So far, we've discussed "tracking analysis," a commonly used method for measuring the sound and vibration of rotating bodies, and "balancing measurement," which is crucial for vibration countermeasures in rotating bodies. This time, our theme is "rotational fluctuation measurement." Rotational fluctuations in rotating bodies are caused by various factors, such as load fluctuations and torque fluctuations, but here we will use tracking analysis technology to measure rotational fluctuations due to torsional vibration phenomena. We will then introduce an example of calculating the angular displacement fluctuation component from these fluctuations.
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Figure 1 Various types of rotation detectors manufactured by Ono Ono Sokki
A rotation sensor is necessary to obtain rotational speed information, and there are various types of rotation detectors, such as those shown in Figure 1. Here, we will use the MP-900/9000 series Electromagnetic Detector.
The detection principle, as shown in Figure 2, involves bringing Electromagnetic Detector close to a gear (magnetic material) attached to a rotating shaft (approximately 0.5 to 1 mm away). As the tip of the detector approaches or moves away from the gear teeth, a change in magnetic resistance occurs, inducing an electromotive force in the internal coil, which is then output. In essence, an AC voltage is generated using the principle of a generator, resulting in a sinusoidal time waveform. As the rotation speed increases, the frequency and amplitude of this waveform increase. In this example (Figure 2), the output of the detector is input to a TM-3100 series digital tachometer to display the rotation speed (unit: r/min).
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Figure 2 shows the MP-9000 Electromagnetic Detector and the TM-3100 digital tachometer.
A digital tachometer, as shown in Figure 2, can measure the average rotational speed. However, to determine the rotational fluctuation component (the amount of change in rotational speed), a common method is to convert the rotational speed information into a voltage-time signal using a high-speed FV converter and then analyze it with a frequency analyzer (FFT analyzer). In the example in Figure 3, a rotary encoder is used as the rotation sensor, and the rotational fluctuation component is measured using this method.
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Figure 3: Example of rotational fluctuation measurement using a high-speed FV converter and FFT analyzer.
The signal input to the high-speed FV converter (hereinafter referred to as FV-1500) can be either a sinusoidal AC signal or a pulse signal (the output of the rotary encoder in Figure 3). Since the sinusoidal signal is converted to a pulse signal at the input stage for processing, the following explanation will assume a pulse signal input.
Since the rotational speed is proportional to the frequency of the pulse signal (the signal from the rotation detector), the FV-1500 outputs a voltage signal corresponding to the reciprocal of one period of the pulse signal (i.e., the instantaneous frequency) each time a pulse signal is received. As shown in the example in Figure 4, it can be seen that a voltage signal proportional to the frequency of the input signal is output.
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Figure 4 Output result of the swept sine wave
Above: Output voltage waveform of FV-1500
Below diagram: Input signal (swept from 10 Hz to 40 Hz in 10 seconds)
As another example, the image below (Figure 5) shows the result of converting a signal obtained by frequency modulating (FM) a sine wave (carrier wave) of a constant frequency (5 kHz) with a 50 Hz signal wave into a voltage signal using the FV-1500.
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Figure 5: Example of high-speed FV conversion of a frequency-modulated signal.
(Original FM wave: 5 kHz, signal wave: 50 Hz)
If a rotating body that is rotating at a constant speed experiences rotational fluctuations, this is physically the same phenomenon as frequency modulation. Therefore, by performing a high-speed FV conversion on the rotational speed information, the rotational fluctuation component can be extracted.
This section outlines important points to consider when using the FV-1500 high-speed FV converter to measure rotational fluctuations. The device's calculation method outputs an analog signal for each pulse, meaning it essentially samples (quantifies) the modulated signal (rotational fluctuation component) with the input pulse and outputs it as an analog signal. Therefore, the input pulse frequency must be greater than the rotational fluctuation frequency per revolution; in other words, a rotation detector that outputs N samples per revolution is required. A 1 P/R rotation detector cannot detect the rotational fluctuation component.
For example, using a 60 P/R rotation detector, it is theoretically possible to determine rotational variation components up to the 30th order (1/2) using the sampling theorem. However, due to waveform distortion, aliasing errors, and the 0th order hold error of the D/A converter, in reality, only about the 6th order component (1/10) is practical.
