Pitch Plunge Research Articles

Design and analysis of structure for milling OCr13 stainless steel with unequal tooth pitch plunge milling tool

Design and analysis of structure for milling OCr13 stainless steel with unequal tooth pitch plunge milling tool

Dynamic responses of a conceptual two-dimensional airfoil in hypersonic flows with random perturbations

This paper investigates a conceptual pitch–plunge airfoil in hypersonic flows with both structural and aerodynamic nonlinearity, meanwhile the effects of irregular fluctuations in the flow and external random loads modeled as a Gaussian white noise are considered. The dynamic responses of the airfoil model especially the bifurcation behaviors are analyzed via the harmonic balance method for the deterministic case. We found the airfoil system undergo both subcritical and supercritical Hopf bifurcations. Subsequently, the effects of stochasticity on the dynamical behaviors of the hypersonic airfoil system are explored in the regimes of subcritical and supercritical Hopf bifurcations depending on the system parameters. Several interesting phenomena are triggered under random perturbations such as intermittency and stochastic transition between the low-amplitude oscillation state and the undesirable high-amplitude oscillation state. These dynamics are further characterized via the probability density function and time history. This work will provide new insights into the safety and reliability design of hypersonic aircraft.

Probabilistic prediction of coalescence flutter using measurements: Application to the flutter margin method

Zimmerman and Weissenburger’s flutter margin method is widely used to estimate the aeroelastic coalescence flutter speed. In contrast to aeroelastic decay rates, the flutter margin exhibits monotonic decay with respect to airspeed rendering it effective in extrapolating the flutter speed using flight test data conducted at pre-flutter airspeeds. This paper reports the generalization of the Bayesian formulation of the flutter margin method by Khalil et al. developed to tackle measurement and modeling uncertainties. This paper improves the predictive performance of the previous algorithm by incorporating the joint prior of aeroelastic modal frequencies and decay rates among airspeeds in order to better estimate the joint posterior of modal parameters using observational data. The modal parameter prior is constructed using the classical two-degree-of-freedom pitch–plunge aeroelastic model whose system matrices (e.g. structural stiffness and damping matrices) vary randomly. Such joint modal parameter prior enforces statistical dependence among posteriors of modal parameters and the associated flutter margins across airspeeds. Numerical studies demonstrate a considerable reduction of uncertainties on the predicted flutter speed obtained from the generalized Bayesian flutter margin method. This improved algorithm can cut cost by reducing the number of flight tests and better assess the uncertainty against aeroelastic flutter.

Experimental assessment of Theodorsen's function for uncoupled pitch–plunge motion

The accuracy of Theodorsen's lift model for pure-pitch, pure-plunge and combined pitch–plunge oscillations of a two-dimensional model is compared with wind tunnel results. The reduced frequency of the oscillation was in the range of 0.01<k<0.15, and the freestream Reynolds number was in the range of 130 000<Re<400 000. The lift response to an uncoupled combined pitch–plunge motion (for which the frequency of pitch and plunge motions were not the same) is discussed using experimental results. The Theodorsen’s lift model is rewritten for the general uncoupled pitch–plunge motions by a linear superposition of all components of the airfoil bound circulation. Both amplitude and phase from the Theodorsen's function are compared with those of the wind tunnel data, and the results are discussed. The Theodorsen’s function is found to be a good estimator for both pure-pitch and pure-plunge motions. It further appropriately estimates the lift amplitude for the case of coupled pitch–plunge motion; however, the prediction is not accurate for the uncoupled pitch–plunge motion. A motion amplitude ratio is defined, which shows the level of aperiodicity of the motion. Discrepancy between experimental and analytical results increase with the reduction of the lift amplitude ratio and with the deviation of frequency ratio from unity.

Pitch–Plunge Equivalence in Dynamic Stall of Ramp Motion Airfoils

Large-eddy simulations (LES) are employed to investigate the pitch–plunge equivalence of an SD7003 airfoil undergoing constant ramp motions at Reynolds number . The equivalence is constructed based on the geometric effective angle of attack according to the quasi-steady thin-airfoil theory. Two rates of descent (or pitch up) are analyzed for different Mach numbers in order to investigate the effects of compressibility on the evolution of the dynamic stall vortex (DSV). During the onset of the DSV and its transport along the airfoil surface, remarkable similarities are found between pitch and plunge in terms of flow topology, aerodynamic loads, and signatures of wall pressure and friction coefficients. However, these flow similarities cease at high-load conditions as the DSV becomes more susceptible to the peculiarities of the airfoil motion, manifested here by different trailing-edge vortices. Employing a correction for the rotation-induced apparent camber effect present in the pitching case, which results from the quasi-steady thin-airfoil theory, improves the agreement between pitch and plunge. However, it is not sufficient to assimilate their disparate trailing-edge systems. Results also demonstrate that the limit angle at which pitch–plunge equivalence remains valid decreases for higher Mach numbers.

