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Vibration Engineering

Rotor Balancing Simulator

When a fan, an impeller or a grinding wheel vibrates, the cause is almost always mass unbalance. Enter the original vibration, the vibration with a trial weight, and the trial-weight data, and the influence-coefficient method computes the exact correction-weight mass and angle by vector arithmetic.

Parameters
Original vibration amplitude A₀
µm
Vibration amplitude at running speed, no trial weight
Original vibration phase φ₀
°
Angle of the vibration peak from the reference pulse
Trial-weight mass mₜ
g
Known mass attached to measure the influence coefficient
Trial-weight mounting angle θₜ
°
Angular position of the trial weight on the rotor
Vibration amplitude with trial weight A₁
µm
Vibration amplitude with the trial weight fitted
Vibration phase with trial weight φ₁
°
Angle of the vibration peak with the trial weight
Results
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Influence coeff. |α| (µm/g)
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Correction-weight mass (g)
—
Correction-weight angle (°)
—
Trial-weight effect (µm)
—
Predicted reduction (%)
—
Balance verdict
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Complex-plane vector diagram and rotor

The polar grid on the left shows the original vibration V0, the with-trial-weight vibration V1, the trial-weight effect V1-V0 and the correction-weight position as arrows. The right shows the spinning rotor with the unbalance and correction-weight positions.

Vibration vector diagram (complex x-y plane)
Vibration before and after correction
Theory & Key Formulas

$$\alpha=\frac{V_1-V_0}{W_t},\qquad W_c=-\frac{V_0}{\alpha}$$

Influence coefficient α and correction weight Wc. V0: original vibration, V1: vibration with the trial weight, Wt: trial weight. All are amplitude-angle-phase vectors, and α is the rotor's influence coefficient in µm of vibration per gram of weight.

$$x=A\cos\theta,\quad y=A\sin\theta,\qquad A=\sqrt{x^2+y^2},\quad \theta=\operatorname{atan2}(y,x)$$

Vector arithmetic is done by converting to rectangular form (x, y), adding, subtracting and dividing complex numbers, then converting back to amplitude A and phase θ normalised to 0–360°.

$$\Delta V = V_1-V_0,\qquad |W_c| = \frac{|V_0|}{|\alpha|},\quad \angle W_c = \angle V_0 + 180^\circ - \angle\alpha$$

ΔV is the trial-weight effect vector. Complex division divides magnitudes and subtracts angles, so the correction-weight angle is the angle of −V0 minus the angle of α.

What is Rotor Balancing?

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I heard a big factory blower keeps humming with vibration and wears out its bearings fast. What actually causes that?
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If the vibration is synchronous with the running speed, the first suspect is unbalance. If the rotor's centre of mass does not sit exactly on the axis of rotation, spinning it throws a rotating centrifugal force that shakes the bearings. Dust on a blade, a repair weld, a tiny machining error — each one shifts the centre of mass just a little, and at speed that becomes a force you cannot ignore.
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OK. So if the centre of mass is off, I can just add a weight on the opposite side — but how do I decide how many grams and where?
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That is where the influence-coefficient method comes in. No guesswork — it gives the exact answer from just three measurements. First, measure the original vibration at running speed — not just the amplitude, but the phase, which tells you when in each revolution the vibration peaks. That makes the vibration a vector. The A₀ and φ₀ sliders on the left are exactly that.
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So we treat the vibration as a vector. What is the trial weight for?
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The trial weight is a move to measure how this particular rotor responds to added mass. You bolt on a weight whose mass and angle you know (mₜ, θₜ) and run it again. The vibration vector changes from V0 to V1. That change, V1−V0, is the vibration the trial weight alone produced, and dividing it by the trial-weight mass gives the rotor's responsiveness — the influence coefficient α, a single complex number: α = (V1−V0)/Wt.
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Once I have the influence coefficient α, is the correction weight just a calculation away?
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Yes — from here it is a single division. To cancel the original unbalance, the vibration that α produces must be exactly opposite and equal in size to V0. So the correction weight is Wc = −V0/α. Complex division just divides the magnitudes and subtracts the angles. The magnitude of Wc is the mass to add, and its angle is the exact mounting position. The vector diagram at the top right shows V0, V1, their difference and the correction direction at a glance.
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Can a single weight really take that large vibration to nearly zero?
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When single-plane balancing works well, in theory the residual vibration goes to near zero. In the field too, a grinding-wheel that was running rough turns silent with one well-chosen correction weight. It does not always land perfectly the first time, though — then you treat the leftover vibration as V0 again and repeat the same calculation. That follow-up step is called a trim balance.

