## J o

This integral value, M, , is called the "generalized mass" of the z'th mode. These relations thus demonstrate that the mode shapes for the string, which is fixed at both ends, form an orthogonal set of functions.

The above development is for a string of constant mass per unit length and constant tension force. It is important to note that it can readily be generalized for nonuniform mass per unit length. In more involved developments for beam bending and torsional deformation, the structural stiffnesses, which are analogous to the tension force in the string problem, may also be nonuniform along the span. Although these quantities may not be taken outside the integrals in such cases, the rest of the development remains quite similar.

### 2.1.4 Using Orthogonality

The property of orthogonality is very useful in many aspects of structural dynamics analysis. As an illustration, consider the so-called homogeneous initial condition problem for the string. In this case there are no external loads on the string, but it is presumed to have an initial deflection shape and an initial velocity distribution. Let these initial conditions be represented as v(x, 0) - /(*),

— (*, 0) - *(*), at where it should be noted that both f(x ) and g(x ) must certainly be compatible with the boundary conditions.

Using Eq. (2.29), these initial conditions can be written in terms of modal representation as v(x, 0) - F> sin {—) - f

Both of these relations will be multiplied by sw.(jjtx/i) and integrated over the length of the string, yielding

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