Purdue Aeroelasticity 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? Purdue Aeroelasticity

Purdue Aeroelasticity Lecture 2 - 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 do we want to learn? Vehicle stability? The origin of unusual features that have not been seen before? How important is simplicity? Purdue Aeroelasticity

Topology Optimization Structural Model Fidelity, Purdue Aeroelasticity Design development uses models that increase in fidelity as more information and detail becomes available AML DRACO OptiStruct ASTROS NASTRAN Concepts Innovation TSO FASTOP FEM II Layout Sizing Optimization A well-defined component FEM I Many choices Topology Optimization Load Paths 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 lift, L pitching Moment MAC spring force Khh torque KTq downward displacement, h Purdue Aeroelasticity

Purdue Aeroelasticity What do you mean by “fundamental?” Aerodynamic models describe math relationships lift, pitching moment and angle of attack lift and pitching moment co-efficients angle of attack, a Purdue Aeroelasticity

Terminology – airfoil center of pressure Choose an origin (point “o” is usually at the leading edge) locate a position x, sum moments center of pressure definition L x MO Center of pressure Purdue Aeroelasticity

We’ll use something called the aerodynamic center Compute change in aero moment with respect to angle of attack L xAC MO Aero center Aerodynamic center is at ¼ chord position for a 2D airfoil with incompressible flow Purdue Aeroelasticity

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 b h Torque=T t Boeing 727 wing box with torsional model outlined in red Purdue Aeroelasticity

Features of the typical section aeroelastic model lift airspeed Single degree of freedom q, the structural twist angle Structure resists twisting with an internal moment (torque) proportional to q Ms=KTq the more you twist the more the structure resists - linearly 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. The term KT is a “torsion spring constant” M q Elastic axis – shear center? Purdue Aeroelasticity

Purdue Aeroelasticity Sum the torsional moments - write them in terms of torsional displacement, q ¼ chord collect terms Shear center Stiffness is defined as the change in the generalized force (in this case the torque about the shear center) caused by a unit change in generalized displacement (in this case twist). The apparent torsional stiffness of the airfoil is the sum of the mechanical and aerodynamic stiffnesses. At dynamic pressures larger than the divergence dynamic pressure, the apparent torsional stiffness is negative, implying that any slight increase in angle of attack of the airfoil will cause it to twist without limit. As the aeroelastic torsional stiffness declines to zero. Shear center offset Purdue Aeroelasticity

Solution for twist angle q Unless the aeroelastic torsional stiffness is positive, there can be no solution to the static equilibrium equation for this airfoil. A negative stiffness gives a negative twist angle for a positive angle of attack input, clearly an impossible answer. What does this model tell us? Why is it useful? How can we add fidelity? Why would we want to? The aeroelastic term qSeCLa/ KT appears. As qSeCLa/ KT approaches unity, q will approach infinity if the input angle of attack is constant. A phenomenon called aeroelastic divergence will happen when the dynamic pressure, q , equals KT/SeCLa Purdue Aeroelasticity

Purdue Aeroelasticity Lift equation Write this equation over a common denominator Purdue Aeroelasticity

Lift expressed a different way The aeroelastic parameter Purdue Aeroelasticity

The final equation for lift Purdue Aeroelasticity

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 Purdue Aeroelasticity