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dx dx

For the special case of uniform beams this relation simplifies to the one-dimensional wave equation

GJ 320 320

Other than the constants that multiply the second partial derivatives, this is the same equation that governed the dynamic behavior of the string. Thus, all the previously discussed properties of standing and traveling waves will exist here as well. However, as will be discussed in detail below, there are more interesting possibilities for the boundary conditions.

To establish these properties the separation of variables method will be applied as

6(x, t) = X(x)Y(t), which, when substituted into the wave equation, yields

Since the dependencies on x and t have been separated across the equality, each side must equal a constant, say —a2, so that

Figure 2.13 Clamped end of a beam.

Two ordinary differential equations then follow from this, namely,

Note the similarity with Eqs. (2.17). For a ^ 0, Eqs. (2.157) have solutions that can be written as

To complete the solution the constants A and B can be determined from the boundary conditions at the ends of the beam, and C and D can be found as a function of the initial beam deflection and rate of deflection. The special case of a — 0 is very important and will be addressed in more detail in Section 2.2.3.

### 2.2.2 Boundary Conditions

There are four different boundary conditions that can be imposed at the ends of the beam for determination of the constants A and B. For any given beam only one boundary condition is required at each end.

### Clamped End

In this case (see Fig. 2.13) the end of the beam is assumed to be "built-in" or "cantilevered" at a wall. As a consequence there will be no rotation due to elastic twist at the beam end, and the boundary condition is

which is identically satisfied when

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