Amplitude Of Limit Cycle Research Articles

Synchronization and limit cycles in a simple contagion model with time delays

We examine a system of N=2 coupled non-linear delay-differential equations representing financial market dynamics. In such time delay systems, coupled oscillations have been derived. We linearize the system for small time delays and study its collective dynamics. Using analytical and numerical solutions, we obtain the bifurcation diagrams and analyze the corresponding regions of amplitude death, phase locking, limit cycles and market synchronization in terms of the system frequency-like parameters and time delays. We further numerically explore higher order systems with N>2, and demonstrate that limit cycles can be maintained for coupled N−asset models with appropriate parameterization.

Maximal canards in a slow–fast Rosenzweig–MacArthur model with intraspecific competition among predators

Using geometric singular perturbation theory, this paper investigates the canard phenomenon of a slow–fast Rosenzweig–MacArthur model. The model incorporates intraspecific competition among predators, assuming predator reproduction occurs much slower than prey. We demonstrate the occurrence of maximal canards between attracting and repelling slow manifolds as a bifurcation parameter varies. Additionally, we derive an analytic expression to approximate the bifurcation parameter value at which a maximal canard occurs. The method employed for this analysis relies on the blowup method. This involves finding a quasihomogeneous blowup map to desingularize the nonnormally hyperbolic point. Subsequently, charts are utilized to express the blowup in local coordinates, calculate local data, investigate the dynamics of the blown-up vector fields, and establish connections across charts. Furthermore, we provide numerical simulations to illustrate the canard explosion phenomenon in the model. Through parameter variation, we observe that the model transitions from a small amplitude limit cycle to small amplitude canard cycles (canards without a head), then to large amplitude canard cycles (canards with a head), and finally to a large amplitude relaxation cycle.

Torsional behaviors of motor rotor of permanent magnet semi-direct drive system for aerial passenger device

Considering the influence of motor shaft torsional damping, the torsional dynamics equations of a permanent magnet semi-direct drive rotor system in an aerial passenger device are derived. The derivation relies on the coupling between the mathematical model and the gear dynamics model of the permanent magnet motor, and the Lagrange–Maxwell equations are used for streamlined analysis. To protect the motor under extreme working conditions, the influence of supercritical Hopf bifurcation caused by different torsional dampings of the motor shaft on the torsional vibration of the motor rotor is investigated. Based on the Hurwitz criterion, the influence of the linear control gain and nonlinear gain of the washout filter on the system bifurcation point and the limit cycle amplitude of the system is analyzed. Then, from the perspective of the system instability and oscillation caused by friction damping change of the drive motor and the complex electromechanical coupling behavior in the permanent magnet semi-direct drive cutting drive system of the aerial passenger device, the system stability domain is defined. The results can be applied to effectively suppress the torsional vibration of the shaft system in the permanent magnet semi-direct drive system of the aerial passenger device, providing a theoretical guidance for stable operation of the system.

A Polynomial System of Degree Four with an Invariant Square Containing At Least Five Limit Cycles

We consider a class of polynomial systems of degree four with four real invariant straight lines that form a square, called this an invariant square, and also that contains in its interior at least five small amplitude limit cycles for a certain choice of the parameters. Moreover, we will obtain the necessary and sufficient conditions for the critical point inside the square to be a center.

A modified generalized harmonic function perturbation method and its application in analyzing generalized Duffing–Harmonic–Rayleigh–Liénard oscillator

A modified generalized harmonic function perturbation method is proposed in this paper. Compared with the classical version of this method, the modified version can execute its procedures pure symbolically without the need to assign any system parameters even for some complicated nonlinear oscillators. This means that the relations between amplitude of limit cycles and system parameters can be derived analytically from the proposed method. Meanwhile, the analytical expression of characteristic quantity of limit cycles can be also obtained. Via these analytical expressions, the evolutional process of limit cycles can be studied quantitatively in amplitude domain. It demonstrates the entire live period of each limit cycle from its generation to bifurcation to destination. To show the feasibility of the proposed method, a complicated oscillator named generalized Duffing–Harmonic–Rayleigh–Liénard oscillator is investigated in this paper. First, the two analytical expressions mentioned above are derived and the global evolution of its limit cycles are analyzed quantitatively. Second, the critical value of homoclinic and heteroclinic bifurcation parameters are also predicted via this two analytical expressions. Moreover, the analytical approximate solutions of both limit cycles and homo-heteroclinic orbits are calculated. To prove the accuracy, all the above results obtained via the proposed methods are confirmed by the Runge–Kutta method, which show a good accordance. Therefore, the proposed method can be considered as an effective modification for a classical perturbation method. It provides another feasible and reliable analytical quantitative method for analyzing global dynamics of strongly nonlinear oscillators.

