One of the main purposes of using an FFT analyzer to measure vibrations is to determine the natural frequencies.
As discussed in the "Fundamentals of Vibration" section (No. 22-25) of this series, natural frequencies
The phase information of the transfer function (frequency response function, FRF) required for evaluation is an important factor in the decision-making process.
This can happen. This time, we will not determine the frequency response function, but rather obtain the phase information from the 1-channel data.
This is one simple method to determine whether the peak frequency point is at an natural frequency.
I will talk.
First, let's start with a review of the fundamentals of vibration. Here, we will quote from "Fundamentals of Vibration - 2".
Masu.
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Figure 1: Example of an external force applied to a one-degree-of-freedom damped vibration system.
In Figure 1, a sinusoidal force (harmonic excitation force) f(t) is applied, and the displacement of the point mass at that time is x(t).
Then, the equation of motion is
mx(t) + cx(t) + kx(t)= F cos(ω t)
The response displacement in the steady state after the initial transient vibrations have disappeared is
................................. （2）
At this time, the amplitude magnification is
X st = F / (k static displacement), let X be the displacement amplitude.
................................. （3）
Also, the phase angle φ is,
................................. （4）
Here,
Natural angular frequency
................................. （5）
Natural frequency
................................. （6）
Damping ratio
................................. （7）
Damped Natural Angular Frequency
................................. （8）
From equation (4), when the excitation force angular frequency is equal to the natural angular frequency, the phase lag is 90 degrees (π/2 radians).
As the value increases, the phase lag approaches 180 degrees (π radians).
Figure 2 shows the amplitude multiplier from equation (3) and the phase lag from equation (4) (here the lag is assumed to be a negative magnitude).
This is illustrated in the diagram. As you can see, the shape of both graphs depends on the value of the damping ratio ζ.
You will understand.
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Figure 2 Amplitude and phase in forced oscillation of a one-degree-of-freedom damped system
Figure 3 is a graph showing the amplitude scaling factor on a logarithmic scale.
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Figure 3: Differences in amplitude magnification due to damping ratio ζ
Thus, the phase of compliance (the transfer function of the response displacement to the excitation force) changes significantly around the natural frequency (resonance point). Specifically, in the frequency region below the natural frequency, the contribution of the restoring force (kx(t)) is dominant, in the higher frequency region, the contribution of the inertial force (mx(t)) becomes larger, and near the natural frequency, the restoring force and the inertial force are balanced, resulting in a region where only the viscous resistance force (cx(t)) contributes. Considering this in terms of the phase of compliance, in the frequency region below the natural frequency, the displacement is in phase with the force, in the higher frequency region, the acceleration is in phase with the force (i.e., the displacement lags the force by 180 degrees), and near the natural frequency, the velocity and force are in phase (i.e., the displacement lags the force by 90 degrees), resulting in a phase graph shape like that shown in Figure 2.
Using this knowledge, a method for measuring the transfer function and then determining the natural frequency is described below.
vinegar.
- Find several peak frequencies from the measured transfer function.
- We then verify that the coherence function measured simultaneously for those points is approximately 1 (at least 0.9 or higher).
- Verify that the phase difference (lag) between those points is 180 degrees.
(Note 1) Normally the range is 0 to -180, but it can also be 180 to 0.
(Note 2) If the frequency resolution is coarse, it may not rotate 180 degrees reliably.
If the transfer function can be measured correctly, it can be confirmed using the method described above, but if the excitation force cannot be obtained,
Let's consider the case where only one channel of the response vibration waveform can be measured.
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Figure 4. Vibration response waveform obtained by exciting the workpiece (Figure (b) above).
As an example, suppose we obtain a vibration response time waveform as shown in Figure 4(b). Normally, the excitation shock waveform data (Figure 4(a)) would also be recorded, the transfer function would be measured, and the natural frequency would be evaluated from the gain and phase information. However, here we will assume that only 1-channel data of the vibration response time waveform (b) is available.
