Aerodynamic Damping Research Articles

Flutter Calculations Using the Unsteady Source and Doublet Panel Method

The source and doublet panel method (SDPM) developed by Morino in the 1970s can model unsteady compressible ideal flow around wings and bodies. In this work, the SDPM is adapted to the calculation of aeroelastic solutions for wings. A second-order nonlinear version of Bernoulli’s equation is transformed to the frequency domain and written in terms of the generalized mode shapes and displacements. It is shown that the pressure component at the oscillating frequency is a linear function of the generalized displacements, velocities, and accelerations and can therefore be used to formulate a linear flutter problem. The proposed approach has several advantages over the usual doublet lattice method (DLM) approach: the exact geometry is modeled (including thickness, camber, and twist effects), the motion of the surface can be represented using all six degrees of freedom, the pressure calculation is of higher order, and the aerodynamic mass, damping, and stiffness terms are calculated explicitly. The complete procedure is validated using experimental data from the weakened AGARD 445.6 wing, a NACA 0012 rectangular wing with pitch and plunge degrees of freedom, and an experimental model of a T-tail, yielding flutter predictions that lie closer to the experimental observations than those obtained from the DLM, particularly in the case of the T-tail.

Experimental Study on Identification of Aerodynamic Damping Matrix for the Tower of an Operating Wind Turbine by Artificial Excitation

ABSTRACTQuantitation of damping is of great significance for the design and condition assessment of wind turbines. The authors' previous theoretical and numerical studies showed that compared with damping ratios, a modal aerodynamic damping matrix can better describe the damping coupling in the fore‐aft (FA) and side‐side (SS) tower motions. In the present study, an improved damping identification method was first proposed to identify this damping matrix with artificial exciters and then verified by using OpenFAST simulations under different excitation frequencies, excitation force amplitudes, and turbulent wind fields. Following the numerical study, a scaled wind turbine model with a geometric scale ratio of 1/75 was carefully designed based on the NREL 5‐MW wind turbine prototype, in which the scaled blade design follows the rule in thrust coefficient similarity. An identification study was performed with this scaled model by a series of wind tunnel tests. The modal aerodynamic damping matrix was identified under steady‐state harmonic excitation in the operating state and compared with the identified results by a free decay method and the theoretical values. The results experimentally confirm the correctness of the aerodynamic damping matrix theory under uniform wind and the feasibility of the improved identification method in practice.

Coupling Vibration Characteristics and Wind-Induced Responses of Large-Span Transmission Lines Under Multi-Dimensional Wind

Transmission lines, crucial for power and urban infrastructure, are vulnerable to wind damage; this paper addresses research gaps in tower-line systems under multi-dimensional wind loads and aerodynamic damping. By incorporating multi-dimensional aerodynamic damping and conducting comprehensive multi-dimensional wind response analysis, it examines parameters like ground roughness and wind attack angles that significantly influence the tower responses, offering a holistic understanding of system behavior under real wind conditions. This study analyzes wind-induced responses of a large-span Chinese transmission line using a finite element model (FEM) with three spans and two towers. This paper conducts modal analyses of a single tower and the tower-line system, comparing their vibration characteristics under one- and multi-dimensional wind loads generated via harmonic superposition methods. Incorporating the multi-dimensional aerodynamic damping, the impact of wind velocity, ground roughness, and wind attack angle on the tower-line system is analyzed through time-history results and gust response factor. The findings reveal that under the premise of multi-dimensional aerodynamic damping, multi-dimensional wind loads significantly amplify responses compared to one-dimensional loads. As wind speed, ground roughness, and wind attack angle increase, responses are elevated, causing complex changes in gust response factors, underscoring the importance of considering multi-dimensional wind loads.

