Limit Cycle Prediction Research Articles

UHBR Open-Test Case Fan ECL5/CATANA: Non-Linear Analysis of Non-Synchronous Blade Vibration at Part-Speed Conditions

Abstract The onset of non-synchronous vibrations has been predicted for a composite open-test-case UHBR fan geometry by time-linearized simulations at part-speed conditions. In the current study, time-accurate simulations with harmonic blade oscillations have been conducted to investigate the non-linear behavior. Results for different operating conditions show the evolution of aeroelastic stability. The lock-in of convected aerodynamic disturbances with structural vibration is analyzed for different vibration amplitudes and frequencies. The simulations show the development of multiple wave numbers, mode scattering, and a non-linear dependency on vibration amplitude. The results indicate the possibility of limit-cycle prediction and the necessity of strongly coupled methods for the assessment of non-synchronous vibration.

Nonlinear analysis of coupled neutronic-thermohydraulic stability characteristics of supercritical water-cooled reactor

The Supercritical Water-Cooled Reactor (SCWR) exhibits significant changes in thermophysical properties of the coolant, but its dynamic characteristics are different from existing reactors, due to the supercritical conditions of the coolant. This necessitates the study of the stability behaviour of SCWR, which requires a dynamic model of the reactor. In this work, a simple unsteady lumped parameter model (LPM) for SCWR has been developed. The LPM includes point reactor kinetics for neutron balance and a two-region model for fuel and coolant thermal hydraulics. The two regions are separated from each other by a boundary that depends on the pseudocritical temperature of the coolant. Regions of stable and unstable operation are identified in the parameter space by linear stability analysis. Bifurcation analysis is carried out to capture the non-linear dynamics of the system. Generalised Hopf (GH) points are located, and parameter ranges for supercritical (soft and safe) and subcritical (hard and dangerous) Hopf bifurcation are identified. The type of transient behaviour of the system for finite (though small) perturbations is predicted based on this analysis. The existence of predicted limit cycles is confirmed by numerical simulation of the nonlinear ODEs, using the shooting technique in case of unstable limit cycles. Furthermore, the effects of geometric, control and neutronics parameters on the stability of the system are studied.

Bifurcations and limit cycle prediction of rotor systems with fluid-film bearings using center manifold reduction

Bifurcations and limit cycle prediction of rotor systems with fluid-film bearings using center manifold reduction

Limit Cycle Prediction for a Multi-Degree-of-Freedom System with a Freeplay Non-Linearity

Limit Cycle Prediction for a Multi-Degree-of-Freedom System with a Freeplay Non-Linearity

Aeroelastic prediction in transonic buffeting flow with data fusion method

The frequency locking in phenomenon will occur in the transonic buffeting flow of the elastically supported airfoil. This phenomenon is usually accompanied by a large oscillation of the structure, eventually leading to the fatigue damage of the structure. Transonic aeroelasticity research based on computational fluid dynamics is extremely time-consuming. At present, the research on nonlinear aeroelastic reduced-order models attracts attention due to its important role in nonlinear limit cycle oscillation amplitude prediction. However, the training process of current nonlinear aeroelastic reduced order model requires massive smooth amplitude-modulated pseudo-random binary signals (SAPRBS), which limits its application in engineering field. This paper proposes a nonlinear aeroelasticity reduced-order model based on the saturated limit cycle oscillation signals (Harmonic Signals) that can be directly obtained in engineering field, providing a new perspective for nonlinear reduced-order models. The model only uses harmonic signals at three frequencies to accurately predict limit cycle oscillation amplitudes within a wide range of the NACA0012 transonic buffeting frequency-locking region. In addition, the model still maintains a certain generalization ability at the extrapolated frequency points.

