Theory of Machine
The natural frequency of free transverse vibrations due to a point load acting over a simply supported shaft is equal to (where δ = Static deflection of a simply supported shaft due to the point load)

0.4985/√δ
0.5615/√δ
0.571/√δ
0.6253/√δ

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Theory of Machine
Crowning on pulleys helps

In decreasing the slip of the belt
In increasing velocity ratio
For automatic adjustment of belt position so that belt runs centrally
Increase belt and pulley life

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