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Timoshenko's Beam Equations
Timoshenko's theory of beams constitutes an improvement over the EulerBernoulli theory, in that it incorporates shear and rotational inertia effects [77]. This is one of the few cases in which a more refined modeling approach allows more tractable numerical simulation; the reason for this is that Timoshenko's theory gives rise to a hyperbolic system, unlike the EulerBernoulli system, for which propagation velocity is unbounded. It is this partially parabolic character of the EulerBernoulli system which engenders severe restrictions on the maximum allowable time step (at least in the case of explicit methods, of which type are all the scatteringbased methods included in this work). For a physical derivation of Timoshenko's system, we refer the reader to [77,146,152,187,188], and simply present it here:

(5.17a) 
As before, represents the transverse displacement of the beam from an equilibrium state, and the new dependent variable is the angle of deflection of the crosssection of the beam with respect to the vertical direction. Here, the quantities , , and are as for the EulerBernoulli Equation (5.1). is the shear modulus (usually called in other contexts) and is a constant which depends on the geometry of the beam. For generality, we assume that all these material parameters are functions of . Losses or sources are not modeled.
Nitsche [131], in his MDWD networkbased approach preferred to use the more fundamental set of four first order PDEs from which system (5.16) is condensed:

(5.18a) 


(5.19a) 

We have introduced here the quantities
is interpreted as transverse velocity, as an angular velocity, as the bending moment, and as the shear force on the crosssection. Each of the subsystems (5.17) and (5.18) has the form of a lossless (1+1)D transmission line system; they are coupled by constantproportional terms, and it is this coupling that gives the Timoshenko system its dispersive character. The EulerBernoulli system (5.4) is recovered in the limit as
and
[131].
This is a symmetric hyperbolic system of the form given in (3.1), with
,
and
Subsections
Next: Dispersion
Up: Applications in Vibrational Mechanics
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Stefan Bilbao
20020122