Investigation of chordwise functionally graded flexural rigidity in flapping wings using a two-dimensional pitch–plunge model

Insect wings are heterogeneous structures, with flexural rigidity varying one to two orders of magnitude over the wing surface. This heterogeneity influences the deformation the flapping wing experiences during flight. However, it is not well understood how this flexural rigidity gradient affects wing performance. Here, we develop a simplified 2D model of a flapping wing as a pitching, plunging airfoil using the assumed mode method and unsteady vortex lattice method to model the structural and fluid dynamics, respectively. We conduct parameter studies to explore how variable flexural rigidity affects mean lift production, power consumption and the forces required to flap the wing. We find that there is an optimal flexural rigidity distribution that maximizes lift production; this distribution generally corresponds to a 3:1 ratio between the wing’s flapping and natural frequencies, though the ratio is sensitive to flapping kinematics. For hovering flight, the optimized flexible wing produces 20% more lift and requires 15% less power compared to a rigid wing but needs 20% higher forces to flap. Even when flapping kinematics deviate from those observed during hover, the flexible wing outperforms the rigid wing in terms of aerodynamic force generation and power across a wide range of flexural rigidity gradients. Peak force requirements and power consumption are inversely proportional with respect to flexural rigidity gradient, which may present a trade-off between insect muscle size and energy storage requirements. The model developed in this work can be used to efficiently investigate other spatially variant morphological or material wing features moving forward.

Data-driven optimization for flutter suppression by using an aeroelastic nonlinear energy sink

Aeroelastic systems are at risk of flutter instabilities which can lead to structural failure. Recently, an innovative design of nonlinear energy sink (NES) has been introduced to suppress flutter instabilities by adding a NES to the flap control surface as a nonlinear energy absorber resulting in a flap-NES. Optimal choice of flap-NES parameters, however, requires a comprehensive characterization of the nonlinear response of the system in the design space. Performing this nonlinear analysis using traditional approaches demands massive theoretical and computational efforts. This work is a foundational study of flap-NES-based flutter control of a typical pitch–plunge section and focuses on obtaining flap-NES parameters using a data-driven optimization algorithm. The optimization is based on a genetic algorithm combined with a recently introduced data-driven bifurcation forecasting method that allows constructing the bifurcation diagram of the system dynamics using limited time series data. Results of this study show that an optimal design of the flap-NES system results in an increased linear flutter speed, reduced post-flutter limit cycle oscillation amplitudes, and improved nature of the bifurcation diagram from subcritical to supercritical.

Active Flutter Suppression of a Two-Dimensional Wing Using Linear Quadratic Gaussian Optimal Control

This paper aims to utilize the modern control theory in designing an active flutter suppression controller for a two-dimensional wing with a flap. In this study, the wing’s pitch–plunge linear aeroelastic response, in addition to the control flap motion, was modeled using Lagrange’s energy method, and Theodorsen’s classic unsteady aerodynamic theory. The derived mathematical model was represented in state-space so the modern control techniques can be implemented. The open-loop system was analyzed, and the flutter speed for a selected experimental-wing model was calculated. For model verification purposes, the open-loop simulation results were compared to an experimental study from the literature. Next, a linear quadratic Gaussian (LQG) compensator with integral action was designed and tuned using MATLAB®simulation environment to reach the best possible performance at a selected speed in the flutter region, where the results have proven excellent performance for this kind of controller in flutter suppression, as the stable flight region has successfully been extended by 29%.

Pitch/Plunge Equivalence for Dynamic Stall of Unswept Finite Wings

Pitch/plunge equivalence for deep dynamic stall of a straight finite wing is considered by employing large-eddy simulations. The flowfields are computed with a well-established time-implicit high-fidelity large-eddy simulation approach. The wing has a moderate aspect ratio of and a NACA 0012 section incorporating a rounded tip. The flow parameters are a freestream Mach number of and chord Reynolds numbers of . Pure pitching and equivalent plunging maneuvers are considered with a reduced frequency of and minimum and maximum effective angles of attack of 4 and 22 deg, respectively. The pitch and plunge dynamic stall cases are compared in detail in terms of the unsteady three-dimensional flow structure, surface pressure, and loading. Differences in the lift and moment coefficient are shown to be correctable, employing a steady thin-airfoil theory for a cambered airfoil that accounts for the rotation-induced apparent camber present in the pitching case. This pitch/plunge equivalence is extendable to the dynamic stall of a swept wing, as demonstrated in another paper by the authors [“Pitch–Plunge Equivalence for Dynamic Stall of Swept Finite Wings,” AIAA Journal (submitted for publication)].