Effects of Aerodynamic Damping and Gyroscopic Moments on Dynamic Responses of a Semi-Submersible Floating Vertical Axis Wind Turbine: An Experimental Study

Abstract Floating vertical-axis wind turbines (VAWTs) offer certain advantages over floating horizontal-axis wind turbines (HAWTs), particularly in terms of the potential to lower the cost of energy. In this study, a 5 MW floating VAWT concept with three straight blades and a semi-submersible hull deployed in a water depth of 42 m was presented. In addition, the experimental setup is introduced, and calibration tests are also performed to validate the physical model system. Subsequently, the aerodynamic damping and gyroscopic moment effects were investigated by wind/wave basin model tests with a scale ratio of 1/50. Results indicate that aerodynamic damping can suppress the fluctuations of the platform's surge and pitch motion at their respective resonance frequencies and tends to increase with wind speed at below-rated wind speed. Additionally, surge-induced and pitch-induced aerodynamic damping hardly affect the wave frequency response. Meanwhile, the surge natural frequency is substantially altered due to the wind loads. The rotating rotor and pitch motion of the platform together excite significant gyroscopic moments, leading to noticeable oscillations in roll motion. Additionally, there is an increasing trend in the gyroscopic moment effect with rotational speed. During normal operation of the floating VAWT, aerodynamic damping and gyroscopic moment together influence hull roll/pitch motions. Overall, this study contributes to providing valuable insights into the motion characteristics of floating VAWTs.

Aerodynamic Analysis of Blade Stall Flutter Prediction for Transonic Compressor Using Energy Method

In this study, stall flutter onset prediction in a transonic compressor is carried out using the (uncoupled) energy method with Fourier transform. As the study is conducted computationally using computational fluid dynamics (CFD)-based simulations, the energy method was employed due to its higher computational efficiency by implementing the one-way FSI (Fluid Structure Interaction) model. The energy method is relatively uncommon for determining the aerodynamic damping and flutter prediction, specifically in blade stall conditions for the 3D blade passages. The NASA Rotor 67 was chosen for the validation of the study due to the availability of a wide range of experimental data. A flutter prediction analysis was performed computationally using CFD for the two-blade passages of the rotor in the peak efficiency and stall regions. Prior to this, the modal analysis on the prestressed blade was conducted, considering the centrifugal effects. The modal analysis provided accurate blade frequency and amplitude, which were the inputs of the flutter analysis. The first three modes of blade resonance were studied with a range of nodal diameters within near-peak efficiency and stall regions. The energy method implemented in this study for the flutter analysis was successfully able to predict the aerodynamic damping coefficients of the first three modes for a range of nodal diameters from the periodic-unsteady solution of the defined blade oscillation within the regions of interest (peak efficiency and stall point). The results of the study confirm the rotor blade’s stability within the near-peak region and, most importantly, the prediction of the flutter onset in the stall region. The study concluded that the computationally inexpensive and time-efficient energy method is capable of predicting the stall flutter onset. In the future, further validations of the energy method and investigations related to flow mechanism of stall flutter onset are planned.

Experimental investigation on vortex-induced vibration characteristics of the dual parallel suspenders

This study investigates vortex-induced vibrations (VIVs) on dual parallel suspenders using wind tunnel experiments and employs a two-degree-of-freedom setup for each circular cylinder to closely simulate actual structural conditions. The VIV responses of dual cylinders exhibit notable deviations and complexity due to aerodynamic interaction. Results indicate a significant shift in the VIV lock-in range towards higher values for dual cylinders, compared to single ones. Both upstream and downstream cylinders demonstrate lower maximum VIV amplitudes than the single-cylinder. The cylinders exhibit effects of negative aerodynamic damping within the VIV process, with the aerodynamic damping becoming more pronounced as amplitude increases. Additionally, the time-dependent phase relationship between the cylinders contributes to these variations in response. Further analysis of surface wind pressure distribution reveals that aerodynamic interference in dual-cylinder systems leads to intricate pressure patterns and modal behaviors.

