Theory of Machine
The equation of motion for a vibrating system with viscous damping is (d²x/dt²) + (c/m). (dx/dt) + (s/m). x = 0. If the roots of this equation are real, then the system will be

Critically damped
Under damped
Without vibrations
Over-damped

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Theory of Machine
Crowning on pulleys helps

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

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