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Models and concepts
Lecture 2 - summary
Models – physical or mathematical? What do we want to learn?
Design development uses models that increase in fidelity as more information and detail becomes available
Modeling and information Features of good models
Low level degrees of freedom – a first step
What do you mean by “fundamental?” Aerodynamic models describe math relationships lift, pitching moment and angle of attack
Terminology – airfoil center of pressure
We’ll use something called the aerodynamic center
Wing twisting is important –we include it by using structural influence coefficients or stiffness
Features of the typical section aeroelastic model
Sum the torsional moments - write them in terms of torsional displacement, q
Solution for twist angle q
Lift equation
Lift expressed a different way
The final equation for lift
Summary-what have we learned?
Nội dung
Models and concepts •How can I understand the problem? What is important and what is not? •How can I discover or identify the fundamental inter-relationships between phenomena? •How can I fix it? x restrict tosmall angle h(t) θ(t) c.g shear center xcg Purdue Aeroelasticity Lecture - summary Aeroelasticity is concerned with interactions between aerodynamic forces and structural deformation – we need models to describe interactions and their effects Features of good models - simple with results that are easy to interpret Aeroelastic models include aerodynamic, inertial and structural features – Structural load/deformation model with bending and twist unswept semi-monocoque wings – Aerodynamic fundamentals – relationship between aero loads and deformation – Aero/structural interaction analysis model – Models are limited but insightful Purdue Aeroelasticity Models – physical or mathematical? What we want to learn? •Vehicle stability? •The origin of unusual features that have not been seen before? •How important is simplicity? Purdue Aeroelasticity Design development uses models that increase in fidelity as more information and detail becomes available AML DRACO OptiStruct Concepts Innovation ASTROS NASTRAN TSO FASTOP FEM II FEM I Many choices Topology Optimization Load Paths Layout Sizing Optimization A well-defined component Structural Model Fidelity, Cost to Change Purdue Aeroelasticity Modeling and information Features of good models Look at how simple models are developed and how they can help us Realistic predictions of physical phenomena – Identify and use significant problem parameters – Discover underlying causes of interactions Minimal math complexity – Algebraic terms for effects (structures, aero ) – Manageability of math task and results – Ease of relating math to experimental results Purdue Aeroelasticity Low level degrees of freedom – a first step x restrict tosmall angle h(t) θ(t) c.g shear center xcg lift, L downward displacement, h spring torque KTθ pitching Moment spring MAC force Khh Purdue Aeroelasticity What you mean by “fundamental?” Aerodynamic models describe math relationships lift, pitching moment and angle of attack lift and pitching moment co-efficients Hypothetical lift curve CL*5 = times the lift coefficient 20 L = qSC L = qSC L α 10 α -10 -20 -20 CL = (0.95343*alpha + 0.0012411*alpha^3 - 0.0000060698*alpha^5 )/5 -10 10 alpha, angle of attack (degrees) M ref = qScCMref 20 angle of attack, α Purdue Aeroelasticity q = ρV 2 Terminology – airfoil center of pressure Choose an origin (point “o” is usually at the leading edge) locate a position x, sum moments L = qSC L = qSC L α α Lx + M o = M ref center of pressure definition L MO x Center of pressure M ref = − Mo xcp = = f (α ) L Purdue Aeroelasticity We’ll use something called the aerodynamic center Lx + M o = M ref Compute change in aero moment with respect to angle of attack ∂ M o ∂M ref ∂L x AC + = ∂α ∂α ∂α ∂ M AC =0 ∂α Aerodynamic center is at ¼ chord position for a 2D airfoil with incompressible flow L xAC Aero center MO ∂ Mo x AC ∂ Mo ∂α =− =− ∂L ∂L ∂α Purdue Aeroelasticity Wing twisting is important –we include it by using structural influence coefficients or stiffness Going from the real world to the virtual world dθ T = GJ dy b h Torque= T θ t 2Gtbh GJ = 1+ h b Boeing 727 wing box with torsional model outlined in red Purdue Aeroelasticity 10 Features of the typical section aeroelastic model Single degree of freedom θ, the structural twist angle Structure resists twisting with an internal moment (torque) proportional to θ Ms=KTθ the more you twist the more the structure resists - linearly airspeed lift e Torsion spring KT θ M The term KT is a “torsion spring constant” θ Purdue Aeroelasticity Elastic axis – shear center? 11 Sum the torsional moments - write them in terms of torsional displacement, θ ¼ chord e Le + M AC = K Tθ Torsion spring KT θ KTθ = qSeCLα ( α o + θ ) + qScCMAC collect terms Shear center Shear center offset ( KT − qSeCLα ) θ = qSeCL α o + qScCMAC α Purdue Aeroelasticity 12 Solution for twist angle θ C c MAC qSeC L α o + α e CL α θ= KT − qSeC L α Purdue Aeroelasticity 13 Lift equation c CMAC qSeC Lα α o + e C Lα L = qSC Lα α o + K T − qSeC Lα Write this equation over a common denominator Purdue Aeroelasticity 14 Lift expressed a different way q= qSeC L L flex α The aeroelastic parameter KT qSC L C c α = α o + q MAC 1− q e CL α Purdue Aeroelasticity 15 The final equation for lift α K − qSeC + qSeC α + c CMAC Lα Lα o T o e CL α L = qSC L α K T − qSeC L α ( ) qScC MAC αo + KT L = qSC L α qSeC L α 1− KT Purdue Aeroelasticity 16 Summary-what have we learned? We have identified aerodynamic, structural and aeroelastic parameters that are important to aeroelasticity – – – – Torsional stiffness Lift curve slope Elastic axis offset distance Aeroelastic parameter depends upon structural stiffness, geometry and aerodynamic co-efficient q= Purdue Aeroelasticity qSeC L α KT 17