Technical Report: Vibration Damping Materials and Their Performance Measurement 6

The top graph below shows the results of zoom analysis performed on five different materials with varying loss coefficients using an FFT analyzer, with the frequency resolution changed. The horizontal axis represents the number of FFT measurement points within -3 dB (_f²-_f₁) using the half-width method. As the zoom magnification is increased, the measured η gradually converges to a nearly constant value. The bottom graph shows the result of redrawing the graph with the error calculated using this value as the true value.

With recent advances in the manufacturing technology of vibration damping materials, the practical application of materials with large loss coefficients is progressing. However, even including ISO standards, the only standardized method for measuring loss coefficients is the old-fashioned half-width method. Here, the half-width method is a method in which the loss coefficient is calculated as η = (f2 - f1)/ _f0 from the resonant frequency f0, which is obtained from vibration response characteristics such as amplitude, velocity, and acceleration in response to excitation force_, and the frequencies _f1 and _f2, which are the frequencies at which the level at the resonant frequency drops by 3 _dB. However, recently, materials have been developed in _which the level difference between resonance and anti-resonance is less than 3 dB, and the half-width method is completely useless for such materials. Furthermore, as a recent technological trend, attempts are being made to determine the loss coefficient from the sharpness of the characteristics at anti-resonance as well as the sharpness of the characteristics at the resonant frequency. For example, instead of the peak (resonance) of the frequency response function obtained from [velocity / force], a value corresponding to the loss coefficient is sought from the characteristics of the trough (anti-resonance). While the relationship between the sharpness of anti-resonance and the value of the loss coefficient at that frequency is not clear, fitting the resonance and anti-resonance characteristics is absolutely necessary for materials where the difference between peaks and troughs is not insignificant.
For the purposes described above, a method has been proposed to measure the loss coefficient by performing curve fitting to the [velocity/force] or [force/velocity] characteristics using a polar-zero model to fit the resonant and anti-resonant characteristics. This document outlines that method.

The polar zero model is a characteristic that expresses the response characteristics at the excitation point using the following equation.

or;

Here, _ηn represents the bluntness of the characteristics at each resonance and anti-resonance (the loss coefficient in the case of resonance). _ωn is the resonance or anti-resonance frequency, and H is a constant (a positive real number). Equations (1) and (2) can express both impedance ([force / velocity]) and mobility ([velocity / force]), but here they represent mobility. The mobility of a cantilever and a beam with both ends free increases and decreases as the frequency approaches zero, respectively, so the mobility of a cantilever corresponds to equation (2), and the mobility of a beam with both ends free corresponds to equation (1). When fitting to impedance characteristics, you can simply take the reciprocal, so you can swap the equations and apply them accordingly. The greatest advantage of the polar-zero model is that the loss coefficient (which we will also call the anti-resonance case for convenience) is explicitly entered, allowing the sharpness of the peaks and troughs to be defined independently. For reference, a mathematical model commonly used in modal analysis is expressed as follows. Here, only the loss coefficient at resonance can be explicitly input.

To fit the peak and valley characteristics with equivalent error, a special error function is needed to express the magnitude of the difference between the measured data and the model characteristics. A commonly used error function λ is:

Here, A(ωm) is the measured characteristic and X(ωm) is the model characteristic. In resonance, the numbers are large, so even the same percentage error contributes significantly, while the error in the anti-resonant characteristic is disregarded. To solve this problem, two error functions have been proposed.

Equation (5) is a method that uses the frequency response function expressed in decibels for fitting. Equation (6) normalizes the measurement using the measured characteristics so that the same error rate in the peaks and troughs contributes to λ with the same magnitude. This makes it possible to simultaneously fit the resonant and anti-resonant characteristics by using these error functions.

The characteristics can be calculated from equation (1) or (2), fitted as if they were measured data, and the accuracy of the algorithm can be verified by checking how close the obtained values are to the true values. Figure 1 shows the results. The thin solid lines with four poles and zero represent the characteristics to be fitted, calculated by providing the resonance and anti-resonance frequencies and corresponding loss coefficients of the Type B pole-zero model in Figure 1. The resonance and anti-resonance frequencies were set to values that correspond to the frequency distribution when the central excitation of a strip-shaped area occurs. The loss coefficients were appropriately assigned values of 0.1, 0.2, and 0.3 to each resonance and anti-resonance. The thick lines represent calculations where the resonance and anti-resonance frequencies were given random errors of plus or minus 10%, and the loss coefficients were uniformly given an initial value of 0.15.
This calculation was performed using a decibel-type error function. When fitting was performed using a decibel-type error function, the parameter values obtained after fitting were accurate to more than 8 decimal places (the converged curve overlaps with the thin line). The advantage of determining the loss coefficient by curve fitting is that the frequency resolution during measurement can be much coarser than that of the half-width method. Even in the graph in Figure 1, fitting is possible even with a resolution so coarse that it appears as a broken line graph in the low-frequency range.

This is the result of creating frequency response functions that cannot be measured using the half-width method with η = 0.7, 1.0, and 1.2 through simulation, and then curve-fitting them using the dB method. It can be seen that with a loss factor of around 0.7, measurements can be taken with high accuracy up to the 6th resonance order, and with loss factors of around 1.0 and 1.2, measurements can be taken with high accuracy up to the 3rd resonance order.

This method involves suspending a node for each vibration mode. Because it suspends the object from a stationary position, it is considered to allow measurement of the smallest loss coefficient, and is therefore widely used for measuring the loss coefficient of individual metal components. A drawback of this method is that the suspension position must be changed for each mode. Excitation is typically performed using a non-contact electromagnetic exciter, and the response is measured using the same exciter as an electromagnetic velocity sensor. (For electromagnetic exciters/detectors, please refer to the equipment used in the cantilever beam method.)

Looking at the relationship between the test method and vibration modes in the figure below, it can be seen that when the length of the test specimen for the cantilever method is half the length of the test specimen for the central excitation method, the vibration modes for resonance in the cantilever method and anti-resonance in the central excitation method are the same. Furthermore, it can be seen that the vibration modes for resonance in the two-point suspension method and the two-point support method are the same as the vibration modes for resonance in the central excitation method.

The following four figures compare the loss coefficients of 1. resonance and anti-resonance of the cantilever beam method and 2. resonance and resonance of the two-point suspension method and the two-point support method. From these, the following can be said.

Cantilever beam method ≈ Central excitation method (anti-resonance)
Two-point suspension, two-point support method ≈ Central vibration method (resonance)

Effective specimen length
Twice the strength of a cantilever beam = Central excitation method