## 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.

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## Comments

I 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 !!!

I think , when you consider the Air case in your simulation. You might use those information of air I think …

Hi 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.

Hi 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?

Dear Dr. Elabbasi,

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

Dr. Lee

Hi 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.

Dear 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)

Dear 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!

Dear 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.