The need for higher-order averaging in the stability analysis of hovering, flapping-wing flight

Taha, Haitham E.; Tahmasian, Sevak; Woolsey, Craig A.; Nayfeh, Ali H.; Hajj, Muhammad R.

doi:10.1088/1748-3190/10/1/016002

Cited by 69 publications

(46 citation statements)

References 39 publications

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“…The aim is to follow an analytical approach (e.g., averaging theorem) rather than a numerical approach (e.g., Floquet theorem). However, the direct averaging approach is not practically useful because it is restricted to the cases of very high excitation frequencies, as shown by Taha et al [1,2]. Geometric control theory provides two possible remedies for such a drawback of the averaging theorem.…”

Section: Geometric Mechanics and Control Toolsmentioning

confidence: 99%

“…Only the final results of the GATare stated here, and the reader is referred to Sec. 4 in [2] for a detailed presentation of the GAT.…”

Section: A Generalized Averaging Theorymentioning

confidence: 99%

“…Taha et al [2] used second-order averaging to study the dynamic stability of hovering insects. Although the flapping frequency of the hawkmoth is about 30 times its natural frequency (i.e., largest frequency of body flight dynamic modes), which suggests the use of direct averaging, Taha et al [2] showed that higher-order averaging is essential to capture the true stability characteristics and the dynamical responses resulting from the aerodynamic-dynamic interactions.…”

Section: A Generalized Averaging Theorymentioning

confidence: 99%

“…ρ 1 , ρ 2 , ρ 3 = components of ρ c w in the wing frame ρ air = air density τ φ = actuating torque in the flapping direction Φ = flapping amplitude φ = back and forth flapping angle ψ, θ, ϕ = yaw-pitch-roll Euler angles defining the body frame with respect to the inertial frame ω i = angular velocity vector of the ith rigid body with respect to the inertial frame ω 1 , ω 2 , ω 3 = components of ω w in the wing frame dynamical system because of the associated two time scales: the time scale of the fast flapping motion and the associated aerodynamic loads, and the relatively slow time scale of the body motion. The interaction between the periodic aerodynamic loads and the body motion may induce some interesting stabilizing mechanisms [1,2]. All of these interesting dynamical behaviors and challenges led to a recent flurry of research on the flight dynamics of FWMAVs.…”

mentioning

confidence: 99%

“…The second major assumption (averaging the dynamics over the flapping cycle) was found to be deficient in the work of Taha et al [1,2] for hovering insects with a relatively small flapping frequency (e.g., hawkmoth and cranefly). They showed that, in spite of the large ratio of the flapping frequency to the natural frequency of the body motion (30 for the hawkmoth and 50 for the cranefly), there is a strong interaction between the system's two time scales that considerably affects the flight stability.…”

mentioning

confidence: 99%

See 4 more Smart Citations

Geometric Control Approach to Longitudinal Stability of Flapping Flight

Taha

Woolsey

Hajj

2016

Journal of Guidance, Control, and Dynamics

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Flapping flight dynamics is appropriately represented by a multibody, nonlinear, time-varying system. The two major simplifying assumptions in the analysis of flapping flight stability are neglecting the wing inertial effects and averaging the dynamics over the flapping cycle. The challenges resulting from relaxing these assumptions naturally invoke the geometric control theory as an appropriate analysis tool. In this work, the full equations of motion governing the longitudinal flapping flight dynamics near hover are considered and represented in a geometric control framework. Then, combining tools from geometric control theory and averaging, the full dynamic stability of hovering insects is assessed. Nomenclature C L α = three-dimensional lift curve slope of the wing c = chord length c = mean chord length d = distance between the wing root and the wing c.g. along the wing longitudinal axis F = vector field representing the dynamics of the pulled back or the adjoint system F i , M i = external force and moment vectors applied on the ith rigid body at its reference point f = drift vector field (uncontrolled dynamics) g = gravitational acceleration g = control vector field h i = angular momentum vector of the ith rigid body with respect to the inertial frame _ h 1 , _ h 2 , _ h 3 = components of _ h w in the wing frame I x , I y , I z = moments of inertia of the wing about the wing axes I y b = moment of inertia of the body about the pitching axis i ν , j ν , k ν = unit vectors in the x, y, and z directions in the ν frame M a = aerodynamic pitching moment at the body c.g. m b , m w , m v = masses of body, wing, and whole vehicle m 0 = wing mass per unit area N a = aerodynamic torque resisting flapping motion of the wing q = generalized coordinates vector R = wing radius (length) R ζ = rotation matrix by an angle ζ about the corresponding axis r cg = spanwise distance between the wing root and the wing c.g. S= area of one wing T, ω = flapping period and angular frequency t = time variable u, w = components of the body velocity vector in the body frame v i = inertial velocity vector of the reference point of the ith rigid body X a , Z a = aerodynamic forces in the body frame x, z = quasi coordinates associated with body velocity components u and w x b , y b , z b = body-fixed frame x h = longitudinal distance from the body c.g. to the wing hinge line x I , y I , z I = inertial fixed frame x w , y w , z w = wing fixed frame α = angle of attack α d , α u = angle of attack during downstroke and upstroke α m = mean angle of attack Δx = normalized chordwise distance from the center of pressure to the hinge location η = pitching angle ϑ = plunging angle μ = rotational damping coefficient ρ c i = vector pointing from the reference point of the ith rigid body to its c.g. ρ 1 , ρ 2 , ρ 3 = components of ρ c w in the wing frame ρ air = air density τ φ = actuating torque in the flapping direction Φ = flapping amplitude φ = back and forth flapping angle ψ, θ, ϕ = yaw-pitch-roll Euler angles defining the body frame with respect ...

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Section: Geometric Mechanics and Control Toolsmentioning

confidence: 99%

“…Only the final results of the GATare stated here, and the reader is referred to Sec. 4 in [2] for a detailed presentation of the GAT.…”

Section: A Generalized Averaging Theorymentioning

confidence: 99%