# Chris Bickford Introduction to Finite Elements HW #5; Problem 1

```Chris Bickford
Introduction to Finite Elements
HW #5; Problem 1
MAPLE:
Using Maple software, the frequencies for the first six modes of vibration in the beam
were calculated. See attached Maple sheets for details.
ABAQUS FEA CODE:
ABAQUS 6.14 software was used to perform finite element analysis (FEA) to obtain an
approximate solution for problem (1). A 3-dimensional eigenvalue analysis was
performed using the Lanczos solver. Two models were analyzed, one with linear
elements and one with quadratic elements. Both models had 108 elements, Figure 1
shows the mesh and second bending mode shape. The results are shown in Table 1.
Both elements gave pretty good approximations for the first and second modes, with the
quadratics out-performing the linear elements. However, after the second mode of
vibration, the quality of the approximation decreased significantly.
Table 1 Summary of FEA Results
Exact Soln
[Hz]
1st Mode
2nd Mode
3rd Mode
4th Mode
5th Mode
6th Mode
58.902
369.134
1033.586
2025.416
5001.569
6985.662
Linear Elements
Freq. [Hz]
% Error
58.353
0.93%
349.57
5.30%
919.85
11.00%
1669.8
17.56%
2542.8
49.16%
3493.3
49.99%
Freq. [Hz]
% Error
59.145
-0.41%
354.48
3.97%
932.63
9.77%
1692.7
16.43%
2578.7
48.44%
3547.5
49.22%
Figure 1 First Bending Mode
```