## Natural Frequencies of Immersed Beams

##### Nagi Elabbasi | April 22, 2014

*Today, we invite guest blogger Nagi Elabbasi of Veryst Engineering to share a modeling example of immersed beams.*

When thin structures such as beams, plates, or shells are immersed in a fluid, their natural frequencies are reduced. The fluid also affects their mode shapes and is a source of damping. This phenomenon affects structures across a wide range of industries and sizes, from micro-scale structures (e.g. MEMS actuators) to larger structures (e.g. ships).

### The Model: An Immersed Cantilever Beam

Today, we will take a look at a model of a cantilever beam immersed in a fluid:

An approximate analytical solution of the form shown below is frequently used to estimate the natural frequencies of the immersed beam. This is estimated based on the *structure-only* natural frequencies, beam geometry, and the ratio of fluid-to-beam densities. The analytical expression approximately accounts for the added mass of the fluid that is displaced by the beam. It does not account for viscous effects.

### A Multiphysics Approach to Determining Natural Frequencies and Mode Shapes

We at Veryst Engineering used COMSOL Multiphysics to determine the natural frequencies and mode shapes of an immersed cantilever beam. Then, we compared the results with the analytical approximation.

We set up the problem as a coupled acoustic-structure eigenvalue analysis. To account for the mass of the fluid, we selected a pressure acoustics formulation, and we accounted for damping due to fluid viscosity by including a viscous loss term. We assumed the fluid space to be sealed. The COMSOL software automatically detects the solid-fluid boundary and applies the necessary boundary conditions at the solid-fluid interface.

### Solution

Below, you can see a table with the first and fourth natural frequencies (in kHz) of beams in vacuum, air, and water:

As expected, the results show that air has a minor effect on the beam, while water reduces the lowest natural frequencies of the beam by about 20%. Also shown in the table is the analytical estimate for a beam immersed in water. The analytical estimate is close to the COMSOL Multiphysics prediction for this relatively standard beam configuration.

Next, we can have a look at a couple of animations of our results.

The first animation depicts the fourth mode of deformation for a beam immersed in water:

*Fourth mode shape of cantilever beam immersed in water.*

The second animation demonstrates fluid pressure contours and fluid velocity arrow plots at a section along the beam, again for the fourth natural frequency of the beam.

*Velocity and pressure contours for fourth beam natural frequency.*

This modeling example involved a simple cantilever to illustrate the concept. However, the coupled structural-acoustic modeling approach used is also applicable to more realistic geometries, such as ship hulls and MEMS actuators.

## Comments

Aimaiti RehemanjiangMay 9, 2014 6:28 amI am very interesting about this simulation. this is very cool simulation. Can you tell some information about the Young’s modulus, Poisson’s ratio for air ?? because in order to add air as environment in Structural Mechanics interface, I have encountered this problem … thank you !!!

Aimaiti RehemanjiangMay 9, 2014 6:30 amI think , when you consider the Air case in your simulation. You might use those information of air I think …

Nagi ElabbasiMay 9, 2014 2:19 pmHi Aimaiti, I am glad you liked the simulation!

You should not use the structural module only for this. You need instead acoustic-structural interaction with the air/fluid modeled as an acoustic medium. In that case the required material parameters for the fluid are the density and the speed of sound. Since we accounted for viscous dissipation then you also need the dynamic viscosity of the fluid.

I believe this simulation without the viscous damping only requires the basic COMSOL Multiphysics package. You need the Acoustics Module if you want to include viscous damping, thermo-acoustic effects or other features like linearized Euler flow.

Danilo D’AgostinoJuly 4, 2014 10:38 amHi Nagi,

It is a very interesting example, but I am wondering which boundary conditions have to be setted besides the fixed constraint at L=0?

Jin LeeJuly 29, 2014 6:11 pmDear Dr. Elabbasi,

I am just wondering how you’ve calculated the correction factor for the analytical solution above. Thanks!

Dr. Lee

Nagi ElabbasiAugust 11, 2014 9:34 amHi Danilo,

The boundary conditions are quite simple. The beam is fixed at one end as you noted, COMSOL takes care of the acoustic-structure boundary, and you specify an appropriate acoustic boundary conditions on the faces of the fluid region. In this example it was a hard sound boundary since it is a sealed space.

Nagi ElabbasiAugust 11, 2014 9:35 amDear Dr. Lee,

We used this reference to calculate the correction factors: CA Van Eysden, JE Sader, Resonant Frequencies of a Rectangular Cantilever Beam Immersed in a Fluid, Journal of Applied Physics, 100 (2006)

Yuanyu YuAugust 19, 2014 8:48 amDear Dr. nagi,

I created a circular plate with air above it using acoustic-structural interaction physics. The circular plate is clamped as the boundary condition. By using the eigenfrequency study, but the result is wrong (with imaginary part, and the value of real part is very small)… I set Linear Elastic Material to the plate.

What is wrong? Thanks!

Nagi ElabbasiAugust 20, 2014 10:51 amDear Yuanyu,

What you did seems right. Try plotting the mode shapes at these reported frequencies and maybe you will spot a wrong boundary condition or material setting. Also try tightening the eigensolver convergence tolerance, and increasing the maximum number of eigenvalue iterations.

Anulekha MajumdarSeptember 18, 2014 3:30 amDear Dr. Nagi

I read your article and I am doing something similar to your work. Since I am new to this software, can you please suggest me how to add damping in the material when doing the eigenfrequency analysis in water. I tried directly adding damping from linear elastic material, but the result is not near the experimental values. Also that during underwater analysis, do we have to add PML (perfectly matched layer) always?

