Bachner B.

A mathematical formulation for linear plane-wave acoustic modelling of complex silencer systems containing multi-port elements was developed. The prevalence of such multi-port elements in automotive silencers – perforated pipe elements and crossovers being typical examples – dictates the need for a simulation tool that facilitates their integration into an acoustic network consisting of lumped and distributed elements. A fundamental aspect of the presented approach evolves from the individual mathematical characterization of lumped and distributed elements: lumped elements are governed by algebraic equations; distributed elements by differential equations. The resulting matrices thus have distinctive forms: matrices similar to the so-called impedance form are used for lumped elements; scattering-matrices for distributed elements. This distinction leads consequently to an object-oriented software design with a set of classes reflecting the unique properties associated with each of the acoustic element classifications. A procedure for the automated assembly of the matrix system for the complete network is introduced, in which the problem is intrinsically reduced to the smallest possible number of unknowns. The solution delivers a vector of travelling pressure waves entering the distributed elements at their connexions. By calculating the pressure waves leaving the distributed elements via the scattering matrices, the sound field for the entire system can be easily obtained. If one is only interested in the transmission behaviour of a complex system, the appropriate transfer matrix can be obtained directly from matrix condensation by eliminating all inner degrees of freedom. Since source terms can be added to both lumped and distributed elements, active systems can also be modelled using the presented approach.