Static Aeroelasticity: MDOF Model

Question 1

Describe the MDOF model for torsional divergence.

Answer 1

The model is composed by several rigid wing portions connected through torsnional springs. For each rigid portion the simple pseudo-2D aerodynamic model is used. No mutual aerodynamic influence between portions is considered.

Then the model is developed using the virtual work principle, where we obtain the system (Ks-qKa) = M0.

Question 2

Describe the Power Method

Answer 2

The power method is used to find the eigenvalue with the maximum modulus of the matrix.
If:
1) A Ε Rnxn and z Ε Rn
2) A possess n eigenvectors linear independent
3) |λ1| > |λ2| >= |λ3|

We apply the series: zk+1 = Azk
We compute an index σκ = zk’Azk/|zk|^2
and we verify that for the limit of σκ going to infinite, the value is equal to λ1.

Question 3

How can the Power Method be applied to the MDOF case?

Answer 3

Since we are interested in the lower divergence pressure qd, we can divide the eq with qd and set λ = 1/qd. But in this case an inversion of the structural stiffness matrix is required. In order to avoid this inversion, we can set c= Kaθi and Ksθi+1 = c. The solution of the second linear system could be obtained without the invesrsion of the structural stiffness matrix. The problem can also be solved using LU decomposition, or Cholesky decomposition.

Question 4

What are the Flexibility Influence functions? How can they be computed?

Answer 4

Cθθ is the flexibility influence function that represents the twist at y due to a unit moment at γ.

1. They can be computed experimentally by applying a moment at yi and measuring the twist at all other sections
   2) They can be computed using VWP, computing θ due to a moment at ξ.

In the end the problem can be formulated using the flexility matrix F (contains all flexibility influence functions (1/ciCLai) - q*F(ei)) (cCLei) = qFmoi

Question 5

What is the Ritz method?

Answer 5

We express the uknown function (displacement) as a complete linear combination of known functions Ni and qi, which are the shape functions and generalized coordinates correspondingly.

The shape functions chosen must belong to a complete set, meaning that in the domain considered, taking a sufficient number of terms, it must be possible to represent the exact solution to any degree of accuracy.

The coefficient of the series qi are a new set of coordinates (ie degrees of freedom used to describe the displacement field)

*If the method is applied to a strong formulation (ie differential) the chosen shape functions must satisfy all BC.
*Each shape function should be p times differentiable where p is the highest spatial derivative present in the formulation
*At least one of the shape functions p-th order derivative should be not-zero.

The Ritz approach has some residual error. The goal is to find θ’, which is the approximate numerical of the exact solution θ. To gind it, it is necessary to minimize the error ε. In order to do, we can use:
1. Collocation
We explicitidly set the error ε = 0 for each yi.
2. Weighted integrals
int(wiiε(y)dy) = 0

Using a weak (integral) formulation, such as VWP, it leads naturally to a weighted integral approach, where the weight functions are the virtual displacements. Here:
*Regularity required to shape functions is lower since the maximum p derivative is lower.
*The shape functions must satisfy boundary conditions on displacements and rotations, often called “geometric” BC.
*The BC on forces and moments, called “natural”, are not necessary. Their satisfaction will be obtained at convergence. However, satisfying also these conditions can speed up convergence.