Fun with Mode Shapes and Eigen Frequencies

(Cantilever Beams - Theoretical, FEA and Experimental Verification)

This weekend, I was trying to learn how to do modal analysis in a new FEA software package. I wanted to gain more confidence with the use of this new FEA package. So, I decided to set up a simple cantilever beam problem in the FEA package and compared its first three modes to theoretical calculations and experiment as a sanity check. It was a fun exercise.

First, the experiment:

I first focused on the experiment as I was constrained to use the resources that were immediately available in my apartment. Not good to leave the house unnecessarily these days. I used a plastic (acrylic) ruler as a cantilever beam and my cell phone as an accelerometer. A C-clamp was used to fix one end of the plastic ruler to the table. The below figure shows the picture of the setup.

Experimental Setup

The cell phone was mounted on the ruler with the help of removable adhesive named Plasti-tak. A free accelerometer app named Resonance was used to measure the natural frequency of the system. The figure below shows that the first mode of the cantilevered ruler + cellphone system occurs at 3.1 Hz.

Experimental Result: Measured 1st Mode = 3.1 Hz (Z Accelerometer)

Next, the FEA

Once I got my experimental results, I decided to simulate the same problem in the FEA world and do modal analysis. The ruler was modeled as is with acrylic material assigned to it. The cellphone was 200 grams in weight. So, in the FEA model, a dummy mass of 200 grams was attached to the ruler.

For such simple problems, one would think that beam elements with a point mass at the end would have sufficed giving a quicker and more computationally efficient FEA model. However, the cellphone is not really a point mass. It has a quite a bit of moment of inertia distributed laterally (in X) which could result in a low frequency torsional vibration mode of the ruler. Therefore, I decided with go with 3D FEA. Tetrahedral meshing was created via 10-node tetrahedral solid elements. Mesh mating was done and a glued contact was created between the interface of the cellphone and plastic ruler. A fixed constraint was put at the other end of the plastic ruler.

The below images show the 1st three mode shapes and eigen frequencies reported by FE

1st Modal Frequency = 3.19 Hz

2nd Modal Frequency = 14.6 Hz

3rd Modal Frequency = 30.7 Hz

From the above animations, it is clear that the 1st mode is governed by the bending stiffness of the cantilever beam in the Y-Z plane, 2nd mode is governed by the torsional stiffness of the ruler about the Y axis, and the 3rd mode is governed by the bending stiffness of the ruler in X-Y plane.

Finally, hand calculations

First Mode Calculation (Bending in Y-Z plane)

Second Mode Calculation (Torsional)

Third Mode Calculation (Bending in X-Y Plane)

This was indeed a quick fun weekend exercise. Although, this is a simple problem, it was exhilarating to see great agreement between theoretical calculations, FEA and experiments. Atleast, I am now a bit more confident of my FEA software package.

:-)