Vibrations 2 - 230320

Question 1

What is the stiffness matrix for Vibrations?

Answer 1

1. Equilibrium condition for the forces in the elements.
2. Hooke’s Law in each element.
3. You get the stiffness matrix, with the stiffness for each spring.

Question 2

Which two types of matrices are there?

Answer 2

Stiffness and mass matrix (lump and consistent mass matrix).

Question 3

What is the relation between angular frequency ω and frequency?

Answer 3

ω = 2pif

Question 4

What are the steps for doing a dynamic calculation of the vibration modes of a system?

Answer 4

1. Find the eigenfrequencies w (angular frequencies).
2. Find the eigenvectors x (mode shapes), for example a rigid body mode, where both elements move at the same time. The problem with this method lies in that the eigenvectors have no fixed amplitude, and instead, any multiple of it can be a vector.

Question 5

How many rigid body modes you have for each degree of freedom?

Answer 5

One rigid body mode for each degree of freedom.

Question 6

Limitations of FE method?

Answer 6

1. No amplitudes can be obtained, only mode shapes.
2. No damping is taken into account, just the springs.

Question 7

What is the modal space? How do you transform the cartesian model into a modal space?

Answer 7

Set up a coordinate system with each eigenvector as the unit vector (something similar to the i, j and k vectors, but with the eigenvectors replacing those).

Question 8

State the eigenvector equation in modal coordinates. Where does it come from?

Answer 8

ΩMX=KX

Question 9

Advantage of modal coordinates:

Answer 9

1. Diagonal: diagonal mass matrices, decoupled equations.
2. Small: The X matrix is a collection of only the modes that are of interest (usually the first 10 or so eigenfrequencies).

Question 10

What is mass normalization?

Answer 10

Convention useful to output numbers. Modal mass matrix is a unit matrix. Otherwise, the solution remains as variables a and b as in the first example.
Used by ANSYS by default.

Question 11

What is stiffness normalization?

Answer 11

Make the stiffness matrix an identity matrix. The problem is that it does not work with unfixed systems.

Question 12

How do you get the static solution of a one degree of freedom system?

Answer 12

Set the equations in matrix form and solve with the inverse of the stiffness matrix.

Question 13

What are the graphs for the Frequency Response Functinon for a SDOF?

Answer 13

(Remember the bee with the sppring and how with low frequencies you had 0 phase angle, high frequenncies, 180 phase angle).

Question 14

What is a rigid body mode?

Answer 14

A vibration mode where all the elements, including the spring move as one whole piece.