Understanding Vector Addition in Rotor Balancing

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Vector addition is the mathematical operation of combining two or more vectors into a single resultant vector. In rotor balancing, vibration is treated as a vector because it carries two pieces of information at once: a magnitude (its amplitude) and a direction (its phase angle). This matters enormously, because separate sources of unbalance combine vectorially, not algebraically — their phase relationships count for just as much as their sizes. A firm grasp of vector addition is therefore what lets an engineer read balancing data correctly and predict how a correction weight will reshape the vibration of the whole rotor system.

1. Why Vibration Must Be Treated as a Vector

The vibration produced by unbalance is a rotating force that repeats exactly once per revolution. Measured at any one sensor location it has two inseparable properties:

• Amplitude: the magnitude or strength of the motion, usually in mm/s, in/s or microns.
• Phase: the angular instant at which the peak occurs relative to a reference mark on the rotor, read in degrees from 0° to 360° and timed from the keyphasor pulse.

Because phase is decisive, vibration amplitudes can never simply be summed. Picture two unbalances that each generate 5 mm/s: the total can be anything from 0 mm/s — if they sit 180° apart and cancel — to 10 mm/s, if they are in phase and reinforce. Everything in between is possible depending on the angle. Only vector addition, which respects both amplitude and phase, gives the right answer.

2. The Mathematical Basis of Vector Addition

A vector can be written in two equivalent forms, and balancing uses both, converting freely between them.

Polar form (magnitude and angle)

Here the vector is an amplitude ఎ at a phase angle θ — for example, 5.0 mm/s ∠ 45°. This is the most natural form for a technician because it maps directly onto what the instrument displays and onto a polar plot.

Rectangular (Cartesian) form (X and Y components)

Here the vector is split into a horizontal (X) and a vertical (Y) component using trigonometry:

• X = A × cos(θ)
• Y = A × sin(θ)

Addition then becomes trivial: sum all the X components, sum all the Y components, and you have the resultant’s components, which can be turned back into polar form whenever a magnitude-and-angle answer is wanted.

A worked example

Take two vibration vectors:

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

Convert each 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

Add the components:

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

Convert 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°. Notice that two vectors of 4.0 and 3.0 mm/s did not add to 7.0; because they were 90° apart they combined to exactly 5.0, the familiar 3-4-5 right triangle. That gap between the naive sum and the true result is precisely why phase cannot be ignored. If you want to combine your own measured vectors without hand arithmetic, the Vibration Phase Angle Calculator performs the conversion and addition directly.

3. The Graphical Tip-to-Tail Method

Vector addition can also be done by drawing, which gives an immediate visual feel for how vectors combine and is easily sketched on a polar plot:

1. Draw the first vector: from the origin, with its length set to the amplitude and its direction set to the phase.
2. Position the second vector: place its tail at the tip of the first, keeping its own correct length and angle.
3. Draw the resultant: a line from the origin to the tip of the second vector is the sum.

This construction is handy for quickly estimating the effect of adding or removing a correction weight, and for sanity-checking the numbers an instrument produces.

4. Practical Application in Balancing

Vector addition is not a side calculation — it is woven through every stage of the balancing workflow.

Combining original unbalance and trial weight

When a trial weight is fitted, the new reading is the vector sum of the original unbalance vibration (O) and the trial weight’s effect (T). The instrument measures (O+T) directly; to isolate T alone it performs vector subtraction: T = (O+T) − O.

Calculating the influence coefficient

The influence coefficient is found by dividing the trial weight’s vector effect by the trial mass, so it too is a vector quantity — an amount of vibration per unit of weight, at a characteristic angle. The Influence Coefficient Calculator automates this single-plane case.

Determining the correction weight

The correction weight vector is the negative (a 180° phase shift) of the original vibration, divided by the influence coefficient. Sized this way, its effect — when vectorially added back to the original unbalance — cancels it, driving the vibration toward zero.

Predicting final vibration

Once the correction is fitted, the expected residual vibration can be forecast by adding the original vibration vector to the calculated effect of the correction. Comparing that prediction with the measured outcome is a powerful quality check on the whole job.

5. Vector Subtraction

Vector subtraction is nothing more than vector addition with the second vector reversed (rotated 180°). To subtract vector B from vector A:

• Reverse B by rotating it 180° — or, in rectangular form, simply negate both its components.
• Add the reversed B to A with ordinary vector addition.

As noted above, this is the operation that isolates a trial weight’s effect, T = (O+T) − O, where O is the original vibration and (O+T) is the reading with the trial weight installed.

6. Common Mistakes and Misconceptions

Most balancing errors that trace back to vector maths fall into three traps:

• Adding amplitudes directly: treating 3 mm/s + 4 mm/s as 7 mm/s ignores phase entirely; as the worked example showed, the true result depends on the angle between them.
• Ignoring phase information: attempting to balance on amplitude alone, with no phase reference, almost never converges to a good result.
• Inconsistent angle convention: mixing clockwise and counter-clockwise conventions, or measuring from the wrong reference, sends correction weights to the wrong position on the rotor.

7. Modern Instruments Handle the Vector Maths

Although understanding the mathematics is essential for any balancing professional, the arithmetic itself is now done automatically by the instrument. A portable analyser such as the Balanset-1A collects amplitude and phase from both channels, carries out every vector addition, subtraction and division internally, displays the results numerically and graphically on polar plots, and reports the final correction weight mass and angular location ready to fit. Yet the underlying theory still earns its keep: an engineer who understands it can verify the instrument’s output, diagnose anomalies when a result looks wrong, and grasp why certain balancing strategies converge faster than others.

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