This time, I'll talk about the attenuation waveform in impact tests.
(1) Damping ratio
In previous discussions on transfer functions, we mainly focused on mass and stiffness, which are the components of vibration, but damping elements are also reflected in the measurement results.
For example, in vibration experiments, impact tests are often performed, and if you observe the time data from these tests, you will generally see that the vibration amplitude decreases as time passes. This is because the generated vibration energy is gradually consumed due to damping within the material (converted into heat due to the displacement of atoms and molecules) and loss of vibration energy due to external damping elements such as dampers.
Vibration damping occurs at a constant rate, with energy being lost at a constant rate. For example, in the case of a simple structure with one degree of freedom (1 mass - 1 stiffness - 1 damping), the damped waveform is wavy, and we can find the ratio of the peak value of the first peak of this waveform to the peak value of the next peak. We can see that the ratio of the peak value of the Nth peak to the peak value of the N+1th peak is the same. This ratio is called the damping ratio.
This damping ratio represents the value of the damping element, which is a component of the vibration model.
As some of you may have noticed, the number of peaks in the vibration's decay is the same regardless of frequency. Therefore, with the same damping ratio, the higher the frequency, the shorter the time it takes for the vibration to decay. (A higher frequency means a shorter time interval between peaks.) In other words, even with the same damping ratio, vibrations decay faster at higher frequencies. Conversely, to quickly bring low-frequency vibration phenomena to a halt, a large damping ratio is required, which can make vibration countermeasures extensive and difficult in some cases.
The attenuation ratio can be determined by using the Hilbert transform function, which is included in devices such as FFT analyzers, to obtain the envelope of the time data, and then calculating the attenuation ratio from its slope.
(2) Bode plot and half-width method of attenuated waveforms
Up until now, we've been considering this in the time domain, but how is this attenuation element applied in the frequency domain?
Let's consider this using a Bode plot of acceleration (force). A signal that does not undergo vibration damping, a sine wave, will have a sharp, mountain-shaped peak in its power spectrum. The phase will be -90 degrees. As damping is applied, we can observe that the peak decreases while its base widens. If the damping is increased further, it will become like a gentle hill and approach flatness.
As for what happens to the peak frequency, as the attenuation increases, the frequency gradually decreases compared to when it was a sine wave, and the phase also shifts in the direction of less than -90 degrees.
Is there a way to determine the attenuation ratio in the frequency domain?
The half-width method is commonly used. This method involves measuring the transfer function, focusing on the peak point (vibration mode, natural frequency) of the transfer function, and dividing the difference between the lower frequency fL (which is exactly half the height of the peak) and the upper frequency fH by the center frequency f0 (peak frequency).
Damping ratio = (fL + fH) ÷ f0
For signals that are close to a sine wave, the attenuation ratio will be small because the frequencies of half the upper and lower peaks are not far apart. However, for data that resembles a gentle hill, the difference is large, resulting in a large attenuation ratio.
Furthermore, since the damping ratio is a ratio of frequencies, it is an element that is not affected by the natural frequency. However, in reality, due to issues such as frequency resolution, it is sometimes impossible to determine the true peak. Therefore, instead of using the half-width method, curve fitting is often performed based on the mode circles of the real and imaginary parts of the transfer function to calculate the damping ratio.
(3) Nyquist plot (mode circle) of the attenuated waveform
A mode circle is a representation of a transfer function in the complex plane (a plane where the x-axis represents real numbers and the y-axis represents imaginary numbers), and is called a Nyquist diagram. Let's consider a damped waveform using the Nyquist diagram of acceleration (acceleration/force).
The eigenvalues of each vibration mode can be represented by complex numbers (the real and imaginary parts at the peak of the transfer function). If we plot these on the complex plane, we can see that normal damping exists only in the left half-plane.
Eigenvalues (with their imaginary parts located symmetrically above and below) exist at the peak of the sinusoidal frequency, where the real value is zero. As damping is applied, the real part of these eigenvalues becomes negative. The more negative the real part becomes, the greater the damping ratio of the vibration.
Reference diagram
The following data shows the results of striking a plate-shaped metal piece with an impulse hammer. The left column of the diagram shows the data for the metal piece alone, while the right column shows the data for the same metal piece with plastic attached using double-sided tape to increase damping.
Waveform of the force of the impact
Acceleration waveform
Acceleration/force (MAG)
phase
Real numbers
Imaginary number
Nyquist diagram
-
The curve does not form a circle due to insufficient frequency resolution. The green dashed line and red dots represent the expected Nyquist plot and resonant frequency points. -
Same as above. Due to the suspension, measurements in the low-frequency range are not performed properly, resulting in a phase shift. Phase correction is necessary, as shown in Figure 4-b.
Acceleration/Force Overlay Display
Overlay Figures 3-a and 3-b above.
Nyquist diagram
By using the zoom function to increase the resolution of Figure 7-a, we obtain the predicted Nyquist line shown in Figure 7-a.
Damping ratio calculated using the half-width method
The damping ratio (Damp) was calculated using the half-width method from the same ZOOM measurement data.
Damping ratio obtained from Hilbert transform
The damping ratio (Damp) obtained from the Hilbert transform in Figure 2-a above is
Relationship
Loss factor (Loss.F) = 2 × Damping ratio (Damp)
Logarithmic decay rate (log.d) = 2π × decay ratio (Damp)
As a side note, is it possible for eigenvalues to lie on the right side of the complex plane?
In fact, when the eigenvalue is located on the right side, it's not damping, but rather a phenomenon where the vibration gradually increases. In other words, when struck, the vibration gradually increases, but this doesn't happen in normal structures. For example, when control is being implemented, there are divergent phenomena such as vibration, and in this case, the eigenvalue is on the right side. One could consider the problem of vibration control as how to move this eigenvalue to a good position in the right half-plane.
The time domain and frequency domain can be confusing, but generally, when there are multiple natural modes (natural vibrations with multiple degrees of freedom), the time domain waveform is a composite of the damping of all vibration frequencies, making the time waveform complex and difficult to calculate. Therefore, it is common to measure the damping ratio for each natural frequency in the frequency domain using the half-width method.
Technical Report
Vibration damping materials and their performance measurement
(Excerpt from the email newsletter issued on September 24, 2004)