Here, we consider the relationship between the rotational speed and angular velocity of a rotating body. Normally, rotational speed refers to the number of rotations per minute, with units of m⁻¹ or r/min. The number of rotations per second is the rotational frequency, with units of s⁻¹ or Hz. In contrast, angular velocity (angular frequency) represents the angle of rotation per second (how many radians), with units of rad/s. Now, if we let the rotational speed be n, the frequency be f, and the angular velocity be ω, then the angle of one rotation is 2π (360 degrees).
This is the relationship. For example, if the rotational speed is 1000 (r/min), the angular velocity is approximately 105 (rad/s).
Based on this background knowledge, we will use the rotational demonstration device shown in Figure 6 to sweep the rotational speed and measure the rotational fluctuation component caused by torsional vibration. The rotating shaft of this device is fitted with a 60 P/R gear, and the rotational speed information is detected by Electromagnetic Detector, which is then quickly converted into an analog voltage signal by the FV-1500 and input to an FFT analyzer to perform tracking analysis. The external sample pulse signal (1 P/R) required for rotational tracking analysis is detected by another sensor.
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Figure 6: Rotational fluctuation measurement using Electromagnetic Detector and a high-speed FV converter.
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Figure 7: Results of stoichiometric tracking analysis of frequency fluctuation components (4th order)
Sweep range: 1000 to 3500 r/min
Figure 7 shows the measurement results. The tracking diagram of the fourth-order component (horizontal axis: rotational speed, vertical axis: frequency) shows that there is a large rotational fluctuation due to torsional vibration resonance around 1643 r/min.
Frequency fluctuation amplitude: 273.4 (mHz)
Angular velocity fluctuation amplitude: 1.718 (rad/s)
Rotational speed fluctuation amplitude: 16.40 (r/min)
Based on these results, we will now determine the amplitude of the angular displacement fluctuation.
Here, I will explain the concept of frequency modulation.
In the following, x(t) is a frequency-modulated time signal.
ω(t) is its angular velocity, ω(s) is the angular velocity of the fluctuating component, a m This is the angular velocity variation of the fluctuating component.
ω c:center angular velocity
ω s: Angular velocity of the fluctuating component
am:角速度変動量
From these relationships, the angular displacement can be calculated by dividing the angular velocity fluctuation by the angular velocity of the fluctuation component. The specific calculation method uses the frequency calculus function (jω operation), which is a feature of the FFT analyzer.
Figure 8 shows the calculation results.
The vertical axis represents angles, but it's based on radians (rad). To convert it to degrees (degrees), you need to multiply by 360. Specifically,
The amplitude of the angular displacement variation component is 397.3 μm x 360 = 0.14 deg (at 1643 r/min).
In this measurement example, we used Electromagnetic Detector and a high-speed FV converter. However, if you measure the tangential velocity v of the disk attached to the rotating demonstration device with a laser in-plane velocity meter, you can also directly detect the angular velocity ω from the relationship ωrv = (where r is the radius of the disk).
Next time, I will explain how torsional vibration measurement is performed using 2 channels.
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Figure 8. Results of stoichiometric tracking analysis of angular displacement variation component (4th order).
Figure 7 shows the results obtained by single integration along the frequency axis.
In conclusion, here's a summary.
- There are various types of rotation sensors that detect the rotation information of a rotating body, but Electromagnetic Detector are easy to use, relatively inexpensive, and widely used.
- A common technique involves converting rotational pulses from Electromagnetic Detector into an analog signal of rotational speed (or angular velocity) fluctuations using a high-speed FV converter, and then analyzing it with an FFT analyzer.
- The rotational fluctuations caused by torsional resonance in a rotating body can be determined by tracking the output of a high-speed FV converter.
- The rotational speed (or angular velocity) fluctuation component can be detected not only by the rotation detector + high-speed FV converter method, but also by a laser in-plane velocometer.
【keyword】
Tracking analysis, balancing measurement, rotational fluctuation measurement, torsional vibration, Electromagnetic Detector, magnetoresistance, induced electromotive force, digital tachometer, rotational speed, high-speed FV converter, rotary encoder, carrier wave, frequency modulation, sampling theorem, aliasing, zero-order hold, angular velocity, angular frequency, tracking diagram, frequency calculus, laser in-plane velocometer
(Excerpt from the email newsletter issued on May 22, 2018)
