Pivector Labs
Insights
Structural Dynamics

Mode Shapes and Eigen Frequencies of a Cantilever Beam

Validating a new modal-analysis workflow on a simple cantilever beam — an acrylic ruler with a cellphone accelerometer — by comparing the first three mode shapes and eigen frequencies across experiment, FEA, and hand calculation.

May 22, 20267 min read

I wanted to build more confidence in using a new FEA software package for modal analysis. To do this, I set up a simple cantilever beam problem and compared the first three modes from FEA against theoretical calculations and experimental measurements.

Experiment

I first focused on the experiment, using resources that were immediately available. I used a plastic acrylic ruler as the cantilever beam and a cell phone as an accelerometer. A C-clamp was used to fix one end of the plastic ruler to the table.

Experimental cantilever beam setup using a C-clamp, acrylic ruler, and cellphone accelerometer
Figure 1. Experimental cantilever beam setup using a C-clamp, acrylic ruler, and cellphone accelerometer.

The cell phone was mounted on the ruler using removable adhesive, Plasti-tak. A free accelerometer app called Resonance was used to measure the natural frequency of the system. The measured first mode of the cantilevered ruler and cellphone system was 3.1 Hz.

Experimental accelerometer result showing a measured first mode frequency of 3.1 Hz
Figure 2. Experimental accelerometer result showing the measured first mode at 3.1 Hz in the Z accelerometer direction.

FEA model

After obtaining the experimental result, I simulated the same problem in FEA and performed a modal analysis. The ruler was modeled using acrylic material properties. The cellphone weighed 200 grams, so a dummy mass of 200 grams was attached to the ruler in the FEA model.

For a simple problem like this, beam elements with a point mass at the end may appear sufficient and would create a quicker, more computationally efficient model. However, the cellphone is not really a point mass. It has a distributed moment of inertia laterally in the X direction, which can result in a low-frequency torsional vibration mode of the ruler. For that reason, I used a 3D FEA model.

Tetrahedral meshing was created using 10-node tetrahedral solid elements. Mesh mating was done, and a glued contact was created between the cellphone and plastic ruler interface. A fixed constraint was applied at the other end of the plastic ruler.

The following figures show the first three mode shapes and eigen frequencies reported by FEA.

FEA animation of the first mode shape of the cantilever beam system
Figure 3. First FEA mode shape. The first modal frequency was 3.19 Hz.
FEA animation of the second mode shape of the cantilever beam system
Figure 4. Second FEA mode shape. The second modal frequency was 14.6 Hz.
FEA animation of the third mode shape of the cantilever beam system
Figure 5. Third FEA mode shape. The third modal frequency was 30.7 Hz.

From the animations, the first mode is governed by the bending stiffness of the cantilever beam in the Y-Z plane. The second mode is governed by the torsional stiffness of the ruler about the Y axis. The third mode is governed by the bending stiffness of the ruler in the X-Y plane.

Hand calculations

First mode: bending in the Y-Z plane

Hand calculation setup for the first mode of the cantilever beam with cellphone mass
Figure 6. First mode hand calculation setup, including beam dimensions, cellphone mass, ruler mass, acrylic material assumption, and bending stiffness formulation.
Hand calculation result for the first bending mode of the cantilever beam
Figure 7. First mode hand calculation result. The cantilever beam with cellphone mass is modeled as a massless spring system, giving a calculated natural frequency of 3.22 Hz.

Second mode: torsional

Hand calculation setup for torsional stiffness of the cantilever beam
Figure 8. Torsional stiffness calculation for the rectangular ruler cross-section, including torsional constant and shear modulus assumptions.
Hand calculation result for the torsional mode of the cantilever beam and cellphone system
Figure 9. Torsional mode hand calculation using the cellphone mass moment of inertia, giving a calculated natural frequency of 15.56 Hz.

Third mode: bending in the X-Y plane

Hand calculation for the third bending mode of the cantilever beam in the X-Y plane
Figure 10. Third mode hand calculation for bending stiffness in the X-Y plane, giving a calculated natural frequency of 31.46 Hz.

Comparison

ModeDominant deformationExperimentFEAHand calc
FirstBending (Y-Z plane)3.1 Hz3.19 Hz3.22 Hz
SecondTorsion (about Y)—14.6 Hz15.56 Hz
ThirdBending (X-Y plane)—30.7 Hz31.46 Hz

Although this is a simple problem, it was encouraging to see strong agreement between theoretical calculations, FEA, and experiment. It also helped build confidence in the FEA software package for modal analysis.

Back to all insights