upport excitation occurs in different cases, e.g., multi-span bridges, vehicles, equipment contained in vibration affected environments, etc. We recognized that in the vast majority of cases the equation of motion can be written in terms of the supports' accelerations only.
Material available for this class: Structural Matrices
In this class, we studied the properties of the structural matrices, sketched the restricted part of FEM that is useful to our purposes and introduced the idea of static condensation.
We defined the mass, damping and stiffness matrices as a mean of writing the equation of dynamic eq. in terms of displacements and their time derivatives.
We found that the eigenvectors are orthogonal not only with respect to M and K but also to an infinity of other matrices, and we wrote down a formula to generate all the matrices for which orthogonality holds.
We introduced the flexibility matrix F to open the way to an understanding of the stiffness matrix K, defined both as K=F-1 and in terms of the force vectors needed to have a series of single unit displacements.
Having observed that inverting F (or even computing F) for slightly complex structures is not easy, we discussed the little section of the finite element method (FEM) that we need for our purposes: special stress was posed on beam elements and the shape function procedure.
The procedure for assembling K was explained and applied to an example.
Mass matrix can be constructed using a consistent approach or by lumping translational masses on the nodes. Advantages and disadvantages of both procedures.
The static condensation procedure, theory and an example.
Material available for this class:
- the slides i used for my lesson
- a ready to print version of the slides.