Aeroelastic-Acoustics Simulation of Flight Systems K. K. Gupta 1 NASA-DRFC, Edwards, CA S. B. Choi 2 Advanced Engineering Solutions, Inc., Ormond Beach, FL A. Ibrahim 3 Norfolk State University, Norfolk, VA 1 Research Engineering Directorate, Associate Fellow, AIAA 2 Research Engineer, Member, AIAA 3 Professor, Department of Engineering, Member, AIAA 11 Introductory remarks • Paper describes a numerical FE based aeroelastic-acoustics analysis methodology • This is followed by solution details of a 2-D airfoil and results correlation with known theoretical solution • Also presented the 3-D wing case and related results pertaining to vibration, steady and unsteady flow (CFD), aeroelastic and aeroelastic-acoustic simulations • Further associated solution results are presented for a numerically simulated unsteady pressure data • Also SPL results from a SOFIA flight sensor data is presented in some detail • Provides a discussion on implementation of these techniques in an existing FE software suitable for solution of complex, practical problems • Discussions and concluding remarks 22 Introductory remarks • Paper describes a numerical FE based aeroelastic-acoustics analysis methodology • This is followed by solution details of a 2-D airfoil and results correlation with known theoretical solution • Also presented the 3-D wing case and related results pertaining to vibration, steady and unsteady flow (CFD), aeroelastic and aeroelastic-acoustic simulations • Further associated solution results are presented for a numerically simulated unsteady pressure data • Also SPL results from a SOFIA flight sensor data is presented in some detail • Provides a discussion on implementation of these techniques in an existing FE software suitable for solution of complex, practical problems • Discussions and concluding remarks 33 Numerical Formulation • Structural (FE) free vibration matrix equation solving for ω and φ && 0 Mu+ Ku = (1) • The aerodynamic data are next computed by solving the Navier-Stokes (FE) equation ∂ν ∂f ∂g j j + + = f j =1,2,3 (2) b ∂t ∂x ∂x j j in which [ ]T ν = ρ ρu ρE i =1,2,3 i • Vehicle equation of motion is then cast into the frequency domain ˆ && ˆ & ˆ ( ) ( ) Mq +Cq + Kq + f t + f t = 0 (3) a I where Mˆ (= ΦTMΦ) is the generalized mass matrix and similarly K ˆ , C ˆ ; f ( t ) is the a ( ) aerodynamic load vector; f t being the generalized impulse force vector I 44 Numerical Formulation (cont’d) ( ) • Generalized impulse force vector f t I • Earlier, the CFD code analysis results were verified with flight test data – Hyper-X vehicle and flight data comparison 55 Numerical Formulation (cont’d) • Equation (3) may then be cast in a state-space matrix form as I 0 q&  0 I  q  0   0  0     −     −   −   = (4) 0 I q&& − Mˆ -1Kˆ − Mˆ -1Cˆ  q& − Mˆ -1 f (t) − Mˆ -1 f (t) a I or & ( ) ( ) ( ) x t = A x t + B f t (5) s st s st where  0 I   0  q ( ) ( ) ( ) A = , B = , f t = f t + f t , x =       (6) st − Mˆ -1Kˆ − Mˆ -1Cˆ st − Mˆ -1 a I s q& and ( ) ( ) ( ) y t = C x t + D f t (7) s st s st in which C = I and D = 0 st st 66 Numerical Formulation (cont’d) • In the presence of sensors, for the most general aeroservoelastic case these equations are converted into zero order hold (ZOH) discrete time equivalent at the k-th step: ( ) ( ) ( ) x k +1 =G x k +H f k (8) s s s s ( ) ( ) ( ) y k +1 =C x k +D f k (9) s s s s in which ( ) ( ) ( ) f k = f k + f k a I and [ ][ ] G = eAs∆t H = eAs∆t − I A−1B (10) s s s s where A and B are A and B having been modified to include sensors s s st st 77 Numerical Formulation (cont’d) • Coupled aeroelastic (AE) model 88 Numerical Formulation (cont’d) • Acoustic frequencies are obtained by performing FFT on computed unsteady aerodynamic pressures • Also the sound pressure level (SPL) for a specified node is computed by first fixing a time band, t and then performing the following calculation using n number of sampling points a) compute average pressure ( ) ∑n P = P n (11) avg i=1 i b) compute the root mean square of pressure ∑n ( )2 P = P − P n (12) rms i=1 i avg c) compute the SPL P SPL(dB) = 20log rms (13) 10 P ref where P = 2 0 × 1 0 − 6 P a for air ref 99 Flowchart of aero-elastic-servo-acoustic analysis Input structural/aero data Steady-state Euler/Navier-Stokes solution (local time stepping) Heat conduction analysis Modal analysis φ's,ω's Form G H C . D s s s s (structural state-space matrices) For each of a number of a specified ρ f =ΦT[pA] a a Impulse load f 1 input for t = t thru t s e Form G s H s C s. Ds Save q , q & . (structural state-space matrices) Solve for q , q & (k+1 step) (k th step) Control law to enforce desired control surface motion Update time modified q,q& u&=Φq& u=Φq If u is large, regrid aero Change velocity B.C. at solid boundary feed u ,u&. Unsteady aerodynamic analysis Euler/Navier-Stokes solution (one-step global time stepping) Calculate Acoustic wave frequencies Aeroelastic-Acoustics, SPL Stop 1100