Order Spectrum — Control Systems/Vibration

Order Spectrum

Control Systems / Vibration

The order spectrum of a rotating machine – MATLAB's orderspectrum over the frames completed so far. A component of a machine sits at a fixed order of its shaft (a multiple of the rotation rate), so it smears across an ordinary spectrum whenever the speed moves. This block measures the signal against shaft angle instead of time: it interpolates the vibration onto a constant-angle grid using the tachometer, transforms every frame of that grid through a window, and publishes the root-mean-square over the frames so far, order by order.

Ports

• x – the vibration, a scalar stream.
• rpm – the shaft speed at the same instant, a scalar stream in revolutions per minute. It is what turns time into angle: the block integrates it (trapezoid by trapezoid) to get shaft revolutions.
• spec – the order spectrum, a column of nfft/8 + 2 entries, where nfft = max(256, Window Length). Entry m is order m·O_max/(nfft/8), the last one just past O_max = f_s/(2·Maximum RPM/60), which is the highest order the sample rate can carry.

Parameters

• Maximum RPM – the highest shaft speed the run will reach. It sets O_max, the angle grid (8·O_max samples per revolution) and the order axis. ⚠ MATLAB takes this from the record (max(rpm)), which a stream has not seen; the block is orderspectrum exactly when this value is the run's maximum. A speed above 3.75× it is clamped, because past that a fifteenth of a sample would owe more than one angle sample.
• Sample Rate (Hz) – f_s, the rate the two streams arrive at.
• Window Length – L, the frame length in angle samples, 4 to 1024. It sets the order resolution, ENBW(window)·8·O_max/L. MATLAB's own default resolution picks 967 for a flat top window (385 for Hann, 350 for Hamming, 256 for rectangular) whatever the speed and rate, which is why 967 is this block's default.
• Window – the taper on each frame: Flat Top (rpmordermap's default, and this block's), Hann, Hamming or Rectangular. Each is divided by its own sum first, as rpmordermap does, so a sinusoid's amplitude is read off directly.
• Overlap Percent – how much of a frame the next one repeats, 0 to 99; the hop is L − min(ceil(percent·L/100), L−1). 50 is MATLAB's default and this block's.
• Amplitude – what an entry means.
  • RMS – the root mean square of the frames' magnitudes; orderspectrum's default and this block's.
  • Peak – the same, scaled by √2 so a sinusoid reads its amplitude.
  • Power – the mean of the squared magnitudes, not its root.
• Sampling Time (s) – zero or less inherits the solver's rate; a positive value runs the block at that period.

What it does per sample

Every input sample is interpolated to 15 points by a 301-tap zero-phase filter – MATLAB's resample(x, 15, 1), which reads 10 samples ahead, so the block runs 10 samples behind the input. The tachometer is interpolated the same way and integrated into revolutions; whenever the shaft passes the next grid angle, one angle sample is interpolated and pushed into the frame. A completed frame is windowed, transformed at nfft and accumulated; the published spectrum is the running root-mean-square, and it holds between frames. Before the first complete frame it is all zeros.

Code export

All ten targets: Python, MATLAB, Java, Rust, C, C++, VHDL, Verilog, SystemVerilog and PLC Structured Text. The 301 interpolation taps and every constant are computed at export time; the windows and the transform's sines and cosines are computed in the emitted core, which keeps the emitted file small whatever the frame length. Every loop has a fixed bound – 15 upsampled points per sample, at most one angle sample each, at most one frame each – so the core's worst case per sample is known at export time. Nothing is a tunable parameter: every setting changes the grid, the window or the accumulators.

The three HDL targets are simulation-only: they carry the arithmetic in real and quantize only at the port boundary. An angular resampler divides by a run-time phase increment and a spectrum spans several decades, neither of which belongs in a Q16.16 datapath. VHDL runs the whole sample in an emitted function over a record of state, because a clocked process has three scratch variables and this needs arrays.

Simulink bridge

No equivalent (Support::None), so nothing crosses in either direction and no parity testbench is owed. orderspectrum and rpmordermap are Signal Processing Toolbox functions, that toolbox ships no Simulink library, and no library block resamples a signal onto shaft angle. Code export verification still covers the block across all ten languages.

Notes

• Stateful and inherently discrete (setDiscreteOnlyBlock(true)): an input delay line, a tachometer delay line, the phase, the angle-domain frame and one accumulator per order, all advancing once per sample. Its period comes from its own "Sampling Time (s)", or the model's global sampling time when that is zero or less.
• It runs 10 samples behind, because the interpolation reads that far ahead. That is resample's own filter, not an approximation of it.
• Three departures from the batch function, all of them stated above: Maximum RPM is a parameter rather than the record's maximum; the answer is the spectrum of the frames completed so far, so it is a running estimate rather than one answer at the end; and the frames MATLAB builds out of a record's last ten samples – where its own resampling pads with zeros – are frames a stream never reaches. Measured on a 600-sample run-up (1200 to 3000 rpm, f_s = 1000): every frame's order map agrees with rpmordermap to 1.0×10⁻¹⁴, the running spectrum over the same frames to 1.1×10⁻¹⁵, and the whole-record answers differ by 0.38–0.60 % where the stream is 1 to 5 frames short of MATLAB's count – and by 1.1×10⁻¹⁵ where it is not.
• ⚠ "Maximum RPM" is not a formality. On that same run – whose speed peaks at 2942.6 rpm – setting the parameter to a round 3000 instead moved the answer by 3.5 % to 24 %, because it is what sets the angle grid and the order axis. Give it the run's maximum, not a nameplate figure.
• An order spectrum needs a run-up: at a constant speed it is a frequency spectrum with a relabelled axis. It is the speed CHANGING that separates a shaft-order component from a fixed-frequency one.
• No state space. An angular resampler followed by a magnitude spectrum is not linear.
• Related blocks. Vibration / Time-Synchronous Average sharpens what is synchronous with the shaft in the time domain and makes the same substitution for the period; Spectral Measurements / Welch PSD is the fixed-frequency counterpart of this measurement.