Technical Report: Frequency Ratio Analysis and Tracking Analysis 1

For rotating machines such as engines and compressors, which must cover a wide range of rotational speeds from low to high, the resonance between the natural frequencies of each component (such as the rotating shaft, gears, brackets, etc.) that make up the rotating machine and the rotational speed becomes the most important issue. In the case of torsional vibrations in large generators, resonance can generate significant excitation energy that exceeds the allowable stress, potentially leading to catastrophic failures. Additionally, in vibration noise countermeasures, it is important to know at which rotational speed the vibration or noise increases, and which components of the rotating machine are causing the vibration or noise. Methods to investigate these issues include "defining the phenomenon that occurs once per revolution as the first-order rotational component, and n times that as the n-th order rotational component, representing the X-axis as the order and the Y-axis as the magnitude of the vibration noise of the order component in 'rotational order ratio analysis'" and "analyzing how the magnitude of the vibration noise of the order component of interest changes with the increase or decrease in rotational speed in 'rotational-tracking analysis'". It should be noted that the term 'ratio' is sometimes added to the rotational order ratio in comparison to rotation, but in this case, the order and the order ratio have the same meaning.

Figure 1 shows a conceptual diagram illustrating the relationship between the "frequency analysis" and "rotational order ratio analysis" (hereinafter referred to as order ratio analysis), as well as the "rotational-tracking analysis" (hereinafter referred to as tracking analysis), measured every 50 r/min increase in rotational speed and displayed in three dimensions. In the three-dimensional display data of frequency analysis, the X-axis represents frequency (Hz), and the components of the same order appear on a diagonal line rising to the right as the rotation increases, while in order ratio analysis, the X-axis represents the order, so the components of the same order appear vertically. This is the main difference between frequency analysis and order ratio analysis during the increase in rotational speed. In tracking analysis, the components of the same order are extracted from each data set, with the X-axis representing the rotational speed and the Y-axis representing the order component. This allows us to see how the order component of interest changes with the increase in rotational speed. In other words, as the rotational speed approaches the resonance frequency of the components, the vibration of the order component of interest gradually increases, reaching a maximum when it coincides with the resonance point, and gradually decreases as the rotational speed surpasses the resonance point. Such data can easily confirm the resonance state of the components. Tracking analysis mainly includes "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 representation of frequency analysis and rotational ratio analysis measured at an increase in rotational speed from 850 r/min, with measurements taken every 50 r/min increase, and how the first component was analyzed through rotational tracking.

The data that forms the basis for fixed ratio tracking is ratio analysis.

In frequency analysis, input signals are sampled at a sampling clock frequency that is 2.56 times the frequency range obtained from the crystal oscillator inside the FFT analyzer. When analyzing the vibrations or noise of a rotating body with varying rotational speeds using this sampling method, the frequency of the sampling clock remains constant, so when the time for one rotation changes (due to different rotational speeds), the number of samples per rotation will vary (see Figure 2).

In contrast, when sampling is performed using a sampling clock synchronized with the rotation speed, such as a signal with 64 pulses per rotation, the number of samples per rotation does not change even if the rotation speed varies (see Figure 3 below).

When vibration and noise signals sampled with a clock synchronized to the rotational speed are subjected to FFT, the unit of the X-axis changes from frequency (Hz) to order. The data displayed as the power spectrum of the order components is referred to as rotational order ratio analysis data.

The first-order component of rotation refers to the component that has one cycle per rotation about a defined axis of rotation, while the second-order component similarly refers to the component that has two cycles per rotation. When converting the order to frequency, for example, the first-order component of a rotating body at 600 r/min can be calculated to be 10 Hz using the following formula. Similarly, at 900 r/min, it becomes 15 Hz.

In this way, the first-order frequency changes with variations in rotational speed; however, when considering it in terms of order units, the expression "the component that appears as one cycle per rotation is referred to as first order" indicates 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 changes in rotational speed represented in a color 3D display. The first figure (Figure 4) presents a degree ratio analysis displayed along the X-axis, while the next figure (Figure 5) illustrates a standard frequency analysis (with the magnitude of the spectrum represented in color, where blue indicates smaller values and transitions to yellow and then red for larger values).

