Quasi-steady Aerodynamic Model Research Articles

Aerodynamic Analysis of Hovering Flapping Wing Using Multi-Plane Method and Quasi-Steady Blade Element Theory

In the design of flapping-wing micro-size air vehicles capable of hovering, wings serve as the primary source of hovering power, making the analysis of aerodynamics and aerodynamic efficiency crucial. Traditional quasi-steady models treat the wings as single rigid plane, neglecting the deformable characteristics of flexible wings. This paper proposes a multi-plane method that, in conjunction with various design parameters of flexible wings in a two-dimensional plane, analyzes their deformation characteristics under the assumption of multiple planes in three-dimensional space, and describes the deformation of wings during flapping. By combining the quasi-steady aerodynamic model, aerodynamic analysis of the deformed wings can be conducted. The relationship between the slack angle, wing flapping position, and wing deformation are analyzed, along with their effects on aerodynamics and aerodynamic efficiency. Experiments validate the deformation patterns of wings during flapping and compare the simulated aerodynamic forces with measured ones. The results indicate that wing deformation can be accurately described by adjusting the parameters in the multi-plane method and that the aerodynamic analysis using this method closely approximates the average lift results. Additionally, the multi-plane method establishes a connection between wing morphology and aerodynamic forces and efficiency, providing valuable insights for aerodynamic analysis.

Wing geometry and kinematic parameters optimization of bat-like robot fixed-altitude flight for minimum energy

In this paper, the geometry parameters (WGP) and kinematic parameters (WKP) of bat wings are optimized by using the revised quasi-steady aerodynamic model and genetic particle swarm optimization algorithm (GPSO) according to the structural characteristics and kinematic law of bat, to meet the requirement of minimum flapping energy and improve the endurance of bat-like robot. For the first time, the dynamic model of bat flapping motion is improved by considering the effects of active deformation, passive flexible deformation, gravity moment, and inertia moment of the wing structure. Based on the power density model, the optimization objective function is established, and the lift-weight ratio (LtoW), Reynolds number (Re), aspect ratio (AR), Strouhal number (St), and boundary constraints of each parameter are added to complete the optimization of WGP and WKP. The optimized aerodynamic forces and power densities are compared with the estimates for actual bats flying at a fixed-altitude, and the results showed that the optimized parameters resulted in more stable aerodynamic forces and reduced energy consumption by half. Finally, the sensitivity analysis of the influence of each parameter on the LtoW and power density is carried out to better understand the effect of each parameter on keeping fixed-altitude flight and reducing energy consumption. This study can provide the theoretical basis for reasonable parameter design of bat-like robot.

A Continuous Time Domain Direct Identification Method for Unsteady Fluid Dynamic Model in Incompressible Flow

To identify the unsteady fluid dynamic model for a solid moving in the fluid, a new identification method that operates in the continuous time domain is proposed. It is illustrated that this new identification method is more accurate and appropriate for identifying unsteady fluid dynamic model in incompressible flow in comparison with the classical frequency domain or discrete domain method. This new method is applied to identify the unsteady fluid dynamic models of a NACA0012 airfoil and the effect of the fluid viscosity on the model parameter is analyzed. The result shows that the added mass in viscous flow is dependent on the input signal frequency and is exactly the added mass in inviscid flow when the frequency approaches infinity. Based on this discovery, the long controversy on the relationship of added mass between viscous and inviscid flow is solved. Then, the enlightenment of this discovery on the identification of both linear and nonlinear unsteady aerodynamic model in incompressible flow is illustrated. At last, the advantage of the new acquired unsteady fluid dynamic model is discussed in comparison with the classical quasi-steady aerodynamic model.

