Vibrating Membrane

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Vibrating Membrane

In the following example you compute the natural frequencies of a pretensioned membrane using the 3D Membrane interface. This is an example of “stress stiffening”; where the transverse stiffness of a membrane is directly proportional to the tensile force.

The results are compared with the analytical solution.

Model Definition

The model consists of a circular membrane, supported along its outer edge.

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Membrane radius, R = 0.25 m

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Membrane thickness, h = 0.2 mm

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Young’s modulus, E = 200 GPa

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Poisson’s ratio, ν = 0.33

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Mass density, ρ = 7850 kg/m3

The outer edge of the membrane is supported in the transverse direction. Two points have constraints in the in-plane direction in order to avoid rigid body motions.

The membrane is pretensioned by in the radial direction with σi = 100 MPa, giving a membrane force T0 = 20 kN/m.

Results and Discussion

The analytical solution for the natural frequencies of the vibrating membrane given in Ref. 1 is:

(1)

The values kij are derived from the roots of the Bessel functions of the first kind.

In Table 1 the computed results are compared with the results from Equation 1. The agreement is very good. The mode shapes for the first six modes are shown in Figure 1 through Figure 6. Note that some of the modes have duplicate eigenvalues, which is a common property for structures with symmetries.

Table 1: Comparison between analytical and computed natural frequencies.
Mode number	Factor	Analytical frequency (Hz)	COMSOL result (Hz)
1	k10 = 2.4048	172.8	172.8
2	k11 = 3.8317	275.3	275.3
3	k11 = 3.8317	275.3	275.3
4	k12 = 5.1356	369.0	369.0
5	k12 = 5.1356	369.0	369.0
6	k20 = 5.5201	396.6	396.7

Figure 1: First eigenmode.

Figure 2: Second eigenmode.

Figure 3: Third eigenmode.

Figure 4: Fourth eigenmode.

Figure 5: Fifth eigenmode.

Figure 6: Sixth eigenmode.

Notes About the COMSOL Implementation

An eigenfrequency simulation with a pre-stressed structure can be simulated in two ways. If stresses are known in advance, it is possible to use an initial stress condition. This is shown in the first study.

In a general case, the prestress is given by some external loading, and is thus the result of a previous step in the solution. Such a study would consist of two steps: One stationary step for computing the prestressed state, and one step for the eigenfrequency. The special study type Prestressed Analysis, Eigenfrequency can be used to set up such a sequence. This is shown in the second study in this example.

Since an unstressed membrane has no stiffness in the transverse direction, it is generally difficult to get an analysis to converge without taking special measures. One such method is shown in the second study: A spring foundation is added during initial loading, and is then removed.

1. A. Bower, Applied Mechanics of Solids, CRC Press, 2010.

Structural_Mechanics_Module/Verification_Examples/vibrating_membrane

Modeling Instructions

From the menu, choose .

In the window, click

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In the window, click

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In the tree, select .

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In the tree, select .

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Global Definitions

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In the window, under click .

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In the window for , locate the section.

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In the table, enter the following settings:


Name	Expression	Value	Description
R	250[mm]	0.25 m	Radius
thic	0.2[mm]	2E-4 m	Thickness
T0	100[MPa]*thic	20000 N/m	Pretension force
E1	200[GPa]	2E11 Pa	Young's modulus
rho1	7850[kg/m^3]	7850 kg/m³	Density
nu1	0.33	0.33	Poisson's ratio
fct	sqrt(T0/(thicrho1))/(2pi*R)	71.853 1/s	Common factor in natural frequencies
f10	2.4048*fct	172.79 1/s	1st natural frequency
f11	3.8317*fct	275.32 1/s	2nd and 3d natural frequencies
f12	5.1356*fct	369.01 1/s	4th and 5th natural frequencies
f20	5.5201*fct	396.64 1/s	6th natural frequency

Cylindrical System 2 (sys2)

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In the window, expand the node.

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Right-click and choose .

Work Plane 1 (wp1)

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In the toolbar, click

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In the window, click .

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In the window for , click

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Work Plane 1 (wp1)>Circle 1 (c1)

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In the toolbar, click

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In the window for , locate the section.

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In the text field, type R.

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In the window, right-click and choose .

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button in the toolbar.

Material 1 (mat1)

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In the window, under right-click and choose .

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In the window for , locate the section.

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In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Young’s modulus	E	E1	Pa	Basic
Poisson’s ratio	nu	nu1	1	Basic
Density	rho	rho1	kg/m³	Basic

Membrane (mbrn)

Thickness and Offset 1

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In the window, under click .

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In the window for , locate the section.

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In the d text field, type thic.

Linear Elastic Material 1

In the window, click .

Initial Stress and Strain 1

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In the toolbar, click

and choose .

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In the window for , locate the section.

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In the N0 table, enter the following settings:


T0	0
0	T0

Prescribed Displacement 1

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In the toolbar, click

and choose .

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Select all four edges.

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In the window for , locate the section.

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Select the check box.

Fixed Constraint 1

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In the toolbar, click

and choose .

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Select Point 1 only.

Prescribed Displacement 2

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In the toolbar, click

and choose .

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Select Point 2 only.

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In the window for , locate the section.

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Select the check box.

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In the window, under click .

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In the window for , locate the section.

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From the list, choose .

Step 1: Eigenfrequency

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In the window, under click .

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In the window for , locate the section.

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Select the check box.

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In the toolbar, click

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In the window, expand the node, then click .

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In the window for , locate the section.

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In the text field, type w.

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In the toolbar, click

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button in the toolbar.

Mode Shape (mbrn)

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In the window, click .

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From the list, choose the first frequency at Hz.

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In the toolbar, click

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From the list, choose the first frequency at Hz.

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In the toolbar, click

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From the list, choose the first frequency at Hz.

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In the toolbar, click

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From the list, choose the first frequency at Hz.

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In the toolbar, click

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In the window for , locate the section.

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From the list, choose .

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In the toolbar, click

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Now, prepare a second study where the prestress is instead computed from an external load.

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In the toolbar, click

to open the window.

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Go to the window.

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Find the subsection. In the tree, select .

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Click in the window toolbar.

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In the toolbar, click

to close the window.

Membrane (mbrn)

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In the toolbar, click

and choose .

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Select all four edges.

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In the window for , locate the section.

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From the list, choose .

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Locate the section. Specify the FL vector as


T0	r
0	phi
0	a

Add a spring with an arbitrary, small stiffness in order to suppress the out-of-plane singularity of the unstressed membrane.

Spring Foundation 1

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In the toolbar, click

and choose .

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Select Boundary 1 only.

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In the window for , locate the section.

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From the list, choose .

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In the kA table, enter the following settings:


0	0	0
0	0	0
0	0	10

Switch off the initial stress, which should not be part of the second study. In the eigenfrequency step, the stabilizing spring support must also be removed.

Step 1: Stationary

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In the window, under click .

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In the window for , locate the section.

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Select the check box.

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Locate the section. Select the check box.

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In the tree, select .

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Step 2: Eigenfrequency

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In the window, click .

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In the window for , locate the section.

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Select the check box.

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In the tree, select and .

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In the toolbar, click

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Mode Shape (mbrn) 1

The eigenfrequencies computed using this more general approach are the same as before, except some small numerical differences.

To make behave as when it was first created, the features added for must be disabled.

Solver Configurations

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In the window for , locate the section.

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Select the check box.

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In the tree, select and .

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