In the degree frequency analysis, as shown in Figure 4, the components of the same degree are displayed vertically along the Y-axis. However, in the frequency analysis shown in Figure 5, the same degree components increase in frequency in proportion to the rise in rotational speed, resulting in a diagonal line that slopes upwards to the right. The yellow-red color in the figures highlights this phenomenon.

In the case of normal frequency analysis, the frequency of the sampling clock should be 2.56 times the analysis frequency range (the maximum frequency to be analyzed). Similarly, for order ratio analysis, a pulse that is 2.56 times the maximum order to be analyzed per rotation is required as the sampling clock.

The maximum analysis frequency of our equipment is 6.25/12.5/25/50/100/200/400/800 times, so the sampling number of 2.56 times will be as shown in the table below. Our tracking analysis function has the capability to automatically divide and multiply the required sampling pulses according to the set maximum analysis frequency by simply setting the number of pulses output per rotation of the rotation detector (e.g., 1 P/R). These sampling pulses are synchronized with the rotation pulses described 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 results in 1000/400 = 2.5 Hz (for an analysis data length of 1024 points), allowing the spectrum to be read every 2.5 Hz. In contrast, for order ratio analysis using an external sampling clock synchronized with rotation, the relationship between the maximum analysis order and its resolution is given by the following formula.

Regarding the order resolution, as explained so far, it is independent of the rotational speed and can be determined by the above equation (1). Next, let's consider converting the order resolution into frequency.

For example, with an analysis data length of 1024 points and a maximum analysis order of 100, when converted to frequency, at a rotational speed of 600 r/min, the order resolution becomes 2.5 Hz, and at 6000 r/min, it becomes 25 Hz, showing that the resolution changes proportionally with the rotational speed. When considering the order resolution in terms of order units, it remains constant regardless of the rotational speed as per the above equation (1), but when converted to frequency units, it changes proportionally with the rotational speed as shown in equations (2) and (3). This is why order ratio analysis is referred to as "constant ratio." On the other hand, in the case of frequency analysis, the frequency resolution remains constant regardless of the rotational speed, hence it is referred to as "constant width."

Conversion formula from order resolution to frequency

In the case of order ratio analysis, aliasing phenomena may occur just like in frequency analysis. Please refer to the notes below for more information on aliasing.

Let's consider this aliasing phenomenon. Now, let the maximum order we want to analyze be M, and the rotational speed be N r/min. The frequency f _x of the maximum order can be obtained using the following equation (4). Additionally, in external sample mode, the sampling frequency f _s is automatically set to 2.56 times the frequency f _x of the maximum order. At that time, the anti-aliasing low-pass filter (digital filter) is also applied in conjunction with the sampling frequency, so aliasing usually does not occur.

Here, we consider the case where the anti-aliasing low-pass filter is fixed and does not vary with the sampling frequency. Let's set the cutoff frequency of the low-pass filter to 1000 Hz. For example, if we want to analyze a maximum order of 25 and the rotational speed of the rotating body is 2400 r/min, the 25th frequency f _x can be derived from the following equation as 1000 Hz, and similarly, f _s /2 is found to be 1280 Hz. At this time, it can be seen that 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 frequency f _x can be calculated from the following formula as 416.7 Hz, and similarly, f _s /2 is found to be 533.4 Hz. In this case, components between 533 Hz and 1000 Hz (low-pass filter) will be subject to aliasing phenomena (folding). Therefore, if signal components such as vibration noise exist in this frequency range, it can be seen that aliasing phenomena will occur while keeping the cutoff frequency of the low-pass filter set at 1000 Hz.

In the tracking analysis function, when performing analysis while varying the rotation speed, a tracking low-pass filter is equipped that changes its cutoff frequency according to the rotation speed.

According to the sampling theorem, it is necessary to sample at a frequency that is at least twice the highest frequency component of the signal f _m, and the frequency that is half of the sampling frequency is called the Nyquist frequency. If the original time signal contains frequency bands f _m that are above the Nyquist frequency, then its frequency spectrum willThe components appear at the frequency positions folded around f _s. This phenomenon is called aliasing (folding), and to avoid it, we have an anti-aliasing low-pass filter that cuts signals above 1/2 f _s.