Amplitude Of Limit Cycle Oscillations Research Articles

Validation of a Methodology to Assess The Flutter Limit Cycle Oscillation Amplitude of Low-Pressure Turbine Bladed-Disks - Part I: Mach Number Effects

Abstract This paper presents a methodology to estimate the vibration amplitude of fluttering low-pressure turbine blades saturated due to friction effects. The study utilizes an analytical model that balances aerodynamic work and dry-friction work. The analytical predictions are compared against experimental results to validate the model. The first part of this paper focuses on the influence of the Mach number on the work balance between aerodynamic and mechanical components. It is observed that the vibration amplitude of low-pressure turbine rotor blades notably increases with higher Mach numbers. In addition, numerical simulations are employed to assess the influence of the Mach number on the critical damping ratio. The results demonstrate that an appropriate scaling of the critical damping ratio with the exit Mach number collapses all the damping versus inter-blade phase angle curves into a single curve. This finding validates the scaling of the aerodynamic damping for different pressure ratios. Unsteady pressure measurements were acquired, carefully post-processed to extract their flutter-induced peak components, and presented in a nodal diameter by nodal diameter basis. The post-processed data were then used to characterize the vibration amplitude observed in the experiments. The trends of the measured unsteady pressure on the casing of a rotating rig and the proposed model with the Mach number for different shaft speeds are in good agreement. The vibration amplitude and the mean unsteady pressure increase with the Mach number and exhibit a maximum with the shaft speed.

High-dimensional nonlinear flutter suppression of variable thickness porous sandwich conical shells based on nonlinear energy sink

Nonlinear energy sink (NES) is widely applied in engineering field due to the advantages of light weight, high robustness, unidirectional energy transfer, rapid and broadband vibration isolation. In this paper, nonlinear energy sink is utilized to suppress vibration of the variable thickness porous sandwich conical shells for the first time, and the high-dimensional nonlinear flutter suppression characteristics of the system of simply supported variable stiffness truncated porous sandwich conical shell coupled NES under aerodynamic force and thermal stress are investigated. By applying the first-order shear deformation theory (FSDT), Hamilton's principle and Galerkin technique, the high-dimensional nonlinear ordinary differential flutter suppression equations of the system appended with NES are established. The accuracy of the theoretical approach is ensured by the comparison of frequency results, while the NES dissipated kinetic energy ratio and the comparison of NES performance with other suppression systems are presented to prove the effectiveness of NES on nonlinear flutter suppression. The time history diagrams and limit cycle oscillation (LCO) amplitude curves, which reflect the high-dimensional nonlinear flutter suppression effect of NES, are obtained by employing the Runge-Kutta method. The effects of aerodynamic pressure, the parameters and positions of single NES, and the positions of parallel NES and series NES on the high-dimensional nonlinear flutter suppression characteristics of the system attached with NES are discussed in depth. Finally, the optimal high-dimensional nonlinear flutter suppression scheme is arrived.

Dynamically Linearized Time-Domain Approach for Limit-Cycle Oscillation Prediction for an Elastic Panel

A nonlinear aeroelastic model for a flexible panel is extended to consider the Euler equation as the aerodynamic model. A highly efficient method is presented to compute the generalized aerodynamic forces from a CFD simulation, and the results are compared to those results previously obtained using full potential flow aerodynamics and the linear piston theory. The flutter onset condition as well as limit cycle oscillation amplitudes are computed for a range of supersonic Mach numbers. Additionally, the effects of a shock wave and a viscous boundary layer (Reynolds-averaged Navier–Stokes) are considered in the computational fluid dynamics simulation and are implemented in the aeroelastic model via the generalized aerodynamic force obtained by the new approach. Conclusions are made based on the use and application of this more complete aerodynamic theory, including cases with shock impingement and near-transonic Mach numbers.