Flow field characteristics and vibration responses of saddle-shaped membrane structures

Elastically mounted flexible membrane roofs exposed to flows are prone to vortex-induced vibrations and even aero-instability due to the strong fluid–structure interaction (FSI). This study is to investigate the FSI mechanism in the saddle-shaped membrane structure over a range of Reynolds numbers and wind directions in laminar flows, by bridging structural vibration responses and flow dynamics. The aeroelastic characteristics of membrane structures, including statistics of displacement responses, oscillation frequency, and oscillation damping ratios, were identified from the perspective of time and frequency domains. Simultaneously, the particle image velocimetry system was employed to visualize the flow features, including velocity vector, turbulence intensity, and vortex evolution in both space and time. The flow modes were further decomposed by proper orthogonal decomposition (POD) to capture the salient aspects of the flow. Three patterns of POD modes are identified, and the first mode plays the dominant role in POD modes. It showed that as the wind Reynolds number increases, the space between the shear layer and membrane surface would be narrowed, and resultantly the vortices turn out smaller in scale and closer in space. This trend leads to an increase in the frequency of vortex shedding and a stronger FSI effect. When the frequency of vortex shedding approaches the fundamental frequency of structures, the vibration of the membrane would be shifted from turbulent buffeting to vortex-induced resonance, featured with lock-in frequency, significant amplified displacement, and negative aerodynamic damping ratio.

Experimental study on vortex-induced vibrations of double-hanger cable: Upstream wake flow induces downstream vibrations

This study is inspired by an experimental observation for vortex-induced vibrations (VIVs) of a double-hanger cable system, in which a smaller vibration of the downwind hanger cable is re-excited over a narrow range of wind speeds beyond the “lock-in” range. This phenomenon is known as the wake vortex-induced vibrations (WVIVs), where the upstream wake flow induces downstream vibrations. To further investigate the characteristics and reasons for WVIVs, a refined wind tunnel test of double-hanger cable was conducted to consider the influence of aerodynamic interaction. The double-hanger cable was modeled by the tandem-arranged spring-mounted cylinders vibrating in two dimensions. The vibration responses of hanger cables were obtained under various wind speeds to reproduce the lock-in phenomenon. In addition, the vibration trajectory, phase relationship, damping ratio, inter-cable correlation, and the wind pressure on the surface of cables were analyzed. Finally, the Stockbridge dampers were designed to suppress the vibrations. The results show that under the aerodynamic interaction of the cables, the onset wind speed of VIVs in the double cables increases, and the downstream cable WVIVs closely follow the VIVs. During the WVIV phase, the downstream cable behavior is characterized by increased negative aerodynamic damping and an inverse displacement correlation between the cables. The phase relationships between the cables are time-varying due to the aerodynamic interaction. The first proper orthogonal decomposition mode of wind pressure dominates the cross-wind of motions and is crucial in vibrations. Stockbridge dampers can effectively reduce the amplitude of VIVs and eliminate WVIVs in cables.

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.

Validation of a Methodology To Assess The Flutter Limit Cycle Oscillation Amplitude of Low-Pressure Turbine Bladed-Disks - Part II: Rotational Speed Effects

Abstract The effect of the operating conditions on the vibration amplitude trends of an isolated low-pressure turbine rotor is described. The study utilizes an analytical model correlating the aerodynamic and dry-friction work introduced in Part I of the paper. In this Part II, the analysis has been extended to incorporate the influence of rotational speed. The force distribution and the penetration length of the fir-tree contact surfaces are key parameters within the heuristic micro-slip model used to characterize the friction forces. These parameters change with rotational speed, consequently influencing the dry-friction work involved in the process. The model is closed with numerical simulations to compute the aerodynamic damping, and it is compared against experimental data gathered from the experimental campaign detailed in Part I. The results demonstrate a significant impact of the shaft speed on flutter vibration amplitude. The vibration amplitude has been observed to reach a maximum near the on-design conditions. The analytical model can correctly capture this trend indicating that the essential physics is retained in it. Nonlinear friction, mistuning and 3D unsteady aerodynamics have shown to play a predominant role to explain the change of vibration amplitude with the shaft speed.

