PRELIMINARIES

The discussion of aeroelasticity requires certain preliminary information on the theory of elasticity, aerodynamics, and mechanical vibrations. There exist a number of excellent textbooks on these subjects. Therefore we shall review only briefly some of the fundamental facts in this chapter and explain the notations and sign conventions that will be used in the text.

The reader is urged, however, to read carefully §§ 1.2 and 1.3, concerning the definitions of shear center, elastic axis, flexural line, etc., because these terms have been used somewhat ambiguously in the engineering literature. In § 1.4 the influence functions are explained, and in § 1.6 the generalized coordinates and Lagrange’s equations are reviewed and illustrated by several examples. These subjects must be understood thoroughly.

Throughout this book a vector will be printed in boldface type, as, for instance, a velocity vector v, a force vector F. A vector in a three – dimensional space has three components, which are indicated by sub­scripts. Thus a force F referred to a system of rectangular Cartesian coordinates x, y, z has three components Fx, Fy, Fz, Sometimes it is more convenient to label the xyz coordinates as x1x2xs coordinates and to indicate Fx as Fu Fy as F2, Fz as F3. The vector F, being specified by the three components F1; F2, F3, may also be identified simply by writing F{ (j = 1, 2, 3).

A relation among several vectors may be expressed either by a single vector equation or by a system of equations expressing the relations among the components of the vector. For example, let a (with components ax, a2, a3) be the acceleration of a particle, m its mass, and F (with com­ponents Fx, F2, F3) the force acting on the particle. Then Newton’s law of motion for this particle can be written either as

F = m&

0)

Fx — таг

F2 = ma2

(2)

F3 = ma3

Equations 2 may be shortened into the following form Fi = ma{ (i = 1, 2, 3)

We shall consider Eqs. 1 and 3 as entirely equivalent expressions.

This notation will be extended to tensor equations and matrix equations by means of multiple subscripts.

PRELIMINARIES Подпись: (4)

One of the most important simplifying conventions in all mathematics is the summation convention: to use the repetition of an index to indicate a summation over the total range of that index. For example, if the range of the index і is 1 to 5, then

If at = a and bt = b are two vectors, the product аф{ is the scalar product of a and b:

Подпись: (5)Подпись: (6)Подпись:a • b = a fit

As another example, if i, j = 1, 2, 3, then з

CijFj = ‘^CljFj = CUF1 + C12F2 + C13F3

i=i

з

F’ijFj =■ ‘^fC3jFj = C3Fl + ад + ад

3

F’SjFj ~ ‘^’CqFj ~ С31Р, + C32F2 + ^33F3

3 = 1

The system of Eqs. 6 may be simply written as

з

ад = lt 2> 3)

3=1

This summation convention will be used in this book.

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