Rotational Unbalance Response
Rotational Unbalance Response: Theoretical Foundations
What is Rotational Unbalance?
Professor, what is rotational unbalance?
It is the centrifugal force generated when the center of mass of a rotating body is offset from the axis of rotation (eccentricity $e$). This centrifugal force excites the structure at the same frequency as the rotational speed.
It's proportional to the square of the rotational speed! The force gets larger at higher speeds.
That's why balancing (reducing imbalance) is crucial for high-speed rotating machinery. The balance quality grade (G grade) is specified in ISO 1940.
Response Analysis
The imbalance force is a synchronous excitation (1st order: $\omega$, 2nd order: $2\omega$, ...). In FEM frequency response analysis:
1. Define the imbalance force $F = me\omega^2$ as a function of rotational speed
2. Calculate the response (displacement amplitude, vibration velocity) at each rotational speed
3. Judge if the response is within allowable limits (ISO 10816, etc.)
The magnitude of the imbalance force is proportional to $\omega^2$, but the response spikes sharply near resonance.
When the input increase of $\omega^2$ and the amplification of resonance overlap, it results in a very large response. Therefore, avoiding critical speeds (natural frequency = rotational speed) is the top priority design consideration.
Summary
Key Points:
- $F = me\omega^2$ — Imbalance force is proportional to $\omega^2$
- Resonance at Critical Speed — Natural frequency coincides with rotational speed
- ISO 1940 Balance Quality — Specification for allowable imbalance amount
- ISO 10816 Vibration Limits — Allowable values for response vibration
Rotor Imbalance Can Be a Problem Even at the Atomic Level
For ultra-precision spindles (used in semiconductor lithography equipment, exceeding 60,000 rpm), the allowable imbalance amount is below Grade G0.4 (ISO 21940-11), numerically below a few g·mm. This corresponds to a mass eccentricity equivalent to a small piece of paper (approx. 80g/m²) about 1cm² in size. In the 1990s, ASML applied this standard to rotating optical components in EUV lithography machines, becoming a catalyst for tightening the balance standards for precision rotating machinery by one level.
Computational Methods for Rotational Unbalance Response
Imbalance Response in FEM
How do you input the imbalance force in FEM?
Nastran
```
$ Imbalance force (rotational speed dependent)
RLOAD2, 100, 200, , , 1.
DLOAD, 300, 1., 1., 100
$ F = meomega^2 → Set force amplitude proportional to ω^2
```
Set the amplitude of RLOAD2 proportional to ω^2 based on imbalance mass and eccentric distance.
Abaqus
```
*STEP
*STEADY STATE DYNAMICS
f1, f2, npoints, 1.
*CLOAD, OP=NEW, AMPLITUDE=unbalance_amp
node, 1, 1.0
node, 2, 1.0
```
Define AMPLITUDE as ω^2. Apply with a 90° phase difference in x and y components (rotating force).
Since the imbalance force rotates, you apply forces with a 90° phase difference in the x and y directions simultaneously, right?
Correct. $F_x = me\omega^2 \cos(\omega t), F_y = me\omega^2 \sin(\omega t)$. In frequency response analysis, input as a complex load $F_x + iF_y$.
Summary
Key Points:
- Input $F = me\omega^2$ as a frequency-dependent load — RLOAD2 (Nastran), AMPLITUDE (Abaqus)
- 90° phase difference in x, y directions — Representation of rotating force
- Coordination with Campbell Diagram — Verification of critical speed avoidance
The Origin of Two-Plane Balancing Dates to 1907
The theory of dynamic two-plane balancing for rotating bodies was formulated by W.E. Dalby in 1907. Today's fully automatic balancing machines (Hofmann, Schenck, etc.) are merely electronic implementations of this two-plane method. The distinction between single-plane and two-plane balancing for automotive tires stems from the mathematical basis in Dalby's theory, which distinguished between "overhung" and "between bearings" cases.
Rotational Unbalance Response in Practice
Practical Work for Imbalance Response
What is the practical workflow for imbalance response analysis?
1. Set Imbalance Amount — Calculate $me$ from ISO 1940 G grade × mass
2. Campbell Diagram — Identify critical speeds
3. Imbalance Response Analysis — Displacement, velocity, acceleration at each rotational speed
4. Compare with Vibration Limits — ISO 10816, API 617, etc.
5. If Balancing is Insufficient → Improve balance quality or add damping
ISO 10816 Vibration Limits
| Group | Vibration Velocity (mm/s) rms |
|---|---|
| Good | < 2.8 |
| Acceptable | 2.8 to 7.1 |
| Warning | 7.1 to 18 |
| Unacceptable | > 18 |
So you evaluate based on vibration velocity.
Vibration velocity is proportional to vibration energy, so it correlates well with structural fatigue and bearing life.
Practical Checklist
Jet Engine Balancing is in Units of 0.01g
The fan blades (each about 4kg) of the GE90 engine (for Boeing 777) are adjusted after assembly to have an imbalance of less than 0.01g. Analysis uses MSC Nastran's ROTORDYNAMICS function to calculate imbalance response and confirms the design condition that critical speeds are at least 30% away from each operating point (takeoff, cruise, landing).
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