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 |
 |
|