Numerical Analysis of Free Play-Induced Aeroelastic Phenomena: A Numerical Approach With Adaptive Step Size Control

This study presents a detailed numerical analysis of nonlinear aeroelastic behavior in a two degree of freedom (DOF) model, focusing on plunge and pitch motions and employing the continuation method (CM) with an adaptive step size control algorithm. The research incorporates free-play nonlinearity at the plunge hinge, a common structural nonlinearity in aeronautics that can induce detrimental limit cycle oscillations (LCOs) during flight. By examining three scenarios—linear response, unhindered plunge motion, and nonlinear stiffness behavior—the study assesses the effects of free play on flutter and LCO phenomena, including discontinuity-induced bifurcations like grazing bifurcation. Additionally, the study explores parameter variation for nonlinear flutter analysis, revealing the dynamics of grazing bifurcation and its impact on LCO behavior. The research also demonstrates the method’s superior accuracy in flutter speed estimation and mode-switching identification, despite higher computational demands. The findings underscore the diminishing influence of nonlinear free-play behavior on LCO amplitude, providing insights with significant implications for aeroelastic design and aircraft safety.

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.

Fluid–structure–surface interaction of a flexibly mounted pitching and plunging flat plate in proximity to the free surface

This experimental study investigates the fluid–structure–surface interactions of a flexibly mounted rigid plate in axial flow, focusing on flow-induced vibration (FIV) response and vortex dynamics of the system within a reduced velocity range of $U^*=0.29\unicode{x2013}8.73$ , corresponding to a Reynolds number range of $Re=518\unicode{x2013}15\,331$ . The plate, with one and two degrees of freedom (DoFs) for pitching and plunging oscillations, is examined at various submerged heights near the free surface. Results show that the plate exhibits divergence instability at low reduced velocities in both 1DoF and 2DoF systems. As the flow velocity surpasses a critical reduced velocity, periodic limit-cycle oscillations (LCOs) occur, increasing in amplitude until a second critical reduced velocity is reached. Beyond this point, LCOs are suppressed, and the plate experiences an increased static divergence angle with further flow velocity increase. The proximity to the free surface significantly influences the FIV response, with decreasing submerged heights leading to reduced LCO amplitudes and a shift of instabilities to higher reduced velocities. Vortex dynamics are analysed using time-resolved volumetric particle tracking velocimetry and hydrogen bubble flow visualisation. The analysis reveals disruptions in the symmetric flow field near the free surface, causing elongation and fragmentation of vortices in the wake of the plate, as well as vortex coupling. Proper orthogonal decomposition (POD) identifies dominant coherent structures, including leading-edge and trailing-edge vortices, captured in the first and second paired modes. On the other hand, higher POD modes capture the interaction of vortices in the wake and near the free surface.

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.

Fundamental investigation into output-based prediction of whirl flutter bifurcations

This paper investigates an approach for predicting whirl flutter bifurcations using pre-flutter output data. The approach leverages the critical slowing down phenomenon, which makes aeroelastic systems recover to equilibrium from disturbances more slowly when closer to the flutter boundary. By quantifying this phenomenon based on output data from pre-flutter transient responses, one can predict the flutter onset and the amplitude of limit-cycle oscillations (bifurcation diagram). The approach is demonstrated for a propeller-nacelle system using output data from transient simulations. Bifurcation diagrams are predicted within a few percentages of reference results using output data at two forward speeds, as low as 25% to 30% below the whirl flutter speed. Extensive sensitivity analyses highlight the accuracy and robustness of predictions for various output data choices and levels of nonlinearity, providing insights for future applications to higher-complexity systems. This work opens a new path to predicting whirl flutter bifurcations during the design of aeroelastic systems prone to this instability.

Application of heat transfer at the combustor wall to attenuate self-excited thermoacoustic oscillations

Self-sustained thermoacoustic oscillations (also known as combustion instability) arise in combustors when pressure and heat release fluctuations couple to produce high-energy sound waves. The need for diverse strategies to control these oscillations will increase as future design moves towards highly efficient, lean combustors. In this work, the impact of heat transfer at the combustor wall on the generation of thermoacoustic oscillations is examined through computational fluid dynamics. Characteristics of the thermoacoustic oscillations generated across cases with a range of thermal boundary conditions are compared to an adiabatic reference case. The amplitude of limit cycle oscillations in a Rijke tube is seen to decrease as wall temperatures increase. A stable combustor with a negative growth rate is observed when the full outer wall is heated to 900 K, and a sound pressure level reduction of 75 dB is achieved. The most efficient stable configuration is achieved when only the inlet duct wall is heated to 1000 K, requiring a minimum of 430 W of input power. The present work introduces an alternative stable combustor design and demonstrates that optimizing the temperature distribution of the combustor wall can effectively dampen thermoacoustic oscillations.

