Distributed Masses
Infinite number of DOF
When the problem leads to a model with lumped masses it is usually convenient to do a FE model and, using static condensation, a dynamic model with a cut down number of degrees of freedom.
When the structure is large, complex and the distributed nature of the masses does not allow for a reduced number of degrees of freedom, we have only the option of using most of the DOF of a refined FE model to perform the dynamic analysis.
When we face a relatively simple problem characterized by distributed masses, we have the alternative of using a continuous model, where the unknown is a function of time and position, leading to a formulation of the equation of motion in terms of partial differential equations.
It is possible to formulate and solve different problems of structural dynamics in terms of partial differential equations, but today we will study only the problem of transveral deflection of beams.
Equation of motion for an infinitesimal beam slice
Earthquake effective load
Free vibration problem
Eigenvalues and eigenfunctions
Forced response, modal superposition
Earthquake response
Material available for this class:
- the slides i used for of my lesson and
- a ready to print version of the same slides, plus the exercise on the moving load.
- a script for plotting the cantilever mode shapes
- a program to prepare the mass modal expansion functions to be plotted with the help of small script.
- another program to prepare the data that figure in the table of the last slide