Frequently Asked Questions about Measurement - Part 15: "Frequency Axis Calculus and Low-Frequency Noise"

In this measurement column, we will address frequently asked questions received by our customer support center.
We are presenting the answers.

There are various types of sensors for measuring vibration, and the displacement depends on the physical quantity being measured.
They can be divided into sensors, speed sensors, and acceleration sensors. Piezoelectric acceleration sensors are contact type.
While relatively easy and inexpensive to measure, the physical quantity that can be measured is acceleration.
In this case, the signal obtained from the acceleration sensor can be subjected to differential and integral calculus of the time-domain waveform or spectral analysis.
This is converted into velocity and displacement using frequency axis differential and integral calculus.

Acceleration signals containing DC (direct current) offset and low-frequency noise can be analyzed using frequency axis differential and integral calculus.
When converted to velocity or displacement, the first line of the spectrum and the low-frequency components may actually be present.
Sometimes, an extremely large value may be displayed.

What is frequency axis differential and integral calculus?

The time waveform a(t) of an acceleration signal with frequency f and amplitude A is given by Equation 1, where t is time.
is.

a(t)=A×cos( 2πft ) Equation 1

The velocity waveform v(t) and displacement waveform x(t) are obtained by single and double integration of the acceleration waveform as shown in equations 2 and 3.
You can request it.

formula 2

formula 3

If we let V be the amplitude of the velocity and X be the amplitude of the displacement, then we can obtain equations 4 and 5 from equations 2 and 3.

formula 4

formula 5

When determining the power spectrum of velocity or displacement, the frequency of each component is
Since the numbers are known, we can use equations 4 and 5 to convert the acceleration values into velocity and displacement values.
This is possible. The frequency axis differential and integral function of our analysis device converts to velocity and displacement using this method.
It is.

About DC (Direct Current) Offset

Analysis devices such as FFT analyzers convert voltage signals into digital data using A/D converters.
Yes, if the input to the analysis device is 0V, the converted digital data should be 0.
However, due to the characteristics of the analog circuit and A/D converter, the value will be slightly off. This amount of deviation
This is called DC (direct current) offset, and is used in our CF-9000 series FFT analyzers and DS-3000.
In series data stations, this is less than 1/1000 of the voltage range.

If there is a DC offset of 1 mV, does this mean that even if the input is 0, a constant signal of 1 mV is being applied?
It will be displayed as follows: Sensitivity is 1 mV/(m/s ² When using an accelerometer, the time-domain waveform
Looking at this, 1 m/s ² It will appear as if a constant acceleration is occurring.

1 m/s ² Figure 1 shows the results of generating a constant acceleration signal and determining its power spectrum.
The frequency range was 50 Hz, the number of sample points was 2048, and the Hanning window function was used for the analysis. FFT
The frame duration is set to 16 seconds, and the frequency resolution to 0.0625 Hz. Power spectrum
This displays a magnified view of the 0 Hz to 2 Hz range. The magnitude of the 0 Hz component is 1 m/s². ² However, due to the effect of the window function (Hanning), the magnitude of the 0.0625 Hz component next to the 0 Hz line is 0.707 m/s². ² This is the result. This is not the actual acceleration that is occurring, but a component that appears due to the DC offset (error).

Figure 2 shows the acceleration power spectrum from the lower part of Figure 1 converted to velocity and displacement power spectra using frequency axis differential integration. The 0 Hz component is zero because it cannot be converted to velocity and displacement. The magnitude of the velocity for the 0.0625 Hz component is 1.801 m/s and 4.585 m. These are not the actual velocity and displacement that occur.

If the analysis device has a slight DC offset (error), it may detect acceleration that is not actually occurring.
The degree value appears. The conversion to velocity and displacement using the frequency axis differential and integral function (Equations 4 and 5) involves 2πf
Because division is involved, the velocity and displacement values for low-frequency components are actually impossible.
It will be a very large value.

One line of the power spectrum of velocity and displacement obtained by integration using frequency axis differential and integral functions.
If very large velocity and displacement values are displayed, it indicates that vibration is actually occurring.
Rather, it is possible that the error in the analysis device was amplified by the frequency axis differential and integral calculations.
It is highly likely.

Our analysis equipment is equipped with a DC cancellation function as one of the time-axis preprocessing functions, and this function
Enabling this feature can reduce the effects of DC offset. However, it can completely eliminate the effects.
It's not possible to make it all zero.

Acceleration-time waveform and spectrum of pendulum oscillation

The acceleration time waveform and power of vibrations measured by attaching an accelerometer horizontally to a pendulum weight
—Shown in Spectrum Figure 3. The frequency range was 50 Hz, and the number of sample points analyzed was 2048.
The FFT frame time is set to 16 seconds, and the frequency resolution to 0.0625 Hz.
The spectrum is displayed with an enlarged view of the 0 Hz to 20 Hz range.

According to the power spectrum, the vibration frequency is 1.313 Hz and its magnitude is 0.199 m/ ^s². The lower limit of the accelerometer's frequency response is 1 Hz, and it is set to AC coupling, so vibrations below 1 Hz cannot be measured. Therefore, components below 0.5 Hz can be presumed to be due to low-frequency noise. The magnitude of the 0.0625 Hz component of the low-frequency noise is 0.108 m/ ^s².

Figure 4 shows the velocity power spectrum and displacement power spectrum of the pendulum oscillation. Both were converted from acceleration using frequency axis differential integration. The magnitude of the 1.313 Hz component is reasonable, with a velocity of 0.0242 m/s and a displacement of 0.0029 m (2.9 mm). The magnitude of the 0.0625 Hz component is 0.2761 m/s for velocity and 0.7030 m for displacement. Because equations 4 and 5 were directly applied to the frequency components observed due to low-frequency noise to convert them to velocity and displacement, the values become impossibly large for pendulum oscillation.

Figure 4 shows the velocity power spectrum (top) and displacement power spectrum (bottom) of a pendulum oscillation.

summary

In this example, we showed a case where, when an acceleration signal containing DC (direct current) offset and low-frequency noise is converted to velocity and displacement using frequency-axis calculus, an impossibly large value is displayed in the first line of the spectrum or in the low-frequency components.

Doubling the number of sample points in the FFT analysis conditions halves the frequency resolution. If remeasurement under conditions that halve the frequency resolution shows that the frequency values at which abnormal velocity/displacement occurs are also halved, then DC offset (error) is likely the cause. In addition, if the power spectrum of acceleration shows low-frequency components below the lower limit of the acceleration sensor's frequency characteristics, or if low-frequency components that should not be emanating from the object are observed, then low-frequency noise may be the cause.

If abnormal values are observed in velocity or displacement, temporarily turn off the frequency axis calculus function, check the power spectrum of acceleration, and investigate the cause.

If the cause of abnormal velocity and displacement values is DC offset or low-frequency noise, the influence is reduced by changing the measurement conditions, or these values are ignored when evaluating the experimental results.

(Excerpt from the email newsletter issued on October 24, 2017)