Dynamic stability and response of morphing wing structure with time-varying stiffness

Abstract Morphing, adaptable or smart structures are being used in mechanical and aerospace applications in recent years. These structures often have the property of time-varying stiffness or inertial properties, which can cause parametric instability issues that are not well understood. This paper examines the dynamic stability and response of a morphing aircraft wing with periodically time-varying structural stiffness. The wing is modeled as a beam with coupled bending-torsion motion, and parametrically excited stiffness. Aerodynamic loads introduce aerodynamic damping and aerodynamic stiffness to the wing structure. The dynamic and aeroelastic equation of motion resembles a coupled, damped Mathieu-type equation but differs with asymmetric damping and stiffness matrices, and symmetric inertial matrix. Further, these equations are functions of airspeed, magnitude and frequency of parametric excitation. Initially, dynamic stability of the wing is analyzed using Floquet theory, and the instability regions are numerically quantified by stability charts. Subsequently, dynamic responses in the stable and unstable regions are investigated with a Floquet-based Harmonic balance method. The findings reveal that, at zero airspeed, the combination instabilities are eliminated by varying the bending and torsional stiffness with equal magnitudes and frequency. However, as airspeed increases, instability regions shift unevenly, leading to the emergence of new instabilities. Furthermore, the response analysis within stable regions uncovers several unfavorable zones for operating the variable stiffness, where response decay is slower. The results clearly show that parametric excitation can cause unusual phenomena that significantly impact the operation of morphing wings with variable stiffness, which needs to be thoroughly investigated for successful implementation.

Vibration control of the straight bladed vertical axis wind turbine structure using the leading-edge protuberanced blades

This study investigates the impact of structural responses resulting from alterations in aerodynamic characteristics caused by incorporating leading-edge protuberance (LEP) blades in vertical-axis wind turbines (VAWTs) under various wind speeds. The current experimental investigation was conducted on a scaled-down straight-blade VAWT with multiple wind speeds at a fixed pitch angle and uniform blade configuration. The unsteady structural excitation caused by the decay of vortices due to the different aerodynamic loads and its effect on the VAWT structure is examined in great detail. The presence of counter-rotating vortices from the LEP profile, however, enhances the aerodynamics of the blades at higher angles of attack. It is crucial to understand the structural characteristics resulting from changes in aerodynamics. The paper presents the effects of the LEP on the structural excitation of the VAWT by analyzing the acceleration and displacement data obtained from the static structure of VAWT for various operational wind speeds. The results showed decreased acceleration and displacement amplitudes for the VAWT with LEP at medium and higher wind speeds. Significant aerodynamic and structural damping was observed in the VAWT with LEP blades. The results obtained were in close agreement with the time and frequency domains of the measured displacement and acceleration from the structure. The bidirectional vibration control was observed for the VAWT with LEP, which was found to have a reduction ratio of 70 %.

Self-excited force characteristics and flow mechanisms of a 5:1 rectangular cylinder in torsional vortex-induced vibration and flutter

In this paper, the self-excited force characteristics, flow rules and vibration mechanisms of a 5:1 rectangular cylinder in torsional vortex-induced vibration (VIV) and flutter were investigated through wind tunnel tests and numerical simulations. The nonlinearity of the self-excited lifting moment is not monotonous but exhibits a peak within a specific reduced wind speed and amplitude range. The amplitude-dependent characteristics of dimensionless work and aerodynamic damping are the primary reasons for non-divergent vibration, which is mainly influenced by the amplitude-dependent phase difference. The reduced wind speed significantly affects the formation zone length and convection speed of the leading-edge vortex, thereby determining the distribution of surface pressure phase difference and dimensionless work. As the reduced wind speed increases, there is a positive and negative alternation in dimensionless work, leading to the occurrence of torsional VIV and flutter. While the torsional amplitude does not affect the distribution of phase difference, it merely influences the formation zone length, resulting in amplitude-dependent dimensionless work and accounting for the non-divergent vibration. The characteristics of vortex convection, pressure phase variation and dimensionless work of the 5:1 rectangular cylinder in torsional VIV and flutter follow similar rules, indicating that the two vibration forms share the same inherent essence.

