Step by Step Methods
Today's class was devoted to the numerical methods for the integration of the equation of motion in the time domain, that we collectively name Step by Step Methods.
We introduced the concept of the state of a linear or non linear system, and the idea of computing the state of the system at the end of a short time step as a mean to compute, step by step, the response of a SDOF system in an approximate manner.
The specific qualities and defects of SbS methods were discussed at some length, especially the cause of errors and the problem of numerical instability.
First of all, we discussed the Central Differences Method, an unstable method that appeals the numerical analyst with its semplicity. The method gives also a taste of the structure of step by step solutions, where the end state is expressed in terms of the initial state and the values of the loadings at the extremes of the step.
The central differences is an explicit method, next we introduced implicit methods, that can be collected under the name of Newmark's Beta Methods, and we studied in particular the Constant and the Linear Acceleration Methods, the Constant Acceleration being remarkable as it is a stable method of integration.
We ended introducing the modified Newton-Raphson method
for the solution of non linear problems, with the
discussion of an elasto-plastic system.
The homework assigned was about an elastoplastic
system subjected to an impulsive loading.
- the slides i used for my lesson
- a ready to print version of the slides,
- a Python script to do analytical integration of the EOM for the homework example,
- the short paper, that was shown in class, that documents the above script
- another Python script that solves numerically, using the modified Newton-Raphson method, problem similar to the homework
- a last Python script to integrate analytically the EOM for the previous problem, so that you can compare the exact solution with the results of the numerical integration.
also something about your homework