Technical Report: Degree Ratio Analysis and Tracking Analysis 1

For rotating machinery such as engines and compressors, which must cover a wide range of rotational speeds from low to high RPMs, the resonance between the natural frequencies of each component (rotating shafts, gears, brackets, etc.) and the rotational speed is the most important issue. In the case of torsional vibrations in large generators, resonance can generate large excitation energy that exceeds the allowable stress, potentially leading to major accidents and destruction. Furthermore, for vibration and noise countermeasures, it is desirable to know at what rotational speed the vibration and noise of the rotating machinery become significant, and which component of the rotating machinery is causing the vibration and noise. Common methods for investigating such phenomena include "rotational order ratio analysis," which defines a phenomenon occurring once per rotation as one period as the first rotational component, and n times that as the nth rotational component, with the X-axis representing the order and the Y-axis representing the magnitude of the vibration noise of the order component, and "rotational-tracking analysis," which analyzes how the magnitude of the vibration noise of the order component of interest changes with increasing or decreasing rotational speed. Note that the term "rotational order ratio" is sometimes used with the addition of "ratio" due to the comparison with rotation, but in this case, order and order ratio have the same meaning.

Figure 1 below is a conceptual diagram showing the relationship between "frequency analysis," "rotational order ratio analysis" (hereinafter referred to as order ratio analysis), and "rotational-tracking analysis" (hereinafter referred to as tracking analysis), measured and displayed in three dimensions at every 50 r/min increase in rotational speed. In the three-dimensional display data of frequency analysis, the X axis represents frequency (Hz), and components of the same order appear on a line that rises diagonally to the right as the rotational speed increases. However, in order ratio analysis, the X axis represents the order, so components of the same order appear vertically. This is the main difference between frequency analysis and order ratio analysis when the rotational speed increases. Furthermore, in tracking analysis, components of the same order are extracted from each data point, and the rotational speed is plotted on the X axis and that order component on the Y axis. This allows us to see how the order component of interest changes with increasing rotational speed. In other words, as the rotational speed approaches the resonant frequency of the component, the vibration of the component of interest gradually increases, reaching a maximum when it coincides with the resonance point, and then gradually decreasing after the rotational speed passes the resonance point. From this data, the resonant state of the component can be easily confirmed. There are mainly two types of tracking analysis: "constant ratio tracking" and "constant width tracking".

Here, we will explain this difference in relation to order ratio analysis and frequency analysis.

This shows a three-dimensional display of frequency analysis and rotational order ratio analysis measured at every 50 r/min increase in rotational speed, starting from 850 r/min, along with the rotational tracking analysis of the first-order component from that data.

The data that forms the basis of constant-ratio degree tracking is degree ratio analysis.

In frequency analysis, the input signal is sampled using a sampling clock with a frequency 2.56 times the frequency range, obtained from a crystal oscillator inside the FFT analyzer. When analyzing vibrations and noise of a rotating body with varying rotational speed using this sampling method, the sampling clock frequency remains constant, so if the time it takes to complete one rotation changes (i.e., the rotational speed changes), the number of samples per rotation changes (see Figure 2 below).

In contrast, if sampling is performed using a sampling clock synchronized with the rotational speed, for example, a signal with 64 pulses per revolution, the number of signal samples per revolution does not change even if the rotational speed changes (see Figure 3 below).

When vibration and noise signals sampled with a clock synchronized to the rotational speed are subjected to an FFT, the unit of the X axis becomes the order (order) rather than frequency (Hz). The data displayed as the power spectrum of the order component is called rotational order ratio analysis data.

The primary rotational component is the component that has one period for each rotation of the rotation axis defined as the reference, while the secondary rotational component similarly has two periods for each rotation. Converting the order to frequency, for example, the primary rotational component of a rotating body rotating at 600 r/min can be calculated as 10 Hz from the following formula. Similarly, at 900 r/min, it is 15 Hz.

As described above, the first-order frequency changes with rotational speed, but when considering it in terms of order units, the order notation "the component that appears as one period per revolution is called the first order" means that it is a unit that is independent of rotational speed. This is the key point of order ratio analysis.

The following two figures show a color three-dimensional display with increasing rotational speed. The first figure, Figure 4, shows an order ratio analysis with the order on the X-axis, while the next figure, Figure 5, shows a normal frequency analysis (the magnitude of the spectrum is represented by color, with blue → yellow → red representing larger values).

