2. Numerical Solution of Ordinary Differential Equations

The equations of motion that emanate from Newton’s laws for particles and rigid bodies are second order ordinary differential equations, and are often nonlinear. In our study of the dynamic response of structural systems we will encounter problems with complex forcing functions (e.g., earthquakes) and nonlinearity in the response (e.g., material yielding). While there are methods available to solve linear ordinary differential equations in a classical sense, many practical problems are not amenable to those techniques. We will rejoice when a classical solution is available (and squeeze as much insight from it as we can), but we will also prepare for the more common situation where numerical analysis of the equations of motion is required. While it may seem like a bit of an interruption to turn our attention to the numerical solution of ordinary differential equations, we do so to lay the groundwork for developing the tools that we will need to make progress with many of the problems of structural dynamics. In this chapter we consider a few of the more common numerical integrators and examine their behavior in some detail.