Amplitude Of Limit Cycle Research Articles

Discrete Adjoint Approach for Nonlinear Unsteady Aeroelastic Design Optimization

Presented is a discrete adjoint approach for computing nonlinear unsteady aeroelastic geometric design sensitivities, which can subsequently be used for nonlinear unsteady aeroelastic geometric design optimization. The methodology is applicable to compressible Reynolds-averaged Navier–Stokes computational-fluid-dynamics solvers and is based on a harmonic balance nonlinear frequency-domain technique for modeling nonlinear aerodynamics in the frequency domain. Automatic differentiation is used to derive the computer code representing the adjoint gradient of the harmonic balance computational-fluid-dynamics solver. Discrete adjoint method airfoil geometric design nonlinear unsteady aerodynamic sensitivities are first computed for a benchmark NLR 7301 airfoil transonic aeroelastic configuration, and results are compared with finite difference computations to demonstrate the accuracy of the methodology. The airfoil surface is mathematically curve-fit using an airfoil parameterization technique, and the discrete adjoint methodology provides the gradient of the flutter onset or limit-cycle oscillation reduced velocity with respect to changes in each of the parameters of the airfoil curve-fit methodology for the cost of a single adjoint solution. This gradient information is supplied to a limited memory Broyden–Fletcher–Goldfarb–Shanno (BFGS) optimization solver, which then determines values for the curve-fit parameters that maximize flutter onset or limit-cycle oscillation reduced velocity. The discrete adjoint nonlinear unsteady aeroelastic geometric design sensitivity computation methodology is then used to redesign the shape of the benchmark NLR 7301 airfoil aeroelastic configuration to optimize the aeroelastic flutter onset reduced velocity or the reduced velocity for a specified limit-cycle oscillation response amplitude.

A nonlinear model for a hanging tubular cantilever simultaneously subjected to internal and confined external axial flows

A full nonlinear model is developed for the dynamics of a hanging tubular cantilever that is simultaneously subjected to internal and external axial flows. These two flows are dependent on each other and are in opposite directions. Also, the external flow is confined over the whole length of the cantilever. A nonlinear equation of motion for the cantilever is obtained via Hamilton's principle to third-order accuracy. The virtual work due to the fluid-related forces acting on the cantilever is derived, to the same accuracy, considering the non-conservative forces associated with the internal flow as well as the inviscid, viscous and hydrostatic forces related to the external one. In addition, a vortex-lift mechanism due to the external flow is considered, and the associated steady hydrodynamic forces are derived and added to the expression of the virtual work. The equation of motion is then discretized and solved using the pseudo-arclength continuation method and a direct time-integration technique. The dynamical behaviour obtained is compared to the one predicted by another linear theoretical model, from the literature, for the same system parameters. The two models are in good qualitative and quantitative agreement with each other in terms of the type of instability, namely flutter, that occurs with increasing flow velocity and its onset. However, the proposed model can also predict quantitative facets of the dynamical behaviour beyond the onset of instability, such as limit-cycle amplitude and frequency. Moreover, the influences of various system parameters are investigated theoretically, namely the degree of confinement of the external flow, gravity, mass ratio, drag coefficient, and the thickness of the tubular cantilever.

Medium amplitude limit cycles in second order perturbed polynomial Liénard systems

We consider polynomial generalized Liénard systems that come from second order polynomial perturbations of a linear center. For these systems we found a generic (open and dense) subset in the space of perturbations of degree 2 l , with l ≥ 1 , such that each associated perturbed generalized Liénard system has at most 2 l − 1 medium limit cycles. Moreover, we show that this upper bound is reached. In addition, some concrete examples are given.

