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Order tracking follows vibration or other signal components according to their ratio to a reference shaft's rotational frequency. That ratio is the rotational order:where $r$ is speed in revolutions per minute and $f,f_{\mathrm{rot}}$ are measured in hertz. A component at three cycles per reference revolution has order 3 even while its ordinary frequency changes. The reference shaft must be specified; changing the reference changes the order value.^[ref]

For an original example, a shaft accelerates from 600 to 1200 revolutions per minute. Its third-order component sweeps from 30 to 60 hertz, while its seventh-order component sweeps from 70 to 140 hertz. A Fourier transform over the entire time record spreads those components over frequency. An unrelated fixed 90-hertz tone instead retains its frequency but changes from order 9 to order 4.5.

Angular resampling expresses the record on a uniform shaft-angle grid. A speed-locked component $\cos (q\theta (t)+\varphi )$ then becomes $\cos (q\theta +\varphi )$, with fixed angular periodicity. An order spectrum or order-versus-speed map can separate that structure from a frequency-domain smear. MathWorks demonstrates this distinction with explicitly simulated helicopter-vibration data.^[ref]

In an apparatus, an accelerometer can record housing motion while a tachometer or rotary encoder records the reference. Tracking an order's amplitude and phase over a run-up can reveal where the machine's response grows, but a large amplitude need not identify the forcing mechanism by itself. Mechanical resonance, load changes and sensor placement can also change the response.

Time-synchronous averaging is related but answers a different question: it estimates a waveform repeatable over the chosen reference period. Order tracking can retain amplitude variation over speed and inspect selected components. Record speed-estimation assumptions, angular resolution and amplitude normalization with either analysis.

A rotational order expresses a signal’s frequency relative to a specified reference shaft’s rotational frequency:Thus $q=1$ means one oscillation per reference revolution and $q=1/2$ means one oscillation per two reference revolutions. A rotational order is dimensionless and need not be an integer. It is especially useful when a machine changes speed: a shaft-synchronous feature can keep the same rotational order while moving through many frequencies in an ordinary frequency spectrum.^[ref]

The reference is essential. Suppose shaft A turns at $600\mathrm{rpm}=10\mathrm{Hz}$ and shaft B at $1500\mathrm{rpm}=25\mathrm{Hz}$. The same $100\mathrm{Hz}$ signal has rotational order $10$ relative to A and $4$ relative to B. Changing the reference has changed the dimensionless label, not the measured vibration. Likewise, a 24-tooth gear turning at $900\mathrm{rpm}$ has a gear mesh frequency of $24\times 15=360\mathrm{Hz}$. Its mating 8-tooth gear turns at $2700\mathrm{rpm}$, giving the same gear mesh frequency but rotational orders $24$ and $8$ relative to the two shafts.

For a component with unwrapped phase $\varphi (t)$ and reference angle $\theta (t)$, the instantaneous version iswhere the reference angular velocity is nonzero and both angles use the same units. A constant rotational order gives $\varphi (t)=q\theta (t)+{\varphi }_0$. This explains why angular resampling makes a speed-following component periodic as a function of shaft angle. A vibration fixed at $100\mathrm{Hz}$ instead changes from rotational order $10$ to $5$ when the reference shaft accelerates from $600$ to $1200\mathrm{rpm}$.

An order tracking measurement therefore pairs a vibration sensor, such as an accelerometer, with a shaft-angle or speed reference, such as a tachometer. An order peak identifies a speed relationship; it does not by itself establish which part is damaged. A resonance can greatly change the amplitude as speed changes even when the excitation’s rotational order stays fixed.

Time-synchronous averaging aligns repeated signal segments to a common reference phase and averages corresponding positions. In rotating machinery, a tachometer or rotary encoder provides a shaft reference. Components repeating at that reference phase remain coherent, while unrelated variations can cancel. MathWorks implements the operation using supplied pulse times and allows the averaging interval to span one or several rotations.^[ref]

For angle-aligned records $x_m(\theta )=s(\theta )+{\eta }_m(\theta )$, the average over $M$ revolutions isIf the noise terms are centered, independent between revolutions and each have variance ${\sigma }^2$ at that phase, the average's noise variance is ${\sigma }^2/M$. In an original example, noise standard deviation 4 signal units falls to 0.5 after 64 independent aligned records. Shared drift or other inter-record correlations invalidate that reduction rule.

Alignment is essential when speed changes. Equal time intervals after a trigger can represent different shaft angles in different revolutions. Angular resampling supplies a common angle grid before averaging. Errors in the reference phase also matter: a rotational order $q$ acquires phase error $q\delta \theta$ from a shaft-angle error $\delta \theta$, making high orders especially sensitive.

Cancellation is not proof that a component was unwanted noise. Consider a half-order component on a continuously increasing shaft angle, $x(\theta )=\sin (\theta /2)$. Consecutive one-revolution records satisfyAn even number of such records averages to zero, although the component is perfectly deterministic and repeats every two revolutions. Averaging over a two-revolution reference interval can retain it. The reference period therefore selects what the procedure regards as repeatable.

A useful bench acquisition records vibration and the reference pulses together, checks missing pulses and sensor timing, and compares individual aligned records with their average. Time-synchronous averaging estimates a repeatable waveform; it does not preserve every transient or every component merely because it is mechanically important.