Routes to Synchronization in a pitch–plunge aeroelastic system with coupled structural and aerodynamic nonlinearities

The present study aims to investigate the synchronization characteristics of a pitch–plunge aeroelastic system subjected to coupled structural free-play and dynamic stall-induced aerodynamic nonlinearities. The semi-empirical Leishman–Beddoes (LB) aerodynamic model is used to capture the dynamic stall phenomena at large angles-of-attack (AoA). A bifurcation analysis, considering the flow-speed as the control parameter, reveals that the free-play nonlinearity is primarily responsible for the dynamical transition in the presence of the attached flow. In this case, a pitch and plunge frequency coalescence takes place through an aperiodic to periodic signature transition. Furthermore, by tracking the flow specific variables estimated using the LB model, it is observed that the onset of the dynamic stall plays a pivotal role in the bifurcation scenario at high flow speed regimes, marking the presence of stall flutter. The framework of synchronization is used to discern the response dynamics at low speed and high speed flow regime by estimating the pitch and plunge modal frequencies and their corresponding phase components. The routes to synchronization are identified to be through phase-locking (frequency coalescence) and through suppression of the plunge mode by the dominance of pitching frequency in the regimes before and after the onset of dynamic stall, respectively. Finally, it is shown that the different synchronization routes can possibly be a useful tool to identify the presence of stall flutter oscillations in a coupled aeroelastic system.

Lift Equivalence and Cancellation for Airfoil Surge–Pitch–Plunge Oscillations

A NACA 0018 airfoil in freestream velocity is oscillated in longitudinal, transverse, and angle-of-attack directions with respect to the freestream velocity, known as surge, plunge, and pitch. The lift-based equivalence method introduces phase shifts between these three motions to construct in-phase sinusoidal components for maximum lift, waveform construction. Lift cancellation is also determined with the exact negative pitch and plunge motion amplitudes found from the equivalence method to achieve out-of-phase wave destruction. Lift cancellation occurs when a combination of these motions is sought to obtain a constant lift magnitude throughout the oscillation cycle. To achieve both equivalence and cancellation of lift, a prescribed pure pitch amplitude through the Theodorsen theory equates the corresponding equivalent plunge amplitude and pitch–plunge phase shift. These Theodorsen, linear superposition findings of pitch–plunge are leveraged toward the Greenberg theory to determine a closed-form, surge–pitch–plunge solution through the addition of a surge–plunge phase shift and optimal surge amplitude for lift cancellation. The lift cancellation surge–pitch–plunge amplitudes define the equivalence amplitude investigated here and theoretically limit the experiment to combinations of the first lift harmonic of the Greenberg theory. The analytical results are then compared with experimental lift force measurements and dye visualization. The normalized lift differences due to unsteady wake and boundary-layer behavior are examined to explore the extents of the Greenberg theory for these cases of lift-based equivalence and cancellation.

Flutter Onset Prediction Based on Parametric Model Estimation

This paper presents a thorough evaluation of three flutter prediction techniques based on the well-understood autoregressive moving-average (ARMA) parametric model. Other than the original stability parameter that is derived from the identified autoregressive coefficients, the flutter margin and the Houbolt–Rainey (H-R) method are also presented and adjusted to provide alternative flutter predictions. From the application aspects, the empirical flutter margin, which is insensitive to the damping estimation, is first proposed. Feasibility of the empirical flutter indictor is demonstrated in this study. Taking advantage of the inherent recursive nature, the recursive least-squares algorithm with forgetting factor is suggested to perform the linear extrapolation iteratively at different flow conditions to facilitate an accurate flutter prediction. To verify the performance of the inspected methods, a linear pitch–plunge aeroelastic system and a dedicated wind-tunnel flutter test of an elastic wing are adopted to carry out the analyses for flutter statistics. While the extensive simulations are easy to implement on the aeroelastic idealization, the challenge of generating an adequate sample size based on the limited experimental data arises. To this end, the concept of surrogate data is first introduced to the flutter statistical analysis. Surrogate datasets generated via the iterative amplitude adjusted Fourier transform algorithm manage to describe the statistical behavior of the evaluated stability parameters. Both numerical and experimental results further substantiate the validity of the ARMA parametric model for aeroelastic modeling and provide alternative flutter prediction parameters with their own scope of application.