In order ratio analysis, as shown in Figure 4, the X-axis represents the order, so components of the same order are displayed on a vertical line. However, in frequency analysis (Figure 5), the frequency of components of the same order increases in proportion to the increase in rotational speed, so they are displayed on a diagonally upward sloping line. You can see this by looking at the yellow and red in the figure.

In typical frequency analysis, the sampling clock frequency is 2.56 times the analysis frequency range (the maximum frequency to be analyzed). Similarly, in order ratio analysis, the sampling clock requires 2.56 times the maximum order to be analyzed per revolution as pulses.

Our equipment has a maximum analysis order of 6.25/12.5/25/50/100/200/400/800, so the sampling number 2.56 times this is as shown in the table below. Our tracking analysis function automatically divides and multiplies the required sampling pulses according to the set maximum analysis order simply by setting the number of pulses output per revolution of the rotation detector (e.g., 1 P/R). These sampling pulses are synchronized with the rotation as explained earlier.

The frequency resolution of frequency analysis using an internal sampling clock is 1/400 of the set frequency range when the analysis data length is 1024 points, and 1/800 when it is 2048 points. For example, if the frequency range is set to 1 kHz, it becomes 1000/400 = 2.5 Hz (analysis data length of 1024 points), and the spectrum can be read every 2.5 Hz. In contrast, in the case of order ratio analysis using an external sampling clock synchronized with rotation, the relationship between the maximum analysis order and its order resolution is given by the following equation.

As explained above, the order resolution is independent of the rotational speed and can be calculated using equation (1). Next, let's consider converting the order resolution to frequency.

For example, if the analysis data length is 1024 points and the maximum analysis order is 100th order, the order resolution can be converted to frequency as shown in the following equation: at a rotation speed of 600 r/min, it is 2.5 Hz, and at 6000 r/min, it is 25 Hz, meaning the resolution changes proportionally to the rotation speed. When considering order resolution in order units, it is constant regardless of rotation speed as shown in equation (1) above, but when converted to frequency units, it changes proportionally to the rotation speed as shown in equations (2) and (3) below, which is why order ratio analysis is called "constant ratio". On the other hand, the frequency resolution in the case of frequency analysis is constant regardless of rotation speed, so it is called "constant bandwidth".

Conversion formula from order resolution to frequency

Aliasing can occur in order ratio analysis, just as it does in frequency analysis. Please refer to the notes section below for more information on aliasing.

Let's consider this aliasing phenomenon. If we let M be the highest order we want to analyze and N be the rotation speed, the frequency _fx of the highest order can be calculated using equation (4). Also, in external sample mode, the sampling frequency _fs is automatically set to 2.56 times the frequency _fx of the highest order. At that time, the anti-aliasing low-pass filter (digital filter) is also applied in conjunction with the sampling frequency, so aliasing does not usually occur.

Now let's consider the case where the anti-aliasing low-pass filter is fixed and not linked to the sampling frequency. Let's assume the cutoff frequency of the low-pass filter is 1000 Hz. For example, if the maximum order we want to analyze is 25th order and the rotation speed of the rotating body is 2400 r/min, then the frequency fx of the 25th order can be found to be 1000 Hz from the following equation, and similarly, f _s /2 can be found to be 1280 Hz. In this case, components above 1000 Hz are cut off by the low-pass filter, and aliasing does not occur.

Next, assuming that the rotation speed decreases to 1000 r/min while keeping the cutoff frequency of the low-pass filter fixed at 1000 Hz, the 25th-order frequency _fx can be calculated as 416.7 Hz from the following formula, and similarly, _fs /2 is 533.4 Hz. In this case, the components between 533 Hz and 1000 Hz (low-pass filter) are the ones that will occur as aliasing (folding back). Therefore, if signal components such as vibration noise are present in this frequency band, it can be seen that aliasing will occur if the cutoff frequency of the low-pass filter is kept set to 1000 Hz.

Therefore, when performing analysis while changing the rotation speed in the tracking analysis function, a tracking low-pass filter is equipped, in which the filter's cutoff frequency changes according to the rotation speed.

According to the sampling theorem, it is necessary to sample a signal at a sampling frequency at least twice the highest frequency component f _m, and the frequency that is half the sampling frequency is called the Nyquist frequency. If the original time signal contains a frequency band f _m above the Nyquist frequency, then its frequency spectrum will includeThe components appear at frequency positions where they are folded around _fs. This phenomenon is called aliasing, and to avoid it, an anti-aliasing low-pass filter is provided to cut out signals above 1/2 _fs.