Understanding Vector Addition in Rotor Balancing

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Definition: What is Vector Addition?

Vector addition is the mathematical operation of combining two or more vectors to produce a single resultant vector. In the context of rotor balancing, vibration is represented as a vector because it has both magnitude (amplitude) and direction (phase angle). Vector addition is fundamental to the balancing process because multiple sources of unbalance combine vectorially, not algebraically, meaning that their phase relationships matter as much as their magnitudes.

Understanding vector addition is essential for interpreting balancing measurements and predicting how correction weights will affect the overall vibration of a rotor system.

Why Vibration Must Be Treated as a Vector

Vibration caused by unbalance is a rotating force that repeats once per revolution. At any given sensor location, this vibration has two critical properties:

• Amplitude: The magnitude or strength of the vibration, typically measured in mm/s, in/s, or microns.
• Phase: The angular timing of when the peak vibration occurs relative to a reference mark on the rotor. This is measured in degrees (0° to 360°).

Because phase information is critical, we cannot simply add vibration amplitudes together. For example, if two unbalances each produce 5 mm/s of vibration, the total vibration could be anywhere from 0 mm/s (if they are 180° out of phase and cancel each other) to 10 mm/s (if they are in phase and reinforce each other). This is why vector addition, which accounts for both amplitude and phase, is necessary.

Mathematical Basis of Vector Addition

Vectors can be represented in two equivalent forms, and both are used in balancing calculations:

1. Polar Form (Magnitude and Angle)

In polar form, a vector is expressed as an amplitude (A) and a phase angle (θ). For example: 5.0 mm/s ∠ 45°. This is the most intuitive form for balancing technicians because it directly corresponds to measured vibration data.

2. Rectangular (Cartesian) Form (X and Y Components)

In rectangular form, a vector is broken down into its horizontal (X) and vertical (Y) components. The conversion from polar to rectangular form uses trigonometry:

• X = A × cos(θ)
• Y = A × sin(θ)

Adding vectors in rectangular form is straightforward: simply add all the X components together and all the Y components together to get the resultant vector’s components. The resultant can then be converted back to polar form if needed.

Example Calculation

Suppose we have two vibration vectors:

• Vector 1: 4.0 mm/s ∠ 30°
• Vector 2: 3.0 mm/s ∠ 120°

Converting to rectangular form:

• Vector 1: X₁ = 4.0 × cos(30°) = 3.46, Y₁ = 4.0 × sin(30°) = 2.00
• Vector 2: X₂ = 3.0 × cos(120°) = -1.50, Y₂ = 3.0 × sin(120°) = 2.60

Adding them:

• X_total = 3.46 + (-1.50) = 1.96
• Y_total = 2.00 + 2.60 = 4.60

Converting back to polar form:

• Amplitude = √(1.96² + 4.60²) = 5.00 mm/s
• Phase = arctan(4.60 / 1.96) = 66.9°

Result: The combined vibration is 5.00 mm/s ∠ 66.9°

Graphical Method: The Tip-to-Tail Method

Vector addition can also be performed graphically on a polar plot, which provides an intuitive visual understanding of how vectors combine:

1. Draw the First Vector: Draw the first vector from the origin, with its length representing amplitude and its angle representing phase.
2. Position the Second Vector: Place the tail (starting point) of the second vector at the tip (end point) of the first vector, maintaining its correct angle and length.
3. Draw the Resultant: The resultant vector is drawn from the origin (tail of the first vector) to the tip of the second vector. This resultant represents the sum of the two vectors.

This graphical method is particularly useful for quickly estimating the effect of adding or removing correction weights and for verifying the results of electronic calculations.

Practical Application in Balancing

Vector addition is used at every stage of the balancing process:

1. Combining Original Unbalance and Trial Weight

When a trial weight is added to a rotor, the measured vibration is the vector sum of the original unbalance (O) and the effect of the trial weight (T). The balancing instrument measures (O+T) directly. To isolate the effect of the trial weight, vector subtraction is performed: T = (O+T) – O.

2. Calculating the Influence Coefficient

The influence coefficient is calculated by dividing the vector effect of the trial weight by the trial weight mass. This coefficient is itself a vector quantity.

3. Determining the Correction Weight

The correction weight vector is calculated as the negative (180° phase shift) of the original vibration divided by the influence coefficient. This ensures that when the correction weight effect is vectorially added to the original unbalance, they cancel each other out, resulting in near-zero vibration.

4. Predicting Final Vibration

After installing a correction weight, the expected residual vibration can be predicted by performing vector addition of the original vibration and the calculated effect of the correction weight. This prediction can be compared to the actual final measurement as a quality check.

Vector Subtraction

Vector subtraction is simply vector addition with the second vector reversed (rotated 180°). To subtract vector B from vector A:

• Reverse vector B by rotating it 180° (or multiply it by -1 in rectangular form).
• Add the reversed vector to vector A using normal vector addition.

This operation is commonly used to isolate the effect of a trial weight: T = (O+T) – O, where O is the original vibration and (O+T) is the measured vibration with the trial weight installed.

Common Mistakes and Misconceptions

Several common errors arise from misunderstanding vector addition in balancing:

• Adding Amplitudes Directly: Simply adding vibration amplitudes (e.g., 3 mm/s + 4 mm/s = 7 mm/s) is incorrect because it ignores phase. The actual result depends on the phase relationship.
• Ignoring Phase Information: Attempting to balance based on amplitude alone without considering phase will almost never result in successful balancing.
• Incorrect Angle Convention: Mixing up clockwise vs. counterclockwise angle conventions or using the wrong reference point can lead to correction weights being placed at incorrect locations.

Modern Instruments Handle Vector Math Automatically

While understanding vector addition is important for balancing professionals, modern portable balancing instruments perform all vector calculations automatically and internally. The instrument:

• Collects amplitude and phase data from sensors.
• Performs all vector addition, subtraction, and division operations.
• Displays results both numerically and graphically on polar plots.
• Provides the final correction weight mass and angular location directly.

However, a solid understanding of the underlying vector mathematics enables technicians to verify instrument results, troubleshoot anomalies, and understand why certain balancing strategies are more effective than others.

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