Technical Report on Vibration-Damping Materials and Their Performance Measurement 6

The graph above shows the results of a zoom analysis conducted with an FFT analyzer on five different materials with varying loss coefficients, measured with varying frequency resolution. The horizontal axis represents the number of FFT measurement points within the half-width method, referred to as -3 dB (f ₂ − f ₁). As the zoom magnification increases, the measured η gradually converges to a nearly constant value. The graph below is a rewritten version that calculates the error using this value as the true value.

What can be said from this graph is that if there are about 15 resolution points within the half-value width of the FFT, the measurement accuracy of the loss coefficient can be said to be around 1%. It can also be stated that zoom analysis is essential when the loss coefficient is small.

The figure below shows the impact of the presence or absence of zoom analysis and mass cancellation on the measurement error of loss coefficients. The measurement error of the loss coefficient is minimized when both zoom analysis and mass cancellation are performed. Notably, when the loss coefficient is below 0.01, the differences in loss coefficients due to the presence or absence of zoom analysis and mass cancellation become particularly pronounced.

Moreover, the impact on the loss coefficient is considerably greater with the presence of zoom analysis than in the case of the absence of mass cancellation.

(Currently, it is considered the most excellent curve fitting method for loss coefficient measurement, and with the approval of Professor Hideo Suzuki from Chiba Institute of Technology, the paper will be published in its original form.)

With the recent advancements in the manufacturing technology of vibration-damping materials, the practical application of materials with a large loss coefficient is progressing. However, as a standardized method for measuring the loss coefficient, there is only the traditional half-width method, even including ISO. Here, the half-width method is a method to determine the resonance frequency f ₀ from the vibration response characteristics such as amplitude, velocity, and acceleration against the excitation force, and to calculate η = (f ₂- f ₁) / f ₀ from the frequencies f ₁ and f ₂ where the level drops by 3 dB at the resonance frequency. However, recently, materials have been developed where 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 characteristics at the resonance frequency as well as the sharpness of characteristics at the anti-resonance. For example, instead of using the peak (resonance) of the frequency response function obtained from [velocity / force], it attempts to derive a value corresponding to the loss coefficient from the characteristics of the trough (anti-resonance). The relationship between the sharpness of the anti-resonance and the value of the loss coefficient at that frequency is not clear, but fitting including the characteristics of both resonance and anti-resonance is absolutely necessary for materials where the level difference between peaks and troughs is not insignificant.
For the above purposes, a method is proposed to measure the loss coefficient by using the extreme zero model to perform curve fitting on the characteristics of [velocity / force] or [force / velocity] to fit the characteristics of resonance and anti-resonance. Here, an overview of this method will be explained.

The extreme zero model is a characteristic that expresses the response characteristics at the point of excitation using the formula below.

or;

Here, η _n represents the characteristic damping at each resonance and anti-resonance (the loss coefficient in the case of resonance). ω _n 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 we will consider them to represent mobility. The mobility of a cantilever beam and a simply supported beam increases and decreases, respectively, as the frequency approaches zero, so the mobility of the cantilever beam corresponds to Equation (2), while the mobility of the simply supported beam corresponds to Equation (1). When fitting to impedance characteristics, one can simply take the reciprocal, so the equations can be swapped and applied accordingly. The greatest feature of the pole-zero model is that the loss coefficient (which we will also conveniently refer to as the loss coefficient in the case of anti-resonance) is included in an explicit form, allowing the sharpness of the peaks and valleys to be defined independently. For reference, the mathematical model commonly used in modal analysis is expressed as follows. Here, only the loss coefficient at resonance can be input in an explicit form.

When trying to fit the characteristics of mountains and valleys with equal error, it is necessary to devise an error function that expresses the magnitude of the difference between the measured data and the model characteristics. The traditional error function λ is often used.

In this context, A(ω m) represents the measured characteristics, while X(ω m) represents the model characteristics. In resonance, the numbers are large, so the same percentage of error contributes significantly, whereas the errors in the characteristics during anti-resonance are considered negligible. To solve this problem, two error functions have been proposed.

Equation (5) is a method for fitting characteristics expressed in decibels of the frequency response function. Equation (6) normalizes the measured characteristics so that the same error rate at peaks and troughs contributes equally to λ. This enables the simultaneous fitting of resonance and anti-resonance characteristics using these error functions.

By calculating the characteristics from equations (1) or (2) and treating them as measured data, we can perform a fit and verify how close the obtained values are to the true values, thus validating the correctness of the algorithm. Figure 1 shows the results. The thin solid line with four poles and zeros represents the resonance and anti-resonance frequencies calculated for the pole-zero model of type B in Figure 1, corresponding to the characteristics being fitted. The resonance and anti-resonance frequencies were assigned values that would yield a frequency distribution during central excitation in a strip shape. The loss coefficients were randomly assigned values of 0.1, 0.2, and 0.3 to each resonance and anti-resonance. The thick line was calculated with resonance and anti-resonance frequencies having a random error of plus or minus 10%, with a uniform initial value of 0.15 for the loss coefficient.
It was calculated using a decibel-type error function. When fitting using the decibel-type error function, the parameter values obtained after fitting were found to be correct to an accuracy of more than eight digits (the curve after convergence overlaps with the thin line). The advantage of determining the loss coefficient through curve fitting is that the frequency resolution during measurement can be much coarser than that of the half-width method. In the graph of Figure 1, it is also possible to fit even if the low-frequency range appears as rough as a broken line graph.

When overlaid with the graph showing the required number of measurement points within the half-width mentioned in the necessity of FFT zoom analysis, it appears as shown in the figure below. With this curve fitting method, if there are about 0.3 measurement points within the half-width, the error can be contained within a few percent.

This is the result of curve fitting using the dB method for frequency response functions that cannot be measured with the half-width method at η = 0.7, 1.0, and 1.2. It is found that with a loss coefficient of about 0.7, measurements can be accurately taken up to the sixth resonance order, while for loss coefficients of 1.0 and 1.2, measurements can be accurately taken only up to the third resonance order.

This is a method of suspending nodes for each vibration mode. It is commonly used for measuring the loss coefficient of metal samples, as it is said to allow measurement from the position that does not move, which traditionally has the smallest loss coefficient. A drawback of this method is that the suspension position must be changed for each mode. Excitation is typically done using a non-contact electromagnetic exciter, and the response is measured using the same exciter as an electromagnetic velocity sensor. (For information on electromagnetic exciters/detectors, refer to the equipment used in the cantilever beam method.)

About the position

This represents the position from the left and right ends as a percentage, assuming the length of the test specimen is 100%. If there are multiple specimens, any point can be suspended.

From the relationship between the testing methods and vibration modes shown in the figure below, it can be seen that when the length of the cantilever beam specimen is half that of the centrally excited specimen, the resonance of the cantilever beam method and the anti-resonance of the centrally excited method have the same vibration mode. Additionally, it can be seen that the resonance modes of the two-point suspension method and the two-point support method are the same as the resonance mode of the centrally excited method.

The following four diagrams compare 1. the resonance of the cantilever beam method and the anti-resonance of the central excitation method, and 2. the resonance of the two-point suspension method and the two-point support method, along with the loss coefficients in both cases. From these, the following can be stated.

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

Effective specimen length
twice the cantilever beam = central vibration method