Response analysis of a pitch–plunge airfoil with structural and aerodynamic nonlinearities subjected to randomly fluctuating flows

This study focuses on numerically investigating the response dynamics of a pitch–plunge airfoil with structural nonlinearity under dynamic stall conditions. The aeroelastic responses are investigated for both deterministic and randomly time varying flow conditions. To that end, a pitch–plunge airfoil under dynamic stall condition is considered and the nonlinear aerodynamic loads are computed using a Leishman–Beddoes formulation. It is shown that the presence of structural nonlinearities can give rise to a variety of dynamical responses in the pre-flutter regime. Next, a response analysis under the presence of a randomly fluctuating wind is carried out. It is demonstrated that the route to flutter occurs via a regime of pre-flutter oscillations called intermittency. Finally, the manifestation of these stochastic responses is characterized by invoking stochastic bifurcation concepts. The route to flutter via intermittency is presented in terms of topological changes occurring in the joint-probability density function of the state variables.

Statistical analysis on the effect of reduced frequency on the aerodynamic behavior of an airfoil in dynamic physical motions

In this paper, an extensive experimental tests were performed to determine the aerodynamic characteristics of an airfoil undergoing static, dynamic pitch, dynamic plunge and dynamic combined pitch and plunge motions for various cases. This paper, however, focuses on the effects of reduced frequencies and mean angles of attack on the surface pressure distribution and on the corresponding lift of the airfoil oscillating in either pure pitch or in combined pitch–plunge motions. The angles of attack variations were set such that the model motion would be ceased lower than the static stall, near the static stall and beyond the static stall angles of attack. All tests were conducted at a constant Reynolds number and at various reduced frequency. The plunging motion which is created by the vertical displacement of the model inside the test section, is transformed to the equivalent angle of attack and is added to the pitch angle of attack for the case of the combined pitch and plunge motion. The differences in the pressure distribution and the corresponding aerodynamic loads as a function of the equivalent angle of attack, represent an extreme dependence of the pitch–plunge motion on the reduced frequency. Pitch oscillation seems to have dominant role in the pitch–plunge oscillation case. For the same situation, same frequency and mean angles of attack, the corresponding aerodynamic lift of the pitch–plunge motion was seen to be much higher than the pitch alone oscillation case.

Chaos control of nonlinear aeroelastic pitch plunge model

Chaos control of nonlinear aeroelastic pitch plunge model

Chaos control of nonlinear aeroelastic pitch plunge model

This paper deals with the control of chaos in a nonlinear aeroelastic pitch-plunge model for aircraft wings. While design of nonlinear controllers to asymptotically stabilise the initial bifurcation exists in the current literature, the primary goal of this paper is to control the chaotic motion using feedback linearisation. For this purpose, a pitch-plunge model with two flaps is considered for the study. The approach proposed for the control of chaos is that of a tracking problem where the system is controlled to a defined limit cycle; thus the chaotic motion is replaced by orbital stability. Using Lie algebra, a feedback linearisation is performed by transforming the nonlinear space to a linear one. The error dynamics is established as the deviation of the chaotic trajectory from that of the desired path and tracking is achieved by reducing the error to zero. The ability of this approach to control chaos is demonstrated for a certain set of parameters of the pitch-plunge model, chosen from the literature. In order to validate the approach followed in this paper, the Rossler system undergoing chaotic motion is tracked to a LCO and results are compared with the literature.

Transition to chaos in the flow-induced vibration of a pitching–plunging airfoil at low Reynolds numbers: Ruelle–Takens–Newhouse scenario

This study focuses on numerically analyzing the transition from periodic to chaotic dynamics in the fluid-elastic response of a 2-dof flexibly-mounted airfoil with chord-wise rigidity. The computational framework is composed of a high fidelity Navier–Stokes solver, weakly coupled with a structural model having geometric nonlinearity represented by cubic order stiffness terms. A low Reynolds number flow regime and a very low structure-to-fluid added mass ratio have been considered to simulate the flying conditions of very light-weight unmanned devices. A bifurcation analysis of the system, in the absence of actuation or control forces, is undertaken with the wind velocity as the control parameter. The route to chaos – identified to be the Ruelle–Takens–Newhouse quasi-periodic route – is established for the first time for a flexible pitch–plunge flapping system. Robust nonlinear time series analysis techniques have been implemented to characterize different complex dynamical states present in the system.

