AAE556 Lecture04Divergence2011

AAE 556 Aeroelasticity Lecture Reading: 2.8-2.12 4-1 Purdue Aeroelasticity Agenda i Review static stability – Concept of perturbations – Distinguish stability from response i Learn how to a stability analysis i Find the divergence dynamic pressure using a “perturbation” analysis 4-2 Purdue Aeroelasticity Perturbed 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 4-3 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 4-4 Purdue Aeroelasticity Lα αo The DOF divergence condition i ( KT − qSeCLα ) ( ∆θ ) = Neutral stability KT = qD SeCLα i KT qD = SeCLα or 4-5 Purdue Aeroelasticity Observations i The equation for neutral stability is simply the usual static equilibrium equation with right-hand-side (the input angle αo) set to zero i The neutral stability equation describes a special case – – only deformation dependent external (aero) and internal (structural) loads are present these loads are “self-equilibrating” without any other action being taken 4-6 Purdue Aeroelasticity Stability investigation i Take a system that we know is in static equilibrium (forces and moments sum to zero) K h   i  Kh   i i  h    − qSC Lα K T  θ  0 0  − 1  h  − 1 0   = qSC Lα α o   + qScCMAC   e  θ  e 1 Perturb the system to move it to a different, nearby position (that may or may not be in static equilibrium)   h + ∆h    − qSC Lα KT  θ + ∆θ  0 0  −1  h + ∆h  (?)  −1   = qSCLα α o   + qScCMAC e  θ + ∆θ  e Is this new, nearby state also a static equilibrium point?  Kh   0  − qSCLα  KT  0 0  −1   ∆h  (?) 0  =    e    ∆θ  0  Static equilibrium equations for stability are those for a self-equilibrating system Purdue Aeroelasticity 0    1  Neutral stability i Neutral stability is only possible if the system is “self-equilibrating.”  Kh   0  − qSCLα  KT  0 0  −1   ∆h  0  =    e   ∆θ  0  i The internal and external loads created by deformation just balance each other i The system static stiffness is zero i We’ll see that this requires that the system aeroelastic matrix become singular (the determinant is zero) Purdue Aeroelasticity The deformations at neutral stability are eigenvectors of the problem i At neutral stability the deformation is not unique (∆θ is not zero - can be plus or minus with indeterminate amplitude) i At neutral static stability the system has many choices (equilibrium states) near its original equilibrium state – wing position is uncontrollable - it has no displacement preference when a load is applied 4-9 Purdue Aeroelasticity For stability, only system stiffness is important This graph shows where the equilibrium point for twist is located M shear center M structure = KT θ Structural Aero overturning resistance M aero = qSeCLα ( α o + θ ) Slope depends on qSCLa Equilibrium point twist θ 4-10 Purdue Aeroelasticity When we perturb the twist angle we move to a different position on the graph One of the moments will be larger than the other/ M structure = KT θ M shear ∆θ center M aero = qSeCLα ( α o + θ ) Equilibrium point twist θ 4-11 Purdue Aeroelasticity The slope of the aero line is a function of dynamic pressure so the line rotates as speed increases This is a plot of the lines right at divergence M aero = qDiv SeCLα ( α o + θ ) M shear Lines are parallel M structure = KT θ center The equilibrium point lies at infinity twist θ 4-12 Purdue Aeroelasticity When the dynamic pressure is larger than the divergence dynamic pressure the crossing point is negative This is mathematics way of telling you that you are in trouble M shear M aero = qSeCLα ( α o + θ ) center M structure = KT θ twist θ 4-13 Purdue Aeroelasticity Let’s examine how aeroelastic stiffness changes with increased dynamic pressure (K T ) − qSeCLα θ = Le + M AC = M SC The standard definition of stiffness is as follows ∆M SC ∂ M SC = = K effective = K e ∆θ ∂θ M sc K effective = KT − qSeCL α twist θ Aeroelastic stiffness decreases as q increases 4-14 Purdue Aeroelasticity As we approach aeroelastic divergence we get twist amplification i Consider the single degree of freedom typical section and the expression for twist angle with the initial load due to αo i neglect wing camber qSeCL α o qα o θ= = KT ( − q ) − q α 4-15 Purdue Aeroelasticity Write this expression in terms of an infinite series qαo θ= 1− q ∞    n qα o  ÷ = qα o 1 + q + q + q + = + ∑ q ÷ n =1    1− q  4-16 Purdue Aeroelasticity The first term is the uncorrected value of twist angle with no aeroelasticity θ = qα o ( + q + q + ) Plot the relative sizes of terms with qbar=0.5 0.75 q bar = 0.5 0.5 the sum of the infinite series is 0.25 4-17 Purdue Aeroelasticity Let’s take a look at the series and explain it as an aeroelastic feedback process θ = qα o ( + q + q + ) θo is the twist angle with no aero load/structural response "feedback" θo = qSeC L α o α KT 4-18 Purdue Aeroelasticity Write the series slightly differently θo = qSeC L α o α KT θ = θ o ( + q + q + ) θ = θ o + qθ o + q θ o + 4-19 Purdue Aeroelasticity The second term is the response to the first term θ1 = q θ o = qSeC L θ o α KT This is the response to angle of attack θo instead of αo …and, the third term θ = q θ o = q θ1 4-20 Purdue Aeroelasticity Conclusion Each term in the series represents a feedback "correction" to the twist created by load interaction ∞ θ = θ + ∑θ n n =1  n θ = θ o 1 + ∑ q   n =1  ∞ Series convergence q