Mechanism of eccentricity influence on 3-DOF aerodynamic stability: New insights into instability evolution, energy harvesting, and vibration control

The inconsistency of mass center and elastic center, i.e., eccentricity, widely exists in various engineering structures and components, including iced conductors, wind turbine blades, airfoils, and cable-supported bridges, and can also be extended to the application of energy harvesting and vibration control. Although the eccentricity influence on aerodynamic stability has already been recognized as important, the underlying mechanism remains rather unclear. This study established a comprehensive framework of explicit eigenvalue solutions, offering in-depth insights into the aerodynamic stability mechanism in a vertical-horizontal-torsional 3-degree-of-freedom linear system under all possible frequency scenarios. These solutions elucidated the coupling mechanism of eccentricity, aerodynamic damping, aerodynamic stiffness, and structural frequencies. Remarkably straightforward criteria to predict the promoting/suppressing effect of eccentricity were proposed, consisting of eccentric angle and aerodynamic forces, which not only align with the traditional principles but also offer a broader application range in terms of structure types and wind speeds. Validations of the proposed solutions were performed for all frequency cases via numerical studies. For instability tendency evaluation, numerical studies on a thin plate showed that eccentricity has a significant impact due to the strong aerodynamic effect and large ratio of width/gyration radius, highlighting the importance of considering small eccentricity errors during experiments. Examination for energy harvesting revealed that a small eccentricity can dramatically enhance power generation efficiency, and the explicit solutions successfully explained the rules governing performance changes against frequency ratio, mass center position, rotation center position, etc. The analysis of a bundled conductor showed that the double pendulum can control galloping because it diminishes the eccentricity effect by shifting the eccentric angle. The proposed explicit solution framework provides a systematic view and new insights into the aerodynamic instability mechanism influenced by eccentricity, serving as a reference for the design and studies of various engineering structures, energy harvesting, and vibration control.

Tower loads characteristics of a semi-submersible floating wind turbine: An experimental study

For a Floating Wind Turbine (FWT), the tower emerges as a critical structure that connecting the Rotor-Nacelle Assembly (RNA) and the supporting platform, facilitating the transmission of both aerodynamic and hydrodynamic loads. This paper deduces the distinct responses of the bending moment generated by the motion of a FWT compared with a fixed wind turbine. Due to the complexity of the tower load responses, a comprehensive experimental investigation was conducted on a semi-submersible FWT. Stiffness-matched and geometry-matched tower models were designed and fabricated. These models accurately simulate the tower's flexibility and windward areas, impacting the global dynamic response. The model test precisely replicates the marine environmental conditions, incorporating the interactions between wind, waves, and current. It is particularly worth noting that, a multi-fans large-scale Wind Generation System (WGS) with rectifier networks was improved and fabricated to provide a reliable wind field. Results indicate that the tower load responses under the conditions with wind exhibit distinct nP component characteristics. Notably, 1P and 3P responses dominate most tower load responses, with the 3P response typically increasing with wind speed. The extreme value of My.b (y-axis moment at tower-base) is approximately 10 times larger than that of My.t (y-axis moment at tower-top), signifying a significant magnitude difference and reflecting the response characteristics of the tower loads. And the responses of My.b and Fx.t (x-axis force at tower-top) at the natural frequency of pitch motion are notably suppressed by the aerodynamic damping effect of wind. Furthermore, Fx.b (x-axis force at tower-base) and My.b exhibit obvious responses at the natural frequency of pitch motion. As the wave height increases, the wave-frequency response of the tower base load also strengthens, but the high-frequency response still accounts for a certain proportion of energy in the tower top load.

Identification of amplitude-dependent aerodynamic damping from free vibration data using iterative unscented kalman filter

This study presents an innovative approach employing an iterative Unscented Kalman Filter (IUKF) for the identification of amplitude-dependent nonlinear aerodynamic damping using wind tunnel free vibration data. The wind-structure interaction system is represented as a single-degree-of-freedom system, with the amplitude-dependent aerodynamic damping modeled as a polynomial function of structural displacement and velocity. The augmented state variables, encompassing structural vibration frequency and polynomial coefficients for aerodynamic damping, are concurrently estimated from free vibration data using the UKF technique. To enhance the robustness of the identification results against variations in initial conditions, the UKF is applied iteratively by assigning the estimated polynomial coefficients as new initial values for the state variables. Validation of the IUKF-based method is performed through a numerical example featuring a typical bridge deck sectional model, as well as experimental data from two spring-suspended sectional models experiencing vertical vortex-induced vibration (VIV) and torsional post-flutter limit cycle oscillation. The feasibility of identifying amplitude-dependent aerodynamic damping for amplitude range not covered by the displacement signal is examined.

