1. Example: Acoustics in a Muffler

2. Introduction The damping effectiveness of a muffler is studied in the frequency range 100─1000 Hz In the low-frequency range, reactive damping prevails. Therefore, resistive damping is not included in the model. The purpose of the model is to show how 3D acoustics can be treated in a fairly complex geometry consisting of several separate sections and pipes divided by thin, but perfectly rigid, walls. Acoustics in a Muffler

3. Geometry Acoustics in a Muffler – Problem Definition Inlet Outlet Resonator chambers

4. Domain Equations Frequency domain acoustics is goverened by a slightly modified Helmholtz’s equation for the acoustic pressure, p. Material properties are density,  0, and speed of sound, c. Note that the density cannot be eliminated from the equation unless it is a global constant Acoustics in a Muffler – Problem Definition

5. Boundary Conditions The natural boundary condition corresponds to inward normal acceleration. Three types are used: Acoustics in a Muffler – Problem Definition Hard wall Incoming plus outgoing plane wave Outgoing plane wave

6. Implementation Different dependent variables are used in the pipes and in the resonator chambers in order to decouple the pressure on the inside of the pipe walls from the pressure on the outside. The continuity of the pressure where the pipes open into the resonator champers is enforced through boundary conditions. The geometry is cut into three parts, not physically connected to eachother, in order to decouple the pressure across the walls separating the chambers. At the openings between the chambers, the pressure is made continuous through use of coupling variables and weak boundary conditions. Acoustics in a Muffler – Implementation

7. Results The plot shows the pressure amplitude and energy flow at 470 Hz Energy is conserved as no dissipation has been included in the model The frequency is close to one of the resonance frequencies of the middle section. Acoustics in a Muffler – Results

8. Results The damping coefficent shows typical ”dips” at the resonance frequencies of the separate chambers. Acoustics in a Muffler – Results