Step by Step Methods
The first part of today's lesson 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.
Generalized SDOF Systems
In the second part, we started yhe study of generalized SDOF systems.
Sometimes an articulated rigid body happens to be a single degree of freedom system,
sometimes we want to approximate the dynamical behaviour of a deformable,
infinite degrees of freedom system with a simple, single degree of freedom
First, we study the behaviour of assemblages of rigid bodies, connected by bilateral, frictionless constraints and by springs and dampers, when the resulting system has one degree of freedom. Under the hypotesis of small displacements we find the instantaneous motion of every piece of the system, and write the equation of equilibrium (including inertial forces) using the principle of virtual displacements. The concept of generalized characteristic is introduced.
A similar procedure can be applied also to deformable bodies,
Materials available for this class:
- the slides for the first part and the second part of the class.
- a ready to print version of the slides, again the first part and the second part,
- for the exercise about piecewise exact integration,
- the interactive notebook, following the link you'll see a static rendering but you're free to download the notebook;
- the corresponding PDF;
- the exercise about the constant acceleration algorithm,