State-Space Representation of Unsteady Aerodynamic Models

Transcription

1 State-Space Representation of Unsteady Aerodynamic Models -./, &! %! $! #! "! x α α k+1 A ERA x = 1 t α 1 α x C L (k t) = C ERA C Lα C L α α α ERA Model fast dynamics k + B ERA t quasi-steady and added-mass contributions k α k input + D ERA α k!! " # $ % & ' ( )*+, Steve Brunton & Clancy Rowley Princeton University 63rd APS DFD November 21, 21

2 Motivation Applications of Unsteady Models Conventional UAVs (performance/robustness) Micro air vehicles (MAVs) Flow control, flight dynamic control Autopilots FLYIT Simulators, Inc. Flight simulators Safety Concerns severe weather Predator (General Atomics) wake vorticity gust disturbances Wake Vorticity Flexible Wing (University of Florida)

3 3 Types of Unsteadiness 1. High angle-of-attack 2. Strouhal number 3. Reduced frequency α > α stall St = Af k = πfc Large amplitude, slow Moderate amplitude, fast Small amplitude, very fast Closely related α eff = tan 1 (πst) Brunton and Rowley, AIAA ASM 29

4 3 Types of Unsteadiness 1. High angle-of-attack 2. Strouhal number 3. Reduced frequency α > α stall St = Af k = πfc Large amplitude, slow Moderate amplitude, fast Small amplitude, very fast Closely related α eff = tan 1 (πst) Brunton and Rowley, AIAA ASM 29

5 3 Types of Unsteadiness 1. High angle-of-attack 2. Strouhal number 3. Reduced frequency α > α stall St = Af k = πfc Large amplitude, slow Moderate amplitude, fast Small amplitude, very fast Closely related α eff = tan 1 (πst) Brunton and Rowley, AIAA ASM 29

6 Candidate Lift Models C L =2πα C L = C Lα α C L = C L (α) C L (t) =C δ L(t)α() + C L = π a ḧ + α 2 2 α Added-Mass t +2π C δ L(t τ) α(τ)dτ α + ḣ + 12 α 12 a Circulatory Wagner s Indicial Response C(k) Theodorsen s Model New models! Model Criteria x T W C x = 1 x 2 x 2 x 2 Captures input output dynamics accurately Computationally tractable fits into control framework x T W O x = 1 x 1 T balancing transformation x 1 x 1 W C = W O = diag(σ)

7 C L = π a ḧ + α 2 2 α Added-Mass Theodorsen s Model +2π α + ḣ + 12 α 12 a Circulatory C(k) k = πfc 2D Incompressible, inviscid model Unsteady potential flow (w/ Kutta condition) Linearized about zero angle of attack Apparent Mass Increasingly important for lighter aircraft Not trivial to compute, but essentially solved force needed to move air as plate accelerates Circulatory Lift Captures separation effects Need improved models here source of all lift in steady flight Theodorsen, Leishman, 26.

8 Bode Plot of Theodorsen k = πfc C L = π 2 Frequency response output is lift coefficient C L a ḧ + α 2 α Added-Mass input is α ( α is angle of attack) Magnitude (db) +2π α + ḣ + 12 α 12 a Circulatory C(k) Low frequencies dominated by quasi-steady forces High frequencies dominated by added-mass forces Crossover region determined by Theodorsen s function C(k) Phase (deg) p/c=. p/c=.1 p/c=.2 p/c=.3 p/c=.4 p/c=.5 p/c=.6 p/c=.7 p/c=.8 p/c=.9 p/c= w (rad c/u)

9 Wagner s Indicial Response Given an impulse in angle of attack, α = δ(t), the time history of Lift is CL(t) δ The response to an arbitrary input C L (t) = t α(t) is given by linear superposition: C δ L(t τ)α(τ)dτ = C δ L α (t) t τ 1 τ 2 τ 3 t!'#$()*+,*)#&")* y δ (t t )!"#$% u(t) Given a step in angle of attack, α = δ(t), the time history of Lift is C S L(t) and the response to an arbitrary input α(t) is given by: y δ (t τ 1 ) C L (t) =C S L(t)α() + t C S L(t τ) α(τ)dτ τ 1 y δ (t τ 2 ) Wagner, Model Summary Reconstructs Lift for arbitrary input Linearized about α = Based on experiment, simulation or theory convolution integral inconvenient for feedback control design τ 2 τ 3 y δ (t τ 3 ) y(t) =y δ u Leishman, 26. &$%#$%

10 Reduced Order Wagner fast dynamics α!"#$%&$'(#)*+,+#))()+-#$$ C L α C L α s + C L x α α k+1 A ERA x = 1 t α 1 α x C L (k t) = C ERA C Lα C L α α α ERA Model k + B ERA t quasi-steady and added-mass k α k input + D ERA α k C Lα s 2 G(s) Brunton and Rowley, in preparation..#$'+)*/#-%$ Model Summary Linearized about α = Based on experiment, simulation or theory Recovers stability derivatives C Lα,C L α,c L α associated with quasi-steady and added mass ODE model ideal for control design

11 Bode Plot - Pitch (LE) 6 4 Magnitude (db) Frequency response input is α ( α is angle of attack) output is lift coefficient C L Pitching at leading edge Phase (deg) QS+AM (r=) ERA r=2 ERA r=3 ERA r=4 ERA r=7 ERA r=9 Model without additional fast dynamics [QS+AM (r=)] is inaccurate in crossover region Models with fast dynamics of ERA model order >3 are converged Frequency (rad U/c) Punchline: additional fast dynamics (ERA model) are essential Brunton and Rowley, in preparation.

12 Bode Plot - Pitch (QC) Frequency response input is α ( α is angle of attack) output is lift coefficient C L Pitching at quarter chord Magnitude (db) Quarter-Chord Pitching 4 Reduced order model with ERA r=3 accurately reproduces Wagner Wagner and ROM agree better with DNS than Theodorsen s model Asymptotes are correct for Wagner because it is based on experiment Model for pitch/plunge dynamics [ERA, r=3 (MIMO)] works as well, for the same order model Phase (deg) Frequency (rad U/c) ERA, r=3 Wagner Theodorsen DNS ERA, r=3 (MIMO) Brunton and Rowley, in preparation.

13 Pitch/Plunge Maneuver Canonical pitch-up, hold, pitch-down maneuver, followed by step-up in vertical position 1.5 Angle (deg) 5 Angle of Attack Vertical Position Height Time CL DNS Wagner ERA, r=3 (MIMO) ERA, r=3 (2xSISO) QS+AM (r=) Time OL, Altman, Eldredge, Garmann, and Lian, 21 Brunton and Rowley, in preparation. Reduced order model for Wagner s indicial response accurately captures lift coefficient history from DNS

14 Summary Reduced order model for Wagner s indicial response - Based on eigensystem realization algorithm (ERA) - Systematic reduced order models based on step-response - Linear input-output system ideal for flight dynamic framework Future Directions Combine ERA models with nonlinear heuristic and POD models - Capture unsteady forces due to vortex shedding and stall Generalize theory to large angle of attack Develop H2 optimal controller to minimize gust disturbance References: Haller, 22 Shadden et al., 25 Leishman, 26. Brunton and Rowley, 21 Wagner, Theodorsen, Brunton and Rowley, AIAA ASM 29 Brunton and Rowley, in preparation. OL, Altman, Eldredge, Garmann, and Lian, 21