Nonlinear Torsional Vibration Analysis and Nonlinear Feedback Control of Complex Permanent Magnet Semidirect Drive Cutting System in Coal Cutters

The semidirect drive cutting transmission system of coal cutters is prone to unstable torsional vibration when the resistance values of its driving permanent magnet synchronous motor (PMSM) are affected by changes in temperatures and tough conditions. Besides, the system has the properties of complex electromechanically coupling such as the coupling between electrical parameters and mechanical parameters. Therefore, in this study, the nonlinear torsional vibration equation was established on the basis of the Lagrange‐Maxwell theory. Moreover, in light of the nonlinear dynamic bifurcation theory, the system stability was analyzed by taking the resistance value of power motor as the bifurcation parameter. In addition, the influence of subcritical bifurcation on the torsional vibration was studied by investigating the necessary and sufficient conditions for dynamic Hopf bifurcation and classifying the bifurcation types. At last, in order to suppress destabilizing oscillation induced by Hopf bifurcation, the nonlinear feedback controller was constructed, with the introduction of feedback from the motor velocity as well as the selection of voltage value on the q shaft as the controlled variable. Meanwhile, the three‐order normal form and controlling parameters of the system were obtained with the aid of the multiple scales method and the harmonic balance method. In this way, the Hopf bifurcation point was transferred to control the stability of Hopf bifurcation and the amplitude of limit cycle, thus guaranteeing reliable and safe operation of the system. The numerical simulation results indicate that the designed controller boosts an ideal controlling effect.

Nonlinear Torsional Vibration Analysis and Control of Semidirect Electromechanical Coupling Transmission System in Shearer

The nonlinear torsional vibration and instability oscillation caused by nonlinear damping in the shearer electromechanical coupling cutting transmission system in shearer driven by the permanent magnet synchronous motor (PMSM) is investigated in this paper. The electromechanical coupling transmission system in the shearer is equivalent to a concentrated mass model for the purpose of establishing the system dynamic model by the Lagrange–Maxwell equation. Then, the Routh–Hurwitz criterion is used to determine the torsional vibration critical point and stability region for the Hopf bifurcation for the cutting transmission system. According to the Routh–Hurwitz stability criterion, the Hopf bifurcation type and the effect of the supercritical Hopf bifurcation in the torsional vibration of the cutting transmission system are analyzed. Furthermore, based on the washout filter, the Hopf bifurcation controller is designed for suppressing the transmission system’s large vibration amplitude and unstable oscillation. In addition, the influences of the linear gain and nonlinear gain on the bifurcation point and the limit cycle amplitude are discussed. Finally, the numerical simulation results indicate the effectiveness of the designed controller. The research achievements can provide a theoretical basis for design or optimize the cutting transmission system of high‐reliability shearer driven by PMSM.

Saddle-node of limit cycles in planar piecewise linear systems and applications

In this article, we prove the existence of a saddle-node bifurcation of limit cycles in continuous piecewise linear systems with three zones. The bifurcation arises from the perturbation of a non-generic situation, where there exists a linear center in the middle zone. We obtain an approximation of the relation between the parameters of the system, such that the saddle-node bifurcation takes place, as well as of the period and amplitude of the non-hyperbolic limit cycle that bifurcates. We consider two applications, first a piecewise linear version of the FitzHugh-Nagumo neuron model of spike generation and second an electronic circuit, the memristor oscillator.

Stability and Limit Cycles of a Nonlinear Damper Acting on a Linearly Unstable Thermoacoustic Mode

The resonant coupling between flames and acoustics is a growing issue for gas turbine manufacturers, which can be reduced by adding acoustic dampers on the combustion chamber walls. Nonetheless, if the engine is operated out of the stable window, the damper is exposed to high-amplitude acoustic levels, which trigger unwanted nonlinear effects. This work provides an overview of the dynamics of this coupled system using a simple analytical model, where a perfectly tuned damper is coupled to the combustion chamber. The damper, crossed by a purge flow in order to prevent hot gas ingestion, is modeled as a nonlinearly damped harmonic oscillator. The combustion chamber featuring a linearly unstable thermoacoustic mode is modeled as a Van der Pol oscillator. Analyzing the averaged amplitude equations gives the limit cycle amplitudes as function of the growth rate of the unstable mode and the mean velocity through the damper neck. Experiments are also performed on a simple rectangular cavity, where the thermoacoustic instability is mimicked by an electro-acoustic instability. A feedback loop is built, through which the growth rate of the instability can be controlled. A Helmholtz damper is added to the cavity and tuned to the mode of interest. The stabilization capabilities of the damper and the amplitude of the limit cycle in the unstable cases are in good agreement between the experiments and the analytical and numerical predictions, underlining the potentially dangerous behavior of the system, which should be taken into account for real engine cases.

