Dynamics of Structures

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.

Rigid Body Assemblages

Sometimes an assemblage of rigid bodies, springs and dampers happens to be a single degree of freedom system.

In this case, the equation of motion can be written determining all the forces acting on the system (using the D'Alembert principle for the determination of inertial forces) in terms of the sole free variable and using the Principle of Virtual Displacements for rigid bodies (for a rigid, virtual displacement the virtual work of the external forces must be equal to zero).

We have seen a number of examples.

Materials available for this class:

• the slides i used for my lesson
• a ready to print version of the slides,
• the notebooks that i used in class,
  • exact response for piecewise linear excitation,
  • constant acceleration algorithm,
  • linear acceleration algorithm,
  • exact integration + numerical integration of the EOM for the EP system described at the end of the slides.
  Please note that the above are links to a site that renders statically my notebooks in your browser. Would you like to interact with the notebooks,
  1. open a notebook using one of the links above
  2. download the notebook using the top-right icon
  3. start the ipython notebook server in the same directory where you have saved the notebook (you can save many notebooks in the same directory).
  What about the ipython notebook server bit in the above sentence? If you're a Mac or Windows user, my suggestion is the Anaconda python distribution, where you can find an installer that fills your hard disk, ahem, that puts all the relevant items in place. When you're done, open a Terminal.app a.k.a. command prompt, cd to the directory with the notebooks and type ipython notebook... Caveat: last time I tried on windows, IE didn't work and I had to use chrome as my browser. Your experience could be different.