Order tracking

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.

Angular resampling converts a signal recorded at known times into values at equally spaced shaft angles. It is a change of sampling coordinate, used in order tracking and time-synchronous averaging. The signal must be accompanied by a reliable monotonic angle history over the interval being transformed.^[ref]

For positive speed $r(t)$ in revolutions per minute, an unwrapped reference angle isChoose ${\theta }_j={\theta }_0+j\Delta \theta$, find times $t_j$ satisfying $\theta (t_j)={\theta }_j$, and estimate $x(t_j)$ by interpolation from the original samples. Uniform time samples generally produce nonuniform angle samples; evaluating at the new $t_j$ is the resampling step. A modulo-$2\pi$ angle must first be unwrapped, rather than inverted as if each revolution restarted time.

An original acceleration model has $r(t)=600+600t$ revolutions per minute, with $t$ in seconds. Taking ${\theta }_0=0$ givesSampling every quarter revolution requires $10t_j+5t_j^2=j/4$, henceThe first quarter-turn intervals are about 0.02470 and 0.02411 seconds: angular spacing stays fixed while time spacing decreases. A third-order sinusoid becomes a fixed sinusoid of three cycles per revolution on the resampled coordinate.

With $N$ equally spaced samples per revolution, the angular Nyquist limit is $N/2$ cycles per revolution. Thus representing a maximum order $q_{\max }$ requires more than $2q_{\max }$ angular samples per turn, with practical margin and appropriate filtering. This does not repair aliasing already present in the original time record. Its sampling rate, anti-aliasing filter and the fastest physical signal frequency also matter.

A once-per-revolution tachometer constrains completed turns but does not directly measure every speed fluctuation within a turn. Inferring the intermediate angle introduces a model; a higher-resolution rotary encoder can supply more information. Stops, reversals, missing pulses and timing offsets need explicit handling. Resampling an uncertain angle history can produce a sharp-looking spectrum without making that angle history accurate.