Elastic Transfer Matirx Method

The Transfer Matrix Method (TMM) has traditionally been a powerful and efficient mathematical tool for analyzing wave propagation phenomena. Metaphorically, this method accurately predicts how waves, like light or sound, pass through multi-layered planar structures (e.g., anti-reflective coatings on eyeglasses) as if precisely calculating stacked 'Lego blocks.' However, this powerful tool has a critical limitation: it is fundamentally based on Cartesian coordinates. This makes it extremely difficult to apply the method to cylindrical geometries with curved surfaces, such as optical fibers that power the internet or turbine blades in high-speed rotating systems. In particular, a research gap has existed in analyzing the vibrations or waves in elastic annular periodic structures—structures with repeating circular ring (annular) patterns—found in high-speed rotating machinery, as conventional TMM has rarely been applied to them.This research directly confronts this limitation by proposing a new theoretical framework that fundamentally reconstructs the conventional TMM to fit cylindrical coordinates.

• 1D Cylindrical TMM: This one-dimensional extension enables the analysis of phenomena where waves propagate from the center outwards (or vice versa) in structures composed of multiple cylindrical layers, much like tree rings.

• 3D Cylindrical TMM: The three-dimensional extension is far more powerful. It means that waves propagating in the axial direction (along the length) of the cylinder, like inside an optical fiber, can be perfectly modeled while simultaneously accounting for periodic patterns repeating in the annular direction (around the circumference). Implementing this requires new algorithms for handling special functions, such as Bessel functions, which are the native language of cylindrical problems.