Sensitivity Analysis of Wing Geometric and Kinematic Parameters for the Aerodynamic Performance of Hovering Flapping Wing

The wing planform and flapping kinematics are critical for the hovering flight of flapping wing micro air vehicles (FWMAVs). The degree of influence of wing geometry and kinematic parameters on aerodynamic performance still lacks in-depth analysis. In this study, a sensitivity analysis was conducted based on the quasi-steady aerodynamic model. Each parameter was investigated independently by using the control variable method. The degree of each variable’s influence on lift, power, and power loading is evaluated and compared. Furthermore, detailed exponential relationships were established between the parameters and the corresponding aerodynamic properties. It is found that, for the geometric parameters, wing area has the greatest influence on lift, and the distribution of area has the most visible effect on aerodynamic power. All geometric parameters are negatively correlated with power loading. For the kinematic parameters, flapping frequency, compared with sweeping amplitude, results in faster lift growth and slower drop in power loading, while their influence on aerodynamic power is nearly comparable. A moderate pitching amplitude with advanced rotation will maximize the lift. For the flapping trajectory, lift and power loading are primarily affected by the shape of the pitching motion rather than the sweeping motion. But the sweeping motion seems to dominate the power consumption. The research in this paper is helpful to understand the effect of each parameter and provide theoretical guidance for the development of FWMAVs.

Whirl Flutter Suppression of Tiltrotor Aircraft Using Actively Controlled Aileron

Whirl flutter of a tiltrotor aircraft is a complex aeroelastic phenomenon and it can result in catastrophic consequences. The deflection of an aileron mounted on a wing has the potential to solve this fatal problem. Whirl flutter suppression using an actively controlled aileron is studied in this study. Firstly, a semi-span aeroelastic model is established for the whirl flutter problem using the Hamilton principle. This model is composed of three parts: a rigid rotor, a rigid nacelle and a flexible wing, and the effect of the aileron deflection on the aeroelastic responses is also taken into consideration through a quasi-steady aerodynamic model. In addition, the accuracy of this aeroelastic model is validated with the results of two different wind-tunnel tests. Then, an LQR controller is developed to control the dynamic deflection of the aileron, and a full-dimensional state observer is built to estimate the state of the time-invariant system of a tiltrotor aircraft. Finally, simulations are carried out using the aeroelastic model and the LQR controller at different flight conditions to study the influence of the aileron deflection on whirl flutter. The simulation results demonstrate that the flutter boundary speed can be improved by 18.1% with the active deflection of the aileron, compared with the uncontrolled condition.

Improved Model for Flexible Flapping Wings: Considering Spanwise Twisting and Bending

Insect wings and biomimetic wings in flapping-wing micro air vehicles (FWMAVs) are flexible and subject to passive deformations, including spanwise twisting and bending. This raises a typical bilateral fluid–structure interaction (FSI) issue, which is conventionally solved based on combined computational fluid dynamics (CFD) and computational solid dynamics (CSD) methods. To reduce the computational cost of this FSI issue while maintaining a reasonable accuracy, a theoretical model with improved adaptability is proposed here. The improvement results from the consideration of spanwise bending: the distribution of which is formulated by a quadratic polynomial. The aerodynamic force is approximated by a predictive quasi-steady aerodynamic model based on the blade element theory. The FSI iteration at a time step is converged within 0.5 s in our model, whereas a traditional CFD–CSD solution takes about 30 s. Compared to our previous model, the current model can better match the experimental measurements of insect wings. Further analysis reveals that considering spanwise bending affects the stiffness design of flexible flapping wings quantitatively. To maintain a high lift efficiency, the structural stiffness of the wing should be appropriately decreased. Our model provides a refined tool for the wing design in FWMAVs and can promote the development of FWMAVs.

Centre of mass location, flight modes, stability and dynamic modelling of gliders

Falling paper flutters and tumbles through air, whereas a paper airplane glides smoothly if its leading edge is appropriately weighted. We investigate this transformation from ‘plain paper’ to ‘paper plane’ through experiments, aerodynamic modelling and free flight simulations of thin plates with differing centre of mass (CoM) locations. Periodic modes such as fluttering, tumbling and bounding give way to steady gliding and then downward diving as the CoM is increasingly displaced towards one edge. To explain these observations, we formulate a quasi-steady aerodynamic model whose force and torque coefficients are informed by experimental measurements. The dependencies on angle of attack reflect the transition from attached to separated flow and a dynamic centre of pressure, effects that prove critical to reproducing the observed motions of paper planes in air and plates in water. Because the model successfully accounts for unsteady and steady flight modes, it may be usefully applied to further problems involving actuated motions, feedback control and interactions with ambient flows.