Figure 5 shows the amplitude and phase of the Fourier spectrum obtained by performing an FFT on the data in Figure 4(b). Here, we will refer to these as the amplitude spectrum (Figure 5(a)) and the phase spectrum (Figure 5(b)), respectively. When we examine the amplitude spectrum in (a), we can see several peak points that appear to be candidates for natural frequencies. We will also check the phase information in (b) in relation to these points.
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Figure 5. Fourier spectrum of vibration response waveform
(a): Amplitude spectrum, (b): Phase spectrum
Now, regarding the basics of phase, the phase of the Fourier spectrum of a 1-channel time waveform represents the initial phase of the cosine wave at each frequency component. If there is a time delay (in this case, the timing of the FFT time window extraction), the phase will lag progressively in proportion to the frequency.
As explained in Figure (b) above, the vertical axis is displayed at ±180 degrees (±200 degrees in this graph to allow for a margin of error), so you can see that the phase lags down to the right, like the teeth of a saw. If it were simply a time lag, it would lag almost linearly (proportional to the frequency on the horizontal axis), but if you look closely, you can see that this linear lag is disrupted in several places.
To make it easier to understand, Figure 6 shows the phase spectrum of the impact waveform of the excited force (Figure 4(a)) and compares it with the other phase spectrum. You can see that the upper figure (phase spectrum of the impact waveform) has a nearly linear phase lag (i.e., a nearly constant delay time regardless of frequency). By subtracting these two phase spectra, we can approximately determine the phase of the 2-channel transfer function. The practical calculation method is shown in Figure 7, which is obtained by ratio (division) of the 2-channel Fourier spectra (complex number data).
As explained earlier, we can see that the phase lags by 180 degrees near the natural frequency, and that the phase lag is always 90 degrees at the natural frequency point, which is the point where the phase lag change is greatest.
Based on these findings, the phase spectrum of the response vibration waveform in Figure 5 (or Figure 6) is plotted on the horizontal axis.
By differentiating with respect to the frequency axis, we can obtain information about the phase change.
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Figure 6 Comparison of phase spectra
Top figure: Phase spectrum of the shock waveform
Figure below: Phase spectrum of the response vibration waveform
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Figure 7. Phase of the ratio of the 2-channel Fourier spectrum (complex number data).
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Figure 8 shows the result of differentiating the phase data of the Fourier spectrum of the impact response waveform.
Figure 8 is a graph showing the result of differentiating the numerical data of the phase spectrum from Figure 4 with respect to frequency using Microsoft Excel© (actually, it's a difference calculation, and the absolute value on the vertical axis is meaningless). It shows that the frequency at the peak point of the amplitude spectrum almost coincides with the point where the slope (derivative coefficient) of the phase lag is largest and negative (the point where the phase lag is -90 degrees). From this, it can be understood that these frequency points are natural frequencies. However, in this example, the anti-resonance point around 3.3 kHz is incorrectly detected due to a large error.
Furthermore, in Figure 8, the numbers (red) at the negative peak points represent the No. in the list in Figure 9.
Yes, they are.
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Figure 9 shows the amplitude of the ratio and its peak list for the 2-channel Fourier spectrum (complex number data).
In conclusion, here's a summary.
- In a transfer function measured for a single-degree-of-freedom vibration system, the natural frequency is at the point where the gain is maximum (peak), and its phase lags by 180 degrees.
- If only the time waveform of the impact response is available, its natural frequency can be estimated from the phase information of its Fourier spectrum.
【keyword】
Harmonic excitation force, amplitude ratio, natural angular frequency, natural frequency, damping ratio, compliance, damping strength
Angular frequency, restoring force, inertial force, viscous resistance, amplitude spectrum, phase spectrum
【reference】
"Introduction to Modal Analysis" by Akio Nagamatsu, Corona Publishing Co., Ltd. (1994)
(Excerpt from the email newsletter issued on May 23, 2017)