A novel approach for flutter prediction of pitch–plunge airfoils using an efficient one-shot method

In this work, the one-shot method previously developed for the solution of limit cycle oscillation problems is extended to predict flutter boundaries of aeroelastic systems. In essence, the one-shot method determines the aeroelastic response of wings and airfoils in a tightly-coupled fashion where both aerodynamic and structural dynamic problems are solved simultaneously using harmonic balance. This approach is superior to the frequency-based techniques previously reported in the literature such that it eliminates the need to sweep over a range of frequencies to determine flutter conditions. For each Mach number of interest, the values of flutter frequency and flutter velocity are determined as part of a single aeroelastic run. A method for identifying appropriate initial conditions is also presented. It is shown that the flutter onset point for given flow conditions can be accurately identified by prescribing a very small pitch amplitude treating flutter prediction as a response problem instead of the classical stability problem. Using this technique, three two-degree-of-freedom aeroelastic models, including a flat plate, the NACA 64A010 airfoil and the supercritical NLR 7301 airfoil, are studied under different flow conditions ranging from low-speed, inviscid flow to transonic, viscous, turbulent flow. The results are verified against reference results from the literature. In addition, two other established flutter methods are implemented in this work for verification purposes, and the efficiency and robustness of the one-shot method are investigated.

Dimensionless modeling and nonlinear analysis of a coupled pitch–plunge–leadlag airfoil-based piezoaeroelastic energy harvesting system

In this paper, an airfoil-based piezoaeroelastic energy harvesting system is proposed with an additional supporting device to harvest the mechanical energy from the leadlag motion. A dimensionless dynamic model is built considering the large-effective-angle-of-attack vibrations causing (1) the nonlinear coupling between the pitch–plunge–leadlag motions, (2) the inertia nonlinearity, and (3) the aerodynamic nonlinearity modeled by the ONERA dynamic stall model. Cubic hardening stiffness is introduced in the pitch degree of freedom for persistent vibrations with acceptable amplitude beyond the flutter boundary. The nonlinear aeroelastic response and the average power output are numerically studied. Limit cycle oscillations are observed and, as the flow velocity exceeds a secondary critical speed, the system experiences complex vibrations. The power output from the leadlag motion is smaller than that from the plunge motion, whereas the gap is narrowed with increasing flow velocity. Case studies are performed toward the effects of several dimensionless system parameters, including the nonlinear torsional stiffness, airfoil mass eccentricity, airfoil radius of gyration, mass of the supporting devices, and load resistances in the external circuits. The optimal parameter values for the power outputs from the plunge and leadlag motions are, respectively, obtained. Beyond the secondary critical speed, it is shown that the variations of the power outputs with those parameters become irregular with fluctuations and multiple local maximums. The bifurcation analysis shows the mutual transitions between the limit cycle oscillations, multi-periodic vibrations, and possible chaos. The influences of these complex vibrations on the power outputs are discussed.

Piezoaeroelastic energy harvesting based on an airfoil with double plunge degrees of freedom: Modeling and numerical analysis

In this paper, a piezoaeroelastic energy harvester based on an airfoil with double plunge degrees of freedom is proposed to additionally take advantage of the vibrational energy of the airfoil pitch motion. The analytical model of the proposed harvester is built, and an equivalent model using the well-explored pitch–plunge configuration is presented. The nonlinear aerodynamics is calculated based on the dynamic stall model. The dynamic response, average power output, energy harvesting efficiency, and cut-in speed (flutter speed) of the proposed harvester are numerically studied. It is demonstrated that the harvester with double-plunge configuration outperforms its equivalent pitch–plunge counterpart in terms of both the power output and energy harvesting efficiency beyond the flutter boundary. In addition, case studies are performed to reduce the cut-in speed and to enhance the energy harvesting efficiency of the proposed harvester, including the airfoil mass characteristics, the configuration, mass, damping, and stiffness characteristics of the two plunge supporting devices, and the load resistances in the external circuits. It is shown that the cut-in speed is greatly reduced by increasing the airfoil mass while tuning the mass eccentricity. The mass of the first (windward) supporting device should be a bit smaller than that of the second one for an applicable cut-in speed and a high-energy harvesting efficiency. Besides, the decrease of airfoil mass moment of inertia or the damping of the supporting devices is shown to be beneficial for the energy harvesting performance. In addition, the optimal location of the first supporting device is found to be at the airfoil leading edge. Decreasing the distance between the two supporting devices reduces the cut-in speed. The load resistances affect the cut-in speed slightly, and optimal values are found to maximize the energy harvesting efficiency.