Frequently Asked Questions about Measurement - Part 14: "Displaying Time Waveforms of Velocity and Displacement"

This measurement column addresses frequently asked questions received by our customer support center and provides answers to those questions.

To quantitatively capture vibration, three physical quantities are generally used: displacement (unit: m), velocity (unit: m/s), and acceleration (unit: m/ ^s²). There are various types of sensors for measuring vibration, and they can be classified into displacement sensors, velocity sensors, and acceleration sensors depending on the physical quantity being measured. Each sensor has its own characteristics and limitations, and it is not always possible to use a sensor that can measure the physical quantity you want to measure.

Piezoelectric acceleration sensors are contact-type sensors, making them relatively easy and inexpensive to use. However, they can only measure acceleration. To measure velocity or displacement, the acceleration signal obtained from the sensor must be converted into velocity or displacement using methods such as numerical integration.

Simply numerically integrating the acceleration time waveform will result in noise interference, preventing the acquisition of accurate velocity and displacement time waveforms. In such cases, the IFFT (Inverse Fourier Transform) function is used. This time, using the DS-0321 FFT analysis function of our real-time Sound and Vibration Analysis System as an example, we will introduce the procedure for displaying velocity and displacement time waveforms using the IFFT function.

Acceleration-time waveform and spectrum of pendulum oscillation

Figure 1 shows the acceleration time waveform and power spectrum of vibrations measured by attaching an accelerometer horizontally to a pendulum weight. The frequency range was 50 Hz, and the number of sample points was 2048. The FFT frame length was set to 16 seconds, and the frequency resolution to 0.0625 Hz. The power spectrum is magnified to show the range from 0 Hz to 20 Hz.

According to the power spectrum, the oscillation frequency is 1.313 Hz, so the period of the pendulum is its reciprocal, 0.762 seconds. The RMS value of the first component of acceleration was 0.199 m/ ^s², and the overall value was 0.250 m/ ^s². Converting these to peak-peak amplitude values, they are 0.563 m/ ^s² and 0.707 m/ ^s², respectively. It can also be seen that the power spectrum contains low-frequency noise components below 0.5 Hz.

Reading the maximum value of both amplitudes (peak-peak) from the time waveform, we get 0.862 m/s ² Therefore, if there are higher-order components or low-frequency noise, the values read from the power spectrum will not perfectly match.
However, the values obtained are generally reasonable.

Dividing the acceleration value by (2 × π × frequency) gives the velocity value, and dividing it by (2 × π × frequency) twice gives the displacement value. The RMS acceleration value of the first component (1.313 Hz) read from the power spectrum was 0.199 m/ ^s², so the RMS velocity value is 24.1 mm/s and the RMS displacement value is 2.92 mm. The peak-to-peak velocity amplitude value is 68.2 mm/s, and the peak-to-peak displacement amplitude value is 8.27 mm.

Note that in the Data Display Settings menu ⇒ Y-axis Scale Settings, the scale is set to AUTO.
Also, in the Data Display Settings menu ⇒ Cursor Settings, under Y-axis cursor settings, the time axis...
The cursor is set to Peak-Peak, and "Show two cursor values" is turned ON.

Numerical integration of time waveforms to obtain velocity-time waveforms and displacement-time waveforms

The DS-0321 FFT analysis function includes a time-domain differential and integral calculus function using numerical calculations. Figure 2 shows the acceleration time waveform of the pendulum oscillation, as well as the velocity time waveform and displacement time waveform obtained using the time-domain differential and integral calculus function. The FFT frame length is 16 seconds, but 8 seconds of that is displayed in an enlarged view.

The cursor values for the velocity-time waveform and displacement-time waveform show clearly abnormal values such as -499 mm/s and -3256 mm. Furthermore, the waveforms are generally bowed. This is because low-frequency noise in the signal has been amplified by integration, and these are not the correct velocity-time waveforms or displacement-time waveforms.

The time-axis calculus function is configured in the analysis settings menu ⇒ time-axis data analysis (Figure 3). Performing a single integral with time-axis calculus on the acceleration time waveform yields the velocity time waveform. Performing a double integral yields the displacement time waveform. DC removal should be turned ON. Turning on integration unit conversion allows you to specify the units for velocity and displacement.

Velocity-time waveform and displacement-time waveform obtained using IFFT function

To cut out the effects of low-frequency noise and obtain velocity-time waveforms and displacement-time waveforms from acceleration-time waveforms, we use frequency calculus and IFFT (inverse Fourier transform) functions.

The acceleration time waveform is first subjected to an FFT to obtain the Fourier spectrum. Dividing each frequency component of the obtained Fourier spectrum by (2 × π × frequency) once or twice yields the Fourier spectra of velocity and displacement. Figure 4 shows the results of performing an IFFT (inverse Fourier transform) on the obtained velocity and displacement Fourier spectra after cutting out components below 1 Hz.

The amplitude values (peak-peak) for both the velocity-time waveform and the displacement-time waveform were 81.8 mm/s and 10.1 mm, respectively, which are reasonable values.

Figure 4 shows the acceleration-time waveform (top), velocity-time waveform (middle), and displacement-time waveform (bottom) of a pendulum oscillation, calculated using IFFT.

The procedure for displaying velocity and displacement time waveforms using the IFFT function is as follows:

1. Go to the Input/Output Settings menu ⇒ Window Function Settings and set the window function to Rectangular.
   The window function setting will take effect after remeasurement.
2. In the Data Display Settings menu, go to Data Settings and display the Fourier spectrum on the graph.
3. In the analysis settings menu, under Frequency Calculus, set it to single integral (when displaying velocity) or double integral (when displaying displacement). Also, turn on unit conversion and set the units for velocity and displacement.
4. In the analysis settings menu ⇒ IFFT calculation, set Bandwidth Limiting to ON and Bandwidth Limiting Method to "Outer Cut," then turn on the IFFT function. Set the lower frequency limit to a value that is not affected by low-frequency noise. If the time waveform is not the expected waveform, adjust the lower frequency limit. Set the upper frequency limit to the same value as the frequency range.

summary

This time, we introduced the procedure for converting the acceleration time waveform into velocity and displacement time waveforms using the IFFT (Inverse Fourier Transform) function. If the velocity and displacement time waveforms obtained by numerical integration of the time waveform do not match the expected waveform, please use the IFFT function.

Please note that the procedure described in this measurement column is for version 2.5 of the DS-3000 series real-time Sound and Vibration Analysis System. The procedure differs in older versions.

(Excerpt from the email newsletter issued on August 29, 2017)