Strength and Mechanics of Materials
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The following are equations which predict
the deflection and mode patterns for beams. They are based
on simple beam theory and are accurate for beams having a
length to depth ratio of the order of ten or more.
The characteristics numbers for the first
three modes of this beam are: 4.73, 7.853 and 10.996.
| Where: |
| y |
= Deflection [in] |
 |
= The characteristic number
for the nth mode and is the root of the
equation  |
| n |
= Mode number |
Frequencies of higher modes for this beam
are given below, in which w is the weight per unit length of
the beam and g = 386 in/sec2.
| Uniform
Bar With Free Ends |
| Illustration |
Amplitude Profile |
Mode |
Natural Circular
Frequency |
|
|
 |
2 Nodes |
|
 |
3 Nodes |
|
 |
4 Nodes |
|
 |
5 Nodes |
|
|