Control of large amplitude limit cycle of a multi-dimensional nonlinear dynamic system of a composite cantilever beam

For the first time, a control strategy based on Fuzzy Sliding Mode Control is implemented in the control of a large amplitude limit cycle of a composite cantilever beam in a multi-dimensional nonlinear form. In the dynamic model establishment of the investigated structure, the higher-order shearing effect is applied, as well as the second-order discretization. Numerical simulation demonstrates that a multi-dimensional nonlinear dynamic system of the investigated structure is demanded for accurate estimation of large amplitude limit cycle responses. Therefore, a control strategy is employed to effectively suppress such responses of the beam in multi-dimensional nonlinear form.

Hunting stability control of high-speed bogie based on active yaw damper

Hunting stability is an inherent property of railway vehicles that determines the operational speed. This paper establishes a half-vehicle model with nonlinear wheel/rail equivalent conicity and interactive forces. Additionally, the dynamic performances of vehicle under various levels of wheel wear with passive suspension are compared and analyzed. Importantly, the investigation delves into the hunting stability of the vehicle system, employing both linear and nonlinear control approaches. The results demonstrate a notable reduction in critical speed during the end-worn period with a passive suspension. However, this reduction can be substantially countered through the application of active control, resulting in a significant speed increase. The implementation of stiffness control raises the frequency of limit cycles, whereas damping control serves to diminish it. Notably, an appropriate linear cubic stiffness control effectively mitigates the amplitudes of limit cycles during instances of instability. Moreover, the control strategy derived from the simplified model is extended to enhance the stability of the entire vehicle system. The research findings hold the potential to offer a promising strategy for the active control of high-speed vehicles, particularly during periods of wheel wear.

Effects of structural geometric nonlinearities on the transonic aeroelastic characteristics of wing

The structural geometric nonlinearity has a great influence on the transonic aeroelastic characteristics of aircraft wings, which is an important problem for the aerodynamic and structural design of wing. In this paper, a CFD/CSD coupling method based on the high precision in-house solver is first established, and the reliability of the CFD and the CFD/CSD coupling method is then verified. Subsequently, the effects of geometric nonlinearity on the transonic aeroelastic characteristics of wing under different dynamic pressures are studied. The results show that: (1) At low dynamic pressure conditions, for geometric linear wing structure, limit-cycle oscillation occurs at various angles of attack, and the limit-cycle oscillation amplitude decreases with the increases of angle of attack. At low angle of attack, the wing deformation characteristics are dominated by first bending and first torsion, and the first torsion weaken as the angle of attack increases. For the geometric nonlinear wing structure, the limit-cycle oscillation occurs only at an attack angle of 0° As the angle of attack increases, the structural displacement eventually exhibits static deformation characteristics. (2) Under high dynamic pressure conditions: the displacement of geometric linear wing structure presents a divergent trend. For geometric nonlinear wing structure, the limit-cycle oscillation amplitude decreases first and then increases with the increase of the angle of attack. At low angle of attack, the deformation characteristics of the wing are dominated by first bending and first torsion. With the increase of the angle of attack, the first torsion weakens, while the second bending strengthen. With the further increase of the angle of attack, the second bending weaken and become co-dominated by the first bending and first torsion. It can be seen from the above results that the aeroelastic characteristics of both the geometric linear structure and the geometric nonlinear structure wings are significantly different. These findings provide valuable insights for the meticulous design of aircraft wings.

Vortex-induced vibrations of wind turbines: From single blade to full rotor simulations

Vortex-induced vibrations (VIVs) of wind turbine blades are a phenomenon that cannot be predicted sufficiently accurate by low fidelity methods. High fidelity methods, such as computational fluid dynamics (CFD) coupled to a structural solver come at a high computational cost. Most studies of the phenomenon have so far focused on analysis of single blades. For these cases, the so-called forced-motion method of prescribing the blade motion according to structural modeshapes has proven effective.In this article, VIVs of the IEA 10 MW reference turbine is studied in a fluid–structure interaction setup, where the structural dynamics of the full turbine are represented in the structural solver and the full rotor aerodynamics are simulated in the CFD solver. Due to the high computational cost this study is limited to only a single case with the rotor in bunny configuration, 80 degrees pitch and 90 degrees yaw. Also the purely structural power dissipation of the full turbine is studied to enable full rotor forced-motion in future work. A blade tip amplitude of roughly 17 meters is reached in the full turbine simulations, which is a significantly larger limit cycle amplitude than what would be reached in a corresponding isolated single blade simulation. The analysis also shows that forced-motion simulations combining three single blades vibrating according to the structural modeshape would be feasible. It is further observed, that the power injection or extraction close to the blade tip may change drastically at very large amplitudes compared to lower amplitudes.

