Lectures 22, 23 Typical dynamic instability problems and test review

AAE 556 Aeroelasticity Lectures 22, 23 Typical dynamic instability problems and test review Purdue Aeroelasticity 22-1 How to recognize a flutter problem in the making Given: a DOF system with a parameter Q that creates loads on the system that are linear functions of the displacements  M1     &&x1   K1  + M   &&x2   0   x1   = Q    K   x2   p21   x1     x1     =  ei ωt   x 2   x   p12   x1      x2  ( ) K = M1 ω 22 = ω − ω ω 12 + ω 22 + ω12ω 22 = Q is a real number If p12 and p21 have the same sign (both positive or both negative) can flutter occur? ω12   −ω + ω 12     Q  − M p21    ( ( ) ∆ = −ω +ω 2 )( K2 M2  Q   − p12       M1   x  = 0   x  0  −ω + ω 22      ( ) ) Q=0 Q not zero Q2 −ω +ω − p12 p21 = M 1M 2 2 The modified determinant  M1 −ω     x1   K1  +   M   x2   ( ∆ = − ω + ω12 ω12 = 2 ω + ω 2 ωn = ± 2 )(   x1    −Q  K   x2   p21 p12   x1  0  r  =      x2  0  ) Q − ω + ω22 − p12 p21 = M 1M K1 M1 ω 22 (ω −ω K2 = M2 ) 2 Q2 +4 p12 p21 M 1M If flutter occurs two frequencies must merge 2 ω + ω 2 ωn = ± 2 (ω −ω ) 2 Q2 +4 p12 p21 M 1M FLUTTER – Increasing Q must cause the term under the radical sign to become zero K1 ω1 = ω 22 = (ω M1 K2 M2 Q =− −ω ) Q2 =− p12 p21 M 1M ??? 2 ( M M ω 12 − ω 22 p12 p21 ) p12 p21 = − ( M M ω 12 − ω 22 ) 4Q For frequency merging flutter to occur, p12 and p21 must have opposite signs If one of the frequencies can be driven to zero then we have divergence ( ∆ = −ω +ω 2 ωn = )( ) Q2 −ω +ω − p12 p21 = M 1M 2 2 ( )( ) ω 12 ω 22 ( )( ) Q2 = p12 p21 M 1M KK Q = p12 p21 2 ∆ = = ω1 ω2 Q2 − p12 p21 M 1M M M 2ω 12ω 22 Q = p12 p21 p12 p21 = M M 2ω12ω 22 Q2 Divergence requires that the cross-coupling terms have the same sign Aero/structural interaction model TYPICAL SECTION What did we learn? L = qSCL α (αo + θ ) V lift e θ GJ KT ∝ span torsion spring KT  qScCMAC   αo +  K T  L = qSCLα   − qSeC Lα    K   T Divergence-examination vs perturbation L= 1− qSCLα qSeCLα αo + KT Kh  1− qSCLα qSeC Lα KT  qScC  MAC    K   T h   − L   =   KT θ  MSC  ∞ = + q + q + q + = + ∑ q n 1− q n=1 Perturbations & Euler’s Test V KT (∆θ ) > (∆L)e lift e θ torsion spring KT .result - stable - returns -no static equilibrium in perturbed state KT ( ∆θ ) < ( ∆L)e result - unstable -no static equilibrium - motion away from equilibrium state KT ( ∆θ ) = ( ∆L)e result - neutrally stable - system stays - new static equilibrium point Stability equation is original equilibrium equation with R.H.S.=0 ∆θ ≠ V θ lift e torsion spring KT (KT − qSeC Lα )= KT = The stability equation is an equilibrium equation that represents an equilibrium state with no "external loads" – Only loads that are deformation dependent are included The neutrally stable state is called self-equilibrating Multi-degree of freedom systems A 2KT 3KT panel panel e b/2 V 5 KT   −2 A b/2 αο + θ2 αο + θ1 shear centers aero centers From linear algebra, we know that there is a solution to the homogeneous equation only if the determinant of the aeroelastic stiffness matrix is zero view A-A −2 θ1   −1  θ1  1   + qSeC Lα    = qSeC Lα αo    θ2  −1   θ  1 Three different definitions of roll effectiveness • Generation of lift – unusual but the only game in town for the typical section • Generation of rolling moment – • contrived for the typical section – reduces to lift generation • Multi-dof systems – this is the way to it • Generation of steady-state rolling rate or velocity-this is the information we really want for airplane performance • Reversal speed is the same no materr which way you it Control effectiveness  q  c  CM δ   1+    qD e CLδ   L = qSCL δ δ o =0 q 1− qD q  c  CM δ 1+ =0   qD e CLδ KT  CL δ   qR = −  ScCLα  CMδ  Lift α0+ θ V MAC t orsion spring KT shear center e δ0 reversal is not an instability - large input produces small output opposite to divergence phenomenon Steady-state rolling motion  qScCMδ v   L = = qSCLα δo − + qSC Lα δ o  KT V Lift α0+ θ V MAC t orsion spring KT shear center e δ0 Swept wings α structural= θ − φ tan Λ qn = qcos2 Λ K1 d f K2 αo Λ V C V cosΛ A b c B