AAE556 v g flutter lecture 32 The Vg method

AAE 556 Aeroelasticity The V-g method g k decreasing V/bωθ mode mode flutter point Airfoil dynamic motion Ma e P=-L θ(t) V xθ aero K θ center T Kh h This is what we’ll get when we use the V-g method to calculate frequency vs airspeed and include Theodorsen aero terms 1.6 1.4 Frequency Ratio (ω / ω ) θ 1.2 0.8 0.6 0.4 0.2 0 0.5 1.5 2.5 Velocity (V/ ω b) θ 3.5 4.5 When we the V-g method here is damping vs airspeed 0.8 0.6 0.4 flutter 0.2 g divergence -0.2 -0.4 -0.6 -0.8 -1 0.5 1.5 2.5 Velocity (V/ ω b) θ 3.5 4.5 To create harmonic motion at all airspeeds we need an energy source or sink at all airspeeds except at flutter i i Input energy when the aero damping takes energy out (pre-flutter) Take away energy when the aero forces put energy in (post-flutter) 2D airfoil free vibration with everything but the kitchen sink &   h && && Mh + Mxθ θ + K h ( g h + g ) + h  = P = − Leiωt ω   ( −ω M + K h ) 1 + i ( g h + g )  h − ω Mxθ θ = P &   θ && && Iθ θ + Mxθ h + Kθ ( gθ + g ) + θ  = M a = M a eiωt ω   ( −ω I θ ) + Kθ 1 + i ( gθ + g )  θ − ω Mxθ h = M a We will get matrix equations that look like this  A B  h / b  0  =     D E   θ  0  m µ= πρ b …but have structural damping that requires that … A(k, ω , g)E(k, ω , g) − B(k)D(k) = The EOM’s are slightly different from those before (we also multiplied the previous equations by µ) B  h / b  0  Each term contains inertial, structural stiffness, structural =     E   θ  0  damping and aero information A D A = µ{1− (ω / ω )[1 + i(gh + g)]} + Lh h B = µ x θ + Lα =- Lh (1 / + a) θ 1  D =µxθ +M h −Lh  +a ÷ 2  E = µ r {1 − ( ω / ω )[1 + i(gθ + g)]} θ − Mh (1 / + a) + Mα − Lα (1 / + a) + Lh (1 / + a) Look at the “A” coefficient and identify the eigenvalue – artificial damping is added to keep the system oscillating harmonically   ωh   A = µ 1 −  ÷ + i ( g h + g fake )  + Lh ω     ( ) We change the eigenvalue from a pure frequency term to a frequency plus fake damping term So what?   ωh 2  ωθ   A = µ 1 −  + ig fakier )  + Lh ( ÷  ÷ ωθ   ω     Ω = (ω / ω )(1 + ig) = Ω + iΩ 2 θ 2 R I The three other terms are also modified  A B   h / b  0  D E   θ  = 0  Each term contains inertial, structural stiffness, structural damping and aero information B = µ x θ + Lα =- Lh (1 / + a) D =+ µ x θ + Mh − L h (1/ + a)   ω   θ E = µ rθ 1 −  ÷ ( + ig )  ω     1  1  1  − M h  + a ÷+ M α − Lα  + a ÷+ Lh  + a ÷ 2  2  2  To solve the problem we input k and compute the two values of Ω2 2  ωθ   ωθ  Ω =  ÷ + ig  ÷ = Ω 2R + iΩ 2I ω  ω  Ω = (Ω ) + i(Ω ) 2 R I Ω = (Ω ) + i(Ω ) 2 R 2 I The value of g represents the amount of damping that would be required to keep the system oscillating harmonically It should be negative for a stable system ω = ω θ / (Ω R )1 g1 = (Ω ) / (Ω ) I R ω = ω θ / (Ω R ) g2 = (Ω 2I )2 / (Ω 2R )2 Now compute airspeeds using the definition of k V1 = bω / k ω = ω θ / (Ω R )1 Remember that we always input k so the same value of k is used in both cases One k, two airspeeds and damping values V = bω / k ω = ω θ / (Ω R ) Typical V-g Flutter Stability Curve g ' = g h + g = gθ + g gh ≈ gθ k decreasing g V/bωθ mode flutter point mode Ω = (ω / ω )(1 + ig′ ) 2 θ Now compute the eigenvectors V1 = bω / k h 2 (bθ / h)1 = −D / E(Ω1 ) ; = (Ω = Ω ) b V = bω / k (h / b θ )2 = − B / A(Ω ) ; θ = (Ω = Ω 22 ) Example Two-dimensional airfoil mass ratio, µ = 20 quasi-static flutter speed VF = 160 ft/sec gθ = g h = 0.03 b = 3.0 ft Example k = 0.32 / k = 3.1250 ω h = 10 rad / sec ωθ = 25 rad / sec Lα = −13.4078− i3.7732 Lh = −0.10371− i40973 Mα = 0.37500 − i3.1250 Mh = 0.50000 The determinant k = 0.32 A = 19.896 − i4.0973 − 3.2Ω B = −11.3767 − i2.5440 D = 2.5311+ i1.22919 E = 9.2380 − i2.3618 − 5.0Ω A E − BD = 16(Ω) + (−129.043+ i28.044)Ω2 + 199.794 − i64 418= Final results for this k value – two g’s and V’s b = 3.0 ft Ω = 4.0326 − i0.87638± 3.0067 − i3.0420 Ω − 4.0326 − i0.87638± (1.9084− i0.79702) Ω12 = 5.9410 − i1.67340 ω = 10.257 rad / sec (ω h = 10 rad / sec) Ω 22 = 2.1242− i0.07936 V1 = 96.157 ft / sec g1 = g + gθ = −0.2817 ω = 17.153 rad / sec (ω θ = 25 rad / sec) V = 160.810 ft / sec g = g + gθ = −0.0374 Final results Flutter g = 0.03