In the previous three installments, we discussed the definition, calculation method, and visualization method of transfer functions. In this fourth installment, we will explain specific measurement examples of transfer functions and how to interpret the data.
Transfer functions have a wide range of applications, including electrical systems, control systems, acoustic systems, and mechanical vibration systems, but here we will introduce a few examples. Also, as we discussed last time, there are two methods for measuring transfer functions: the FRA method and the FFT method. Our DS-3000 series servo analysis software DS-0342 supports both methods, so we mainly use this software for measurements.
Furthermore, in measuring transfer characteristics, the signal source that excites the transfer system is crucial. In this system, the signal source is changed depending on the measurement method, as shown in Table 1.
Table 1 Measurement Method and Signal Source
| method | signal source |
| FRA Law | Sine wave sweep (log/linear) |
| FFT method | Random, Swept Sign Pseudo-Random, Impulse |
First, let's look at an example of measuring the frequency characteristics of an amplifier or filter, which are electrical components. A typical connection example is shown in Figure 1.
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Figure 1. Example connection for measuring filter characteristics.
Figure 2 shows the frequency response of a filter measured by the FRA method using the connection method shown in Figure 1. The top left is the Bode plot (gain and phase), the top right is the Nyquist plot, the bottom left is the time waveform monitor (overlay of Ch1 and Ch2), and the bottom right is the instantaneous spectrum of the time waveform (overlay of Ch1 and Ch2).
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Figure 2. Example of the frequency characteristics of a filter measured by the FRA method.
From the measurement results;
- This filter is a bandpass filter.
- The lower frequency limit is 100Hz, and the upper frequency limit is 5kHz.
- The passband gain is 20 dB (10 times).
- The gain and phase characteristics, and the fact that the phase is rotated 360 degrees (2π), indicate a 4th-order Butterworth filter (±24dB/oct).
It is presumed that this is the case.
Next, as an example of acoustic measurement, we will measure the frequency response of a typical speaker system.
The measurement environment was a semi-anechoic chamber, and a sound level meter was used as the acoustic sensor, as shown in Figure 3. The results are shown in Figure 4. To obtain a flat frequency response, the frequency weighting of the sound level meter was set to the Z-weighting characteristic.
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Figure 3: Example of frequency response measurement of a loudspeaker.
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Figure 4: Example of frequency characteristics of a speaker system
These results show that this speaker system has a nearly flat frequency response from 60 Hz to 10 kHz. The multiple phase rotations are due to a time delay of approximately 3 ms caused by the approximately 1 m distance between the speaker and the microphone. Figure 5 shows the result after correction using the "phase rotation calculation correction" function.
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Figure 5 Frequency characteristics of the speaker system with delay time corrected.
Next is an example of an electroacoustic system in which the impedance of a single speaker is measured. Since this is a measurement of the internal impedance of the speaker unit, it is necessary to measure the voltage driving the workpiece and the current flowing through it. As shown in Figure 6, a shunt resistor (r in Figure 6) is connected in series and the current waveform is sensed. Also, as can be seen from this wiring, the signal output and Ch 2 input terminals must be isolated. The value of the shunt resistor r is 0.5 Ω.
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Figure 6: Example of connection for measuring the impedance of a speaker unit.
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Figure 7 Impedance characteristics of the speaker unit
The rated impedance of a speaker is defined as "the value that is minimum in the frequency band higher than the lowest resonant frequency (f0)." Therefore, from the measurement results in Figure 7, we can see that the resonant frequency (f0) of this speaker is approximately 158 Hz and the impedance is approximately 6 Ω.
Next, let's look at its application to mechanical vibration systems.
To measure the vibration response characteristics of a mechanical structure, it is necessary to excite the workpiece, and there are usually two methods: the hammering methodandthe exciter method. The exciter method requires a signal source to excite the mechanical transmission system as shown in Table 1, and can be measured using both the FRA method and the FFT method. In contrast, the hammering method uses a force sensor called an impulse hammer to excite the mechanical transmission system, and can only be measured using the FFT method. (This measurement uses the FFT analysis software DS-0321.)
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Figure 8. Example of a flat plate being vibrated by the hammering method.
As shown in Figure 8, the excitation signal from the impulse hammer is input to Ch 1, and the response acceleration signals from Accelerometer are input to Ch 2 and Ch 3. The system is then excited several times using the trigger function, and the average is calculated.
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Figure 9: Excitation signal and response vibration when a flat plate is excited.
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Figure 10 shows two transfer functions with excitation point P1 and response points P1 and P2.
Figure 9 shows a typical time signal and its spectrum from a hammering test. Figure 10 shows the transfer function obtained by triggering with the excitation signal of Ch 1 and averaging it five times. Here, the excitation point is P1, two acceleration PUs are used, and the response points are P1 and P2, and two transfer functions were measured simultaneously.
In the 1.6 kHz frequency band, there are two resonant frequencies, and from the phase information, it can be inferred that the 856 Hz component is the torsional vibration mode and the 1.032 kHz component is the bending vibration mode.
(For reference, a diagram of the mode animation obtained from vibration mode analysis using multi-point measurements has been added.)
Finally, here is an example of a vibration test using a vibrator.
As shown in Figure 11, the electronic circuit board is excited with a sinusoidal sweep at a constant acceleration (10 m/s², approximately 1 G), and the response vibration velocity of any component on the board is determined non-contact using a laser Doppler vibrometer to determine its vibration response characteristics.
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Figure 11 Example of measuring resonant frequency using a vibrator
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Figure 12. Measurement results of vibration test (Left figure: velocity/acceleration; Right figure: acceleration/acceleration <first derivative of the left figure>)
The right-hand figure in Figure 12 shows the measured results. Since a velocity sensor is used for the response vibration, this represents the response characteristics of velocity vibration for constant acceleration. By taking the first derivative of this transfer function, the acceleration/acceleration response characteristics can be obtained.
Finally, here's a summary.
- Transfer functions have applications in a wide range of fields, including electrical systems, control systems, acoustic systems, and mechanical vibration systems.
- Two methods for measuring transfer functions are the FRA method and the FFT method.
- In measuring transfer functions, the signal source that excites the transfer system is crucial, and the appropriate signal is selected from those shown in Table 1 depending on the measurement method.
- There are two methods for measuring mechanical vibration systems: the hammering method and the exciter method.
- The hammering method does not require a synchronized signal source from the measuring instrument; it uses an impulse hammer with a trigger function for measurement. By performing multi-point measurements, vibration modes can also be confirmed.
- In vibration testing using a vibrator, it is possible to perform sinusoidal sweep excitation while keeping the vibration acceleration value constant, enabling vibration testing under consistent conditions.
【keyword】
FRA method, FFT method, Bode plot, Nyquist plot, bandpass filter, Butterworth filter, sound level meter, frequency weighting, Z characteristics, impedance, shunt resistance, rated impedance, resonant frequency, hammering method, exciter method, impulse hammer, trigger, torsional vibration mode, bending vibration mode, vibration mode analysis, mode animation, laser Doppler vibrometer
(Excerpt from the email newsletter issued on May 21, 2015)
