# Structural Dynamics

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

—Leonardo da Vinci

The purpose of this chapter is to convey to students a small introductory portion of the theory of structural dynamics. Much of the theory to which the students 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 da Vinci quotation, a proper use of this foundation enables the advance of technology.

Structural dynamics is a broad subject, encompassing determination of natural frequencies and mode shapes (i. e., the so-called free-vibration problem), response due to initial conditions, forced response in the time domain, and frequency response. In the following discussion, we deal with all except the last category. For response problems, if the loading is at least in part of aerodynamic origin, then the response is said to be aeroelastic. In general, the aerodynamic loading then will depend on the structural deformation, and the deformation will depend on the aerodynamic load­ing. Linear aeroelastic problems are considered in subsequent chapters, and linear structured dynamics problems are considered in the present chapter. Other impor­tant phenomena, such as limit-cycle oscillations of lifting surfaces, must be treated with sophisticated nonlinear-analysis methodology; however, they are beyond the scope of this text.

The value of structural dynamics to the general study of aeroelastic phenomena is its ability to provide a means of quantitatively describing the deformation pattern at any instant in time for a continuous structural system in response to external loading. Although there are many methods of approximating the structural-deformation pattern, several of the widely used methods are reducible to what is called a “modal representation” as long as the underlying structural modeling is linear. The purpose of this chapter is to establish the concept of modal representation and show how it can be used to describe the dynamic behavior of continuous elastic systems. Also included is an introductory treatment of the Ritz and Galerkin methods, techniques that use assumed modes or similar sets of functions to obtain approximate solutions in
a simple way. Indeed, both methods are close relatives of the finite element method, a widely used approximate method that can accurately analyze realistic structural configurations. Only the basics of applying the finite element method to beams are covered herein; details of this method are in books that offer a more advanced perspective on structural analysis, several of which are listed in the references.

The analytical developments presented in this chapter are conceptually similar to the methods of analysis conducted on complete flight vehicles. In an effort to maintain analytical simplicity, the continuous structural configurations to be exam­ined are all uniform and one-dimensional. Although such structures may appear impractical relative to conventional aircraft, they exhibit structural dynamic prop­erties and representations that are essentially the same as those of full-scale flight vehicles.