AAE 556 Aeroelasticity Lecture 15 Finite element subsonic

AAE 556 Aeroelasticity Lecture 15 Finite element subsonic aeroelastic models I like algebra Algebra is my friend. Purdue Aeroelasticity 15 -1

Lift computation idealizations Purdue Aeroelasticity 15 -2

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 • Lift curve slope in degrees • e = Oswald’s efficiency factor Purdue Aeroelasticity 15 -3

Lift Purdue Aeroelasticity 15 -4

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 Purdue Aeroelasticity 15 -5

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 Purdue Aeroelasticity 15 -6

Strip theory gives different results Source: G. Dimitriadis, University of Liege Purdue Aeroelasticity 15 -7

Background • • Gray and Schenk NACA TN-3030 1952 Adapted for composites 1978 Purdue Aeroelasticity 15 -8

Paneling - idealization requirements and limitations Purdue Aeroelasticity 15 -9

Panel aero model finding the lift distribution pi=r. VGi Purdue Aeroelasticity 15 -10

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 Purdue Aeroelasticity 15 -11

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. Purdue Aeroelasticity 15 -12

Aerodynamic relationship Solving for vortex strengths allow us to approximate the lift distribution wing Relationship between local angle of attack and segment lift values. Purdue Aeroelasticity 15 -13

Aero matrix equation development. Matrix is square, but not symmetrical ai=arigid + qstructural + acontrol Matrix elements are 2 D lift curve slope functions of wing planform geometry Purdue Aeroelasticity 15 -14

Structural idealization Purdue Aeroelasticity 15 -15

Each panel has its own FBD and panel geometry Purdue Aeroelasticity 15 -16

Put them all together to get the static equilibrium equations – this is where the aeroelastic interaction occurs local angles dynamic pressure lift on each element Purdue Aeroelasticity 15 -17

Wing Geometry Purdue Aeroelasticity 15 -18

Flexible and Rigid lift distributions (M=0. 5) areas under each curve are equal Purdue Aeroelasticity 15 -19

Flexible and Rigid lift distribution (M=0. 6) Purdue Aeroelasticity 15 -20

Rigid and flexible roll effectiveness (pb/2 V) MRev= 0. 55 Purdue Aeroelasticity 15 -21

Rigid wing and flexible wing Purdue Aeroelasticity 15 -22

Divergence Mach number Divergence Mach No. = 0. 590 Purdue Aeroelasticity 15 -23

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 do some homework exercises to calculate divergence, lift effectiveness and control effectiveness. • You will compare the trends previously derived Purdue Aeroelasticity 15 -24