Limit Cycles and Invariant Curves in a Class of Switching Systems with Degree Four

In this paper, a class of switching systems which have an invariant conicx2+cy2=1,c∈R, is investigated. Half attracting invariant conicx2+cy2=1,c∈R, is found in switching systems. The coexistence of small-amplitude limit cycles, large amplitude limit cycles, and invariant algebraic curves under perturbations of the coefficients of the systems is proved.

Closed-form estimations of the bistable region in metal cutting via the method of averaging

Machine tool vibrations in turning processes are analyzed by taking into account the nonlinearity of the cutting force characteristics. Unstable limit cycles are computed for the governing nonlinear delay-differential equation in order to determine the bistable technological parameter region where stable stationary cutting and large-amplitude machine tool vibrations coexist. Simple closed-form formulas are derived for the amplitude of limit cycles and for the size of the bistable region considering a general cutting force characteristics. The analytical results are determined by the method of averaging, which can be used to treat the nonlinearities without their third-order approximation. The results are confirmed by numerical continuation and using Melnikov’s integral.

An experimental study into the effect of injector pressure loss on self-sustained combustion instabilities in a swirled spray burner

Combustion instabilities depend on a variety of parameters and operating conditions. It is known, especially in the field of liquid rocket propulsion, that the pressure loss of an injector has an effect on its dynamics and on the coupling between the combustion chamber and the fuel manifold. However, its influence is not well documented in the technical literature dealing with gas turbine combustion dynamics. Effects of changes in this key design parameter are investigated in the present article by testing different swirlers at constant thermal power on a broad range of injection velocities in a well controlled laboratory scale single injector swirled combustor using liquid fuel. The objective is to study the impact of injection pressure losses on the occurrence and level of combustion instabilities by making use of a set of injectors having nearly the same outlet velocity profiles, the same swirl number and that establish flames that are essentially identical in shape. It is found that combustion oscillations appear on a wider range of operating conditions for injectors with the highest pressure loss, but that the pressure fluctuations caused by thermoacoustic oscillations are greatest when the injector head loss is low. Four types of instabilities coupled by two modes may be distinguished: the first group features a lower frequency, arises when the injector pressure loss is low and corresponds to a weakly coupled chamber-plenum mode. The second group appears in the form of a constant amplitude limit cycle, or as bursts at a slightly higher frequency and is coupled by a chamber mode. Spontaneous switching between these two types of instabilities is also observed in a narrow domain.

Prediction of limit cycle amplitudes in thermoacoustic engines by means of impedance measurements

This paper deals with the prediction of the frequency and the amplitude of self-sustained oscillations generated in thermoacoustic prime movers, which are compared to measurements. A specially designed, high amplitude, acoustic impedance sensor was developed to perform measurements of the input impedance of a thermoacoustic core, as a function of the heating power supplied to the device, of the frequency, and of the amplitude of acoustic forcing. Those measurements can then be used to predict the spontaneous generation of acoustic oscillations and their saturation up to a steady-state. Those predictions were successful for various acoustic loads connected to the thermoacoustic core. Moreover, the measurements of acoustic impedance as a function of the amplitude of acoustic oscillations are compared to a model based on the linear thermoacoustic theory, and this comparison provides insights into the processes controlling the saturation of acoustic oscillations. The experimental procedure described in this paper can also have practical value, since it provides an empirical way, in principle, to optimize the coupling between the thermoacoustic core and the load so that the potential efficiency of thermoacoustic energy conversion is maximized.

