## Clamped Free Beam

Figure 2.16 Inertially restrained end of a beam.

finite angular acceleration of the end. Therefore, de d2e

so that

From the functional form of Y(t) as established from the separation procedure, it can be noted that

Substitution into the preceding condition yields

GJX'(l)Y(t) = a2βIcX(i)Y(t), (2.171) Pip which requires that pIpX\i) = a2IcX(l). (2.172)

As above, the reader should verify that the same type of boundary condition at the other end would yield pIpX'( 0) = -a2IcX( 0). (2.173)

2.2.3 Example Solutions for Mode Shapes and Frequencies

In this section we consider several examples of the calculation of natural frequencies and mode shapes of vibrating beams in torsion. We begin with the clamped-free case. Next, we consider the free-free case, illustrating the concept of the rigid-body mode. Finally, we consider a case that requires numerical solution of the transcendental characteristic equation, a beam clamped at its root and restrained with a rotational spring at its tip.

Example 5: Solution for Clamped-Free Beam

To illustrate the application of these boundary conditions, consider the case of a uniform beam that is clamped at x β 0 and free at x β I, as shown in Fig. 2.17.

The boundary conditions for this case are

Figure 2.17 Schematic of clamped-free beam undergoing torsion.

Figure 2.17 Schematic of clamped-free beam undergoing torsion.

Recall that the general solution was previously determined as

where X and Y are given in Eqs. (2.158). For a ^ 0 the first of those equations has the solution

It is apparent that the boundary conditions lead to the following: X(0) = 0 requires B = 0,

If A β 0 a trivial solution will be obtained, such that the deflection will be identically zero. Since a ^ 0, a nontrivial solution requires that cos(af) = 0. (2.178)

This is called the "characteristic equation," the solutions of which consist of a denumerably infinite set called the "eigenvalues" and are given by

The Y(t) portion of the general solution can be observed to have the form of simple harmonic motion, as indicated in Eq. (2.170), so that the natural frequency is

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