The method of the Poincaré map for compressional and torsional waves in composite rods

A generalized Poincaré map is introduced and used to find the compressional and torsional normal-mode frequencies and wave amplitudes of a one-dimensional rod composed by N coaxial cylinders of different length and radius. The method is approximately 100 times faster and more stable than the transfer matrix method. The Poincaré map method is used in a rod composed of two locally periodic systems, each one with a different length of the unit cell. This system can be used as an almost perfect mechanical noise filter.