Deflection of Uniform Bar with Free Ends Mechanical Vibration

Strength and Mechanics of Materials
Accelerometers About and Application

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.

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