Solve a Single-Plane Balance Yourself: A Live Influence-Coefficient Solver

Our field-balancing explainer walks the whole job — confirming the fault is really unbalance, sizing and fastening the trial weight, reading the result against an ISO 21940 grade. What it could not do on a static page is hand you the rotor. This post does: further down is a live influence-coefficient solver. You give it the three things a balancing instrument measures, and it gives you the two things a balancing instrument reports — a correction mass and an angle.

The model is stated up front so there is no mystery: a linear, single-plane rotor at constant speed. Linear means doubling a weight doubles its effect; single-plane means one correction plane is enough (thin discs, fans, flywheels); constant speed means every run happens at the same RPM, because the rotor's response to unbalance — the influence coefficient — is only valid at the speed where it was measured. Inside that box, the whole job is three vectors in and one weight out.

Every number in this job is an amplitude plus an angle, so the convention has to be nailed down before anything else. The solver's is simple: angles are measured in degrees from the once-per-revolution reference mark — the strip of reflective tape the tachometer sees — and positive angles run counter-clockwise, viewed from the measurement end. Phase readings and weight positions use the same zero and the same direction. That is the entire convention.

It matters because most 'the maths is wrong' arguments in balancing are convention bugs, not arithmetic. Instruments differ in whether displayed phase leads or lags, and rotors get viewed from either end. Whichever convention your instrument uses, the discipline is identical: one zero, one positive direction, applied to every phase reading and every weight angle, written on the job sheet. Flip either one halfway through and the correction lands mirror-imaged — confidently, and wrongly.

Enter the reference run O (amplitude and phase), the trial weight (mass and angle), and the trial run O+T. The plot draws O and O+T from the centre, joins their tips with the derived effect vector T, and marks −O — the direction the correction's effect must push. The panel below the plot reports T, the influence coefficient H, the correction mass and angle with the trial weight removed, and the predicted residual once the correction is rounded to something you could actually fit.

Two ways to play. Drag the orange dot around the rim: it is the trial weight, and out of the box the trial-run reading is simulated live from a hidden rotor, so the reading and the solution move as the weight does. Or press Randomise a job for a fresh rotor with a different hidden influence coefficient, and practise reading the answer. Type your own numbers into the O+T boxes at any point and the solver switches to trusting your readings instead. If the trial effect is too small to trust, it will tell you — the same 'make the trial matter' rule the field job has.

Each reading is a complex number: amplitude at an angle. Step one is complex subtraction, T = (O+T) − O, tip to tip on the plot — the trial weight's own effect, isolated. Step two divides that effect by the trial weight as a vector, H = T ÷ (m∠φ), giving the influence coefficient: what one gram at zero degrees would do to this rotor, in these bearings, at this speed. Step three is the payoff, W = −O ÷ H: the mass and angle whose predicted effect lands exactly opposite the original reading. Complex division is just dividing magnitudes and subtracting angles — the polar chart methods of the paper era were this same arithmetic done graphically.

The drag behaviour now explains itself. Moving the trial weight rotates the ∠φ term, so the solved correction rotates with it, holding a fixed offset from wherever the trial weight sits — the offset and the mass ratio are what the two runs actually measured. And note what the predicted residual is: the model grading its own homework. It only reflects rounding the correction to a fittable mass and angle; real rotors add nonlinearity, speed drift and measurement noise, which is why the field procedure ends with a verification run, not a prediction.

A representative job, not a specific customer: a direct-coupled blower at 2,950 RPM reads 8.0 mm/s at 40° on the reference run. A 25-gram trial weight bolted at the 0° mark moves the reading to 9.3 mm/s at 358.3° — the amplitude went up, which is fine; the vector moved by over 6 mm/s, and that is what solves the coefficient. Those are the solver's pre-loaded defaults, so you can check its working: the effect T comes out near 6.3 mm/s at 300°, the influence is about 0.25 mm/s per gram, and the correction lands at roughly 32 grams at 280° with the trial weight removed.

The predicted residual is under 0.1 mm/s — and on a real machine you would trust that exactly as far as the verification run confirms it. Now do what the page version of this job cannot: re-enter the same readings but claim the trial weight was at 90° instead of 0°, and watch the correction swing by the same 90°. Same vectors, different attribution, different answer — a five-second demonstration of why the weight angle gets written down before the run, not remembered afterwards.

The solver waives nothing that matters at the machine. Same speed, every run — reference, trial, correction, verification — because the influence coefficient dies the moment the speed changes. Trial weight off, or on, never halfway: the correction differs for each case, so decide before fitting, tag the rotor, and write it down. And fasten every weight so it cannot fly off — centrifugal force grows with speed squared, and a modest trial mass at radius pulls outward with tens of kilograms of force at ordinary machine speeds. Bolt it, clamp it, or weld it; never tape it; stand clear of the rotor's plane during run-up.

If the vector arithmetic felt quick, the free primers at 101.tieraonline.in take it at beginner pace — the Balancing & Alignment 101 course covers phase, conventions and the influence-coefficient method from first principles. TIERA 101 is a free primer, not an accredited ISO certification. For formal, examined analyst training there is the TCAT programme on our services page, with proctored examinations at exams.tieraonline.in.

The solver on this page is the arithmetic; a field job also needs the measurement done right. The TVIB TB 210 balancing module runs this exact sequence guided step by step — tachometer or laser phase reference, prompted trial-weight runs, automatic correction weight and angle for single-plane and two-plane rotors, and a balancing report for the maintenance file. It runs on any PhonoVibe DAQ; the two-channel PhonoVibe D, 24-bit with IEPE sensor power and USB-powered from the laptop, makes a natural field kit.

To practise beyond a browser, the TMFSS simulator generates controlled, repeatable unbalance on a benchtop — set a known heavy spot, balance it out, get the trial weight deliberately wrong, and see the consequences with nothing at stake.