Wave analysis and control of uniformly varying two-dimensional ladder-network structures

Abstract

This study investigates the wave analysis and control of two-dimensionally (2D) connected ladder networks. These networks describe the out-of-plane vibration of 2D arrayed masses, according to the mechanical–electrical analogy. One typical correspondence to continuous systems is the out-of-plane vibration of membranes. In the analysis, the emphasis is on the properties of the secondary constants–propagation constants and characteristic impedances (admittances)–as analytic functions of the Laplace transform variable s. The system can be treated as a cascade connection of layers comprising the lateral direction elements, varying uniformly by a constant ratio along the longitudinal direction. In addition, the system dynamics can be described by a first-order recurrence relation in the Laplace transform domain. The characteristic polynomial defining the propagation constants can be defined independently of the layer position, even for the uniformly varying case. It can be decomposed into second-order polynomials characterized by a characteristic constant. We demonstrate that, above a threshold, all the constants are different and real, revealing several properties of the secondary constants and transformation matrix required for the wave analysis and control. These properties are also useful for a rational implementation of the impedance matching controller. Numerical examples illustrate the derived results and effectiveness in vibration control.