2 - Structural Dynamics

Summary

O students, study mathematics, and do not build without foundations. …

Leonardo da Vinci

The purpose of this chapter is to convey to the student a small, introductory portion of the theory of structural dynamics. Much of the theory to which the student will be exposed in this treatment was developed by mathematicians during the time between Newton and Rayleigh. The grasp of this mathematical foundation is therefore a goal that is worthwhile in its own right. Moreover, as implied by the above quotation, a proper use of this foundation enables the advance of technology.

The field of structural dynamics addresses the dynamic deformation behavior of continuous structural configurations. In general, load–deflection relationships are nonlinear, and the deflections are not necessarily small. In this chapter, to facilitate tractable, analytical solutions, we restrict our attention to linearly elastic systems undergoing small deflections, conditions that typify most flight vehicle operations. It should be noted, however, that some level of geometrically nonlinear theory is necessary to arrive at a set of linear equations for strings, membranes, helicopter blades, turbine blades, and flexible rods in rotating spacecraft. Among these problems, only strings are treated herein. Indeed, linear equations of motion for free vibration of strings cannot be obtained without initial consideration, and subsequent careful elimination, of nonlinearities. Finally, there are other important phenomena, such as limit-cycle oscillations in lifting surfaces, that must be treated with sophisticated nonlinear analysis methodology; but they are beyond the scope of this text.