AAE 556 Aeroelasticity Lecture 6 – Multi-DOF systems Reading: Sections 2-13 to 2-15 Purdue Aeroelasticity

Purdue Aeroelasticity Homework for Friday? Problem 2.3 Purdue Aeroelasticity

Multi-degree-of-freedom (MDOF) systems Develop process for analyzing MDOF systems Define theoretical stability conditions for MDOF systems Reading - Multi-degree-of-freedom systems – Section 2.14 Purdue Aeroelasticity

Purdue Aeroelasticity Here is a 2 DOF, segmented, aeroelastic finite wing model - two discrete aerodynamic surfaces with flexible connections used to represent a finite span wing (page 57) Torsional springs fuselage wing tip wing root Torsional degrees of freedom Purdue Aeroelasticity

Purdue Aeroelasticity The two twist angles are unknowns - we have to construct two free body diagrams to develop equations to find them Structural restoring torques depend on the difference between elastic twist angles Wing root Internal shear forces are present, but not drawn Wing tip Double arrow vectors are torques Purdue Aeroelasticity

Torsional static equilibrium is a special case of dynamic equilibrium Arrange these two simultaneous equations in matrix form Purdue Aeroelasticity

Purdue Aeroelasticity Strain energy Purdue Aeroelasticity

Purdue Aeroelasticity Summary The equilibrium equations are written in terms of unknown displacements and known applied loads due to initial angles of attack. These lead to matrix equations. Matrix equation order, sign convention and ordering of unknown displacements (torsion angles) is important Purdue Aeroelasticity

Problem solution outline Combine structural and aero stiffness matrices on the left hand side The aeroelastic stiffness matrix is Invert matrix and solve for q1 and q2 Purdue Aeroelasticity

Purdue Aeroelasticity The solution for the q’s requires inverting the aeroelastic stiffness matrix Purdue Aeroelasticity

The aeroelastic stiffness matrix determinant is a function of q The determinant is where When dynamic pressure increases, the determinant D tends to zero – what happens to the system then? Purdue Aeroelasticity

Purdue Aeroelasticity Plot the aeroelastic stiffness determinant D against dynamic pressure (parameter) Dynamic pressure parameter determinant The determinant of the stiffness matrix is always positive turning the air on reduces its size Purdue Aeroelasticity

Solve for the twist angles created by an input angle of attack ao Purdue Aeroelasticity

Twist deformation vs. dynamic pressure parameter Unstable q region panel twist, qi/ao divergence Outboard panel (2) determinant D is zero Purdue Aeroelasticity

More algebra - Flexible system lift Set the wing lift equal to half the airplane weight Purdue Aeroelasticity

Purdue Aeroelasticity Lift re-distribution due to aeroelasticity (originally presented on slide 13) Observation - Outer wing panel carries more of the total load than the inner panel as q increases Purdue Aeroelasticity

Purdue Aeroelasticity MDOF Divergence In general we have matrix relationships developed from EOM’s These can be converted into perturbation relationships Purdue Aeroelasticity

The aeroelastic stiffness matrix determinant is a function of q The determinant is where When dynamic pressure increases, the determinant D tends to zero – divergence occurs Purdue Aeroelasticity

Determinant D plotted against dynamic pressure parameter This nth order determinant is called the stability determinant or the characteristic equation Purdue Aeroelasticity

Purdue Aeroelasticity Eigenvectors 1 2 Purdue Aeroelasticity

Twist deformation vs. dynamic pressure parameter Unstable q region panel twist, qi/ao divergence Outboard panel (2) determinant D is zero Purdue Aeroelasticity