Dynamic Stability and Flight Control of Biomimetic Flapping-Wing Micro Air Vehicle

This paper proposes an approach to analyze the dynamic stability and develop trajectory-tracking controllers for flapping-wing micro air vehicle (FWMAV). A multibody dynamics simulation framework coupled with a modified quasi-steady aerodynamic model was implemented for stability analysis, which was appended with flight control block for accomplishing various flight objectives. A gradient-based trim search algorithm was employed to obtain the trim conditions by solving the fully coupled nonlinear equations of motion at various flight speeds. Eigenmode analysis showed instability that grew with the flight speed in longitudinal dynamics. Using the trim conditions, we linearized dynamic equations of FWMAV to obtain the optimal gain matrices for various flight speeds using the linear-quadratic regulator (LQR) technique. The gain matrices from each of the linearized equations were used for gain scheduling with respect to forward flight speed. The reference tracking augmented LQR control was implemented to achieve transition flight tracking that involves hovering, acceleration, and deceleration phases. The control parameters were updated once in a wingbeat cycle and were changed smoothly to avoid any discontinuities during simulations. Moreover, trajectories tracking control was achieved successfully using a dual loop control approach. Control simulations showed that the proposed controllers worked effectively for this fairly nonlinear multibody system.

Aerodynamic investigation on shifted-back vertical stroke plane of flapping wing in forward flight

In order to properly understand aerodynamic characteristics in a flapping wing in forward flight, additional aerodynamic parameters apart from those in hover—an inclined stroke plane, a shifted-back stroke plane, and an advance ratio—must be comprehended in advance. This paper deals with the aerodynamic characteristics of a flapping wing in a shifted-back vertical stroke plane in freestream. A scaled-up robotic arm in a water towing tank was used to collect time-varying forces of a model flapping wing, and a semi-empirical quasi-steady aerodynamic model, which can decompose the forces into steady, quasi-steady, and unsteady components, was used to estimate the forces of the model flapping wing. It was found that the shifted-back stroke plane left a part of freestream as a non-perpendicular component, giving rise to a time-course change in the aerodynamic forces during the stroke. This also brought out two quasi-steady components (rotational and added-mass forces) apart from the steady one, even the wing moved with a constant stroke velocity. The aerodynamic model underestimated the actual forces of the model flapping wing even it can cover the increasingly distributed angle of attack of the vertical stroke plane with a blade element theory. The locations of the centers of pressure suggested a greater pressure gradient and an elongated leading-edge vortex along a wingspan than that of the estimation, which may explain the higher actual force in forward flight.

Combined selection of the dynamic model and modeling error in nonlinear aeroelastic systems using Bayesian Inference

We report a Bayesian framework for concurrent selection of physics-based models and (modeling) error models. We investigate the use of colored noise to capture the mismatch between the predictions of calibrated models and observational data that cannot be explained by measurement error alone within the context of Bayesian estimation for stochastic ordinary differential equations. Proposed models are characterized by the average data-fit, a measure of how well a model fits the measurements, and the model complexity measured using the Kullback–Leibler divergence. The use of a more complex error models increases the average data-fit but also increases the complexity of the combined model, possibly over-fitting the data. Bayesian model selection is used to find the optimal physical model as well as the optimal error model. The optimal model is defined using the evidence, where the average data-fit is balanced by the complexity of the model. The effect of colored noise process is illustrated using a nonlinear aeroelastic oscillator representing a rigid NACA0012 airfoil undergoing limit cycle oscillations due to complex fluid–structure interactions. Several quasi-steady and unsteady aerodynamic models are proposed with colored noise or white noise for the model error. The use of colored noise improves the predictive capabilities of simpler models.

