AAE556 Lecture 15 Finite element subsonic aeroelastic models

AAE 556 Aeroelasticity Lecture 15 Finite element subsonic aeroelastic models I like algebra Algebra is my friend Lift computation idealizations Everything you wanted to know about aerodynamics but were afraid to ask • Lift per unit length l(y) changes along the span of a 3-D wing • The 2-D lift curve slope is not the same as the 3-D lift curve slope ao CLα = 57.3ao 1+ π eAR • Lift curve slope in degrees • e = Oswald’s efficiency factor Lift L = ∫ l ( y ) dy = q ∫ caoα ( y ) dy = qSC Lα α o l ( y ) = qcaoα ( y ) = ρV aoα ( y ) w = ρV ao = ρVao w V  α ( y) ao ∫ c  ao ∫ cα ( y ) dy αo  CLα = = Sα o S  ÷dy  Aerodynamic strip theory • Wing is sub-divided into a set of small spanwise strips • The lift and pitching moment on each strip is modeled as if the strip had infinite span • There is no aerodynamic interaction • There is limited or no aerodynamic influence between elements Paneling methods • The wing is replaced by a thin • • surface This surface is replaced by a finite number of elements or panels with aerodynamic features such as singularities There is an aerodynamic influence coefficient matrix with interactive elements Strip theory gives different results Source: G Dimitriadis, University of Liege Background • • • • Gray and Schenk NACA TN-3030 1952 Adapted for composites 1978 Paneling - idealization requirements and limitations Panel aero model finding the lift distribution pi=ρVΓi Lifting line wing model Horseshoe vortices with varying strength bound at 1/4 chord points Downwash matching points at 3/4 chord trailing vortices extending to infinity The wing can be unswept or have non-constant chord Panel aerodynamics interacts because of downwash (angle of attack) matching Vortex influence decays with distance Each horseshoe vortex creates a flow field around it The 3/4 chord downwash is affected by every other vortex on the wing The vortex strengths must be adjusted so that all conditions are satisfied Aerodynamic relationship Solving for vortex strengths allow us to approximate the lift distribution wing Relationship between local angle of attack and segment lift values [ ]{ } {α i } = Aij p j q Aero matrix equation developmentMatrix is square, but not symmetrical [ ]{ } {α i } = Aij p j q  Aij =   4clα [ ] αi=αrigid + θstructural + αcontrol   S ij  [ ] Matrix elements are 2D lift curve slope functions of wing planform geometry Structural idealization Each panel has its own FBD and panel geometry Put them all together to get the static equilibrium equations – this is where the aeroelastic interaction occurs { αi } = { α r } + { θs } + { αc } local angles =  Aij  { p j } q dynamic pressure { θ s } = Cij  { p j } lift on each element Wing Geometry Flexible and Rigid lift distributions (M=0.5) areas under each curve are equal CLα Rigid = 4.87 CLα flex = 13.85 M Div = 0.59 M = 0.50 Flexible and Rigid lift distribution (M=0.6) Rigid and flexible roll effectiveness (pb/2V) MRev= 0.55 Rigid wing and flexible wing C Lα Divergence Mach number Divergence Mach No = 0.590 Summary • Use of bound vortices creates a math model that can predict subsonic high aspect ratio wing lift distribution • This model has been incorporated into a MATLAB code that you will use to some homework exercises to calculate divergence, lift effectiveness and control effectiveness • You will compare the trends previously derived