Natural Frequencies of Elastic
Cantilevered Beams
By Chris Klobedanz
Final Project
MANE-4240: Introduction to Finite Elements
Professor Ernesto Gutierrez-Miravete
Introduction:
The purpose of this project is to evaluate the effectiveness of finite element modeling for
the determination of cantilever beam natural frequencies. The finite element systems were built
in both Comsol and Ansys, and the frequencies were evaluated for three different mesh
configurations of varying complexity. The exact solutions were determined in Maple using the
equations from section 5.11 of “Vibrations of Elastic Bodies” by Stephen Timoshenko. To
illustrate the diversification of the modeling tools, three different homogeneous beam crosssectional areas were investigated: square; circular; and I-beam. The results from the finest mesh
configurations of each program were compared to the exact solutions in order to evaluate the
accuracy of the converged solutions.
Background:
The natural frequency is defined as the frequency at which a system oscillates and it is an
important characteristic for several reasons. If you know the natural frequency, you can
understand how the system vibrates and what kinds of waves it will generate. Being able to
calculate this response allows for the creation and control of specifically designed waves in the
system with strategic forced responses.
For our purposes in dealing with a beam model, knowing the natural frequencies can be
crucially important in civil engineering applications, as one example. If the forced response is in
tune with the natural frequency, resonance can occur; with resonance, each cycle of force adds to
the amplitude and oscillation of the system. If we consider our simplified cantilever beam
example as a representation of a bridge or a building, resonance could have a destructive result if
unaccounted for.
System Description:
This project evaluates a 1 m long cantilever steel beam with a 0.01 m2 cross-sectional
area that is subjected to no external forces. The material properties and beam area dimensions are
shown below:
𝐸 = 2𝑒11 𝑃𝑎
𝜈 = 0.3
2
𝜌 = 7800 𝑘𝑔/𝑚3
𝐼𝑥 = 8.3333 ∗ 10−6
𝐼𝑥 = 7.9577 ∗ 10−6
𝐼𝑥 = 8.0208 ∗ 10−5
Analytic Solution:
The cantilever beam problem is a simplified example which is a building block for more
complex analyses. The problem models a horizontal beam with a constant cross-sectional area
that is completely fixed at one end and is free at the other. In “Vibrations of Elastic Bodies”,
Timoshenko outlines the method for analytically determining the natural frequencies of this
system.
By starting with the Euler-Bernoulli Beam Theory:
𝐸𝐼
𝑑4𝑦
𝑑2𝑦
+
𝜌𝐴
=0
𝑑𝑥 4
𝑑𝑡 2
and incorporating appropriate boundary conditions:
𝑎𝑡 𝑥 = 0 → 𝑦 = 0,
𝑑𝑦
=0
𝑑𝑥
𝑑2𝑦
𝑑3 𝑦
𝑎𝑡 𝑥 = 𝐿 → 2 = 0, 3 = 0
𝑑𝑥
𝑑𝑥
the nth natural frequency is calculated in the following equation:
𝜔𝑛
𝑓𝑛 =
2𝜋
where:
𝐸𝐼
𝜔𝑛 = 𝑘𝑛2 √
𝜌𝐴𝐿4
3
and kn is determined as the root of :
cos(𝑘𝑛 𝐿) cosh(𝑘𝑛 𝐿) = −1
kn can also be approximated for initial guesses as:
𝑘𝑛 𝐿 =
(2𝑛 − 1)𝜋
2
The first three natural frequencies should resemble the figure below:
Comsol Modeling:
The Comsol program is an easy-to-use program that allows a finite element system to be
built very quickly and can adjust for numerous variable conditions. To model the subject
cantilever beam systems, a three dimensional eigenfrequency analysis is necessary. After
inputting the geometry, material properties, and boundary conditions, a modal analysis can be
run and the first several natural frequencies are determined. A mesh extension study is then
completed by analyzing the results for an extra coarse, normal, and extra fine mesh
configuration.
Sample meshes of the subject systems are shown below:
Extra Coarse Block
Normal Cylinder
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Extra Fine I-Beam
Ansys Modeling:
The Ansys program is a slightly more intricate finite element tool, which takes longer
than Comsol to use, but is still able to efficiently model complicated systems and analyze results
that would be significantly tougher to analyze analytically. For the sake of this project, a modal
analysis is run using 1D linear elastic beam elements and inputting the system properties (crosssectional area, area moment of inertia, beam height, and material properties) and boundary
conditions. A mesh extension study is then completed by analyzing the results for 5, 10, and 20
elements. The natural frequencies of the system are collected by listing the results in the general
postprocessor. By plotting the deformed shape of the system, the torsional frequencies can be
found (those with no y-displacement) and excluded from the results.
An illustration of a system set up for 20 elements is illustrated below:
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Results:
Sample plots of the deformed systems, subjected to their natural frequencies, are shown
below:
2nd Natural Frequency Block
3rd Natural Frequency Cylinder
4th Natural Frequency I-Beam
A sample output of the natural frequencies for a 20-element square cross-section beam is
shown below. Sets 1, 2, 4, 5, 7, and 8 refer to the first six natural frequencies. The other sets
analyze torsion, which was not considered for this project.
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Sample plots of the deformed systems, subjected to their natural frequencies, are shown
below. The beam deformations are similar for all three beam cross-sectional areas:
1st Natural Frequency
3rd Natural Frequency
6th Natural Frequency
The results of this project are summarized in the table below:
For the given systems, the first six natural frequencies (if applicable), are determined
analytically, through Comsol models, and through Ansys models. The finite element
approximations are compared to the analytic solutions; this was only done for the finer meshes
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because the coarser mesh results had not converged yet and therefore did not fully evaluate the
performance capabilities of the programs.
Conclusion:
From the results above, several conclusions can be drawn about the performance of the
subject finite element programs in relation to the depth of the result being estimated (3 rd natural
frequency versus 6th, for example) and to the complexity of the system. It is apparent that both
Comsol and Ansys are strong tools for estimating the lower natural frequencies of a system, but
they struggle to converge to the analytic solution for the higher natural frequencies. For the first
natural frequency, both Comsol and Ansys converged to within 5 percent of the exact value;
however, the magnitude of the error began to rapidly increase with each subsequent frequency.
Both programs also seemed to estimate more accurately with the less complicated geometrical
configurations as well; the results for the cylinder and block models are far more accurate
(especially for the higher natural frequencies) than the I-beam model.
In comparing the two programs to one another, it appears that Ansys is slightly more
accurate at predicting the analytic solution than Comsol; for example, in the block model, Ansys
is still within 11.5% of the exact value of the 6th natural frequency, whereas Comsol exceeds that
error on the 4th natural frequency. This pattern is consistent throughout most of the project
samples.
It is very possible that both of these programs are capable of converging to more accurate
results through the use of more in-depth options in the construction of the model (using different
elements, using a different analysis technique offered by the program, inputting more
information, etc.). However, in evaluating how Comsol and Ansys fare as finite element tools,
both are able to attain a high level of accuracy with relatively little time having to be spent on
calculations. Side by side, with both programs inputting the same minor amount of information,
Ansys evaluated cantilever beam natural frequencies more accurately, but it also has the trade-off
of taking longer to model than Comsol. Overall, both programs serve as good alternatives to
going through the rigor of deriving an exact solution.
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