On Aerodynamic Models for Flutter Analysis: A Systematic Overview and Comparative Assessment

This work reviews different analytical formulations for the time-dependent aerodynamic load of a thin aerofoil and clarifies numerical flutter results available in the literature for the typical section of a flexible wing; inviscid, two-dimensional, incompressible, potential flow is considered in all test cases. The latter are investigated using the exact theory for small airflow perturbations, which involves both circulatory and non-circulatory effects of different nature, complemented by the p-k flutter analysis. Starting from unsteady aerodynamics and ending with steady aerodynamics, quasi-unsteady and quasi-steady aerodynamic models are systematically derived by successive simplifications within a unified approach. The influence of the aerodynamic approximations on the aeroelastic stability boundary is then rigorously assessed from both physical and mathematical perspectives. All aerodynamic models are critically discussed and compared in the light of the numerical results as well, within a comprehensive theoretical framework in practice. In all cases, results accuracy depends on the aero-structural arrangement of the flexible wing; however, simplified unsteady and simplified quasi-unsteady aerodynamic approximations are suggested for robust flutter analysis whenever the wing’s elastic axis lies ahead of the aerofoil’s control point.

Effects of side ratio on energy harvesting from transverse galloping of a rectangular cylinder

Rectangular cylinders have been identified as ideal bluff bodies for galloping-based energy harvesting, while a systematic study of the effects of the side ratio (i.e., the ratio between the cylinder width and the cylinder height) remains unavailable. This study investigates the influences of the side ratio on the piezoelectric energy harvesting from the transverse galloping of a rectangular cylinder based on a representative electro-aero-mechanical model, in which the aerodynamic force is calculated by the quasi-steady theory. The existing experimental aerodynamic coefficients for rectangular cylinders with side ratios of 0.62–3.0 are utilized as inputs of the quasi-steady aerodynamic force model. The influences of the side ratio and load resistance on the onset velocity, displacement, and power output of the galloping-based energy harvester are investigated. The results show that the onset velocity of galloping is dependent on the load resistance while this dependency becomes less significant with increasing the natural frequency. The onset galloping velocity of the energy harvester decreases with increasing the side ratio, and the lowest onset velocity is achieved by a rectangular cylinder with a side ratio of around 2.50. The largest vibration amplitude is achieved by the cylinder with a side ratio of 1.62 or 2.0 at low flow velocities, while the largest vibration amplitude is always achieved by the square cylinder at high flow velocities. Therefore, the side ratio of a galloping-based energy harvester should be designed according to its working environment: the optimal side ratio is around 1.0 if the harvester is expected to work at relatively high reduced flow velocities, while the optimal side ratio is around 1.62–2.0 if the harvester is expected to be effective at relatively low reduced flow velocities. These conclusions can provide references for designing galloping-based energy harvesters with rectangular cylinders as bluff bodies.

Performance Analysis and Optimization of a Vertical-Axis Wind Turbine with a High Tip-Speed Ratio

In this work, the aerodynamic performance and optimization of a vertical-axis wind turbine with a high tip-speed ratio are theoretically studied on the basis of the two-dimensional airfoil theory. By dividing the rotating plane of the airfoil into the upwind and downwind areas, the relationship among the angle of attack, azimuth, pitch angle, and tip-speed ratio is derived using the quasi-steady aerodynamic model, and aerodynamic loads on the airfoil are then obtained. By applying the polynomial approximation to functions of lift and drag coefficients with the angle of attack for symmetric and asymmetric airfoils, respectively, explicit expressions of aerodynamic loads as functions of the angle of attack are obtained. The performance of a fixed-pitch blade is studied by employing a NACA0012 model, and influences of the tip speed ratio, pitch angle, chord length, rotor radius, incoming wind speed and rotational speed on the performance of the blade are discussed. Furthermore, the optimization problem based on the dynamic-pitch method is investigated by considering the maximum value problem of the instantaneous torque as a function of the pitch angle. Dynamic-pitch laws for symmetric and asymmetric airfoils are derived.

