Eigenmodes in a Muffler

In this example, you compute the propagating modes in the chamber of an automotive muffler. The geometry is a cross section of the chamber in the Absorptive Muffler example.

The purpose of the model is to study the shape of the propagating modes and to find their cutoff frequencies. As discussed in the documentation of the Absorptive Muffler example, some of the modes significantly affect the damping of the muffler at frequencies above their cutoff. In this model, you study modes with cutoff frequencies up to 1500 Hz.

Model Definition

The muffler chamber has a race track shaped cross section, as seen in Figure 1. In this model, the chamber is considered to be hollow and field with air at atmospheric pressure.

Figure 1: The model geometry.

The wave numbers and mode shapes through a cross section of the chamber are found as the solution of an eigenvalue problem for the acoustic pressure p:

where ρ0 is the density, c the speed of sound, κz the out-of-plane wave number, and ω = 2π f the angular frequency. For a given angular frequency, only modes such that κz2 is positive can propagate. The cutoff frequency of each mode is calculated as

Results and Discussion

The model finds five propagating modes, whose characteristics are summed up in the table here below.


Cutoff frequency (Hz)	Characteristics
0	Plane wave
635	Antisymmetric with respect to x, symmetric with respect to y
1210	Symmetric with respect to x, antisymmetric with respect to y
1240	Symmetric with respect to x and y
1467	Antisymmetric with respect to x and y

For a muffler with a centered tube leading into the chamber, the first mode that is symmetric with respect to both the x-axis and the y-axis is propagating when the frequency is higher than 1240 Hz. Figure 2 shows this mode, which for an infinitely long chamber occurs at 1240 Hz.

Figure 2: The chamber’s first fully symmetric propagation mode. The plot shows the real part of the acoustic pressure.

Acoustics_Module/Automotive/eigenmodes_in_muffler

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Geometry 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Square 1 (sq1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 150.

Locate the section. From the list, choose .

Circle 1 (c1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 75.

Locate the section. In the text field, type -75.

Circle 2 (c2)

In the toolbar, click

In the window for , locate the section.

In the text field, type 75.

Locate the section. In the text field, type 75.

Union 1 (uni1)

In the toolbar, click

and choose .

Click in the window and then press Ctrl+A to select all objects.

In the window for , locate the section.

Clear the check box.

Click

Click the

button in the toolbar.

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

By default, the boundaries of the geometry will be considered to be sound hard walls. No other physics settings are needed.

Mesh 1

In this model, the mesh is set up manually. Add a free triangular mesh and then proceed to the solver settings.

Free Triangular 1

In the toolbar, click

Study 1

Step 1: Mode Analysis

In the window, under click .

In the window for , locate the section.

Select the check box. In the associated text field, type 8.

Select the check box. In the associated text field, type 20.

The free-space propagation mode has an out-of-plane wave number equal to omega/c = 27.5 rad/m. With these settings, the solver returns the 8 modes with propagation constants closest to 20 rad/m first in the list.

In the text field, type 1500[Hz].

This setting makes the software look for propagating modes with cutoff frequencies up to 1500 Hz.

In the toolbar, click

Results

Acoustic Pressure (acpr)

The solver has found the free-space mode and all other propagating modes. There is a total of 5 different propagating modes. Because the waves can propagate both into and out of the modeling plane, each mode gets reported twice, with positive and negative out-of-plane wave numbers.

For the positive out-of-plane wave numbers, it holds that the higher the mode, the lower the wave number. However, the solver does not stop at zero. Because you asked for more than the 5 existing propagating modes, you get additional modes with imaginary out-of-plane wave numbers. This indicates that they are evanescent. The default plot shows the acoustic pressure distribution for a mode with a wave number of -17.95i rad/m.

In the window for , locate the section.

From the list, choose .

In the toolbar, click

Click the

button in the toolbar.

This is the lowest fully symmetric mode, which is shown in Figure 2. You can compute the cutoff frequency of this mode using the expression in the model introduction. In order to refer to the speed of sound in air, use an arbitrary point in the geometry for this evaluation.

Point Evaluation 1

In the toolbar, click

In the window for , locate the section.

From the list, choose .

In the list, select .

Select Point 3 only.

Locate the section. In the table, enter the following settings:


Expression	Unit	Description
sqrt(acpr.omega^2-acpr.kz^2acpr.c^2)/(2pi)	rad/s

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