Derived Ritz Vectors
MDOF Numerical Integration
Multiple Support Excitation
Today a sort of miscellanea to close the topic of MDOF systems.
Derived Ritz Vectors
The Derived Ritz Vectors follows from a combination of matrix iteration and Gram-Schmidt orthogonalisation.
Remarkable properties of DRVs are:- orthogonalisation is needed only with respect to the two previous DRVs
- one of the two sweeping coefficients needed was already computed as a normalization factor in the preceding step
- the reduced, Ritz coordinates eigenproblem can be easily formulated in canonical form with respect to a tridiagonal matrix
- the equation of motion for Ritz coordinates can be written with a tridiagonal mass matrix, a unit stiffness matrix and a load vector that has only one coefficient different from zero
- even for complex load shape, the expansion of load shape in terms of DRVs requires only a few lower terms, removing the requirement for static correction that, for these loadings, is needed in modal superposition procedure
Algorithms for the Numerical Integration of MDOF systems
Main problem in numerical integration of the equation of motion for a MDOF system is the numerical stability of the algorithm used. In short, only unconditionally stable algorithms can be used.
It turns out that -formally- the algorithms are identical to the algorithms discussed in the context of SDOF systems, with identical numerical coefficients and dimensional scalar factors substituted by corresponding matrices.- the constant acceleration algorithm is unconditionally stable, we discussed a pseudocode listing that implements the c-a algorithm
- a variation on the linear acceleration algorithm, introduced by Wilson and named Wilson's theta method, is unconditionally stable
Multiple Support Excitation
Multiple support 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:
- the slides i used for today lesson,
- a ready to print version of the slides,
- the program used to compute the table of error norms, for the eigenvector and DRV expansions of different load vectors,
- the program to compute the influence matrix for the first example about multiple support excitation (note that you can extract, from the first part of the program, a method to compute the flexibility matrix of a statically determined structure),
- the program used to compute the response for a multiple support excitation (note that the program contains an ad hoc implementation of the assemblage of the global stiffness matrix, valid for straight, multisupported beams),
- a Python notebook exemplifying the static correction procedure and
- a Python notebook exemplifying the generation of a DRV sequence.