RENSSELAER POLYTECHNIC INSTITUTE
Modal Analysis
MANE6960 – Advanced Topics in Finite Elements
José A. DeFaria
07/23/2015
DeFaria 1
Introduction/Summary/Inputs
The study of the modes of vibration of structural components is an important practical problem since
provisions must be made during design to avoid undesirable resonance frequencies. This study will
investigate the modes of vibration of simple beams and plates.
For the beam case, the beam will be assumed to be simply supported (pinned on both ends) and have
the following dimensions:
L =
W =
H =
1m
0.1 m
0.1 m
For the plate case, the plate will be assumed to be simply supported (pinned on all four ends) and have
the following dimensions:
L =
W =
H =
1m
1m
Varies (0.1, 0.25, 0.5 m)
The material properties of steel were used as follows:
ρ
E
v
=
=
=
7850 kg/m3
2.0x1011 Pa
0.3
To determine if the frequencies found by finite element analysis are correct, they will be compared with
textbook solutions from Roark’s Formulas for Stress and Strain.
For the beam, the natural frequency is found by equation 1a of Table 16.1:
𝑓1 =
6.93 𝐸𝐼𝑔
√
2𝜋 𝑊𝑙 3
1
3
−4 9.81𝑚
6.93 (2𝑒11 𝑃𝑎)(12 ∗ 0.1 ∗ 0.1 𝑚 )( 𝑠 2 )
𝑓1 =
= 503.36 𝐻𝑧
√
2𝜋 (7850 𝑘𝑔 ∗ 1 𝑚 ∗ 0.1 𝑚 ∗ 0.1 𝑚) (1 𝑚)3
𝑚3
For the plate, the natural frequency is found by using the UMASS Lowell resource:
𝜔𝑛 = 𝐵√
𝐸𝑡 3
𝜌𝑎4 (1 − 𝑣 2 )
DeFaria 2
(2𝑒11 𝑃𝑎)𝑡 3
𝜔𝑛 = 5.70√
= 30160.21𝑡
𝑘𝑔
(7850 3 )(1 𝑚)4 (1 − 0.32 )
𝑚
For the plate thicknesses used in this evaluation, this results in:
Plate Thickness
0.10 m
0.25 m
0.50 m
Frequency
953.75 Hz
3770.0 Hz
10663 Hz
Governing Equation and Variational Formulation
The equation governing the undamped transverse vibration of a beam is:
𝜕 2 𝑢 𝐸𝐼 𝜕 4 𝑢
+
=0
𝜕𝑡 2 𝜌𝐴 𝜕𝑥 4
The pinned (simply supported) ends of the beam can be considered homogenous Direchlet boundary
conditions:
𝑢(0, 𝑡) = 0; 𝑢(𝐿, 𝑡) = 0;
The variational formulation of the problem is:
(𝑣, 𝑢𝑡 ) + 𝐴(𝑣, 𝑢) = (𝑓, 𝑣)
DeFaria 3
Creating the Model
Beam
To construct the beam in Abaqus, a sketch was created using the cross-sectional dimensions of the
beam. This was then extruded for 1 m, or the length. An image of the beam is shown at below.
The material properties of steel were then defined and applied to the model. This includes the density,
modulus of elasticity and Poisson’s ratio. To conduct the model analysis a linear perturbation procedure
with a frequency step was created.
No loads were required for the model, but the pinned boundary condition is required. The pinned
boundary conditions sets all displacements (u1, u2, u3 or ux, uy, uz) to zero. This condition was applied
to both ends of the beam and can be seen as orange dots in the image below.
Only quadratic elements (C3D8) will be used in this model, as previous studies have shown they are
more accurate than linear elements. Hex elements with sizes of 0.1, 0.05, 0.025, and 0.01, resulting in
10, 80, 640, and 10000 elements, respectively, will be examined to observe convergence. Additionally,
three different Eigensolvers will be used to compare the results: Lanczos, Subspace, and AMS.
DeFaria 4
Plate
To construct the plate in Abaqus, a sketch was created using the cross-sectional dimensions of the plate.
This was then extruded for 0.1 m, or the thickness. Plates of thicknesses of 0.25 and 0.5 meters were
also evaluated. An image of the plate is shown at below.
The material properties of steel were then defined and applied to the model. This includes the density,
modulus of elasticity and Poisson’s ratio. To conduct the model analysis a linear perturbation procedure
with a frequency step was created.
No loads were required for the model, but the pinned boundary condition is required. The pinned
boundary conditions sets all displacements (u1, u2, u3 or ux, uy, uz) to zero. This condition was applied
to all ends (the thinner sides) of the plate and can be seen as orange dots in the image below.
Only quadratic elements (C3D8) will be used in this model, as previous studies have shown they are
more accurate than linear elements. Hex elements with sizes of 0.1, 0.05, and 0.025 will be used to
observe convergence. Although 0.01 m sized elements were used in the beam analysis, it would result in
too many elements for this plate analysis and will not be used in fear of crashing the computer
resources. The number of resultant elements for each plate thickness is shown below.
