Mathematical Methods in Interdisciplinary Sciences

13Structural Static and Vibration Problems

M. Amin Changizi¹ and Ion Stiharu²

¹Mohawk Innovative Technology, Albany, NY, 12205, USA

²Concordia University, Montreal H3G 1M8 QC, Canada

13.1 Introduction

Although statics and dynamics were defined as disciplines in the classical mechanics about three centuries ago, there are many problems under the above two disciplines that have still not been accurately or fully solved so far. Moreover, the new applications in technologies enforced development of methods that are capable to yield the necessary results for accurate design. Along with time, the accuracy of the solutions has improved, and new phenomena came into the attention of the community. For an example, the simple problem of vibrating cantilever is solved making assumptions that one might not agree with. The stiffness of the cantilever is based on the ratio of the applied force and the deflection of the beam. Such establish value, although comfortable to obtain, may not be agreeable with specific calculations as the tip deflection may not represent the location of interest. The same issue is with the mass of the beam, which, when fixed in one point, may not reflect the same mass in motion as a free beam. On the other hand, the influence of the field force exerted by the electrostatic attraction was of no concern till the microsystems came up.

As past century engineering would accept results within 5% error, such a value is not conceivable now, and the one accepted now will ...