Dynamics of a non-linear Jeffcott rotor in supercritical regime

This work is concerned with the prediction and simulation of limit cycles of a nonlinear Jeffcott rotor. Geometrical nonlinearity (related to large flexural displacements) in both stiffness and internal damping is considered. Classical instabilities due to linear internal damping occurring in the supercritical range are modified by nonlinear effects and lead to oscillating limit cycles at a radius whose value is found analytically as a function of the rotating speed. If the rotor is additionally excited by unbalance, the response spectrum also contains the excitation frequency, and the bifurcation leads to quasi-periodic limit cycles. The dynamics of this system is studied numerically via the harmonic balance method, path continuation and bifurcation tracking. In particular, an original procedure for switching to a branch of quasi-periodic solutions is proposed. To preserve the integrity of the machine post bifurcation, one then considers the possibility of constraining the transverse displacement of the shaft. Such possible rotor–stator interaction generates three-frequency quasi-periodic oscillations. Frequency response curves show that the quasi-periodic solutions created by the bouncing rotor are mainly driven by the friction coefficient.

Detection of subcritical Hopf and fold bifurcations in an aeroelastic system via the Describing Function method

The paper deals with Limit Cycle Oscillations (LCOs) in a Pitch & Plunge model of the wing dynamics, where the stiffness on the plunge displacement is assumed to be an odd fifth-order polynomial. First, it is shown that the model dynamics can be equivalently described via a Lur’e system, i.e., the feedback interconnection between a linear time-invariant subsystem, whose transfer function depends on the wind speed, and a nonlinear memoryless one. Then, the Describing Function (DF) method is used to establish the existence of Predicted Limit Cycles (PLCs) whose amplitude can be analytically expressed in terms of the wind speed, once the intersections of the Nyquist diagram of the linear subsystem with the real axis are obtained. The related bifurcation diagram permits to predict that a stable equilibrium point and two LCOs (one unstable and one stable) coexist for wind speeds smaller than the flutter velocity, at which a subcritical Hopf bifurcation occurs, and larger than the value at which a cyclic fold bifurcation is detected. The closeness of each PLC to the related LCO is measured via the so-called distortion index, which also admits a direct expression in terms of the wind speed. The comparison of the predicted bifurcation diagram with that obtained via numerical simulations confirms the quite good accuracy of the PLCs obtained via the DF method, especially when the distortion index is small.

Swirler effects on combustion instabilities analyzed with measured FDFs, injector impedances and damping rates

The influence of the injection system on combustion instabilities is investigated on a laboratory-scale combustor equipped with a single injector that is weakly-transparent to acoustic waves. The combustor is fed with liquid heptane delivered as a spray by a hollow cone atomizer. Experiments are carried out with three swirlers having similar geometries but different pressure losses and swirl numbers. Self-sustained oscillations (SSOs) corresponding to these swirlers feature differences in oscillation frequency and amplitude for a given chamber length. These observations do not match with standard modeling predictions. Therefore, a low-order analytical model is derived, representing the effect of the acoustically weakly-transparent injection system using an impedance at the injector outlet. This quantity and the flame describing function (FDF), both determined experimentally, are used together with damping rate estimates as model inputs. It is shown that the FDF can only be determined by suitably selecting the position for the measurement of incident velocity disturbances at the injector outlet and that plenum-based velocity measurements cannot be used for this purpose. It is also assumed that the OH*-chemiluminescence intensity can be used as a proxy for the heat release rate. This admittedly strong assumption for spray flames is discussed in detail and justified by showing that the equivalence ratio modulations are relatively weak for the particular spray flames considered in this study. Results from the model indicate that the injector impedance (that depends on the swirler characteristics) shifts the classical bands of instability and modifies the growth rate magnitude compared to a generic combustor with an acoustically transparent injector. Using the proposed model, the stability of the system can be rated along with a prediction for growth rate and frequency of oscillation. Predictions generally agree with experimental observations with some limitations. The model combined with damping rate estimates is finally used to predict limit cycle oscillation amplitudes with the aid of the FDF framework.

