Mechanics Problems from the Functional Analysis Viewpoint

Abstract

In the past, an engineer could calculate mechanical stresses and strains using a pencil and a logarithmic slide rule. Modern mechanical models, on the other hand, are nonlinear, and even the linear models are complicated. Numerical methods in structural dynamics cannot be applied without computers running specialized programs. However, a researcher should have a solid grasp of the equations that underlie a numerical model and the types of results that can be expected. New models appear in mechanics on a regular basis. Some of these, when written out in detail, can span multiple pages and are clearly beyond pencil-and-paper approaches. Although functional analysis does not provide a detailed picture of the results to be expected from a complicated model, it can answer questions regarding whether the problem is mathematically well-posed (e.g., whether a solution exists and is unique). It may also indicate whether the results can be obtained by a general computer program or whether a special program, based on a knowledge of the general properties of the model, is required. In other words, functional analysis can provide valuable insight even to those who rely heavily on numerical approaches.