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Transverse Motion of the Ideal Beam
Consider a thin beam, or rod, aligned parallel to the axis. We will be interested in the transverse motion of the beam, which we will assume to be restricted to one perpendicular direction; we will call the deflection of the beam from the axis . The relevant material parameters of the beam are the mass density , the crosssectional area , Young's modulus and , the moment of inertia of the beam about the perpendicular axis. The material parameters are, in general, allowed to be slowlyvarying functions of . Under the further assumptions that the beam deflection is small, and that the beam crosssection remains perpendicular to the socalled ``neutral axis'', it is possible to arrive at the EulerBernoulli equation [77]:

(5.1) 
Notice that this equation contains a fourth order spatial derivative, resulting from the fact that the beam provides its own restoring stiffness, proportional to its curvature, in marked contrast to the equation for a string, which requires externally applied tension in order to support wave motion. In particular, it does not result from the elimination of variables in a hyperbolic system (see §3.2). If the material properties of the beam do not vary spatially, then (5.1) reduces to the more familiar form

(5.2) 
where
. In what follows, however, we will deal with the more general case.
Subsections
Next: Phase and Group Velocity
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Stefan Bilbao
20020122