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# Applications in Vibrational Mechanics

In the previous chapters, we dealt with the numerical integration of systems of PDEs that were all, in some sense, generalizations of the wave equation. In the case where the material parameters ( and in the case of the transmission line) have no spatial variation, this amounts to saying that wave propagation in these media is dispersionless; a plane wave travels at a fixed speed, regardless of its wavelength. We now turn to sets of equations which fundamentally engender some degree of dispersion, namely those describing the motion of stiff systems such as beams, plates and shells. As a result, we move toward the use of mechanical quantities, as opposed to electrical, but the analogy should be clear. We will show how waveguide and wave digital filter principles can be used in order to obtain numerical solutions of such equations.

We will develop these algorithms in order of increasing dimensionality of the system; first we look at classical beam dynamics, and then proceed to the more modern beam theory devised by Timoshenko. We then look at stiff plates, and in particular the thick plate formulation of Mindlin, and then at two types of cylindrical shell theories, first the so-called membrane shell, then the thick shell system of Naghdi and Cooper. Finally, we look at the general (3+1)D system which describes the motion of a linear elastic solid.

It will be necessary to introduce several new techniques in order to develop numerical methods for such systems, which are considerably more complex than the transmission line test problems which we examined in the previous chapters. First, although these systems are all (with the exception of Euler-Bernoulli beam system) symmetric hyperbolic, we will need to make use of non-reciprocal circuit elements in order to model some asymmetric couplings that occur. Second, we may have to perform some additional initial work on these same systems in order to symmetrize them, as they are not always symmetric hyperbolic in their commonly encountered forms. Third, in some cases we will be forced to make use of vector-valued wave variables and scattering junctions [131] (see §2.3.7).

A full technical summary of this chapter appeared in §1.3.

Subsections

Next: Transverse Motion of the Up: Wave and Scattering Methods Previous: Scattering Networks for Maxwell's
Stefan Bilbao 2002-01-22