A geometrically nonlinear shear deformable beam model for piezoelectric energy harvesters

An electromechanical model for beam-like piezoelectric energy harvesters based on Reissner’s beam theory is developed in this paper. The proposed model captures first-order shear deformation and large displacement/rotation, which distinguishes this model from other models reported in the literature. All governing equations are presented in detail, making the associated framework extensible to investigate various piezoelectric energy harvesters. The weak formulation is then derived to obtain the approximate solution to the governing equations by the finite element method. This solution scheme is completely coupled, and thus allows for two-way interaction between mechanical and electrical fields. To validate this model, extensive numerical examples are implemented in the linear and nonlinear regime. In the linear limit, this model produces results in excellent agreement with reference data. In the nonlinear regime, the large amplitude response of the piezoelectric beam induced by strong base excitation or fluid flow is considered, and the comparison of results with literature data is encouraging. The ability of this nonlinear model to predict limit cycle oscillations in axial flow is demonstrated.

Modeling of the nonlinear flame response of a Bunsen-type flame via multi-layer perceptron

This paper demonstrates the ability of neural networks to reliably learn the nonlinear flame response of a laminar premixed flame, while carrying out only one unsteady CFD simulation. The system is excited with a broadband, low-pass filtered velocity signal that exhibits a uniform distribution of amplitudes within a predetermined range. The obtained time series of flow velocity upstream of the flame and heat release rate fluctuations are used to train the nonlinear model using a multi-layer perceptron. Several models with varying hyperparameters are trained and the dropout strategy is used as a regularizer to avoid overfitting. The best performing model is subsequently used to compute the flame describing function (FDF) using mono-frequent excitations. In addition to accurately predicting the FDF, the trained neural network model also captures the presence of higher harmonics in the flame response. As a result, when coupled with an acoustic solver, the obtained neural network model is better suited than a classical FDF model to predict limit cycle oscillations characterized by more than one frequency. The latter is demonstrated in the final part of the present study. We show that the RMS value of the predicted acoustic oscillations, together with the associated dominant frequencies are in excellent agreement with CFD reference data.

Nonlinear study on a rigid rotor supported by herringbone grooved gas bearings: Theory and validation

The nonlinear behavior of a rigid rotor supported by herringbone grooved journal gas bearings (HGJBs) was investigated in this study. The two-dimensional narrow groove theory (2D-NGT) was adopted to model the HGJBs. A set of integrated rotor-bearing state equations were built by coupling the rotor motion equations and the bearing Reynolds equation. An implicit integrator with adaptive time step method was used to solve those state equations continuously. Two low-stability HGJBs were implemented to experimentally demonstrate and analyze the appearance of self-excitation motions. The theoretical model was successfully validated by the experimental data on predicting the onset speed of the sub-synchronous vibration of the HGJB-rotor system and the whirl frequency ratio. The predicted limit cycle amplitude increases as the speed increases until the rotor contacts with the bearing surface, which leads to a bearing failure. Forward conical mode dominates the self-excited motion during the whole speed range of self-excited motion. The prediction shows that the HGJB-rotor system can still operate in a stable, even though the rotor is installed vertically, i.e., without static load on the bearings. This is a distinct advantage in comparison to plain bearings. As the static load applies on the bearings increases, the onset speed of sub-synchronous vibration increases as well. For the investigated rotor-bearing system, an increase of the onset speed of sub-synchronous vibration from 36 krpm to 75 krpm is predicted as the static load increases from 0 to 4 times of the rotor weight. This indicates an increased HGJB stability with increased static load. The rotor orbits show complex shapes when the imbalance excitation is considered in the simulation. Both synchronous frequency and whirl frequency are shown in the spectral analysis. Moreover, the speed range of self-excited motion reduces from [38,48] krpm to [40,42] krpm as the imbalance increases from 0 to 40 mgmm.