Flutter and Limit-Cycle Oscillations of a Panel Using Unsteady Potential Flow Aerodynamics

The potential aerodynamic theory is implemented to model the flow over a flexible panel. The present study compares the more complete unsteady version of the potential flow theory with its simplification known as the linear piston theory for different Mach numbers and the two- and three-dimensional aerodynamic models. The piston theory is “local” in space and time because it assumes the pressure at a spatial point and time only depends on the panel deformation at the same point and time. The full potential flow model includes the effect of the past history of the panel deformation and the spatial distribution of the panel deformation on the pressure at any instant in time and at any point in space. Aeroelastic analysis is made to trace the flutter onset critical condition based on the limit cycle oscillation amplitudes, and the results are compared with the more traditional implementation using piston theory. Conclusions are made based on the use and application of this more complete aerodynamic theory, particularly for near-transonic and hypersonic flow regimes. Subsonic results are also presented in this study.

Study of Nonlinear Aerodynamic Self-Excited Force in Flutter Bifurcation and Limit Cycle Oscillation of Long-Span Suspension Bridge

This article establishes a nonlinear flutter system for a long-span suspension bridge, aiming to analyze its supercritical flutter response under the influence of nonlinear aerodynamic self-excited force. By fitting the experimental discrete values of flutter derivatives using the least squares method, a polynomial function of flutter derivatives with respect to reduced wind speed is obtained. Flutter critical value is determined by the linear matrix eigenvalues of a state-space equation. The occurrence of a supercritical Hopf bifurcation in the nonlinear system is determined by the Jacobian matrix eigenvalues of the state-space equation and the system’s vibrational response at the critical state. The vibrational response of the supercritical state is obtained through Runge–Kutta integration, revealing the presence of stable limit cycle oscillation (LCO) and unstable limit cycle oscillation in the system, and through analyzing the relationship between the LCO amplitude and wind speed. Considering cubic nonlinear damping and stiffness, the effects of different factors on the nonlinear flutter system are analyzed.

Shimmy Suppression of an Aircraft Nose Landing Gear Using Torsional Nonlinear Energy Sink

Abstract Shimmy motion may occur for an aircraft nose landing gear (NLG) during its takeoff, landing, and taxiing procedure, which is an undesirable motion and should be suppressed. Here, a novel torsional nonlinear energy sink (NES) based on the cam-roller mechanism is proposed and applied in the NLG to improve its shimmy performance. The torsional NES is connected with the upper and lower struts of the NLG and its mechanical characteristic is studied. A seven-dimensional dynamic model of the NLG coupled with NES is built, which considers the NLG torsional and lateral motions, and the nonlinear tire model. The numerical continuation method is applied to analyze its shimmy performance, the stable area and limit cycle oscillation (LCO) amplitude are acquired, further compared with those of the original NLG without using the NES. The results show that as the NES is used in the NLG, the torsional shimmy dominated unstable area reduces significantly and the lateral shimmy dominated unstable area decreases slightly, which results in the increase of the stable area of the NLG; the maximum LCO amplitudes for the NLG torsional and lateral shimmy become smaller, the forward speed ranges as the torsional and lateral shimmy occur become narrower, which indicates that the stable forward speed ranges of the NLG become wider; the NLG shimmy performance improves as the NES has larger torsional inertia and damping, and smaller torsional linear stiffness. Thus, the NES can suppress the NLG shimmy motion and enhance its shimmy performance effectively.

