Structural Dynamics: Modes

Question 1

What is the eigenmode and eigenvalue?

Answer 1

The eigenmode is a vector that multiplied by K is equal to itself multiplied by matrix M and a constant λ = ωi^2. The eigenvalue is the value that allows to scale Μφ to Κφ.

Considering a space-state representation, the eigenvector is a vector that multiplied by A, is equal to itself mutiplied by a constant λ.

Question 2

What kind of normalization of the eigenvectors or modal shapes are used?

Answer 2

Modal shapes are defined unless for a constant. Several normalization are often used:
1) Maximum displacement Max(φi) = 1.0
2) Modal mass φi^T M φi = 1

Τhe first one is used when you want to compare modal shapes independently of how much energy they collect. The modal mass, in some way, is an index of the kinetic energy associated with the mode. The lower the modal mass, the lower will be the kinetic energy.

Question 3

What are some properties of eigenmodes?

Answer 3

1) Orthogonality with respect to mass matrix:
φi^T M φj = 0
2) Orthogonality with respect to sitffness matrix
φi^T K φj = 0.

Question 4

How can the system be decoupled?

Answer 4

Instead of using generalized coordinates used to define the problem, modal coordinates are used: q = Φz, with Φ = [φ1, φ2…]
Therefore:
MΦz_dot_dot + KΦz = 0 and by multiplying with the transpose of the eigenmodes:
μi(zi_dot_dot + ωi^2*zi) = 0. The N-dimensional second differential system is decomposed in N independent scalar linear second order differential equations.

Also for the state-space model, the matrix of eigenvectors is an orthogonal matrix and Λ = Φ^Τ A Φ. Ultimately zi_dot = λzi.

Question 5

What are coincident eigenvalues, algebraic multiplicity and geometric multiplicity?

Answer 5

Coincident eigenvalues typically appear when there are symmetries in the structures under analysis.

Algebraic multiplicity is the number of times an eigenvalue is repeated.

Geometric multiplicity is the dimension of the eigenspace Εv, associated with an eigenvalue λ

If the algebraic multiplicity is equal to the geometric multiplicity, the matrix is still diagonalizable because all eigenvectors associated with the same eigenvalue are linearly independent.

Question 6

Discuss the rigid body modes.

Answer 6

The rigid body movement in a certain direction is associated to a null eigenvalue because it does not imply any deformation (elastic deformation is null. φi^TKφi = 0.

In general, φi^TMφj is different of zero when i,j = 1,…6 where 6 are the rigid body modes of a free flying aircraft.

Question 7

How can the rigid body modes be decoupled for the system?

Answer 7

If the initially computed rigid body modes are not linearly independent, it is possible to compute those who diagonalize the matrices through the GRAM-SCHMIDT orthogonalization.

If φj^TMφi not zero, define a modified j mode shape:
φj’ = φj - αφi, so that φj’^T M φi = 0 and compute α.

Question 8

How is the solution computed for initial conditions?

Answer 8

The general solution is obtained as superimposition of φie^jωit and φie^-jωit for all i.

q = Σ φi(Aicos(ωit) + Bisin(ωit))

Question 9

What is the Rayleigh quotient?

Answer 9

Consider a generic shape function ui compatible with the kinematic BC. Eigenvectors are the basis of the solution, theefore ui can be expressed as a linear combination of them, ui = Φα.

Rayleigh’s quotient is the equivalent of the ratio between the elastic integral and the inertial one in the continuous case. Through the use of the Rayleigh quotient, it is possible to obtain an approximation of the higher eigenvalue. The closer is u to an eigenvector, the closer will be the approximation of the eigenvalue. The higher is the separation between the eigenvalues, the faster the convergences.

If the approximated eigenvector u has an error of the order O(δu), then the eigenvalue approximated with the Rayleigh quotient R(θ) has an error of order O(δu^2).

Question 10

How can a group of eigenvalues/eigenvectors be computed?

Answer 10

Using a Taylor expansion, we get to the point that:
Κa = λiMα. Computing the eigenvalues of the reduced and approximated mass and stiffness matrix, it is possible to otbain an approximation of eigenvalues.

The Bloch-Stodola block iteration can be used:
1. Start with an initial guess for Φ0 that is of size [Nxp], where p is the number of eigenvalues we want to approximate.
2. for k=0,…m
ΚΦk+1 = MΦκ (calculate Φk+1)
3. K = Φk+1^T K Φk+1 and M = Φk+1^T M Φk+1 calculate new stiffness and mass matrices)
4. Solve Κα = λΜα (eigenvalue problem, solve for λi,αi)
5. φnewi = Φk+1αi