COMSOL 6.4 - Vibrations of a Disk Backed by an Air-Filled Cylinder

Vibrations of a Disk Backed by an Air-Filled Cylinder

The vibration modes of a thin or thick circular disc are well known, and it is possible to compute the corresponding eigenfrequencies with an arbitrary precision from a series solution. The same is true for the acoustic modes of an air-filled cylinder with perfectly rigid walls. A more interesting question to ask is: What happens if the cylinder is sealed in one end not by a rigid wall but by a thin disc? This is the question addressed in this tutorial.

This application uses the Acoustic–Shell Interaction, Frequency Domain interface, which is available if you have both the Acoustics Module and the Structural Mechanics Module.

Model Definition

The geometry is a rigid steel cylinder with a height of 255 mm and a radius of 38 mm. One end is welded to a heavy slab, while the other is sealed with a steel disc only 0.38 mm thick. The disc is modeled using shell elements with the outer edge of the disc fixed. The acoustics in the cylinder is described in terms of the acoustic (differential) pressure. The eigenvalue equation for the pressure is

where c is the speed of sound and ω = 2π f defines the eigenfrequency, f.

A first step is to calculate the eigenfrequencies for the disc and the cylinder separately and compare them with theoretical values. This way you can verify the basic components of the model and assess the accuracy of the finite-element solution before modeling the coupled system. When computing the decoupled problem, the acoustic domain is completely surrounded by sound hard boundaries. In the coupled analysis, the boundary at the disc instead has the accelerations of the disc as boundary conditions. At the same time, the acoustic pressure supplies a load on the disc. Such a coupling is set up automatically by the Acoustic–Structure Boundary multiphysics coupling.

Results and Discussion

To be able to study the effects of the coupling, we first look at the solution of the uncoupled problem. Figure 1 through Figure 4 show the two first uncoupled structural and acoustic modes.

Figure 1: First structural mode displayed by displacement magnitude.

Figure 2: Second structural mode displayed by displacement magnitude..

Figure 3: First acoustic mode displayed by pressure isosurfaces.

Figure 4: Second acoustic mode displayed by pressure isosurfaces.

In Ref. 1, D.G. Gorman and others have thoroughly investigated the coupled model at hand, and they have developed a semi-analytical solution verified by experiments. Their results for the coupled problem are presented in Table 1, together with the computed results from the COMSOL Multiphysics analysis.

Table 1: Results from semi-analytical and COMSOL Multiphysics analysIs and experimental Data.
Dominated by	Semi-analytical (Hz)	Computed (Hz)	Experimental (Hz)
str/ac	636.9	637.1	630
str/ac	707.7	707.6	685
ac	1347	1347	1348
str	1394	1396	1376
ac	2018	2018	2040
str	2289	2292/2304	2170
str/ac	2607	2623	2596
ac	2645	2646	–
str/ac	2697	2697	2689
ac	2730	2730	2756
ac	2968	2968	2971

As the table shows, the computed eigenfrequencies are in good agreement with both the theoretical predictions and the experimentally measured values. The table also states whether the modes are structurally dominated (str), acoustically dominated (ac), or tightly coupled (str/ac). The eigenfrequency precision is generally better for the acoustically dominated modes.

Most of the modes show rather weak coupling between the structural bending of the disc and the pressure field in the cylinder. It is, however, interesting to note that some of the uncoupled modes have been split into one covibrating and one contravibrating mode with distinct eigenfrequencies. This is, for example, the case for modes 1 and 2 in the FEM solution.

In Figure 5, the first coupled mode is shown in terms of disc displacements and air pressure. The coupling effect can be clearly displayed using a plot of pressure gradients, as in Figure 6.

Figure 5: Disc deformation and pressure slice plot for the first coupled mode.

Figure 6: Disc deformation and pressure gradient slice plot for the first coupled mode.

Reference

1. D.G. Gorman, J.M. Reese, J. Horacek, and D. Dedouch, “Vibration Analysis of a Circular Disk Backed by a Cylindrical Cavity,” Proc. Instn. Mech. Engrs., Part C, vol. 215, no. 11, pp. 1303–1311. 2001.

Notes About the COMSOL Implementation

Specify the part of the physics for which to compute the uncoupled eigenvalues by selecting the physics interface in the eigenfrequency study.

This model uses the Acoustic–Shell Interaction, Frequency Domain interface. If you decide to couple Pressure Acoustics, Frequency Domain to Shell manually, be careful when selecting the sign of the coupling terms so that they act in the intended direction. You should specify the acceleration in the inward normal direction for the pressure acoustics domain, which in this case is the positive z-acceleration of the disc. The acceleration is denoted wtt as it is the second time-derivative of the variable w. The pressure on the shell can be given using global directions, so that a positive pressure acts as a face load in the negative z direction. This is handled automatically by the Acoustic–Structure Boundary multiphysics coupling.

Acoustics_Module/Verification_Examples/coupled_vibrations

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 .

Cylinder 1 (cyl1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 38.

In the text field, type 255.

Click

Shell (shell)

In the window, under click .

In the window for , locate the section.

Click

Select Boundary 3 only.

Thickness and Offset 1

In the window, under > click .

In the window for , locate the section.

In the d0 text field, type 0.38[mm].

Fixed Constraint 1

In the toolbar, click

and choose .

Select Edges 2, 3, 7, and 10 only.

Materials

Acoustic Material

In the window, under right-click and choose .

In the window for , type Acoustic Material in the text field.

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


Property	Variable	Value	Unit	Property group
Density	rho	1.2	kg/m³	Basic
Speed of sound	c	343	m/s	Basic

Structural Material

Right-click and choose .

In the window for , type Structural Material in the text field.

Locate the section. From the list, choose .

Select Boundary 3 only.

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


Property	Variable	Value	Unit	Property group
Young’s modulus	E	210[GPa]	Pa	Young’s modulus and Poisson’s ratio
Poisson’s ratio	nu	0.3	1	Young’s modulus and Poisson’s ratio
Density	rho	7800	kg/m³	Basic

Mesh 1

In this model, the mesh is set up manually. Proceed by directly adding the desired mesh component.

Free Quad 1

In the toolbar, click

and choose .

Select Boundary 3 only.

Size

In the window, click .

In the window for , locate the section.

Click the button.

Locate the section. In the text field, type 10.

In general, 5 to 6 second-order elements per wavelength are needed to resolve the waves. For more details, see Meshing (Resolving the Waves) in the Acoustics Module User’s Guide. The mesh size in this model corresponds to resolving the wavelength with about 9 elements at 4000 Hz, the maximum eigenfrequency studied in the model. The wavelength in air at 4000 Hz is 8.6 cm.

Click