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Next: Chapter 6: Conclusions and Up: Summary and Results Previous: Chapter 4: Digital Waveguide

Chapter 5: Applications in Vibrational Mechanics

In Chapters 3 and 4, scattering structures are developed to numerically integrate certain simple systems of PDEs, in particular the transmission line equations and the parallel-plate system. In this chapter, we show how the same ideas can be applied to another set of systems, namely those describing the mechanical vibrations of elastic solid media under various conditions. These systems can become quite complex, compared with the simple transmission-line test problems, but as we will see, circuit representations can be developed as before, though several new techniques must be introduced. We look at such systems in order of increasing dimensionality, loosely following the organisation of the text by Graff [77]. Liberal use is made of the unifying result of §4.10 in order to develop wave digital and digital waveguide simulation networks in a parallel fashion for these systems. We first examine the simplest stiff distributed system, the classical, or ideal Euler-Bernoulli beam in §5.1, mainly in order to indicate the difficulties inherent in designing scattering methods for systems which are not symmetric hyperbolic (though we show that it is indeed possible). We then turn to the modern (and much more suitable, in the scattering context) Timoshenko beam theory, which was first treated by Nitsche in [131], and present a variety of distinct scattering methods in §5.2, while indicating the relevant differences, especially with respect to stability. We also apply the system balancing approach (introduced in §3.12) to the Timoshenko beam in §5.2.6 in order to show that it is possible to drastically reduce the computational requirements in certain cases, and take an extended look at boundary conditions in §5.2.4. Then follows a look at stiff plate theory, and in particular the two-dimensional analogue of the Timoshenko beam, called the Mindlin plate, in §5.4. Here, due to the couplings between the variables, we are forced to make use of vector scattering elements, which were introduced in §2.3.7 for this very purpose. Boundary conditions for waveguide networks for the Mindlin plate are dealt with in detail in §5.4.2. We next spend some time examining network representations for two cylindrical shell models, first the membrane shell in §5.5.1, and then the more modern model of Naghdi and Cooper in §5.5.2. Finally, for completeness sake, we revisit in §5.6 Nitsche's MDKC for the full three-dimensional elastic solid dynamic system [131]; as for all the systems in this chapter, we show the alternative network form suitable for DWN discretization in §5.6.1. In keeping with the more applied flavor of this chapter, we also present simulation results for Timoshenko's beam system and the Mindlin plate in §5.2.5 and §5.4.3, respectively, under both uniform and spatially-varying material parameter conditions.


next up previous
Next: Chapter 6: Conclusions and Up: Summary and Results Previous: Chapter 4: Digital Waveguide
Stefan Bilbao 2002-01-22