A   Kφ    −tb  − Q Kθ   −te b  φ b   Qα o     θ  = cosΛ   e   e B C Divergence bt   ∆ = Kθ Kφ + Q Kθ − Kφe   Kθ  e   c   Kφ  tan Λ crit = 2    c b  Kθ  2.0 nondimensional divergence dynamic pressure Seao qD =   b  K  tan Λ  cos Λ  1−   θ   e K    φ  nondimensional divergence dynamic pressure vs wing sweep angle 1.5 sweep back sweep f orward 1.0 5.72 degrees 0 -0 -1.0 b/c=6 e/c=0.10 Kb/Kt=3 -1.5 -2.0 -90 -75 -60 -45 -30 -15 15 30 sweep angle (degrees) 45 60 75 90 Lift effectiveness lift eff ect iveness vs dynamic pressure 2.0 lif t ef f ect iveness unswept wing 1.5 unswept wing divergence 1.0 15 degrees sweep 30 degrees sweep 0 50 10 150 20 250 dynamic pressure (psf ) 30 350 Flexural axis x ref e r enc e ax is Λ θ E = θ − φ tanΛ β y Flexural axis - locus of points where a concentrated force creates no stream-wise twist (or chordwise aeroelastic angle of attack) θE = The closer we align the airloads with the flexural axis, the smaller will be aeroelastic effects How to recognize a flutter problem in the making Given: a DOF system with a parameter Q that creates loads on the system that are linear functions of the displacements  M1     &&x1   K1  + M   &&x2   0   x1   = Q    K   x2   p21   x1     x1     =  ei ωt   x 2   x   p12   x1      x2  ( ) K = M1 ω 22 = ω − ω ω 12 + ω 22 + ω12ω 22 = Q is a real number If p12 and p21 have the same sign (both positive or both negative) can flutter occur? ω12   −ω + ω 12     Q  − M p21    ( ( ) ∆ = −ω +ω 2 )( K2 M2  Q   − p12       M1   x  = 0   x  0  −ω + ω 22      ( ) ) Q=0 Q not zero Q2 −ω +ω − p12 p21 = M 1M 2 2 If flutter occurs two frequencies must merge 2 ω + ω 2 ωn = ± 2 (ω −ω ) 2 Q2 +4 p12 p21 M 1M FLUTTER – Increasing Q causes the term under the radical sign to be zero K1 ω1 = ω 22 = ( M1 ω 12 K2 M2 Q =− − ω 22 ) Q2 = −4 p12 p 21 M1 M ( M M ω 12 − ω 22 p12 p21 ) p12 p21 = − ( M M ω 12 − ω 22 ) 4Q For frequency merging flutter to occur, p12 and p21 must have opposite signs If one of the frequencies is driven to zero then we have divergence ωn =  M1     &&x1   K1  +  M   &&x2   0   x1   = Q    K   x2   p21 ( )( ) 2 ∆ = = ω1 ω2 ( )( ) ω 12 ω 22 Q2 = p12 p21 M 1M KK Q = p12 p21 p12   x1      x2  Q2 − p12 p21 M 1M M M 2ω 12ω 22 Q = p12 p21 p12 p21 = M M 2ω12ω 22 Q2 Divergence requires that the cross-coupling terms are of the same sign Fuel line flutter A hollow, uniform-thickness, flexible tube has a mass per unit length of m slugs/ft and carries liquid fuel with density ρ to a rocket engine The fuel flow rate is U ft/sec through a pipe cross-section of A The tube is straight and has supports a distance L apart, the tube bending displacement is approximated to be  πy   2π y  w (y,t ) = φ1 sin  + φ sin  L L  φ1 φ2 Unknown amplitudes of vibrational motion The free vibration frequencies when the fluid is not flowing are:  π  EI ω =  L mo  2π EI ω 22 =   L mo mo = m + ρA Fluid flow creates system coupling, but through the velocity, not the displacement  ρ AU && φ1 −   mo L  &  ρ AU  π   φ1 = ÷φ2 +  ω1 −  ÷÷ ÷ mo  L      ρ AU && φ2 −   mo L  &  ρ AU  2π   φ2 = ÷φ1 +  ω2 −  ÷÷ mo  L  ÷    Find the divergence speed Estimate the flow speed that flutter occurs, if it occurs Divergence is found by computing the determinant of the aeroelastic stiffness matrix ∆ aesm  ρ AU φ&&1 −   mo L  &  ρ AU  π   φ1 = ÷φ2 +  ω1 −  ÷÷ ÷ m L    o    ρ AU φ&&2 −   mo L  &  ρ AU  2π   φ2 = ÷φ1 +  ω2 −  ÷÷ ÷ m L    o    ρAU  π   ρAU  2π    ω2 − =0 =  ω1 −        m L m L     o o      L   U1 = ρA  π  mo ω12   L   U2 =  ρA  2π  mo ω 22  π  EI U div =    L  ρA Assume that coupling leads to flutter and find an estimate of the merging point  ρ AU  π   φ&&1 +  ω1 − φ1 =  ÷÷  ÷ mo  L     ρ AU  2π 2  φ&&2 +  ω2 − φ2 =  ÷÷  ÷ mo  L    Harmonic motion?  2   2  2  ρ AU π ρ AU π    2  −ω + ω − m  L   −ω + ω − m  L   =  o    o    The frequencies are approximated  2   2  2  π  2π   2 ρAU ρAU      = − ω + ω − − ω + ω −       mo  L    mo  L      ω = ω 12 2 ρAU π    − mo  L   ρAU 2π 2   ω = ω2 − mo  L  2 2 ρAU  π  ρAU  2π  ω −   ≅ ω2 −   mo  L  mo  L  F 2 F 2  π  EI mo L U ≅ ω2 − ω1 = 5  ρA π  L  ρA F ( )