So far, we've discussed the "fundamentals of vibration measurement" over three installments. This time, we'll cover part 4 and
Next, we'll talk about damping ratio.
Damping ratio ζ plays an important role in the behavior of vibrations. free vibration (1st time) Will it vibrate?
It affects whether or not vibration occurs, or how quickly the vibration stops if it does occur. forced vibration (Second time) resonance frequency
wavenumber This affects the magnitude of the peak in Vibration transmittance (3rd time) at the resonant frequency
This affects the magnitude of the peak and the transmission rate in frequency bands beyond the resonant frequency.
In this (fourth) session, we actually performed vibration measurements using an FFT analyzer and determined the damping ratio ζ from the results.
I will explain how to do it.
The parameters used to express decay vary depending on the application field, but first, let's consider the logarithmic decay rate.
Here's how to find it.
The logarithmic decay rate δ is defined as the natural logarithm of the ratio of adjacent amplitudes of damped waveforms of free vibration.
Why the natural logarithm? Well, as shown in Figure 1, the amplitude of free vibration is exponential.
Because it will decrease numerically.
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Figure 1 Damped Free Vibration Waveform
The damped free vibration waveform x (t) in Figure 1 is,
.................................(1)
ωn = Natural angular frequency ζ: Damping ratio
ωd = ωn√1 -ζ²: Damped natural angular frequency
: Damping period
This is the result.
In Figure 1, the logarithmic decay rate δ is
.................................(2)
Here, if ζ << 1
δ = 2πζ .................................(3)
Next, to determine the logarithmic decay rate δ, first, we use the Hilbert transform technique to find the envelope of the decayed waveform, and then draw a graph with time on the horizontal axis and amplitude on the vertical axis in logarithmic scale. (Figure 2)
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Figure 2 shows the envelope of the attenuated waveform, with the vertical axis displayed in dB.
Note that the logarithmic decay rate δ is the slope with respect to the decay period Td, so the dB expression is in common logarithm.
With this in mind, from the slope of the line in Figure 2,
.................................(4)
Here,
that's why
.................................(5)
FFT analyzer (DS-3000 series) Delta search function Using this, from equation (5), logarithmic decay
We will calculate the rate δ.
Furthermore, from equation (3)
.................................(6)
The logarithmic decay rate δ can be calculated from the measured data and equation (5), and the decay ratio ζ can be determined from equation (6).
Figure 3 shows an example where the damping time waveform of free vibration was measured by hammering, and then the logarithmic damping rate was determined, followed by the damping ratio. Note that this figure uses the band-limiting function of the Hilbert transform to obtain the damping waveform for a single resonant frequency.
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Figure 3: Example of determining the logarithmic decay rate and decay ratio from decay time waveform and envelope data.
Next, the damping ratio is determined from the resonance peak of the frequency response function using the half-width method.
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Figure 4: Diagram illustrating the half-width method.
Let G(f) be the frequency response function (compliance) of a 1-degree-of-freedom system.
.................................(7)
Here, nf: natural frequency, ζ: attenuation ratio
In Figure 4, the value of the resonant frequency fn is -3dB from G(f)max (1/2 in power, 1/√2 in amplitude).
Let f1 and f2 be the points where this occurs, and let Δf = f2 - f1.
Since f1 and f2 can be expressed in modern terms as fn ± ∆f / 2, the ratio with fn is
.................................(8)
Substituting this value into equation (7) and taking the reciprocal,
.................................(9)
Also, the maximum value
In modern times, (7) formula is more
Therefore, substituting into equation (9)
...............................(10)
(10) Transform equation
............................... (11)
Since the power uses half the bandwidth, the method of calculating the attenuation ratio using equation (11) above is called the half-power bandwidth method.
In addition to the half-width method, the mdB method can also be considered. In this case,
...............................(12)
Here,
Incidentally, when the level is 1 dB, k = 1.97, and when it's 2 dB, k = 1.31.
The loss factor, used as an evaluation parameter for vibration damping materials, can also be determined using the full width at half maximum (FWHM) method. In damped vibrations with hysteretic damping, a complex modulus K' that generates a restoring force proportional to the vibration displacement is defined, with the real part being K1 and the imaginary part being K2.
K′ = K1 + jK2 = K1(1 + jη )= K1(1 + j tanδ)............................... (13)
Here,
............................... (14)
This is the result. (Figure 5) This η is called the loss coefficient. The complex modulus K' includes both the spring constant k and the viscous damping coefficient c in a one-degree-of-freedom system model. As can be seen from Figure 5, the loss coefficient is also called the tandelta.
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Figure 5 Complex modulus of elasticity and loss coefficient
From the half-width method, the loss coefficient η can be calculated using the following formula:
............................... (15)
............................... (16)
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Figure 6. Example of measured damping ratio and loss coefficient using the half-width method.
Figure 6 shows an example of determining the attenuation ratio and loss coefficient using the half-width method from the FRF (frequency response function) measured by the hammering method using an FFT analyzer.
Table 1 below summarizes the relationships between various parameters that represent the damping characteristics.
Table 1 Interrelationships between various parameters
Finally, here's a summary.
- Damping ratio plays an important role in the behavior of vibrations, such as in free vibration, forced vibration, and vibration transmittance.
I am doing it. - The logarithmic damping rate can be calculated from the damped free vibration waveform, and then the damping ratio can be determined from that.
It will come. - The attenuation ratio can be determined from the frequency response function data using the half-width method.
- The loss coefficient used to evaluate damping materials can also be determined using the half-width method.
【keyword】
Damping ratio, free vibration, forced vibration, resonant frequency, vibration transmittance, logarithmic damping ratio, natural angular frequency, damped natural angular frequency, damping period, Hilbert transform, envelope, delta search function, half-power bandwidth method, loss factor, complex modulus of elasticity, spring constant, viscous damping coefficient, tandelta, tandel, hammering method, Q value
【reference】
- "Introduction to Modal Analysis," by Akio Nagamatsu, Corona Publishing Co., Ltd. (1994)
- Ono Sokki Technical Report: "Coefficients Representing Damping Characteristics"
(Excerpt from the email newsletter issued on January 21, 2016)