Aeroelastic model and analysis of an active camber morphing wing

Morphing aircraft structures usually introduce greater compliance into aerodynamic sections, and therefore will affect the aeroelasticity with the potential risk of increased flutter. A low-fidelity model of an active camber morphing wing and its aeroelastic model are developed in order to investigate the potential critical speed by exploiting its chord-wise dimension and flexibility. Such a model may be used for conceptual design, where low fidelity models are used to explore and optimise a wide range of configurations. The morphing camber concept is implemented using a continuous representation of a two-segment structure with a rigid segment and a deformable part. The aeroelastic model is developed based on both steady and unsteady aerodynamic models, so that different parameters can be easily modified to examine changes in the flutter solutions. Of particular interest are the ratio of the morphing segment length to the chord, and its relative stiffness, as such morphing camber is potential operated using the deformable part as a flap. By comparing the results of the quasi-steady and unsteady aerodynamic models, it is shown that the quasi-steady aerodynamic model gives a more conservative prediction of the flutter speed. In addition, responses in phase space are simulated to show the fundamental aeroelastic behaviour of the morphing camber wing. It is also shown that the active compliant segment can be used to stabilise the morphing aircraft by using feedback control. This paper provides a system-level insight through mathematical modelling, parameter analysis and feedback control into dynamics applications of morphing camber.

Optimized flapping wing dynamics via DMOC approach

Outstanding aerial capabilities that insects present in nature inspire researchers to undertake a challenge to develop a flapping aerial vehicle with performances unmatched by any manmade object. However, the complex aerodynamic phenomena crucial for the insect flight are not easily understood, let alone modeled and utilized for flight. The researchers managed to develop a quasi-steady aerodynamic model capable of capturing the most important aspects of a fruit fly-like insect flight, while still being efficient enough to allow for the usage in the flapping mechanism optimization loop. This experimentally justified quasi-steady model is used in the paper as a building block for creating a novel optimization algorithm, based on the discrete mechanics and optimal control framework. When compared to the conventional approaches to design optimization, this framework includes the natural description of the energy cost function, while incorporating the physical laws in the form of a discrete Lagrange–d’ Alembert equations inherently in optimization constraints. This leads to the discrete description of the inherently continuous problem, allowing the algorithm to search for the optimal solutions in the whole domain. In other words, in contrast to the conventional approaches involving the assumption on the function family and subsequent optimization on the parameters of that function type, this approach is not constrained by the user input and is capable of yielding any solution that respects the physical laws. As presented by the numerical test cases, optimizing the flapping patterns of a fruit fly-like aerial vehicle in standstill hovering leads to both effective and robust optimization tool.

A Modified Quasisteady Aerodynamic Model for a Sub-100 mg Insect-Inspired Flapping-Wing Robot

This study proposes a modified quasisteady aerodynamic model for the sub-100-milligram insect-inspired flapping-wing robot presented by the authors in a previous paper. The model, which is based on blade-element theory, considers the aerodynamic mechanisms of circulation, dissipation, and added-mass, as well as the inertial effect. The aerodynamic force and moment acting on the wing are calculated based on the two-degree-of-freedom (2-DOF) wing kinematics of flapping and rotating. In order to validate the model, we used a binocular high-speed photography system and a customized lift measurement system to perform simultaneous measurements of the wing kinematics and the lift of the robot under different input voltages. The results of these measurements were all in close agreement with the estimates generated by the proposed model. In addition, based on the model, this study analyzes the 2-DOF flapping-wing dynamics of the robot and provides an estimate of the passive rotation—the main factor in generating lift—from the measured flapping kinematics. The analysis also reveals that the calculated rotating kinematics of the wing under different input voltages accord well with the measured rotating kinematics. We expect that the model presented here will be useful in developing a control strategy for our sub-100 mg insect-inspired flapping-wing robot.