Impact of Non-Linear Electronic Circuits and Switch of Chaotic Dynamics

Switch-mode power supply is an extremely non-linear system that can inevitably exhibit unpredictable behavior. These control laws may be insufficient for nonlinear systems because they are not robust when the requirements on the dynamic characteristics of the system are strict [10]. Control laws that are insensitive to parameter variations, disturbances, and nonlinearities must be used. In this paper, we have tested the method of the first harmonic, used to analyses servo controls with a nonlinear element, and to predict certain non-linear behaviors. It mainly allows predicting the limit cycles, but also the jump phenomena, the harmonics as well as the responses of non-linear systems to sinusoidal inputs. We apply this method for the prediction of limit cycles and the determination of their amplitude and frequency. We take as an example a Boost converter controlled by current [4]. This system is chaotic when the duty cycle is more significant than 0.5: we then eliminate the chaos by applying the slippery mode command (for the ripple of the output voltage, for the current ripple of the inductance and switching frequency) when the output is periodic (duty cycle less than or equal to 0.5). In this article, we assess that established approach provides the best outcomes: it appears that the preference between the classical mode and the sliding mode depends heavily on the variance domain of the parameters E, R, and Iref.

Impact of Non-Linear Electronic Circuits and Switch of Chaotic Dynamics

Switch-mode power supply is an extremely non-linear system that can inevitably exhibit unpredictable behavior. These control laws may be insufficient for nonlinear systems because they are not robust when the requirements on the dynamic characteristics of the system are strict [10]. Control laws that are insensitive to parameter variations, disturbances, and nonlinearities must be used. In this paper, we have tested the method of the first harmonic, used to analyses servo controls with a nonlinear element, and to predict certain non-linear behaviors. It mainly allows predicting the limit cycles, but also the jump phenomena, the harmonics as well as the responses of non-linear systems to sinusoidal inputs. We apply this method for the prediction of limit cycles and the determination of their amplitude and frequency. We take as an example a Boost converter controlled by current [4]. This system is chaotic when the duty cycle is more significant than 0.5: we then eliminate the chaos by applying the slippery mode command (for the ripple of the output voltage, for the current ripple of the inductance and switching frequency) when the output is periodic (duty cycle less than or equal to 0.5). In this article, we assess that established approach provides the best outcomes: it appears that the preference between the classical mode and the sliding mode depends heavily on the variance domain of the parameters E, R, and Iref.

Robustness of event-based discrete PI controllers. A sampled describing function approach

In this paper we present a study about the robustness of event-based discrete PI controllers. Our approach is based on Sampled Describing Function technique, which has been used to characterize the non-linear effect of the Symmetric Send on Delta (SSOD) sampling strategy on the control loop. Through several examples this technique is proven to produce accurate results in predicting limit cycle oscillations and to provide guidelines to avoid them, either by detuning the controller or increasing the sampling period.

Input dependent Nyquist plot for limit cycle prediction and its suppression using fractional order controllers

In practical systems having separable hard nonlinearities, sustained oscillation is detected at the steady state response due to the presence of stable limit cycles. An optimization problem is proposed to tune the fractional order controller parameters for system with multiple-nonlinearity to suppress the limit cycle magnitude in addition to meet the desired closed loop specifications. To extend the applicability of the existing Nyquist plot for predicting this limit cycle, an input dependent Nyquist plot is proposed in this paper. A servo system with backlash and relay nonlinearities is considered as a case study for limit cycle prediction with obtained fractional order controllers using an input dependent Nyquist plot. The predicted limit cycle information is compared with optimization results, input dependent root locus and are validated through closed loop simulations. Further, the robustness of the designed controllers are tested under system parameter uncertainty, disturbance and measurement noise conditions.

Refined analytical approximations to limit cycles for non-linear multi-degree-of-freedom systems

This paper presents analytical higher order approximations to limit cycles of an autonomous multi-degree-of-freedom system based on an integro-differential equation method for obtaining periodic solutions to nonlinear differential equations. The stability of the limit cycles obtained was then investigated using a method for carrying out Floquet analysis based on developments to extensions of the method for solving Hill’s Determinant arising in analysing the Mathieu equation, which have previously been reported in the literature. The results of the Floquet analysis, together with the limit cycle predictions, have then been used to provide some estimates of points on the boundary of the domain of attraction of stable equilibrium points arising from a sub-critical Hopf bifurcation. This was achieved by producing a local approximation to the stable manifold of the unstable limit cycle that occurs.The integro-differential equation to be solved for the limit cycles involves no approximations. These only arise through the iterative approach adopted for its solution in which the first approximation is that which would be obtained from the harmonic balance method using only fundamental frequency terms. The higher order approximations are shown to give significantly improved predictions for the limit cycles for the cases considered. The Floquet analysis based approach to predicting the boundary of domains of attraction met with some success for conditions just following a sub-critical Hopf bifurcation.Although this study has focussed on cubic non-linearities, the method presented here could equally be used to refine limit cycle predictions for other non-linearity types.

