Applications in Vibrational Mechanics

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.

- Transverse Motion of the Ideal Beam
- Phase and Group Velocity
- Finite Differences
- Waveguide Network for the Euler-Bernoulli System
- Boundary Conditions in the Waveguide Network

- Timoshenko's Beam Equations
- Dispersion
- MDKC and MDWDF for Timoshenko's System
- Waveguide Network for Timoshenko's System
- Other Waveguide Networks for Timoshenko's System
- Boundary Conditions in the DWN
- Simulation: Timoshenko's System for Beams of Uniform and Varying Cross-sectional Areas
- Improved MDKC for Timoshenko's System Via Balancing

- Longitudinal and Torsional Waves in Rods
- Plates
- Maximum Group Velocity
- MDKCs and Scattering Networks for Mindlin's System
- Boundary Termination of the Mindlin Plate
- Simulation: Mindlin's System, for Plates of Uniform and Varying Thickness

- Cylindrical Shells

- Elastic Solids