## Structural Dynamics

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.

Structural dynamics is a broad subject, covering such things as determination of natural frequencies and mode shapes (the so-called free-vibration problem), response due to initial conditions, forced response in the time domain, and frequency response. In the following we will 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 will then depend on the structural deformation, and the deformation will depend on the aerodynamic loading. As with the structural dynamics, the aerodynamic forces driving the system may be nonlinear. Linear aeroelastic problems are considered in later chapters.

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. It is the purpose of this chapter 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 make use of mode shapes or a similar set of functions to obtain approximate solutions in a simple way. Both methods are indeed close relatives of the finite element method, a widely used approximate method that can accurately analyze realistic structural configurations. The finite element method is not covered herein, but details of this method can be found in books that offer a more advanced perspective on structural analysis, several of which are listed in the bibliography.

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 in relation to conventional aircraft, they exhibit structural dynamic properties and representations that are essentially the same as those of full-scale flight vehicles.

### 2.1 Uniform String Dynamics

To more easily understand a mathematical description of the mechanics associated with the structural dynamics of continuous elastic systems, the classical "vibrating string problem" will first be considered. Although the string can be described by a simple second-order partial differential equation in one dimension, it is typically descriptive of the more complex linearly elastic systems of aerospace vehicles. Once the fundamental concepts are covered for the string, other components will be treated that are more representative of these vehicles. Although the free vibration of a string can be analyzed using equations of motion that are of the same form as that of uniform beam extensional and torsional vibrations, the string is chosen as our first example primarily because, in contrast to these other structures, string behavior can be so easily visualized. Moreover, by this time in their undergraduate studies, most students have had some exposure to the solution of string vibration problems.

### 2.1.1 Equations of Motion

A uniform string of initial length ¿o will be stretched in the x-direction between two walls separated by a distance I > £0. The string tension T will be considered high and the transverse displacements v(x. t) will eventually be regarded as small. At any given instant this system can be illustrated as in Fig. 2.1. To describe the dynamic behavior of this system the forces acting on a differential length dx of the string can be illustrated by Fig. 2.2. It may be noted that the tension and slope, 6, at the right end of the differential element have been represented as a Taylor series expansion of the values at the left end. Because the string segment is of a differential length that can be arbitrarily small, the series will be truncated by neglecting terms of the order of dx2 and higher.

Two equations of motion can be formed by resolving the tension forces in the x- and y-directions and setting the resultant force on the differential element equal to its mass

(m dx) times the acceleration of its mass center. Thus, the equations of motion are

3 d2u

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