“Adde parvum parvo magnus acervus erir” Ovid Purdue Aeroelasticity AAE556 Static Stability “Adde parvum parvo magnus acervus erir” Ovid (“add little to little and there will be a big pile.”) Purdue Aeroelasticity

Purdue Aeroelasticity Lecture 3 – summary Aeroelasticity is concerned with interactions between aerodynamic forces and structural deformation Develop simple static aeroelastic model with pitch (torsion) and plunge (bending) Section 2.4 Purdue Aeroelasticity

Purdue Aeroelasticity Reading topics 2.6 Lifting generation-flexible surface 2.7 Example problem – work it through by hand 2.8 Using simple results 2.9 Load factor 2.10 Simple model – 1 degree-of-freedom-emphasis on stiffness, not strength 2.10.2 – Stability definition – essential 2.11 Example problem using perturbation concept 2.12. Analysis example showing when stability is obvious and when it is not 2.13 Compressibility Purdue Aeroelasticity

Aero/structural interaction model Requirements - simplicity manageability realism TYPICAL SECTION airspeed lift Notice what this model does and does not do. It ignores the bending deflection and can't predict wing stress or twist anywhere on a real wing. It does approximate the interaction between airloads and the structural twist and point out how and where problems may occur. It is left to the reader how to relate the torsional spring to real life structural features and this is not easy to do. The model does give us a general idea about when we need to crank up the advanced analysis to analyze more accurately the load deflection interaction. It also identifies the importance of the offset between two reference points - the shear center and the aero center and shows us why we need to understand exactly the meaning of these two points. We can't produce a number for the lift (like 500.2 pounds) because the angle of attack including flexibility is an unknown in the beginning. The lift will depend on the angle of attack we input, but it also depends on how much the airfoil rotates because of flexibility. This is a statically indeterminate problem and requires that we write an equation for the relationship between loads and deflection. One degree of freedom Purdue Aeroelasticity

Lift and the aeroelastic parameter Purdue Aeroelasticity

Lift equation with wing flexibility Purdue Aeroelasticity

Two degree of freedom aeroelastic model (Section 2.4) Goal - add bending deformation (plunge) to the simple 1 dof model Displacement, h, plunge at the shear center Airspeed, V twist, q Plunge is resisted by spring, Kh Twist is resisted by spring, KT +h Purdue Aeroelasticity

Sum torsional moments about shear center Static equilibrium equations Forces, moments and the importance of mechanics to the effort The problem unknowns are h and q +h Sum forces Sum torsional moments about shear center Structural stiffness matrix Loads, measured at shear center Purdue Aeroelasticity

Purdue Aeroelasticity Write the aerodynamic loads in terms of h & q We use matrix methods – that’s our theme Idealized wing section lift Twisting moment, at wing shear center, positive nose-up Purdue Aeroelasticity

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 & q Aeroelastic stiffness matrix Purdue Aeroelasticity

Solution for wing deflections, h & q Divide by KT to get nondimensional terms Invert 2x2 matrix Get BHM Purdue Aeroelasticity

Purdue Aeroelasticity Wing displacements plunge +h twist Purdue Aeroelasticity

Purdue Aeroelasticity New goals Define structural static stability Concept of perturbations Distinguish stability from response Learn how to do stability analysis Find the wing divergence dynamic pressure using a “perturbation” analysis Purdue Aeroelasticity

Purdue Aeroelasticity Math Summary 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 Systems of simultaneous equations can be written (and solved) in matrix form. Static equilibrium aeroelastic equations yield two important matrices Structural stiffness matrix – symmetrical if you do it right Aerodynamic stiffness matrix – aero people will not recognize this term These matrices are added together to form the aeroelastic stiffness matrix Purdue Aeroelasticity

Euler’s static stability criterion "A system in static equilibrium is neutrally (statically) stable if there exist nearby static equilibrium states in addition to the original static equilibrium state.” Stability - the tendency of a system (structural configuration) to return to its original equilibrium state when subjected to a small disturbance (perturbation) Leonard Euler 1707-1783 Advisor-Bernoulli Student-LaGrange “Read Euler, read Euler, he is our master in everything" Laplace Purdue Aeroelasticity

The perturbed structure 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. We need to apply these above words to equations so that we can put the aeroelastic system to a mathematical test Purdue Aeroelasticity

Stability investigation Given a system that we know is in static equilibrium (forces and moments sum to zero) Add a disturbance to perturb the system to move it to a different, nearby position (that may or may not be in static equilibrium) Is this new, nearby state also a static equilibrium point? Write static equilibrium equations and see if forces and moments balance Purdue Aeroelasticity

Purdue Aeroelasticity Perturbed 1 dof airfoil In flight this airfoil is in static equilibrium at the fixed angle q but what happens if we disturb (perturb) it? There are three possibilities Purdue Aeroelasticity

Purdue Aeroelasticity Example Perturb the airfoil when it is in static equilibrium To be neutrally stable in this new perturbed position this equation must be an true Purdue Aeroelasticity

Perturbation possibilities KT(Dq)>(DL)e statically stable because it tends to return no static equilibrium in the perturbed state KT(Dq)<(DL)e statically unstable motion away from original position KT(Dq)=(DL)e system stays in perturbed position new static equilibrium point Dq Euler test has found a neutral stability condition Purdue Aeroelasticity

Static stability investigation is “stiffness based” Neutral stability means this relationship must be zero (2 states) so... Not zero condition at neutral stability static equilibrium displacement (Dq) is not unique Purdue Aeroelasticity

Purdue Aeroelasticity Observations The equation for neutral stability is simply the usual static equilibrium equation with right-hand-side (the input angle ao) set to zero. 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 Absum! (I’m outta here) Purdue Aeroelasticity