Limit cycles of a Liénard system with symmetry allowing for discontinuity

This paper presents new results on the limit cycles of a Liénard system with symmetry allowing for discontinuity. Our results generalize and improve the results in [32, Theorem 1 and 2] or the monograph [33, Chapter 4, Theorem 5.2]. The results in [33] are only valid for the smooth system. We emphasize that our main results are valid for discontinuous systems. Moreover, we show the presence and an explicit upper bound for the amplitude of the two limit cycles, and we estimate the position of the double-limit-cycle bifurcation surface in the parameter space. Until now, there is no result to determine the amplitude of the two limit cycles. The existing results on the amplitude of limit cycles guarantee that the Liénard system has a unique limit cycle. Finally, some applications and examples are provided to show the effectiveness of our results. We revisit a co-dimension-3 Liénard oscillator (see [22,34]) in Application 1. Li and Rousseau [22] studied the limit cycles of such a system when the parameters are small. However, for the general case of the parameters (in particular, the parameters are large), the upper bound of the limit cycles remains open. We completely provide the bifurcation diagram for the one-equilibrium case. Moreover, we determine the amplitude of the two limit cycles and estimate the position of the double-limit-cycle bifurcation surface for the one-equilibrium case. Application 2 is presented to study the limit cycles of a class of the Filippov system.

Bifurcation diagram of coupled thermoacoustic chaotic oscillators.

A thermoacoustic chaotic oscillator is a fluid system that presents thermally induced chaotic oscillations of a gas column. This study experimentally reports a bifurcation diagram when two thermoacoustic chaotic oscillators are dissipatively coupled to each other. The two-parameter bifurcation diagram is constructed by varying the frequency mismatch and the coupling strength. Complete chaos synchronization is observed in the region with a frequency mismatch of less than 1% of the uncoupled oscillator. In other regions, synchronization between quasiperiodic oscillations and that between limit-cycle oscillations and amplitude death are observed as well as asynchronous states.

Flame macrostructures and thermoacoustic instabilities in stratified swirling flames

The present article investigates the correlation between flame macrostructures and thermoacoustic combustion instabilities in stratified swirling flames. Experiments are carried out in a laboratory scale longitudinal test rig equipped with the Beihang Axial Swirler Independently-Stratified (BASIS) burner, a novel double-swirled combustion system developed by adapting an industrial lean premixed prevaporized (LPP) combustor. At first, the flame macrostructures are investigated and discussed for various total equivalence ratios (ϕtotal) and stratification ratios (SRs). Depending on operating conditions, three different flame types are stabilized in the combustor: two attached flames comprising a stratified flame and a V-shaped flame (V-flame), as well as a lifted flame. Thermoacoustic instabilities are then investigated. The amplitude of the oscillations is found to be more sensitive to SR than the ϕtotal. Large amplitude limit cycles are found for low and high values of SR, for which the V-flame and the lifted flame are observed in the combustor, respectively. The flame dynamics are also investigated using local Rayleigh index maps. It is found that for both the lifted flame and V-flame, the major driving force comes from the flame-to-wall impingement region. Coherent structures associated with flame wrinkling are found along the flame brushes of the V-flame. On the contrary, the stratified flame is found to be more thermo-acoustically stable. Finally, incompressible Large Eddy Simulations is used to obtain the flame responses to forcing at 300 Hz, which is very close to the frequencies at which limit cycle oscillations occur. The results show that the global heat release rate response of the stratified flame exhibits a significant phase shift compared to the responses of the other two flame types, and this is the most likely cause of thermoacoustic stabilization.

