## Gj Y

The general solutions to these equations can be written as X(x) — ax + b, Y(t) — ct + d.

The arbitrary constants, a and b, in the spatially dependent portion of the solution can again be determined from the boundary conditions. For the present case of the free-free beam the conditions are

Because both conditions are satisfied without imposing any restrictions on the constant b, this constant can be anything, which implies that the torsional deflection can be nontrivial for a — 0. From X(x) with a — 0 it is apparent that the corresponding value of 6 will be independent of the coordinate x. This means that this motion for a — 0 is a "rigid-body" rotation of the beam.

The time-dependent solution for this motion, Y(t), is also different from that obtained for the elastic motion. Primarily it can be noted that the motion is not oscillatory; thus, the rigid-body natural frequency is zero. The arbitrary constants, c and d, can be obtained from the initial values of the rigid-body orientation and angular velocity. To summarize the complete solution for the free-free beam in torsion, a set of generalized coordinates can be defined by oc

The first three elastic mode shapes are plotted in Fig. 2.20. The zero derivative at both ends is indicative of the vanishing twisting moment there. The natural

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