The solution of linear mechanical systems in terms of path superposition. (15th February 2017) - Magrans, Francesc Xavier; Poblet-Puig, Jordi; Rodríguez-Ferran, Antonio
Abstract: We prove that the solution of any linear mechanical system can be expressed as a linear combination of signal transmission paths. This is done in the framework of the Global Transfer Direct Transfer (GTDT) formulation for vibroacoustic problems. Transmission paths are expressed as powers of the transfer matrix. The key idea of the proof is to generalise the Neumann series of the transfer matrix – which is convergent only if its spectral radius is smaller than one – into a modified Neumann series that is convergent regardless of the eigenvalues of the transfer matrix. The modification consists in choosing the appropriate combination coefficients for the powers of the transfer matrix in the series. A recursive formula for the computation of these factors is derived. The theoretical results are illustrated by means of numerical examples. Finally, we show that the generalised Neumann series can be understood as an acceleration (i.e. convergence speedup) of the Jacobi iterative method. Abstract : Highlights: A definition of path is provided. The possibility of expressing the solution of all linear problems in terms of paths is proven. The final result is a generalisation of the Neumann series. The results are illustrated with numerical examples. The proposed approach is related with the Jacobi iterative method for linear systems.