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.

Vibration Formula

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

Vibration Mode

Amplitude Profile 1

2 Nodes

Natural Circular Frequency 1

Amplitude Profile 2

3 Nodes

Natural Circular Frequency 2

Amplitude Profile 3

4 Nodes

Natural Circular Frequency 3

Amplitude Profile 4

5 Nodes

Natural Circular Frequency 4