By Chris Klobedanz

MANE-4240 Final Project

Natural Frequencies of Elastic

Cantilevered Beams

Purpose• Evaluate the Natural Frequencies of a Steel

Cantilever Beam System• Compare the Results of Beams with Different

Cross-Sectional Areas: Square, Circle, I-Beam

• Compare the Exact Solution to Finite Element Models in Ansys and Comsol

• Compare Different Mesh Configurations within the Finite Element Models

Analytic Solution• Use Euler-Bernoulli Beam Theory:

• Incorporate Boundary Conditions:

• Determine the Natural Frequencies fn: where

• kn is determined as the root of:

Comsol Models• Build a 3D Eigenfrequency Model and Compare

Extra Coarse, Normal, and Extra Fine Meshes:

Extra Coarse Block Normal Cylinder

Extra Fine I-Beam

Ansys Models• Build a 1D Linear Elastic Model with 3-Noded

Beam Elements• Run a Modal Analysis• Compare 5, 10, and 20 Element Meshes

20 Element Model

Results

Comsol

Ansys

Analytic

* The Deformed Beam Models look similar for block, cylinder, and I-beam models

Results (cont.)

Conclusions• Comsol and Ansys are very accurate at estimating

the lower order natural frequencies, but get increasingly more inaccurate at each subsequent frequency

• Comsol and Ansys are more accurate at estimating the natural frequencies of less complicated models. The block and cylinder models had much less error than the I-beam models

• Ansys had more accurate results than Comsol at the expense of taking more time to build the models

• With more input, it is possible that both programs could converge to more accurate solutions