Discrete mechanics and optimal control optimization of flapping wing dynamics for Mars exploration

Mars exploration is currently in the focus of scientific community interests. The attempts for efficient exploration will probably include the unmanned aerial vehicle in the near future to explore the places inaccessible by rovers. Since the Mars atmospheric conditions render the conventional rotary and fixed wing aerial vehicles inefficient, the insect-type flapping concept emerged as a promising solution. This is due to the fact that insects on Earth fly efficiently at the same values of Reynolds number that the aerial vehicle for Mars exploration would exhibit. The paper proposes the novel design optimization algorithm for development of insect-type aerial vehicle capable of flight in Martian atmosphere. The optimization procedure utilizes the novel flapping pattern optimization based on a quasi-steady aerodynamic model, combined with the discrete mechanics and optimal control framework. A test case of a flapping vehicle with fruit fly-like wings, performing a standstill hovering on Mars, is analyzed in detail. The fruit fly wing is scaled with a wide range of uniform scaling factors and optimized for hovering on Mars in the conditions of bioinspired Reynolds number range. Apart from the single optimal combination for the standstill hovering with fruit fly-like wings, the algorithm also found different efficient flapping patterns for a wide range of scaling factors, providing directions for design of flapping aerial vehicles for Mars.

Comparative study on analytical and computational aerodynamic models for flapping wings MAVs

ABSTRACTA range of quasi-steady and unsteady aerodynamic models are used to predict the aerodynamic forces experienced by a flapping wing and a detailed comparison amongst these predictions in provided. The complexity of the models ranges from the analytical potential flow model to the computational Unsteady Vortex Lattice Method (UVLM), which allows one to describe the motion of the wake and account for its influence on the fluid loads. The novelty of this effort lies in a modification of the predicted forces as a generalisation of the leading edge suction analogy. This modification is introduced to account for the delayed stall mechanism due to leading edge flow separation. The model predictions are compared with two sets of independent experimental data and with computational fluid dynamics (CFD) simulation data available in the literature. It is found that both, the modified analytical model and the UVLM model can be used to describe the time history of the lift force, in some cases with better results than a high-fidelity CFD model. The models presented here constitute a useful basis for the aerodynamic design of bioinspired flapping-wings micro-air vehicles.

Aeroelastic interactions and trajectory selection of vortex gusts impinging upon Joukowski airfoils

The dynamically-coupled interactions of vortex gusts encountering a symmetric Joukowski airfoil on linear elastic supports is formulated analytically and evaluated numerically using a time-dependent conformal mapping. The Brown and Michael framework models the unsteady shedding of vorticity from the airfoil into the wake, and the aeroelastic motion of the airfoil is analyzed using quasi-steady, apparent mass, and fully-unsteady aerodynamic models. Special attention is paid to the influences of the strength of the incident vortex and the airfoil natural frequency on the initial upstream placement of a vortex gust that achieves direct impingement. These results are weakly sensitive to the initial vortex position in the limits of either large or small structural natural frequency, and the initial vertical vortex position to achieve impingement changes monotonically with the vortex strength. A comparison of the numerical model with available experimental vortex gust measurements over stationary airfoils highlights the appropriate use of the point-vortex model for inviscid vortex–airfoil interaction problems without significant vortex strength decay, which occurs for predominantly viscous interactions of close vortex–airfoil encounters.

Lyapunov-based control and trajectory tracking of a 6-DOF flapping wing micro aerial vehicle

In this work, a high-fidelity nonlinear and time-periodic six degrees-of-freedom dynamic model of a flapping wing micro aerial vehicle has been developed. The model utilized Lagrange’s equations for quasi-coordinates. A quasi-steady aerodynamic model that includes the effect of leading-edge vortex and rotational circulation was implemented. Averaging technique was used to obtain a time-invariant dynamic model for the purpose of designing a trajectory tracking control law. The sensitivity of aerodynamic forces and moments to the control parameters, namely: flapping frequency, flapping bias and stroke plane angles, was determined exploiting control derivatives. The developed controller has two loops. The outer-loop, employing integral sliding mode controller, generates desired cycle-averaged aerodynamic forces and moments. The inner loop employs a novel control allocation method. The proposed inner-loop controller provides a closed-loop feedback solution that ensures that the chosen control parameters provide the desired aerodynamic forces and moments. Simulation studies indicate that the proposed controller applied on the high-fidelity six degrees-of-freedom dynamic model successfully achieves trajectory tracking.