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 n^{th }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/sec^{2}.
Uniform
Bar With Free Ends 
Illustration 
Amplitude Profile 
Mode 
Natural Circular
Frequency 


2 Nodes 


3 Nodes 


4 Nodes 


5 Nodes 