Impact of acoustic wave propagation characteristics on aerodynamic damping of rotor blades in a transonic axial compressor

The aeroelastic stability associated with the acoustic wave propagation characteristics of transonic rotor blades was numerically investigated in this study. The influence of the vibration frequency on the aerodynamic damping of the first bending mode was primarily considered at different speeds, including both work and stall points. We found that the state of acoustic wave propagation associated with aeroelastic instability is closely related to rotational speed. At low speeds near stall points, the risk of aeroelastic instability is confined to the upstream cut-off state. However, at high speeds near stall points, aeroelastic instability may occur in both the downstream cut-off state and the acoustic wave propagation state of the upstream cut-off frequency, further expanding the range of the acoustic wave propagation state in which aeroelastic instability can arise. The research findings show that for suction surfaces, aerodynamic work is affected not only by acoustic wave propagation characteristics, but also by shock waves, radial flow, and reflux in the flow field. However, for pressure surfaces, the acoustic wave propagation characteristics play a significant role. When aeroelastic instability occurs, negative damping predominantly arises from the pressure surface. To investigate why lower-order modes are more prone to aeroelastic instability, specific simulations were conducted for the first bending and twisting modes under different operating conditions in the downstream cut-on state. When the vibration frequency significantly exceeds the downstream cut-off frequency, the blade phase is minimally influenced by acoustic wave propagation characteristics, and the rotor is aeroelastically stable in these phases. Simultaneously, there is an approximately linear increase in the unsteady pressure amplitude with increasing vibration frequency, and the aerodynamic work is predominantly influenced by the unsteady pressure amplitude.

Study of aerodynamic damping of a crosswind-excited circular cylinder and its application to full-scale flexible structures

The accurate assessment of aerodynamic damping is essential when estimating the impact of aero-elastic effects on crosswind-excited line-like structures. This study aims to determine the crosswind aerodynamic damping of a circular cylinder by curve-fitting to the response probability density function. The generalized Van der Pol oscillator model is employed to depict the nonlinearity associated with amplitude-dependent aerodynamic damping. The initial section of this study presents the theoretical foundation of the identified approach used in this research. Subsequently, the efficacy of the identification methodology is validated by means of a stochastic simulation of the crosswind response of a circular cylinder. Moreover, identification of nonlinear aerodynamic damping is further accomplished by the utilization of an aero-elastic model wind tunnel test. The identified nonlinear aerodynamic damping is then utilized to determine the response statistical quantities for a total of 16 chimneys. The determined aerodynamic damping provides a satisfactory assessment of the peak response, which is crucial for design considerations.

Synchronized shock wave and compliant wall interactions: Experimental characterization and aeroelastic modeling

The static and dynamic interaction of a normal shock wave (upstream Mach number 1.35) with a compliant wall is characterized experimentally by schlieren visualizations and an optical displacement sensor. Depending on the location of the shock wave along the compliant wall, three different regimes of interaction are found: large-amplitude synchronized regime, small-amplitude synchronized regime and unsynchronized regime. The regime of large-amplitude synchronized oscillations is found for shock locations close to the mid-point of the compliant wall along the streamwise direction; at this location, the coupled system locks to the second vibration frequency of the structure. Three regimes of small-amplitude synchronized oscillations are found depending on the shock position. When the shock is located upstream the center of the compliant wall, the shock may oscillate either periodically at the frequency of the first vibration mode or quasi-periodically with highest amplitudes at the three frequencies of the vibration modes. When the shock is located downstream the center of the compliant wall, the shock oscillates periodically at the frequency of the third vibration mode. Finally, close to the trailing edge of the compliant wall, the shock oscillation is not synchronized with the compliant wall which oscillates with a very small amplitude. An empirical model is proposed to investigate the energy exchange between the flow and the compliant wall during the limit cycle oscillations. A negative aerodynamic damping – and, hence, the possibility of a limit cycle – is observed when a sufficiently extended separation is considered in the model for the pressure distribution at the wall.

Neural network based position control of an underactuated flapping wing aircraft considering the aerodynamic damping

Neural network based position control of an underactuated flapping wing aircraft considering the aerodynamic damping