
The step-by-step method that will be developed further here is known as the piecewise-exact method. This involves approximating the applied loading as a series of straight-line segments over the duration of a time-step.

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	src="/images/courses/fundamentals-of-structural-dynamics-with-python/the-piecewise-exact-method-concept/img1.jpg"
	alt="The force-time history sampled at regular intervals (top) and the piecewise-linear approximation of the force (bottom). | EngineeringSkills.com"
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	caption="Fig 1. The force-time history sampled at regular intervals (top) and the piecewise-linear approximation of the force (bottom)."
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The exact analytical solution for the position and velocity of the mass at the end of a time-step is determined by solving the equation of motion over the duration of the step. Therefore over the duration of a time-step, we are actually performing a linear dynamic analysis. The flexibility in this method is achieved by virtue of the fact that we can take any loading and simply divide it into a series of linear segments.

Furthermore, if necessary we could also alter the dynamic characteristics of the dynamic system between subsequent time-steps. This approach offers us maximum flexibility and is a tool well worth knowing how to use.

It's important to appreciate that potential error is introduced if the force is not sampled at sufficient frequency. The error is easily reduced by reducing the time-step between load samples such that the piecewise-linear approximation closely matches the original force-time history.

The concept is really quite simple and like many numerical solution techniques, the mathematical operations performed over a single time-step are relatively straightforward. We'll develop the detail of this method in the next lecture.


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