Comparison of pressure-based flame describing functions measured in an annular combustor under self-sustained oscillations and in an externally modulated linear combustor

One important aspect in the analysis of combustion instabilities in multiple-injector annular systems is that of representing such oscillations with a linear array of injectors. This issue is considered in the present work by comparing flame dynamics in a self-sustained oscillations situation in the annular configuration MICCA-Spray, with experiments carried out in TACC-Spray, a rectangular combustor equipped with five injectors externally modulated by driver units. In the annular system, the coupling mode is azimuthal, and in the linear array, the flames are submitted to a transverse acoustic field. An original method is used to change the limit-cycle pressure oscillations amplitude in the annular system and favor a standing mode with a well-defined nodal line position. It consists in using different arrangements of two types of injectors, leading to various azimuthal staging patterns. The range of pressure oscillation amplitudes is swept in TACC-Spray by changing the external acoustic forcing level. It is thus possible to compare the flame dynamics in forced and self-sustained situations using these two experimental facilities. This is done by examining flame describing functions (FDFs) based on downstream pressure fluctuations measured in the two configurations for different pressure oscillation amplitudes. The trends observed in the two systems are in good agreement, and there is a reasonably good match between the values of the gains and phases determined in the two configurations. The pressure-based FDFs are determined for flames located at different positions with respect to the acoustic mode. It is found that the transverse or azimuthal velocity component has only a weak influence on the pressure-based FDF, the strongest impact being observed in the neighborhood of the velocity antinode. This study demonstrates the ability of a linear-array setup modulated by a transverse mode to reproduce conditions of azimuthal coupling leading to self-sustained limit cycles in an annular combustor.

A polynomial system of degree four with an invariant triangle containing at least four small amplitude limit cycles

In this work, the existence of a polynomial system of degree four with an invariant triangle containing at least four small-amplitude limit cycles is proved. This result improves the result obtained in [2].

Multi-Parameter Practical Stability Region Analysis of Wind Power System Based on Limit Cycle Amplitude Tracing

The existing division of system stability region does not fully consider the influence of nonlinear characteristics of the system. To address the challenge that it is difficult to inscribe the stability region of wind power grid-connected systems accurately and quantitatively, this paper proposes a nonlinear system multi-parameter practical stability region analysis method based on limit cycle amplitude tracking. Firstly, the nonlinear dynamic model of multi-machine system with doubly fed induction generator (DFIG) is established, and the bifurcation diagram of the system is obtained by nonlinear analysis of the model. Then, the limit cycle-parameter diagram is obtained by tracking the limit cycle of the Hopf bifurcation point of the system, and the problem of how the system oscillation evolves with the change of operating parameters is analyzed. Afterwards, the single parameter practical stability region of the nonlinear system is obtained by combining the system equilibrium manifold. Finally, the multi-parameter practical stability region of the nonlinear system is obtained by the three-dimensional bifurcation map with the system time delay, DFIG output active power and reactive load as the bifurcation parameters. The division of the stability region has important guiding significance for the parameter adjustment in the actual operation of the system.

