AAE556 Lecture 33 V-g Method revisited Purdue Aeroelasticity Final EOM’s for forced response h  Fo A   + B θ = mb b ω is known because we pre-select it ω Lh 2 A=− − ω + ωh µ ω B = −ω xθ − µ 2  1    Lα −  + a  Lh      Purdue Aeroelasticity Moment equilibrium equation h  D  ÷+ Eθ = b 2 ω ω 1  D = −ω xθ − Mh + Lh  + a  µ µ 2  2 ω ω   E = −ω rθ2 + ωθ2 rθ2 + Mh + a − Mα µ 2  µ ω2   ω 1  + Lα  + a  − Lh  + a  µ 2  µ 2  Purdue Aeroelasticity Theodorsen’s method The system is self-equilibrating h  Fo A  ÷+ Bθ = =0 mb b ω Lh ω ω =− − + ω µ ω ω ATM BTM ω2 ω2 = − xθ − ω µω h  1    Lα −  + a ÷Lh      Purdue Aeroelasticity Moment equilibrium equation h  D  ÷+ Eθ = b ω ω ω 1  = − xθ − Mh + L + a÷ 2 h ω µω µω 2  DTM ETM 2 ω 2 ωθ2 ω 1  ω = − rθ + rθ + M h  + a ÷− M α ω ω ω µ 2  ω µ ω 1  ω 1  + Lα  + a ÷− Lh  + a ÷ ω µ 2  ω µ 2  2 Purdue Aeroelasticity Eigenvalue Equation of Motion #1  h  h h ω 1   2 −ω − ω xθθ + ωh −  Lh +  Lα −  + a ÷Lh  θ b b µ  b  2   Divide by ω  ÷=  ωh2 h  h  h 1   − − xθ θ + −  Lh +  Lα −  + a ÷Lh  θ b ω b µ b  2    ÷=  Include structural damping  ωh2  h h 1 h  1   − − xθ θ +  ÷( + ig ) −  Lh +  Lα −  + a ÷Lh  θ b b µ b  2   ω  Purdue Aeroelasticity  ÷=  Equation #2, moment equilibrium    h ω h 2 2 −ω xθ  ÷− ω rθ θ + ωθ rθ θ −  M θθ θ + M θ h ÷ = µ  b b M θθ 1  1  = M α −  + a ( Lα + M h ) +  + a  Lh 2  2  Divide by ω Mθ h 1  = −  + a ÷Lh 2  h  ωθ2 1 h − xθ  ÷− rθ θ + rθ θ −  M θθ θ + M θ h ÷ = ω µ b b Include structural damping h  ωθ2 1 h − xθ  ÷− rθ θ + ( + ig ) rθ θ −  M θθ θ + M θ h ÷ = ω µ b b Purdue Aeroelasticity Matrix equations  ωh2  h  h   ωθ2  h 1   − − xθ θ +  ÷( + ig )  ÷ −  Lh +  Lα −  + a ÷Lh  θ b 2   ω   ωθ  b µ  b   ÷=  h  ωθ2 1 h − xθ  ÷− rθ θ + ( + ig ) rθ θ −  M θθ θ + M θ h ÷ = ω µ b b     ω h  ωθ   ÷  h b   −  ÷( + ig )  ωθ     + x ω   2   θ   θ r θ     Lh +  µ  M θ h xθ  h  b 2 rθ   θ     1    h  L − + a ÷Lh    b  0   α 2      =    0  θ      M θθ  Purdue Aeroelasticity The eigenvalue problem  ωh2   ω   ÷  h b   −  +  ÷( + ig )  ωθ   ω  θ   xθ 2   rθ   θ   Lh +  µ  M θ h  ωθ2    h   ω ÷   Ω  b  =  h   x    θ    θ  rθ    xθ   h  b 2 rθ   θ     1    h  L − + a ÷Lh    b  = 0   α 2           θ    M θθ  xθ   Lh + rθ  µ   M θ h  1     h   Lα −  + a ÷Lh     b           θ  M θθ Purdue Aeroelasticity Another look at it This should be easy for a  ωθ2    h   ω ÷   Ω  b  =  h   x    θ    θ  rθ     xθ   Lh + rθ  µ   M θ h th grader with MATLAB  1     h   Lα −  + a ÷Lh     b           θ  M θθ   ωθ2   Lh   ωθ2   Lα Lh     −  + a ÷÷  1 + ÷  ÷ xθ + h    ωh ÷ µ   ωh   µ µ 2      h     b Ω  b=   1   Mθ h   M θθ   θ     θ    x + r + ÷  θ   rθ  θ µ ÷ r µ     θ  10 Purdue Aeroelasticity The flutter problem – a complex eigenvalue with flutter frequency and airspeed unknown a = elastic axis location (shear center) ωh = bending-torsion frequency ratio ωθ S xθ = θ = dimensionless static unbalance mb rθ = dimensionless radius of gyration about SC µ =density ratio ω = frequency k=reduced frequency 11 Purdue Aeroelasticity