On the design of an electromagnetic aeroelastic energy harvester from nonlinear flutter

A new energy harvester by coupling the electromagnetic induction and the pitch vibration of a rigid wing is built up in this paper. It is aimed: (1) to harvest energy from the pitch limit cycle oscillation (LCO) of the wing due to the preloaded free-play nonlinearity; (2) to introduce a theoretical analysis scheme based on the equivalent linearized method into the design of this harvester. With the equivalent linearized method, the domains of the single stable LCO and double stable LCOs are respectively obtained. Combining the analytical and numerical solutions, the single stable LCO along with the stable limit cycle amplitude greater than its corresponding unstable one is recognized as the better mode for harvesting, since the larger limit cycle domain is induced and the more energy are yielded. Based on such chosen mode, analyses of varying parameters are conducted with respect to the plunge stiffness, pitch stiffness, distance of elastic axis from center of gravity, distance of geometric center from elastic axis, load resistance and magnetic flux density. Meanwhile, three indicators are applied to reveal their effects on the harvesting performances: (1) the size of limit cycle domain, (2) the onset velocity of LCO, and (3) the energy output.

Experimental Investigation of Self-Excited Combustion Instabilities in a Lean, Premixed, Gas Turbine Combustor at High Pressure

Self-excited combustion instabilities in a high pressure, single-element, lean, premixed, natural gas (NG) dump-combustor are investigated. The combustor is designed for optical access and instrumented with high frequency pressure transducers at multiple axial locations. A parametric survey of operating conditions including inlet air temperature and equivalence ratio has been performed, resulting in a wide range of pressure fluctuation amplitudes (p′) of the mean chamber pressure (pCH). Two representative cases, flames A and B with p′/pCH=23% and p′/pCH=12%, respectively, both presenting self-excited instabilities at the fundamental longitudinal (1L) mode of the combustion chamber, are discussed to study the coupling mechanism between flame-vortex interactions and the acoustic field in the chamber. 10 kHz OH*-chemiluminescence imaging was performed to obtain a map of the global heat release distribution. Phase conditioned and Rayleigh index analysis as well as dynamic mode decomposition (DMD) is performed to highlight the contrasting mechanisms that lead to the two distinct instability regimes. Flame interactions with shear layer vortex structures downstream of the backward-facing step of the combustion chamber are found to augment the instability magnitude. Flame A engages strongly in this coupling, whereas flame B is less affected and establishes a lower amplitude limit cycle.

Non-linear dynamics of thermoacoustic eigen-mode interactions

The natural eigen-modes of a combustion system are a strong function of its geometric design, temperature profile, and boundary conditions. Many practical systems have multiple, closely-spaced natural mode frequencies. While several prior studies have noted the nonlinear interactions that occur when multiple, linearly unstable modes are present, the nature of these interactions for closely-spaced modes has not been analytically treated. This paper treats this problem theoretically by utilizing a Galerkin expansion of the Euler equations, and the method of averaging to derive a set of amplitude equations for the modal amplitudes. In cases where the frequency spacing is large, many of the oscillatory terms have short time-scales and average out. However, in the case of closely-spaced modes, terms oscillating at the frequency difference between the closely-spaced frequencies correspond to long time-scales and, thus, remain after the averaging process. Results show that frequency spacing between the modes has significant impacts on limit-cycle amplitudes and their stability, even in the case of a static non-linearity. In addition, the conditions under which both modes can co-exist are a strong function of the frequency spacing. There are also certain conditions where one mode is completely suppressed by the other, even if both modes have positive linear growth rates; non-intuitively, the mode that is suppressed could be the one with the larger growth rate. The conditions under which one mode is suppressed are also a strong function of the frequency spacing parameter. Finally, for conditions in which one mode is suppressed, the limit cycle amplitude of the other mode is independent of the frequency spacing. Taken together these results show the difficulty in experimentally inferring the linear stability/instability of (non-dominant) combustor modes based upon “steady state” data taken during limit cycle conditions.

