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Deflection of Uniform Bar with Free Ends Mechanical  Vibration

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

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