The solution of vibroacoustic linear systems as a finite sum of transmission paths

Linear systems are frequently encountered in low, mid and high vibroacoustics modelling of mechanical built-up structures. It has recently been proved that the solution to those systems can be always factorized as an infinite (weighted) Neumann series summation, which accounts for signal transmission through paths connecting system elements. The key to path expansion relies on the concept of direct transmissibility. In this work, we explore some additional theoretical aspects of transmissibility-based transmission path analysis (TPA), which is known to constitute a valuable tool to remedy noise and vibration problems. In particular, we show that it is also possible to expand the solution of a matrix linear system as a finite summation of transmission paths. Furthermore, our goal is to provide mathematical and physical insight into such path factorization. As regards the former, we exploit the relationship between graph theory and matrix algebra to interpret the terms appearing in the series expansion as combinations of open and closed paths in a graph. In what concerns the second, two benchmark examples are addressed that benefit from the graph theory outcomes. The first one consists of a mass-damping-stiffness system representative of vibroacoustic modelling at low frequencies. A relation is established between the relative weights of the paths, the global system resonances and the resonances of complementary systems, which contain elements not belonging to the paths. The second example involves a statistical energy analysis (SEA) model made of connected plates. The meaning of the relative weights of open paths in the finite expansion for energy transmission between SEA subsystems is analyzed and compared to the results of infinite SEA path factorization.