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# Timoshenko's Beam Equations

Timoshenko's theory of beams constitutes an improvement over the Euler-Bernoulli theory, in that it incorporates shear and rotational inertia effects . 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 Euler-Bernoulli system, for which propagation velocity is unbounded. It is this partially parabolic character of the Euler-Bernoulli 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 scattering-based 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 cross-section of the beam with respect to the vertical direction. Here, the quantities , , and are as for the Euler-Bernoulli 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 , in his MDWD network-based 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 cross-section. 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 constant-proportional terms, and it is this coupling that gives the Timoshenko system its dispersive character. The Euler-Bernoulli system (5.4) is recovered in the limit as and .

This is a symmetric hyperbolic system of the form given in (3.1), with , and Subsections   Next: Dispersion Up: Applications in Vibrational Mechanics Previous: Free End
Stefan Bilbao 2002-01-22