Influence Coefficient

An influence coefficient is a complex field-balancing coefficient that relates a known trial unbalance to the measured change in vibration vector. For a single-plane balance at one measurement point:

where \alpha is the influence coefficient, U_t is the trial unbalance and \Delta A is the change in measured vibration vector:

A_0 is the vibration vector before the trial weight, and A_1 is the vibration vector after the trial weight. The vectors include both amplitude and phase, so the coefficient has magnitude and angle.

Engineering Role

Influence coefficients let engineers convert a trial-weight response into a correction-weight instruction. They are used when a rotor has a repeatable 1x vibration vector, phase reference is available, and the machine can safely run with a trial weight installed.

The coefficient is local to the test condition. It depends on machine speed, measurement point, balance plane, trial-weight radius, phase convention, sensor direction, filtering, load, temperature and rotor dynamic state. A coefficient measured at one speed or support condition may not be valid at another speed, especially near a critical speed or in a nonlinear machine.

For multi-plane balancing, the same idea becomes a matrix relation:

where each \alpha_{ij} relates unbalance in correction plane j to vibration response at measurement point i. This is why two-plane balancing requires controlled vector bookkeeping, not only adding mass at the largest vibration location.

Trial-Weight Quality and Linearity

The trial unbalance should be known as a mass-radius product:

where m_t is trial mass and r_t is installation radius. A trial weight that is too small gives a vibration change hidden inside repeatability scatter. A trial weight that is too large can overload the machine, trip protection or drive the rotor into a nonlinear response region.

A practical data-quality screen compares the vector change with repeatability scatter:

where \sigma_A is the observed repeat-run scatter of the 1x vibration vector at the same speed and setup. A large R_{\Delta} does not prove that balancing is safe, but a small value means the coefficient is dominated by noise, phase jitter, speed drift or process variation.

Linearity should also be checked. The trial run should produce a coherent vector change at the same order, with stable phase reference, comparable load and no new broadband vibration, rub, looseness or bearing temperature rise. Near a critical speed, a small speed change can rotate the phase rapidly, so the coefficient must be tied to the exact speed and ramp or dwell condition used for the calculation.

Worked Example: Single-Plane Correction from a Trial Weight

A fan is balanced at one speed using one correction plane. A trial unbalance is installed at the zero-degree reference:

The original 1x vibration vector is:

After the trial run, the vector is:

Convert the difference using vector subtraction:

With the stated phase convention:

The influence coefficient is:

so:

and:

The correction unbalance that cancels the original vector is:

Magnitude:

Angle:

If the correction radius is:

the required correction mass is:

Engineering comment: the calculation is only valid if the trial weight produced a coherent linear vector change. The result should be implemented with the same angular convention, rotation direction and reference mark used during the trial. After correction, the machine must be rerun and validated; the coefficient is not a certificate of acceptable vibration.

Distinction from Related Terms

Influence coefficient is not unbalance response. Unbalance response describes how the machine dynamics amplify or phase-shift residual unbalance across speed. An influence coefficient is an empirical or model-derived coefficient used for balancing at a specified condition.

Influence coefficient is not a trial weight. The trial weight is the known input. The influence coefficient is the measured response per unit trial unbalance.

Influence coefficient is not a correction weight. The correction weight is calculated from the original vibration vector and the influence coefficient.

Influence coefficient is not a general machine constant. It changes if speed, support condition, operating load, filter, sensor location, balance plane or phase reference changes.

Validation and Common Mistakes

A defensible influence-coefficient calculation states the speed, operating load, balance plane, correction radius, trial weight mass, trial angle, whether the trial weight remains installed, sensor location, vibration unit, filter, phase reference, rotation direction and sign convention.

Common mistakes include:

• using scalar vibration amplitudes and ignoring phase;
• installing a trial weight too small to produce a reliable vector change;
• balancing when phase is unstable, response is nonlinear or the machine is crossing a critical speed;
• mixing degrees measured with and against rotation;
• using an influence coefficient from a different speed, plane or sensor location;
• forgetting whether the trial weight remains installed when calculating correction mass;
• accepting a correction without rerunning the machine and checking vibration, phase, bearing temperature and repeatability.