Inclusion of higher harmonics in the flame describing function for predicting limit cycles of self-excited combustion instabilities

The Flame Describing Function (FDF) is widely used to model non-linear thermo-acoustic phenomena, e.g. limit cycle oscillations in a combustor. The FDF is a weakly non-linear approach, because it accounts for the amplitude dependence of the flame response, but besides that relies on quasi-linear assumptions. In particular, it neglects the excitation of higher harmonics – a typical non-linear feature, which may play a major role in certain thermo-acoustic systems. Consequently, the FDF may provide inaccurate or incomplete results in such cases.In this study, we propose an efficient way to include higher harmonics of the flame response by an extended FDF, which includes additional transfer functions that relate higher harmonics of the heat release rate to the forcing velocity. The extended FDF is also a weakly non-linear approach, and requires the same effort for determination as the standard FDF. This paper shows how to determine the extended FDF and how to employ it for the prediction of limit cycle oscillations. The proposed concept is applied to predict and analyse the limit cycle of a laminar premixed burner for which the standard FDF delivered inaccurate results. Results obtained with the extended FDF show good agreement with fully compressible numerical simulation of the same configuration. Thus, the extended FDF proves its ability to provide accurate predictions in situations where higher harmonics play an important role in thermo-acoustic limit cycles.

Quantum correlations and limit cycles in the driven-dissipative Heisenberg lattice

Driven-dissipative quantum many-body systems have attracted increasing interest in recent years as they lead to novel classes of quantum many-body phenomena. In particular, mean-field calculations predict limit cycle phases, slow oscillations instead of stationary states, in the long-time limit for a number of driven-dissipative quantum many-body systems. Using a cluster mean-field and a self-consistent Mori projector approach, we explore the persistence of such limit cycles as short range quantum correlations are taken into account in a driven-dissipative Heisenberg model.

Nonlinear systems - describing functions analysis and using

The paper presents a basic description and examples of the use of so called descriptive functions, allowing analysing the influence of inherent and indispensable components of all mechatronic systems mechanical subsystems - so called hard nonlinearities. These parts “causing” - in addition to the centrifugal and Coriolis generalized forces- the nonlinearity of the system, can be analysed by the abovementioned method from the point of view of their origin and the estimation of the basic parameters of their frequent consequences - so-called limit cycles. After a short introduction, which introduces and explains describing functions using the example of a nonlinear system taken from the literature, some of the socalled hard nonlinear subsystems (such as the mechanical chain of robots) are shown to be used. The paper is the first part of a more extensive description analysis of nonlinear systems concept using these functions in order to enable analysis and prediction of limit cycles.

Analysis and Suppression of Limit Cycle Oscillation for Transmission System With Backlash Nonlinearity

This paper mainly deals with the prediction and suppression of limit cycle oscillation existed in the two-mass system with nonlinear backlash. To improve position control precision of the drive system, the full-closed-loop feedback scheme is considered. Then, a comprehensive and definite prediction of limit cycle is given via the describing function method. In order to compensate backlash nonlinearity, the state feedback control method is employed. To overcome the insufficient application of the state feedback control in the complex nonlinear system, a novel design concept is proposed. The influence of backlash on the nonlinear system is substituted with an additional compensation component to the control signal, making the system approach a linear one. Following this, a pole placement scheme is designed, which aims to establish a feedback structure that can make the system equivalent to a rigid system. In such a case, the limit cycle oscillation can be eliminated; thus, the position control precision can also be enhanced. In all cases, the validity of the proposal is verified by experimental results.