Theory of Order Analysis

The order analysis computed in Ansys Sound: Analysis and Specification is based on the Synchronous Order Analysis method.

Starting from the temporal waveform (the original signal as samples vs. time) and its associated synchronous RPM Profile, the signal is transformed to obtain a new version of its evolution as a function of an equally spaced vector of number of cycles. In other words, the signal is resampled at a tachometric constant step, meaning a constantly sampled number of revolutions vector.

In what follows, "one revolution" (as used in the term "revolutions per minute", or "rpm") is referred to as "a cycle".

The signal transformation is made as follows:
  1. Consider a signal (named "S") as a function of time, that has a constant sampling frequency. This signal may be an acoustic sound, a vibration, etcetera. Also consider the signal's associated synchronous RPM profile (named "RPM").
  2. From the RPM, the starting times of each cycle are known. Build a vector (named "C") of the number of cycles, including regular subdivisions of the cycle. If "N" is the number of regular subdivisions of the cycle (the number of points per cycle), then the vector C will have N times per cycle. This means that every N points, a new cycle is starting. Thus, from vector C and the RPM, the times associated to C values are known.
  3. Knowing vector C and signal S, a synchronous resampling is performed. The signal S is resampled toward this vector of N number of points per cycle. An anti-aliasing filter is used during the process if needed. This new signal is named "RS", and it contains the amplitudes (from signal S) corresponding to the times associated to the cycle numbers (the vector C).
Note: If you calculate the spectrum (the FFT) of the signal RS, it gives the "order spectrum", which is the representation of the level as a function of the order number.
The signal RS is the base signal used to display and calculate several elements in Ansys Sound: Analysis and Specification:
Note: The settings of the synchronous sampling can be set in the General Settings (ansys.com), and can also be changed from the calculation menu, when the display is already one of these two representations.

The "order vs. time" and "order vs. rpm" displays are colormap displays that are directly issued from the spectrogram of the synchronous sampling signal RS. The spectrogram of RS is calculated using a short-time Fourier transform (the same as the other spectrograms used in Ansys Sound: Analysis and Specification).

The two parameters that control this spectrogram calculation are:

  • Highest order for order view: This parameter (named "Omax") is the maximum order rank that you intend to be able to analyze. It has an influence on the number of cycles N that will be used for the synchronous sampling. The software uses the value N = 2 x Omax + 1, so that the Nyquist theorem is fulfilled for analyzing order Omax.

    Hence, if you are interested in analyzing the orders up to the 20th one, then 41 points per cycle will be used for the sampling.

  • Resolution for order display: This parameter is the resolution of the analysis on the orders axis, in percent of the order.

    Due to the Heisenberg uncertainty principle, the finer the Order analysis resolution is, the coarser the resolution of the axis of cycles number is. Inversely, the coarser the resolution is on the order axis, the finer the resolution is on the cycles axis.

    For example, the image on the left shows the "order vs. rpm" display with a resolution of 4% of order. The image on the right shows the "order vs. rpm" display with a resolution of 0.5% of order.

The Individual order level is calculated from this STFT. The above-mentioned parameters have an influence on the result of this calculation.

The Order analysis width also has influence. The order analysis width is a parameter used for the estimation of the "level vs. cycle" number of a specific order. For example, if you want to estimate the level of the order #2, using an order analysis width of 20% of order, the calculation of the level will be done from the STFT of RS, by integrating (that is, summing) all the level points contained in the order band that goes from 2-0.1 to 2+0.1 (that is, the band of orders around order 2: [1.9, 2.1]).

Obviously, the shape of the resulting curve depends on the parameter Order analysis resolution in percent of order (as stated before for the colormap). The finer this resolution is, the smoother the curve is (but the level is estimated using more cycles), and inversely the coarser the resolution is, the more erratic the shape of the curve is (but the level is estimated using fewer cycles). For more information, refer to Displaying Order Levels in a Graph (ansys.com).

For example, in the top curve, the level of order #2 has a resolution of 2% of order, and in the bottom curve the level of the same order #2 has a resolution of 0.5% of order.

Note: The RPM Profile you use also has an effect on the resolution of the colormap you can produce, and therefore on the order analysis. The figure below shows that the resolution in RPM is better on the colormap when the RPM is evolving slowly, compared to the parts where the RPM is evolving quickly.
This is simply because the more gradual the change in the RPM, the more information is available to estimate an accurate level for each RPM point. The same is valid for time points, as all is calculated when the signal is resampled to create equally spaced cycles.