Investigation of nonlinear and transitional characteristics of flutter varying with wind angles of attack for some typical sections with different side ratios

This study conducted a comprehensive investigation into the flutter responses of three closed sections, namely two streamlined box girders and a 5:1 rectangular cylinder, which possess different side ratios. Different types of flutter were measured during the tests. The nonlinear and transitional characteristics of the flutter were thoroughly discussed. The effects of the wind angle of attack and side ratio on the reduced critical wind speed, limit cycle oscillation amplitude and the transition behaviors of the flutter were investigated. The complex modal motion information of the three systems was discussed. The main factors contributing to the transition of flutter, and the dynamic mechanisms of the occurrence of different flutter manifestations, were investigated. This was achieved by examining the evolution of their modal damping ratios with the vibration amplitude and wind speed. Finally, an examination was conducted to evaluate the impact of the mechanical damping ratio on the transition behavior of flutter as well as its effectiveness in mitigating flutter responses for the considered sections. Results showed that the various manifestations of the flutter are mainly ascribed to the gradually evolved modal damping ratio, where aerodynamic damping plays the most important role in producing different types of limit cycle oscillations and transitional behaviors of flutter. Moreover, the modal damping ratio slowly transitions with the wind angle of attack, wind speed, or the side ratio of a section, instead of experiencing any mutation, despite the observation of some distinct types of flutter during the tests.

Geometrically nonlinear effects in wing aeroelastic dynamics at large deflections

This paper investigates geometrically nonlinear effects due to nonlinear kinematics and follower aerodynamics in the aeroelastic dynamics of a very flexible wing. The test case is the Pazy wing, a benchmark wing model for geometrically nonlinear aeroelastic studies involving large deflections in low-speed flow. The work builds on a low-order model of the wing composed of a geometrically nonlinear beam coupled with potential flow thin airfoil theory. Nonfollower aerodynamics underestimates static deflections by up to 10%, whereas linear kinematics overestimates them by up to 50%. At low static deflections, the wing hump mode flutter mechanism is driven by three-dimensional unsteady aerodynamic effects. As static deflections increase, the flutter onset and offset speeds are driven by curvature effects that can only be captured by nonlinear kinematics, which reduce the natural frequencies associated with torsion and in-plane bending motions. Follower aerodynamics shrinks the hump mode instability region and shifts it to 1–4% lower flow speeds with no change in the flutter frequency, the qualitative trends, and the amplitude of limit-cycle oscillations. Overall, nonlinear kinematics is the primary geometrical nonlinearity influencing the wing aeroelastic behavior: the modal and stability scenarios with and without follower aerodynamics collapse onto the same curves when presented as functions of the tip deflection. However, geometrical nonlinearities alone miss the subcritical nature of the hump mode flutter mechanism, which is attributed to aerodynamic stall and its interaction with large deflections. This work expands the growing literature on the aeroelastic dynamics of very flexible wings and will ease future studies considering aerodynamic nonlinear effects.

Experimental Aeroelastic Investigation of an All-Movable Horizontal Tail Model with Bending and Torsion Free-Plays

In this study, an experimental investigation is performed on a scaled, all-movable horizontal tail to study the aeroelastic behaviors induced by multiple free-plays. The dynamic response in wind tunnel tests is measured by strain gauges, an accelerometer, and a binocular vision measurement system. The obtained results indicate that the present aeroelastic system exhibits highly nonlinear characteristics and undergoes two independent limit cycle oscillations (LCOs) induced by bending free-play and torsion free-play, respectively. Further, various parametric studies are conducted to evaluate the effects of the free-play angles, angle of attack, flow velocity, and gust excitation on the LCOs. It is found that the value of free-play angle has no significant effect on the critical flow velocity which leads to the occurrence of LCOs. The amplitude and frequency of LCOs increase with the increasing free-play angle and flow velocity. Moreover, the horizontal tail experiences high-order harmonic resonances when LCOs appear. Finally, the stability of limit cycles is analyzed based on the gust excitation experiment. Overall, compared to an all-movable horizontal tail with single free-play, the multiple free-plays system exhibits more complex dynamic behaviors. In this paper, the measured results of the scaled model, which has a similar mass distribution and stiffness distribution as actual aircraft, may be valuable for predicting such LCOs induced by multiple free-plays, and providing a reference for the design of all-movable horizontal tail to prevent LCOs.