Thanks in advance

Anulekha Majumdar

Jean CacheuxJune 18, 2015 11:36 amHi,

I am trying to use this model with COMSOL 4.3. Do you know if it is possible ? Cause I can’t see any frequency shift when putting the cantilever in liquid. Plus I would expect a loss in the deformation amplitude, which I can’t observe neither. Could you share the file you’re using for the simulation ?

Thank you,

Jean

Nagi ElabbasiJune 21, 2015 10:52 pmHi Jean,

I see no reason for this not to work with COMSOL 4.3. If you don’t get a frequency shift try artificially increasing the fluid density to see if the lack of shift is physical or a modeling error. Also you will not necessarily get a reduction in deformation amplitude in an eigenfrequency analysis since the mode shapes are normalized. If you do a harmonic frequency analysis then you see a vibration amplitude reduction.

Regards,

Nagi

Waqas WaheedJune 24, 2015 1:59 amHi,

Can you please mention the dimensions of the cantilever? Did you also calculate its Q-factor?

Thanks

Waqas

Ryan TancinJune 30, 2015 2:30 pmHi,

My version of COMSOL 4.4 only has the Pressure Acoustics module, not the acoustic structure interaction, is it still possible to do this simulation with that module? Thanks in advance.

Ryan

Nagi ElabbasiJune 30, 2015 3:44 pmHi Ryan,

You probably only see Pressure Acoustics because you don’t have the Acoustics Module. See the Product Specification webpage for more details: http://www.comsol.com/products/specifications/acoustics/

Regards,

Nagi

Marco TrAugust 19, 2015 10:30 amHi,

I am trying to define the mechanical damping in my analysis with comsol 4.4. I followed a tutorial where it is recommended to follow these steps:

1 -On the Physics toolbar, click Attributes and choose Mechanical Damping.

2 -In the Settings window for Mechanical Damping, locate the Damping Settings

section.

3 -From the Damping type list, choose Isotropic loss factor.

4- From the ηs list, choose User defined. In the associated text field, type 0.001.

But I cannot find either Attributes or Mechanical damping on my software.

Could anyone help to tackle that issue?

May thanks

Nagi ElabbasiAugust 19, 2015 10:49 amHi Marco, that may be because you don’t have the right module. You need the Structural Mechanics module to include material damping. See the Product Specification page for more details: http://www.comsol.com/products/specifications/structural-mechanics/. Regards, Nagi

Levin DieterleJanuary 18, 2016 9:33 amHi,

I’m very interested in this model. Would it be possible to make the COMSOL file available, to see how the interfaces are set. Thanks.

Regards,

Levin

Nagi ElabbasiJanuary 18, 2016 1:52 pmHi Levin, orry I cannot share the model file but if you have specific questions I will try to answer them. Regards, Nagi

murat gelFebruary 17, 2016 4:07 amHi, Is the model available for download?

Nagi ElabbasiFebruary 17, 2016 9:19 amHi Murat, sorry I cannot share the model but I suggest you check out COMSOL Application ID 14731 titled Vibrating Micromirror with Viscous and Thermal Damping. It highlights many of the features used in this immersed beam model. Regards, Nagi

Javier RiveroMay 6, 2016 8:55 amHi Nagi Elabbasi,

I’m interested in the solution of eigenvalue problems involving fluid flows and when I try to solve a very simple case, I don’t make it work. Eigenvalues are all NaN. I’m pretty sure it is because in the eigenvalue problem COMSOL doesn’t perturbe the local inertia term and this term is the only one appearing in the mass matrix. Does your computation takes into account this term?? Even if you don’t I should give some results because you have time derivative anywhere else, but in my case I’m interested in hydrodynamical modes. Regards, Javier.

Nagi ElabbasiMay 6, 2016 9:25 amHi Javier, I can’t think of a good reason why you’re getting NaN eigenvalues. The mass matrix is getting a contribution from the beam itself, which comes from the solid mechanics model. It also gets contribution from the fluid which is straightforward for COMSOL to calculate and does not need a perturbation analysis. Regards, Nagi

Deepak SinghJuly 20, 2016 4:33 amSir,

I want to calculate the natural frequency of a plate which is coupled with water only on one side,

so should I try fluid-structure interaction or acoustic-structure interaction?

Satthiyaraju MJanuary 9, 2017 9:02 amDear Nagi, I have solved only using solid mechanics_eigen frequency study. Defined BC’s are cantilever beam with surrounding medium water. I have created block around the cantilever beam and applied water properties and fixed all sides of the outer block. I got values of eig freqs. But I don’t know whether it’s correct or not? I could not get the proper mode shapes…. Please rely how to model the same. thank you very much.

Nagi ElabbasiJanuary 9, 2017 9:28 amDear Satthiyaraju,

One way to check if your model set up is correct is to either use a stiff beam material or a low density fluid material (air) for validation. The results should be very similar to the structural-only analysis in this case, both in terms of frequencies and mode shapes. After that you can go back to the properties material/fluid properties and compare against analytical frequency estimates if possible, like the one I used above. Note also that there may be additional acoustic-dominant modes that you have to ignore when looking at the beam modes shapes.

Vinit KumarJanuary 14, 2017 5:34 amDear Nagi,

It is an interesting simulation. I was trying it, but, I was wondering the eigenfrequencies returned by Comsol were complex. I am curious to know how we can interpret complex eigenfrequencies.

Thanks!

Nagi ElabbasiJanuary 19, 2017 10:59 amDear Vinit,

If the eigenfrequencies are complex that’s due to damping. The ratio of the imaginary to real part of the eigenfrequency is roughly equal to the damping ratio (if the imaginary part is small compared to the real part). In this model you can add damping to the structure or the fluid, or both. Fluid damping requires a more accurate fluid formulation as explained in the discussion above.

Deepak SinghJanuary 20, 2017 8:08 amSir,

I want to calculate the natural frequency of a plate which is coupled with water only on one side,

how to find?