Unbalance force in rotors lead to premature machine failure. Excess unbalance forces cause bearings and seals to fail prematurely. Excessive vibration can cause other problems. It is therefore important to precision balance rotors.
We can define the unbalance force in two separate ways. First, Force (F) equals the unbalance mass (m) times the radius of the mass from the center of rotation (r) times the rotational speed (w) squared.
F = mrw2
Because the unbalance force is proportional to the rotational speed squared, it is especially important to precision balance high speed rotors.
When diagnosing unbalance with vibration, we are essentially saying that if the unbalance force goes up, the vibration will also go up. It is important to note however that both an increase in unbalance mass AND and an increase in rotational speed will cause the vibration to increase in amplitude. This is why it is so important to always test machines as close to the same running speed as possible and trend the data. Otherwise it is difficult to know if increases in vibration are unbalance related or just due to a higher shaft speed.
Unbalance Force and Eccentricity
Another way to calculate the unbalance force (F) is to multiply the rotor mass (M) by the rotor eccentricity (e) times the rotational speed (w) squared.
What is happening here is the rotor wants to rotate around it’s center of mass, not its geometrical center. To visualize this, think about throwing a uniform wooden pole through the air so it flips end over end. It rotates around its geometric center. Now imagine throwing a hammer through the air the same way. Think about the motion it will create – it will not rotate around the geometrical center but around the head, eq the heavy part.
The rotor eccentricity (e) is the distance between the geometric center and the center of mass. This also explains why the rotor vibrates. If it was perfectly balanced and rotated around its geometric center, it wouldn’t vibrate!
Since we have two equations for the unbalance force, we can say they are equal.
F = Mew2 = mrw2
Me = mr
We are talking here about “rigid rotors” or machines that are running well below their first critical speed. Once we go over the first critical speed, things change.
The formulas above will be useful to consider when we discuss balance standards and balance quality (in a future post.) If you are interested in this topic and would like to learn more, the best way is to sign up for one of my vibration courses: https://www.zencovibrations.com/events/ and get certified in accordance with ISO 18436-2
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