An Improved Mickens Iterative Method for Nonlinear Aeroelastic analysis of Airfoils with an External Store

This paper aims to propose a modified Mickens iteration method for analyzing the flutter stability of the nonlinear airfoil with an external storage system. Unlike the regular Mickens approach, the modified form is established by substituting previous frequencies of the limit cycle oscillation (LCO) into unknown ones. Then the algebraic equations governing the LCO amplitude and frequency could keep linear, which might improve the calculation efficiency. The numerical example shows the LCOs obtained easily and accurately. Furthermore, the phase planes of the LCOs are investigated with the wind speed varying. The approach could be applicable to investigate mechanical systems with complicated nonlinearities.

The energy extraction potential from pitch-heave coupled flutter

An experimental investigation of a pitch-heave fully-passive system undergoing large amplitude coupled flutter induced limit cycle oscillations (LCO) is conducted. The main objective of this paper is to report and document the large aerodynamic efficiencies that can be experimentally obtained from coupled flutter. This is in contrast to a number of investigations in the energy harvesting literature which i. report relatively very low efficiencies for coupled flutter, and ii. emphasise the case of stall flutter and the related phenomenon of dynamic stall. The elastic axis of the set-up is located at 27% chord. The main parameters varied are the airspeed (Reynolds number) and stiffness coefficient in heave, as expressed by the frequency ratio ranging from below to above one. For a frequency ratio ω¯=ωhωθ=1.22, amplitudes up to 80 deg in pitch and 0.6 heave-to-chord ratio are observed at airspeeds close to the flutter speeds. In conjunction with favourable phase differences, these large LCO amplitudes occurring at low speeds lead to aerodynamic efficiency values hovering around 40%. It is also observed that maximum efficiency calculated solely from the heave motion is 50%. This is a consequence of the negative work being done on the pitch by the aerodynamic moment, which can be related back to the inertial coupling and positive phase difference between pitch and heave. For the vast majority of our observations, the aerodynamic moment does negative work, whereas the lift does positive work.

Modulation and Annihilation of Aeroelastic Limit-Cycle Oscillations Using a Variable-Frequency Disturbance Generator

Nonlinear aeroelastic limit-cycle oscillations (LCOs) have become an area of interest due to both detrimental effects on flying vehicles and use in renewable energy harvesting. Initial studies on the interaction between aeroelastic systems and incoming flow disturbances have shown that disturbances can have significant effects on LCO amplitude, with some cases resulting in spontaneous annihilation of the LCO. This paper explores this interaction through wind-tunnel experiments using a variable-frequency disturbance generator to produce flow disturbances at frequencies near the inherent LCO frequency of an aeroelastic system with pitching and heaving degrees of freedom. The results show that incoming disturbances produced at frequencies approaching the LCO frequency from below produce a cyclic growth-decay in LCO amplitude that resembles interference between multiple sine waves with slightly varying frequencies. An aeroelastic inverse technique is applied to the results to study the transfer of energy between the pitching and heaving degrees of freedom as well as the aerodynamic power moving into and out of the system. Finally, the growth-decay cycles are shown to both excite LCOs in an initially stationary wing and annihilate preexisting LCOs in the same wing by appropriately timing the initiation and termination of disturbance generator motion.

On the Influence of Nonlinear Inertial Forces on the Limit Cycle Oscillations of an Inextensible Plate in a Supersonic Axial Flow

Abstract Recent results in the literature highlight the impact of nonlinear inertial forces on the post-flutter limit cycle oscillation (LCO) characteristics of highly deflected structures in supersonic axial flow. The current investigation examines how the ability to passively modulate nonlinear inertial forces may alter the overall aeroelastic response. The structural model is a one-dimensional nonlinear inextensible plate subject to nonlinear aerodynamic forces in accordance with a new, geometrically modified third-order Piston Theory. For the linear aeroelastic case, we find that nonhomogeneous mass distribution elicits discontinuous increases in the critical Mach number for flutter and several flutter mode-switching phenomena that are not observed when mass is added homogeneously. The existence of several different flutter mode mechanisms as a function of a concentrated mass location leads to different post-flutter LCO amplitude behavior. This is found to transition the underlying nonlinear structural dynamics to either be stiffening (when lower-order modes merge) or softening (when higher-order modes merge), which in turn alter the influence of nonlinear aerodynamic forces. We also address discrepancies in LCO amplitude trends due to the nonlinear inertial forces previously reported in the literature.

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.