Frequently Asked Questions

The influence-coefficient method corrects rotor unbalance using measurements alone. First, the original vibration V0 (amplitude and phase) is measured at running speed. Next, a trial weight of known mass and angle is attached and the machine is run again to measure vibration V1. The change caused by the trial weight, V1-V0, divided by the trial-weight vector gives the influence coefficient alpha, which captures how much the rotor vibrates per gram of added mass. The trial weight is essential because it is the only way to measure alpha directly.
Once the influence coefficient alpha is known, the correction weight Wc that cancels the original unbalance follows from a single complex division: Wc = -V0 / alpha. Complex division divides the magnitudes and subtracts the angles. The magnitude of Wc is the mass to add, and the angle of Wc is where to fit it. This tool converts the vibration vectors to rectangular form (x = A cos theta, y = A sin theta), performs the arithmetic, and converts back via A = sqrt(x^2 + y^2) and theta = atan2(y, x).
Single-plane balancing, handled by this tool, works for axially thin disc-like rotors such as fans, grinding wheels and pump impellers: one correction weight in one plane is enough. When a rotor is long in the axial direction, the unbalance appears as a couple (moment), and two-plane balancing is required, correcting both ends simultaneously. Applying only a single-plane correction to a long rotor can improve one end while making the other worse.
Phase is the angle that tells you when, within one revolution, the vibration peaks. It is measured from a reference pulse generated by a photo-tachometer or key-phasor detecting a reflective mark on the shaft. Amplitude alone cannot tell you the direction of the unbalance, so phase information is essential. Only with amplitude and phase together does vibration become a vector, which is what makes the influence-coefficient arithmetic possible.

Real-World Applications

Blowers, pumps and compressors: Large factory fans, ventilation blowers and pump impellers usually have axially thin disc-like rotors, which are exactly where single-plane balancing works best. Because unbalance grows over time from dust build-up, blade wear and repair welds, vibration is measured periodically and a correction weight is added — or excess mass ground away — using the influence-coefficient method.

Grinding wheels and machine-tool spindles: A grinding wheel's centre of mass shifts easily with dressing and moisture absorption, and even a small unbalance shows up directly as chatter marks (waviness) on the machined surface. Many grinders have movable balance weights built into the wheel flange, and their positions are set with the influence-coefficient approach to hold surface finish and dimensional accuracy.

Turbine, generator and motor rotors: Steam-turbine and generator rotors are long and heavy and properly need two- or multi-plane balancing, but on-site quick corrections still use the influence-coefficient method one plane at a time. Because phase changes sharply near a critical speed, it is important to measure and correct at the actual running speed.

Condition monitoring and predictive maintenance: On equipment under continuous vibration monitoring, tracking the amplitude and phase of the running-speed (1× ) component detects growing unbalance early. When the phase stays roughly constant while the amplitude rises, it is a classic unbalance signature, and the timing of an influence-coefficient correction can be planned in advance.

Common Misconceptions and Pitfalls

The most common mistake is assuming "vibration always means unbalance". Most running-speed (1×) vibration is indeed unbalance, but misalignment, a bent shaft, bearing looseness and resonance also produce synchronous vibration. Adding a weight by the influence-coefficient method will not remove those root causes. Before correcting, look at the phase behaviour and the spectrum (for example whether a 2× component is present) to confirm that unbalance really dominates.

Next, the belief that "the influence coefficient α is a fixed property of the machine". α is the response of the whole system — rotor, bearings, foundation and running speed — and it changes significantly when the speed changes, especially near a critical speed. Reusing an α measured at one speed for a different speed or a different machine throws off the correction-weight angle. This tool assumes three vectors measured under identical conditions, so all measurements must be taken at the same running speed and the same location.

Finally, do not be complacent about "a small trial-weight effect V1−V0 being acceptable". If the trial weight barely changes the vibration (ΔV is tiny), the numerator of α = ΔV/Wt is small, and small measurement errors are greatly amplified into α, making the correction-weight mass and angle inaccurate. The rule is to choose a trial weight large enough to change the original vibration by a meaningful amount — aim to change it by at least 30%. This tool also flags a warning when ΔV is extremely small.

How to Use

  1. Enter the original vibration amplitude (µm) and phase angle (°) measured from an unbalanced rotor using a proximity or accelerometer sensor.
  2. Apply a known trial mass (grams) at a specified angular position (°) on the rotor and measure the resulting vibration amplitude and phase shift.
  3. The simulator calculates the influence coefficient |α| in µm/g, then uses modal decomposition to determine the correction-weight mass and optimal installation angle for residual vibration below ISO 20816 acceptance limits.

Worked Example

A 500 rpm cooling fan shows original vibration of 45 µm at 120° phase. A 25 g trial mass placed at 0° reduces amplitude to 38 µm at 135°. The simulator computes influence coefficient |α| = 0.28 µm/g. For final balance to 5 µm (ISO Grade G2.5), correction mass = 142 g at 287° achieves 89% predicted reduction. Trial-weight effect = 7.2 µm vector. Residual imbalance after correction approximately 4.8 µm.

Practical Notes

  1. Measure vibration in the radial direction perpendicular to rotor axis; axial measurements require separate balancing planes and modal analysis.
  2. Trial mass should be 10–20% of estimated rotor mass to avoid nonlinear effects; use adhesive weights (0.5–5 g increments) for precision adjustment.
  3. Phase reference must be consistent—mark rotor keyway or keystone at TDC and synchronize sensor trigger with once-per-rev signal.
  4. Multi-plane balancing (two or more correction planes) becomes mandatory if residual imbalance exceeds 150% of original after single-plane correction.