OCTOBER 19-20, 2012 - YMCA University of Science & Technology

Proceedings of the National Conference on
Trends and Advances in Mechanical Engineering,
YMCA University of Science & Technology, Faridabad, Haryana, Oct 19-20, 2012
The analytical model was solved to obtain the Natural Frequency from the analytical expression. It gives the
following first three Natural Frequency of 512.5, 3212 and 8994 Hz. The first three Natural Frequencies for a
one element FE model are determined as 514.65, 5035.9 and 15741Hz, indicating huge differences between the
second and third Natural Frequency of analytical model and one element FE model. This is due to the gross
approximation in the FE model. The percentage error comes out to be 0.41776 %, 36.21796% and 42.86 % in
first, second and third Natural Frequency respectively.
The first three Natural Frequencies for a two element FE model are determined as 512.47, 3225.5 and 10828 Hz
and are fairly close to the analytical model. The percentage error comes out to be -.0058%, 0.41854 and 16.93%
in first, second and third Natural Frequencies respectively.
The first three Natural Frequencies for a ten element FE model using ANSYS are determined as 512.23, 3198.8
and 8109.8 Hz and are much closer to the analytical model. The percentage error comes out to be -0.05271%, -
0.41265 and 0.9679% in first, second and third Natural Frequencies respectively. This clearly establishes that as
the number of elements is increased the percentage error is reduced further.
Therefore a FE model with large number of elements is desirable to obtain accurate results. The first three
Natural Frequencies for a ten element FE model using ANSYS are determined as 512.23, 3198.6 and 8905.5 Hz
and are much closer to the analytical model. The percentage error comes out to be -0.05271%, -0.41579% and
0.99377% in first, second and third Natural Frequency respectively.
As seen from the above, the error for the first natural frequency is fairly constant and is around -0.05 % but for
the second natural frequency the error is quite high for one element FE model but is quite less for the higher
number of nodes. Same trend can be seen for the third natural frequency of higher number of nodes.
Table 2: Percentage error in different FE models
MOD NO.
1-
ELEMENT
BEAM
2-
ELEMENT
BEAM
10-
ELEMENT
BEAM
50-
ELEMENT
BEAM
100-
ELEMENT
BEAM
150-
ELEMENT
BEAM
200-
ELEMENT
BEAM
250-
ELEMENT
BEAM
1 0.41776 -0.00585 -0.05271 -0.05271 -0.05271 0.74561 -0.05271 0.74561
2 36.21796 0.41854 -0.41265 -0.41579 -0.41579 0.38147 -0.41579 0.38147
3 42.86259 16.93757 -0.96769 -0.99377 -0.99377 -0.1949 -0.99377 -0.19495
4. Conclusion
The exact solution obtained from the differential equations gives us the basis to correlate the results obtained
using the FE solution. As the table.1 shows that one element FE solution gives very crude results as expected
because the error involved in analysis is more and it further increases as the modes are increased. As the number
of nodes are increased the results tends to improve further but it shows a wavy pattern because of rounding off
errors involved in adding different beam elements.
The following conclusions may be drawn from the results thus obtained.
1. A huge amount of errors may be involved in the FE analysis if element selection is not proper thus
needs a considerable experience for making the choice of selecting the elements.
2. The time involved in analysis, which adds to the cost of analysis and processing time also depends
upon the choice of the elements.
3. For complicated or huge structures as used in mechanical and civil engineering the analytical solution
may not be available or may be too complex to obtain.
Therefore it may be concluded that for complex structures or real structures the choice of elements in FEA has a
great impact on results and should be carefully chosen.
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