AAE556Lecture03Stability in Aircraft

AAE556 Static Stability “Adde parvum parvo magnus acervus erir” Ovid (“add little to little and there will be a big pile.”) Purdue Aeroelasticity Lecture – summary i Aeroelasticity is concerned with interactions between aerodynamic forces and structural deformation i Develop simple static aeroelastic model with pitch (torsion) and plunge (bending) – Section 2.4 Purdue Aeroelasticity Reading topics i 2.6 Lifting generation-flexible surface i 2.7 Example problem – work it through by hand i 2.8 Using simple results i 2.9 Load factor i 2.10 Simple model – degree-of-freedom-emphasis on stiffness, not strength i 2.10.2 – Stability definition – essential i 2.11 Example problem using perturbation concept i 2.12 Analysis example showing when stability is obvious and when it is not i 2.13 Compressibility Purdue Aeroelasticity Aero/structural interaction model i Requirements - L = qSC L (α o + θ ) – simplicity – manageability – realism α airspeed lift e Torsion spring KT θ TYPICAL SECTION One degree of freedom Purdue Aeroelasticity Lift and the aeroelastic parameter   α o + qScCMAC KT  L = qSC L  α qSeC L α  1−  KT         q= qSeC L α KT Purdue Aeroelasticity Lift equation with wing flexibility L = qSCLα α o + qSCL0 CL = α C Lα 1− q CL0  C MAC = q   1− q Purdue Aeroelasticity  c     e  Two degree of freedom aeroelastic model (Section 2.4) Goal - add bending deformation (plunge) to the simple dof model Airspeed, V Displacement, h, plunge at the shear center Plunge is resisted by twist, θ spring, Kh Twist is resisted by spring, KT +h Purdue Aeroelasticity Static equilibrium equations Forces, moments and the importance of mechanics to the effort The problem unknowns are h and θ +h Sum forces Sum torsional moments about shear center Kh 0   h   − L   =   KT  θ  M SC  Structural stiffness matrix Loads, measured at shear center Purdue Aeroelasticity Write the aerodynamic loads in terms of h & θ We use matrix methods – that’s our theme Idealized wing section lift Twisting moment, at wing shear center, positive nose-up L = qSC Lα (α + θ ) M SC = M AC + Le M SC = qcSCMAC + qSeC Lα (α + θ ) −L  0 − 1 h  − 1 0   = qSC Lα    + qSC Lα α o   + qScCMAC    0 e  θ  e 1 M SC  Purdue Aeroelasticity Aeroelastic static equilibrium equation Introducing the aeroelastic stiffness matrix constructed out of thin air Wing static equilibrium written in terms of unknown displacements, h & θ Kh 0   h  0 − 1  h  − 1 0   = qSC Lα    + qSC Lα α o   + qScCMAC     Kθ  θ  0 e  θ  e 1 Kh 0   h  0 − 1 h  − 1 0   − qSC Lα    = qSC Lα α o   + qScCMAC     K T  θ  0 e  θ  e 1 Aeroelastic stiffness matrix  Kh 0   − qSCL  α KT  10 Purdue Aeroelasticity 0 −1 0 e    Solution for wing deflections, h & θ Divide by KT to get nondimensional terms qSC Lα Kh   KT KT  h  qSC Lα α o =     qSeC Lα θ KT     1− KT   Invert 2x2 matrix Get BHM  K  TK h h  qSC Lα α o  =   KT  θ     − qSC Lα − 1 qScCMAC  + KT e  Kh   qSeCLα  − 1 qScCMAC 1− KT    + KT  e   qSeCLα 1− KT   K  TK h       11 Purdue Aeroelasticity − qSC Lα 0   1  Kh   qSeC Lα  0 1− KT     1  qSeC Lα 1− KT  Wing displacements qSC Lα α o plunge h= +h 1− Kh qSeC Lα KT qScCMAC − KT  qSC Lα  Kh   − qSeC Lα  KT  qScCMAC + KT        − qSeC Lα    K T   qSeC Lα α o twist θ= 1− KT qSeC Lα KT 12 Purdue Aeroelasticity       New goals i Define structural static stability – Concept of perturbations – Distinguish stability from response i Learn how to stability analysis i Find the wing divergence dynamic pressure using a “perturbation” analysis 13 Purdue Aeroelasticity Math Summary Kh   i  h    − qSC Lα  K T  θ  0 0  − 1  h  − 1 0 = qSC α + qScC    Lα o MAC   e  θ  e   1 Static equilibrium plays an essential role in aeroelastic analysis (surprise, surprise…) – Static equilibrium equations are statically indeterminate (equilibrium depends on knowledge of force/deflection relationship) – Multi-degree-of-freedom systems have as many equations of equilibrium as degrees of freedom i Systems of simultaneous equations can be written (and solved) in matrix form i Static equilibrium aeroelastic equations yield two important matrices – – – Structural stiffness matrix – symmetrical if you it right Aerodynamic stiffness matrix – aero people will not recognize this term These matrices are added together to form the aeroelastic stiffness matrix 14 Purdue Aeroelasticity Euler’s static stability criterion i "A system in static equilibrium is neutrally (statically) stable if there exist nearby static equilibrium states in addition to the original static equilibrium state.” i Stability - the tendency of a system (structural configuration) to return to its original equilibrium state when subjected to a small Leonard Euler disturbance (perturbation) 1707-1783 Advisor-Bernoulli Student-LaGrange “Read Euler, read Euler, he is our master in everything" Laplace 15 Purdue Aeroelasticity The perturbed structure i Static stability analysis considers what happens to a flexible system that is in static equilibrium and is then disturbed – If the system tends to come back to its original, undisturbed position, it is stable - if not - it is unstable i We need to apply these above words to equations so that we can put the aeroelastic system to a mathematical test 16 Purdue Aeroelasticity Stability investigation Kh    h    − qSC Lα  K T  θ  0 0  − 1  h  − 1 0 = qSC α + qScC    Lα o MAC   e  θ  e   1 i Given a system that we know is in static equilibrium (forces and moments sum to zero) i Add a disturbance to perturb the system to move it to a different, nearby position (that may or may not be in static equilibrium)  Kh  i i   h + ∆h    − qSCLα KT  θ + ∆θ   −1  h + ∆h   −1  e  θ + ∆θ  = qSCLα α o  e  + qScCMAC  equilibrium   point?    Is this new, nearby state also a static Write static equilibrium equations and see if forces and moments balance 17 Purdue Aeroelasticity 0    1  Perturbed dof airfoil i In flight this airfoil is in static equilibrium at the fixed angle θ but what happens if we disturb (perturb) it? ∆L = qSCLα ( ∆θ ) lif t + perturbation lif t ∆θ αo+θ MS=KT(θ+∆θ) torsion spring KT V i There are three possibilities 18 Purdue Aeroelasticity Example i Perturb the airfoil when it is in static equilibrium i To be neutrally stable in this new perturbed position this equation must be an true (K T ) ( − qSeCL θ + KT − qSeCL α α ) ( ∆θ ) = qSeC 19 Purdue Aeroelasticity Lα αo Perturbation possibilities i KT(∆θ)>(∆L)e – – i T − qSeCL α no static equilibrium in the perturbed state KT(∆θ)