Element size
thickness
0.1
0.05
0.025
0.1 m
100
250
500
0.25 m
800
2000
4000
0.5 m 6400 16000 32000
Since the plate thickness will be varied, only the Lanczos eigenvalues will be used for the plate model.
DeFaria 5
Results/Discussion/Conclusion
Beam
The plots below show the first and second modes of the beam using Lanczos Eigenvalues (the default)
and 10,000 elements. The frequencies of the first and second modes are equal because they are the first
mode in the horizontal direction and the first mode in the vertical direction. This was checked with every
eigenvalue solver and at every mesh level and was verified to be true within two decimal places of
accuracy. This matches up with the textbook solution which would expect these two modes to be equal
considering the geometry of the beam.
Top: 1st mode (vertical) frequency = 491.68 Hz; Bottom: 2nd mode (horizontal) frequency = 291.68 Hz;
DeFaria 6
The table below shows the results of the mesh refinement for each of the three eigenvalues examined.
The results converge on approximately 492 Hz which is approximately 2% off from the textbook solution
for the natural frequency obtained using Roarks. Considering this small deviation from the textbook
solution, and since all of the values converged to approximately the same number, there is a high
degree of confidence in the results. Additionally, the similar numbers yielded by the three different
Eigenvalues show there is little difference in their selection in this case.
# of
Elements
10
80
640
10,000
Textbook
Solution
503.36 Hz
503.36 Hz
503.36 Hz
503.36 Hz
Lanczos
Eigenvalue
458.42 Hz
499.58 Hz
494.27 Hz
491.68 Hz
Subspace
Eigenvalue
458.42 Hz
499.58 Hz
494.27 Hz
491.68 Hz
AMS
Eigenvalue
458.42 Hz
499.83 Hz
494.81 Hz
492.37 Hz
For information, the 3rd, 4th, 5th, and 6th modes are shown below (using Lanczos and 10,000 elements):
Mode: 3
Frequency: 1261.4 Hz
Mode: 4
Frequency: 1261.4 Hz
Mode: 5
Frequency: 1434.2 Hz
Mode: 6
Frequency: 2282.4 Hz
DeFaria 7
Plate
The plots below show the natural frequency of the plate using Lanczos Eigenvalues and 1000 μm3 sized
elements. The plots depict the natural frequency of the 0.1m, 0.25m, and 0.5m thick plate. The natural
frequency increases as the plate thickness increases, which matches what is expected using the
equations.
0.10 m thick
Frequency=
817.33 Hz
0.25 m thick
Frequency=
1481.8 Hz
0.50 m thick
Frequency=
1865.9 Hz
DeFaria 8
The table below shows the results of the mesh refinement for each of the three plates examined. For all
three plates, the results converge, although they do not converge on the textbook values. In searching
through numerous textbooks for solutions of the natural frequency of plates, each textbook provided a
different formula. There are some differences between how the load is applied in each formula, but the
textbook results provided below are the closest to those obtained via FEA. The convergence values are
similar to the textbook solution for a thickness of 0.10 m. For the other thicknesses, the analysis values
and the actual values are very different. This could be because as the plate thickness increases, it
behaves less like a plate (assumed to be thin), and the equations no longer hold.
Plate
Thickness
0.10 m
0.25 m
0.50 m
1000 μm3
Elements
817.33 Hz
1481.8 Hz
1865.9 Hz
125 μm3
Elements
800.15 Hz
1475.0 Hz
1860.4 Hz
15.625 μm3
Elements
797.12 Hz
1472.4 Hz
N/A
Textbook
Solution
953.75 Hz
3770.0 Hz
10663 Hz
For information, the 2nd, 3rd, 4th, and 5th modes are shown below (using Lanczos and 1000 μm3 sized
elements):
Mode: 2
Frequency: 1568.7 Hz
Mode: 3
Frequency: 1568.7Hz
Mode: 4
Frequency: 2202.8 Hz
Mode: 5
Frequency: 2609.3 Hz
DeFaria 9
References
Anderson, R., Irons, B., & Zienkiewicz, O. (1968). Vibration and Stability of Plates Using Finite Elements.
International Journal of Solids Structures, 1031-1055.
Cosby, A., & Gutierrez-Miravete, E. (2014). Finite Element Analysis Conversion Factors for Natural
Vibrations of Beams. ASME 2014 International Mechanical Engineering Congress and Exposition
(pp. 1-6). Montreal: ASME.
Dassault Systemes Simulia Corp. (2013). Abaqus/CAE User's Guide. Abaqus 6.13. Providence, Rhode
Island.
Gutierrez-Miravete, E. (2015). Index. Retrieved July 6, 2015, from Advanced Topics in Finite Elements:
http://www.ewp.rpi.edu/hartford/~ernesto/Su2015/ATFE/
University of Massachusetts Lowell Modal Analysis and Controls Laboratory. (2015). Natural Frequencies
for Common Systems. IES Seminar (pp. 1-6). Lowell: University of Massachusetts.
Young, W. C., & Budynas, R. G. (2002). Roark's Formulas for Stress and Strain. New York: McGraw-Hill.
Zienkiewicz, O. (2013). Finite Element Method: Its Basis and Fundamentals. Butterworth-Heinemann Ltd.