This measurement column addresses frequently asked questions received by our customer support center and provides answers to those questions.
When analyzing the time waveform of sound or vibration using FFT and observing the power spectrum, it can be confusing to find values that are significantly smaller than expected, or to see changes in the obtained values when the number of samples or frequency range is altered, even if the operating conditions of the rotating machinery remain unchanged. Furthermore, even with steady-state vibrations and sounds, the obtained values can change when the frequency range or analysis conditions are altered.
This section will discuss the causes related to the time waveform of a signal and the measurement/analysis conditions (characteristics of FFT analysis).
Please also refer to this.
Fundamentals of Digital Measurement - Part 8: "Time Window Length and Spectral Resolution"
Fundamentals of Digital Measurement - Part 9: "Various Time Waveforms and Spectra"
Waveforms with changing frequency (e.g., when the rotational speed of a rotating machine is increasing)
Since the rotational speed is changing, the frequency synchronized with the rotation is also changing.
For a sine wave with a constant frequency of 100.0 Hz, as shown in Figure 1, the power spectrum will only have a value at that frequency.
The time waveform in Figure 1 has an amplitude of 14.14 m/s², so the effective value is 10.0 m/s².
The magnitude of both the 100.0 Hz signal and the overall magnitude is 10.0 m/s².
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Figure 1
What happens to the time waveform in Figure 2 below?
The amplitude remains constant, but the frequency changes from low to high.
The amplitude is 14.14 m/s², so the effective value of this waveform is 10.0 m/s².
The frequency changes from approximately 130 Hz to 480 Hz in 0.8 seconds.
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Figure 2
Since the amplitude of the single-wave amplitude is 14.14 m/s², we would ideally want the power spectrum of each frequency component to be 10.0 m/s², but this is not possible with FFT analysis. The power spectrum is a time wave.
Each of the frequency components contained in the shape Mean squared Because it is equivalent to that.
Because the frequency is changing, the relative proportions of each frequency component will be lower during the 0.8 s period.
During the time when it is not present, the magnitude of that frequency component is 0. The average result over 0.8 s is the presence distribution.
It will become smaller depending on the size.
Here are the results. (FFT at 1 kHz, 2048 points) The overall value is 10.0 m/s².
In Figure 3, the second screen from the top shows the FFT with the time window being a Hanning window, and the third screen shows the FFT with the rectangular window being used.
The distribution of the power spectrum is clearly influenced by time variations and the shape of the time window.
This is what it appears to be.
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Figure 3
Hanning window weighted waveform
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Figure 4
In the case of the Hanning window, the weighting is maximized at the central time (0.4 s). (Figure 4)
The magnitude of the frequency components present in that region is larger. In a rectangular window, the weight is always 1, so they are distributed with the same magnitude. (This is due to the effect of waveform cropping.)
The magnitude at 300 Hz is 1.17 m/s² with a Hanning window and 0.598 m/s² with a rectangular window. The Hanning window corrects for the reduction due to the window function, so it yields a higher value than the rectangular window, but it is still far from 10.0 m/s².
The amplitude of the time waveform is 14.14 m/s², but the resulting power spectrum value is small.
This means that each frequency exists for only a short time during 0.8 seconds.
The magnitude obtained from the power spectrum is equivalent to the mean square of the time waveform for each frequency component. If the frequency remains the same for 0.8 seconds, the average will not decrease. However, if the frequency changes and the time spent at each frequency during 0.8 seconds is short, the average will decrease by the proportion of time spent at each frequency. If the frequency change is larger, the magnitude of each frequency component will decrease even further.
Measurement conditions that change the ratio of the FFT time length to the existence time of frequency components include the setting of the number of sample points and the setting of the frequency range.
What happens when we increase the sample size from 2048 points to 4096 points?
The amplitude of the waveform is constant at 14.14 m/s², so the effective value is 10.0 m/s². The change in frequency increases as the time length increases from 0.8 s to 1.6 s. The proportion of time occupied by a particular frequency component decreases to half of the original time length. Since the value of a frequency component is the mean square of the value over the time length, it decreases proportionally to the reduction in proportion. The overall value remains unchanged at 10.0 m/s².
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Figure 5
The magnitude of the effective value was halved compared to 2048 points.
The same applies when the frequency range is lowered. Lowering the frequency range increases the duration accordingly.
A similar phenomenon, where numerical values change depending on measurement and analysis conditions, also occurs with random waveforms.
The difference in values due to time duration also appears in the case of random signals, where the power spectrum is continuously distributed. In the case of random waveforms, this is due to differences in frequency resolution (Δf).
Even with the same signal, the value changes depending on whether the number of samples is different (even if the frequency range is the same), or whether the frequency range is different (even if the number of samples is the same).
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Figure 6
The duration of the measurement will vary depending on the number of samples and the frequency range, which will affect the resulting numerical values. During measurement, it is necessary to record the operating conditions of the device, as well as the measurement and analysis conditions.
(Excerpt from the email newsletter issued on January 23, 2019)
