TABLE OF CONTENTS The free vibration of a beam in transverse bending motion is often referred to as transverse vibration. This type of motion differs from the transverse string dynamics and beam torsional dynamics in that the governing equations of motion are of a different mathematical form. Although these equations are different, their solutions are obtained in a similar manner and exhibit similar physical characteristics. It should also be observed that, whereas most aerospace structures will experience combined or simultaneous bending and torsional dynamic behavior, we have here chosen certain configuration variables to uncouple these types of motion.
As in the case of torsion the beam will initially be treated as having nonuni-form properties along the x axis. The x axis will be taken as the line of the individual cross-sectional neutral axes associated with pure bending in and normal to the plane of the diagram in Fig. 2.24. For simplicity, however, we will only consider uncoupled bending in the x-y plane, thus excluding initially twisted beams from the development. The bending deflections are denoted by v( x, t) in the y direction. The x axis is presumed to be straight, thus excluding initially curved beams. We will continue to assume for now that the properties of the beam allow the x axis to be chosen so that bending and torsion are both structurally and inertially uncoupled. Finally, the transverse beam displacement, v, will be...
