Unbalanced mass is calculated indirectly through its effect on a rotating system. Here's how it's approached:
Understanding Unbalance
The key concept is rotor unbalance (U). This is the product of the unbalanced mass (m) and its distance from the center of rotation, the radius (r). The reference states that "The unbalance mass m times its radius r equals U Rotor unbalance". Thus:
- *U (Rotor Unbalance) = m r**
Where:
- U = Rotor unbalance (typically measured in gram-millimeters or ounce-inches)
- m = Unbalanced mass (typically measured in grams or ounces)
- r = Radius of the unbalanced mass from the center of rotation (typically measured in millimeters or inches)
Calculating Mass Eccentricity
While we often don't directly know the mass m or radius r separately, we can determine a useful quantity called mass eccentricity (e) or specific unbalance, as stated in the reference. This is a measure of unbalance independent of the rotor's total mass and is defined as:
- e = U / M
Where:
- e = Mass eccentricity (typically measured in millimeters or inches)
- U = Rotor unbalance (m * r)
- M = Total rotor mass (typically measured in grams or ounces)
Practical Insights and Solutions
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Direct measurement limitations: Directly determining the specific unbalanced mass and its radial distance separately is difficult in most practical scenarios. Vibration analysis techniques using specialized equipment such as balancing machines are used to indirectly measure the unbalance (U) rather than the unbalanced mass and radius individually.
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Balancing: The goal is to reduce unbalance (U) by adding or removing mass at strategic locations. This usually involves:
- Measuring vibration: Instruments measure the vibration caused by unbalance.
- Determining the location and magnitude of the unbalance (U): Specialized machines and algorithms pinpoint the position where mass needs to be adjusted.
- Correcting the unbalance: Weights are added or removed at the identified locations, reducing vibration and achieving a balanced rotor.
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Mass Eccentricity as a key metric: The reference also points out that mass eccentricity e is the displacement of the mass center from the bearing center. By using mass eccentricity, we evaluate unbalance by considering it relative to the rotor's overall size and mass, allowing comparison between different rotors.
Example
For instance, a fan blade has some imbalance. Measuring it may indicate that U = 1000 gram-millimeters. If the fan blade weights M = 200 grams, then e = 1000 / 200 = 5 millimeters. This tells us the center of mass is 5 mm away from the center of rotation, a more useful measure for balancing than just knowing the absolute value of U.
Summary Table
| Term | Symbol | Definition | Typical Units | How it's Calculated |
|---|---|---|---|---|
| Rotor Unbalance | U | Product of unbalanced mass and radius | gram-millimeters or ounce-inches | m * r |
| Unbalanced Mass | m | The mass causing the unbalance | grams or ounces | Not typically directly calculated, derived from U |
| Radius of Unbalance | r | Distance of unbalanced mass from center of rotation | millimeters or inches | Not typically directly calculated, derived from U |
| Mass Eccentricity | e | Displacement of mass center from bearing center, specific unbalance. | millimeters or inches | U / M |
| Total Rotor Mass | M | Total mass of the rotating object | grams or ounces | Directly measured |
