated the estimated growth rate by adjusting plasma elongation and distance to the inner wall of the vessel, and this regulation was shown to reliably avoid vertical displacement event disruptions (i.e. uncontrolled vertical oscillations) of the plasma. This work presents these experimental results, including the dynamic performance and the effectiveness of the control technique. Details are presented on the training of the neural network model, including concerns such as hyperparameter tuning and uncertainty quantification. Additionally, the methodology for embedding the neural network into the control system is discussed.