Vibration Calculation

It depends on where your rotational axis is in relation to the mass. The minimum vibration would be if the axis is through the center of gravity (static balance) and on either the axis of maximum or minimum moment of inertia (dynamic balance).

Force = mass x acceleration. If you center of gravity is distance r from the axis, acceleration is r x ω², where ω is 2 pi x RPM.

https://en.wikipedia.org/wiki/Angular_acceleration

"All rotating shafts, even in the absence of external load, will deflect during rotation. The unbalanced mass of the rotating object causes deflection that will create resonant vibration at certain speeds, known as the critical speeds. The magnitude of deflection depends upon the following:

(a) stiffness of the shaft and its support
(b) total mass of shaft and attached parts
(c) unbalance of the mass with respect to the axis of rotation
(d) the amount of damping in the system

In general, it is necessary to calculate the critical speed of a rotating shaft, such as a fan shaft, in order to avoid issues with noise and vibration."

"There are two main methods used to calculate critical speed-the Rayleigh-Ritz method and Dunkerley's method. Both calculate an approximation of the first natural frequency of vibration, which is assumed to be nearly equal to the critical speed of rotation. The Rayleigh-Ritz method is discussed here. For a shaft that is divided into n segments, the first natural frequency for a given beam, in rad/s, can be approximated as:

where g is the acceleration of gravity, and the are the weights of each segment, and the are the static deflections (under gravitational loading only) of the center of each segment. Generally speaking, if n is 2 or higher, this method tends to slightly overestimate the first natural frequency, with the estimate becoming better the higher n is. If n is only 1, this method tends to underestimate the first natural frequency, but the equation simplifies to:

where is the max static deflection of the shaft. These speeds are in rad/s, but can be converted to RPM by multiplying by .

Static deflections for several types of uniform-cross-section beams can be found here. If a beam has multiple types of loading, deflections can be found for each, and then summed. If the shaft diameter changes along its length, deflection calculations become much more difficult.

The static deflection expresses the relationship between rigidity of the shaft and inertial forces; it includes all the loads applied to the shaft when placed horizontally.^[1] However, the relationship is valid no matter what the orientation of the shaft is.

Critical speed depends upon the magnitude and location of the shaft unbalance, the length of the shaft, its diameter, and the kind of bearing support. Many practical applications suggest as good practice that the maximum operating speed should not exceed 75% of the critical speed; however, there are cases that require speeds above the critical speed to work correctly. In such cases, it is important to accelerate the shaft through the first natural frequency quickly so that large deflections don't develop."

https://en.wikipedia.org/wiki/Critical_speed