The Effect of Hydrogen on Nonlinear Flame Saturation

Abstract We investigate the effect of increasing levels of hydrogen enrichment on the nonlinear response and saturation of premixed bluff-body stabilized methane/hydrogen flames submitted to acoustic forcing. The thermal power is kept approximately constant to preserve the nozzle velocity while increasing the flame speed through hydrogen enrichment. The flame describing function (FDF) is measured for a fixed frequency and three hydrogen–methane blends ranging from 10% to 50% by power, corresponding to 25% to 75% by volume. We show that when the flame is forced at the same frequency at similar power and bulk velocities, increasing levels of hydrogen enrichment increase the saturation amplitude of the flame. To provide insight into the flame dynamics responsible for the change in the global nonlinear response and saturation amplitude, the flames were investigated using high-speed imaging in combination with OH planar laser-induced fluorescence (OH-PLIF) at a range of forcing amplitudes. At lower hydrogen concentrations, the flame is stabilized along the inner shear layer and saturation in the heat release rate (HRR) occurs at lower forcing amplitudes due to large-scale flame–vortex interactions causing flame annihilation as observed in several previous studies. At increased levels of hydrogen enrichment, distinctly different flame dynamics are observed. In these cases, the flame accelerates and propagates across to the outer shear layer, which acts to suppress large-scale flame annihilation during roll-up of both the inner and outer shear layers. This results in a coherent increase in flame surface area with forcing amplitudes significantly increasing the saturation amplitude of the flame. These results show that high levels of hydrogen increase the amplitude response to acoustic forcing leading to higher saturation amplitudes. This suggests that substituting natural gas with hydrogen in gas turbines increases the risk of much higher limit-cycle amplitudes if self-excited instabilities occur.

Time Series and Spectral Analysis of Thermoacoustic Oscillations for Propane-Oxyfuel Combustion in a Swirl-Stabilized, Nonpremixed Combustor.

In this study, the time series data for sound pressure and heat release fluctuations were analyzed to establish the stability window of the propane-oxyflames under different operating conditions. The CO2 dilution level and the combustor power density were varied at a fixed global equivalence ratio to investigate the thermoacoustic instability of the flames. The phase difference between the fluctuations and the instantaneous Rayleigh index reveals whether the heat release and pressure fluctuations are coupled, which can result in the amplification of the sound pressure fluctuations. Results showed a negative Rayleigh index and uncoupled fluctuations at low (<40%) and high (>60%) CO2 dilution with coupled fluctuations and sound pressure amplification at intermediate CO2 dilution levels. A peak of varying magnitude appeared in the frequency domain at 465 Hz for both the heat release and pressure fluctuations in the coupled mode. The Strouhal number at different CO2 concentrations revealed a range of vortex-shedding frequencies (300-1000 Hz), suggesting that the coupled mode is vortex-induced. Phase space reconstruction for the sound pressure fluctuations was carried out and it is observed that although the pressure fluctuations are amplified in the coupled mode, limit cycle amplitudes have not been reached. The recorded coupling and uncoupling of the oscillations associated with flame-vortex interactions at certain CO2 concentrations provides a valuable insight on the combustor's dynamics and toward the development of nonpremixed-oxy-flames combustors.

On the use of frictional dampers for flutter mitigation of a highly flexible wing

The prospect of aeroelastic instability, leading to large-amplitude limit cycle oscillations, is of particular concern for the highly flexible wings of solar high-altitude long-endurance (HALE) aircraft. Among the different strategies of vibration control, passive mitigation through friction dampers stands out as a promising solution in this context. While this concept has been proposed previously, related studies systematically use simple two-degrees-of-freedom airfoil models. In this paper, a more realistic structural model involving a three-dimensional nonlinear beam description of a full wing is considered. A friction damper is installed at a specific location on the wing span, and the response to aerodynamic loads is investigated numerically. Comparisons of critical speed and limit cycle amplitudes with and without the damper suggest improved vibration mitigation with respect to a linear viscous damper.

Bifurcation of Limit Cycles and Center in 3D Cubic Systems with Z3-Equivariant Symmetry

This paper focuses on investigating the bifurcation of limit cycles and centers within a specific class of three-dimensional cubic systems possessing Z3-equivariant symmetry. By calculating the singular point values of the systems, we obtain a necessary condition for a singular point to be a center. Subsequently, the Darboux integral method is employed to demonstrate that this condition is also sufficient. Additionally, we demonstrate that the system can bifurcate 15 small amplitude limit cycles with a distribution pattern of 5−5−5 originating from the singular points after proper perturbation. This finding represents a novel contribution to the understanding of the number of limit cycles present in three-dimensional cubic systems with Z3-equivariant symmetry.