Dynamics and nonlinear feedback control for torsional vibration bifurcation in main transmission system of scraper conveyor direct-driven by high-power PMSM

The main transmission system of a scraper conveyor direct-driven by the high-power permanent magnet synchronous motor (PMSM) is taken as a study object. With the effect of the nonlinear friction torque caused by the nonuniformity of the transported coal quality in the operation process considered, the torsional vibration bifurcation mechanism and the corresponding control measures for the main transmission system of the scraper conveyor are investigated. Firstly, based on the Lagrange–Maxwell principle, the global electromechanical-coupling dynamic models for the main transmission system of the scraper conveyor are constructed. Secondly, by the Routh–Hurwitz stability criterion, the Hopf bifurcation characteristics of the main transmission system are analyzed to reveal the influence of supercritical bifurcation and subcritical bifurcation on the torsional oscillation of the transmission shafting. Thirdly, in order to suppress the system unstable oscillation caused by the Hopf bifurcation, the motor speed is fed back to construct the nonlinear state feedback controller for the quadrature axis current of the PMSM by the $$I_{d}=0$$ vector control strategy. Similarly, on the basis of the Routh–Hurwitz criterion, the influence of the linear feedback coefficient in the nonlinear state feedback controller on the system bifurcation position is discussed. Meanwhile, by the central manifold theory and canonical form theory, the effect of the square and cubic nonlinear feedback coefficients on the Hopf bifurcation type of the torsional vibration and the amplitude of the stable limit cycle are investigated. Finally, the numerical simulation results show the effectiveness of the designed controller.

Forecasting critical points and post-critical limit cycles in nonlinear oscillatory systems using pre-critical transient responses

A unique method for forecasting Hopf bifurcations and the complete post-bifurcation dynamics of nonlinear oscillatory systems is proposed. The method is model-less and uses measurements of the system only in the pre-bifurcation regime. The forecasts are three-dimensional bifurcation diagrams using a few measurements of the system response to perturbations in the pre-bifurcation regime. To demonstrate the forecasting method and its advantages, it is employed to predict bifurcation diagrams of a 3-DOF fluid-structural system. Surrogate measurement data is obtained from simulations and used as input for the forecasting algorithm. Results show that the bifurcation diagrams are forecasted accurately in both supercritical and subcritical bifurcations. The method accurately forecasts the bifurcation point, the bifurcation type and the post-bifurcation limit cycle amplitudes despite the fact that only pre-bifurcation regime data are used for forecasting.

On the effects of structural coupling on the supersonic flutter and limit cycle oscillations of transversely reinforced panels

Panel flutter is an aeroelastic phenomenon that can critically affect aircraft skin in supersonic flight. The majority of works published on this subject treat each skin panel as an isolated structural element. In reality, however, aircraft skin is usually built as large panels stiffened by stringers and frames. Thus, the multiple subpanels sitting between stiffeners are structurally coupled rather than isolated, and therefore can interact during flutter. In the present work, a reinforced panel is simulated as a rectangular plate reinforced by an elastic beam that “subdivides” the panel into two square subpanels (bays). The panel is modeled as a Mindlin–vonKarman plate, and the stiffener as a nonlinear eccentric Timoshenko beam. The problem is discretized through the Finite Element Method, and the resulting nonlinear equations of motion are solved via numerical time-marching. This approach is a higher-fidelity extension to the methodology employed by the authors in previous works, where the stiffeners were idealized as immovable simple supports – the so-called multibay model. The present results are compared to those from both the multibay model and the single-panel model. Each bay in the stiffened panel is identical to the reference single panel. Results are produced for several stiffener cross-sectional aspect ratios, r. Linear flutter boundary results show that, as r increases, the critical dynamic pressure asymptotically approaches that of a single panel. Nonlinear post-flutter analyses reveal the occurrence of jump discontinuities in the limit cycle amplitude diagrams. Similarly to what has been seen in the literature for multibay panels, the jumps are related to the nonlinear structural coupling between neighboring bays, and occur when there is a shift in the flutter mechanism. Furthermore, it is shown that reducing the cross-sectional aspect ratio postpones not only the onset of flutter but also the jumps, which can be avoided for sufficiently small r. Important engineering design guidelines can be established from the present study, as the multibay and single-panel models are shown to be generally conservative regarding linear behavior, but potentially unsafe in the post-flutter regime.