Rotating unbalance

Rotating unbalance is the uneven distribution of mass around an axis of rotation. A rotating mass, or rotor, is said to be out of balance when its center of mass (inertia axis) is out of alignment with the center of rotation (geometric axis). Unbalance causes a moment which gives the rotor a wobbling movement characteristic of vibration of rotating structures.
Types of balance.
Static balance.
A static balance (sometimes called a force balance) occurs when the inertial axis of a rotating mass is displaced from and parallel to the axis of rotation. Static unbalances can occur more frequently in disk-shaped rotors because the thin geometric profile of the disk allows for an uneven distribution of mass with an inertial axis that is nearly parallel to the axis of rotation. Only one plane receives balance correction.
where U = balance, m = mass, r = distance between unbalance and the centre of the object
Couple balance.
A couple balance occurs when a rotating mass has two equal unbalance forces that are situated 180° opposite each other. A system that is statically balanced may still have a couple unbalance.
Couple unbalance occurs frequently in elongated cylindrical rotors.
where d = distance between the two unbalance forces along the rotation axis.
Dynamic balance.
In rotation an unbalance when the mass/inertia axis does not intersect with shaft axis then it is called dynamic balance. Combination of static and couple balance is dynamic unbalance. It occurs in virtually all rotors and is the most common kind of unbalance. It can be fixed by correcting the weight on at least two planes.
How to correct or compensate balance.
The measurement of existing vibration and calculation of the change of mass required is typically carried out using some form of balancing machine.
Grades.
ISO 21940 classifies vibration in terms of G codes. Unfortunately, it is the theoretical value assuming the rotor was spinning in free space so it does not relate to actual operating conditions. Rotors of the same type having permissible residual specific unbalance value eper, varies inversely with the speed of the rotor.
where ω = angular velocity (radians per second) eper = permissible residual specific unbalance
This constant is quality grade G. Balance Grades are used to specify the allowable residual imbalance for rotating machinery. The ISO 1940 standard defines balance grades for different classes of machinery. A rotor balanced to G2.5 will vibrate at 2.5 mm/s at operating speed if rotating in a suspended state with no external influences.
Where Uper = balance tolerance (or) residual imbalance
Important formulas.
Where F = force due to unbalance, U = unbalance. ω = angular frequency. e = specific unbalance. m = mass. r = distance between unbalance and the axis of rotation of the object.