NORMAL COORDINATES

The last two examples of the preceding section point out a very im­portant fact: By introducing the undamped free-vibration modes of a structure as the basis of generalized coordinates, the equations of motion can be simplified.

The theory of small free oscillations of an elastic body about an equi­librium configuration has been well developed. In particular, it suggests that, if фп{х) represents the oscillation mode* associated with a frequency <яп, and if the frequency spectrum is so arranged that <an > con_.j, {n > 1), then an arbitrary disturbed configuration u can be represented by a series

CO

n(x) = ‘S q„<f>n(x) (1)

£—4 71 = 0

with

4n = J Mx) n(x) • фп(х) dr(x) I J p(x) фп(х) • ф„(х) dr(x) (2)

* That is, the amplitude of the displacement from the equilibrium configuration at the point x. u and фп may be regarded as vectors having three components, each a function of the position vector x, if a three-dimensional elastic body is considered. In Eq. 2 et seq., the product u(x) • ф„(х) denotes the scalar product of u(x) and"^„(xl. Similarly ф„(х) ■ фт(х) is a scalar product.

J p(*) &(x) • фп(х) dr(x)

where p(x) represents the density of the body and dr(x) represents an element of volume at x, the integration being taken over the entire body. Moreover, фп(хJ are orthogonal and can be normalized so that

We shall call фп(х) the normal modes and qn the normal coordinates. Then the kinetic energy and the elastic strain energy can be expressed, respectively, as

Т=^тпЧп

(4)

V = 2

n = 0

The constants mn are called generalized masses:

= J p(*) Фп(х) ■ фп(х) dr(x) (5)

The generalized force is

Qn = $ F(x) • ф„(х) dr(x) (6)

where the integral is again taken over the entire body, F(x) being the force acting at x.

It is clear that, when T and V are expressible in the form of Eq. 4, the inertia and elastic “couplings” between the various generalized coordinates are absent. The equations of motion can be reduced to a set of inde­pendent equations, each containing one qn, provided that all Q„ are independent of the coordinates qn.

We shall not discuss the methods of calculating vibration modes and frequencies in this book.*