The Lippmann–Schwinger equation and renormalization for transmission path analysis in discrete mechanical systems

Abstract

The dynamics of mechanical structures are often described by linear algebraic systems of the form Ax=f. At high frequencies, A may represent the coupling loss factor matrix in a Statistical Energy Analysis (SEA) model, whereas at low frequencies, it may correspond to the dynamic stiffness matrix of a system of oscillators. While such systems admit a Neumann series solution at high frequencies-where the terms can be interpreted as energy transmission paths of increasing order-this series typically fails to converge at low frequencies, rendering its physical interpretation unclear. In this work, we recast the system within the framework of the Lippmann-Schwinger equation and express the solution as a series in powers of a transmission matrix T, defined as the product of the system’s bare Green function and coupling matrix. To achieve convergence, we introduce a multi-parameter product renormalization scheme. We show that, with a suitable choice of parameters based on the eigenvalues of T, a finite expansion is obtained involving powers up to TN−1, where N is the system's dimension. That is, the expansion includes at most the longest open transmission paths between elements. In doing so, we recover-through purely algebraic methods-a result previously derived using considerations from graph theory.