Lyapunov-based robust optimal control for time-delay systems with application in milling process

In this paper, we propose an optimal delay-independent robust controller based on the Lyapunov–Krakovskii theorem for the milling process as a time-delay system in the presence of varying axial depth of cut and different parametric uncertainty, such as stiffness, damping, etc. Milling is widely used in the manufacturing processes for the production of complex-shaped workpieces with high accuracy. However, chatter is a self-excited vibration that may cause adverse effects, such as tool damage, poor surface quality, excessive noise, etc. The dynamic model of the milling process is considered a two-dimensional time-delay system. A nonlinear programming with linear matrix inequality constraints is solved in order to obtain the controller gain where the objective function is corresponding to the norm-2 of controller gain, and the constraints guarantee robust stability. Using the semi-discretization method, stability lobes are shown in both uncontrolled and controlled plants to illustrate the improvement of the stable region via the proposed controller. Bifurcation phenomena have been improved with this controller by postponing the adverse effects to the higher values of the axial depth of cut, reducing the amplitude of limit cycles, and changing the type of bifurcation. Finally, we will compare the proposed controller with an intelligent controller in order to show the efficiency of the proposed method. It is shown that the proposed controller has improved the integral absolute error index by about 3.65 times compared with the intelligent controller.

Estimation of limit cycle amplitude after onset threshold of thermoacoustic Stirling engine

This study estimates the amplitude saturation process of spontaneous pressure oscillations in the thermoacoustic Stirling engine, which harnesses a stack-screen regenerator, after the engine onset. To observe spontaneous initiations leading to finite amplitudes of sound in the engine, the engine is studied experimentally to see how the acoustic pressure fluctuates in the time domain. To describe this, the nonlinear effects of the flow resistance of the stacked-screen regenerators, empirically proposed as an effective radius reff whose size decreases with increasing Reynolds number, are incorporated into the framework of Rott’s linear thermoacoustic theory for depicting the evolution of oscillations from initiation to a steady state. The numerical calculations are based on the coupling between pressure oscillations and the imaginary part of a complex angular frequency which is obtained numerically from the characteristic equation derived from the transfer matrices modeling the engine. For the scenarios of steady oscillations, the measured results are elucidated with calculations by consciously adjusting r0 which is involved in reff and estimates the threshold of thermoacoustic instability. By plotting with the adjusted r0, the measured stability limits expressly agree with the stability curve calculated from Rott’s theory. This work improves Rott’s frequency-domain theory to the temporal description of the thermoacoustic Stirling engine, from start-up to limit cycle oscillation.

Study on Hopf bifurcation types and dominant parameters of grid‐connected voltage source converter considering non‐linear saturation elements

AbstractSubsynchronous oscillation events of grid‐connected voltage source converter (VSC) pose a potential risk to the security and stability of power systems. It is particularly challenging to reveal the subsynchronous oscillation mechanism precisely because the grid‐connected VSC consists of non‐linear components such as saturation. The describing function and the generalized Nyquist criterion are used in this paper to investigate the bifurcation types and dominant parameters of grid‐connected VSC. First, the condition of a supercritical Hopf bifurcation is investigated, which generates subsynchronous oscillations. Then, the dominant parameters of supercritical Hopf bifurcation are deduced, including the grid strength, the active power reference, or the proportional coefficient of phase‐locked loop. Finally, it is examined how the saturation's limit value impacts the limit cycle's amplitude and stability. The findings demonstrate that a system's stability depends on whether its closed‐loop frequency characteristic curve intersects the describing function's negative reciprocal curve. When the two curves intersect, the supercritical Hopf bifurcation appears, and changing the limit value can affect the limit cycle's amplitude. However, if the two curves do not intersect, altering the limit value cannot cause the supercritical Hopf bifurcation, and the subsynchronous oscillation will not happen. Variations in grid strength, active power reference, or proportional coefficient of phase‐locked loop will significantly affect the Nyquist curve, determining if supercritical Hopf bifurcation appears.

Stability Analysis of Nonlinear Rotating Systems Using Lyapunov Characteristic Exponents Estimated From Multibody Dynamics

Abstract The use of Lyapunov characteristic exponents to assess the stability of nonlinear, time-dependent mechanical systems is discussed. Specific attention is dedicated to methods capable of estimating the largest exponent without requiring the Jacobian matrix of the problem, which can be applied to time histories resulting from simulations performed with existing multibody solvers. Helicopter ground resonance is analyzed as the reference application. Improvements over the available literature are: the problem is formulated in physical coordinates, without eliminating periodicity through multiblade coordinates; the rotation of the blades is not linearized; the problem is modeled considering absolute positions and orientations of parts. The dynamic instability that arises at some angular velocities when the isotropy of the rotor is broken (e.g., caused by the failure of one lead-lag damper, a design test condition) is observed to evolve into a large amplitude limit cycle, where the usual Floquet–Lyapunov analysis of the linearized time-periodic simply predicts instability.