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Technical Report: Vibration Damping Materials and Their Performance Measurement (Part 8)

22. Window function used when measuring the loss coefficient

When measuring the loss coefficient using a window function, the frequency response function is expected to be deformed. The figure below shows the lower limit of measurement for the loss coefficient when each window function is applied.

From the previous diagram, the following can be said:

  1. The FLAT-TOP window is not suitable for measuring loss coefficients.

  2. The square wave window is ideal for measuring loss coefficients, except when the signal is input at the worst-case resolution and the midpoint of the resolution.

  3. Traditionally, it has been said that "when the signal is asynchronous, use the Hanning window function." However, when measuring the loss coefficient (when determining η at a value greater than -13 dB from the peak), the square wave window can accurately measure even the smallest loss coefficient. However, if coherence decreases when using a rectangular window, use a tapered window.

  4. The closer the resonance point is to the left side of the FFT screen, the greater the influence of the window function. For example, when using the Hanning window without zooming, the lower limit of measurement for the 10th line of the FFT is η = 0.15.

  5. The fact that the peak of the frequency response function waveform is on the left side of the screen indicates that low-frequency resonance and anti-resonance were measured in a high-frequency range. Moving the peak to the right means (1) lowering the frequency range or (2) performing frequency zoom, thus confirming the necessity of zoom analysis.

  6. Another benefit of zoom analysis is that, since it lengthens the time frame, it reduces the distortion of the impulse response. Furthermore, zoom analysis increases the frequency resolution, so a reduction in errors in this area can also be expected. It is thought that increasing the number of FFT points has a similar effect. Recently, there have even been programs that calculate 25,600 lines of spectrum using 65,536 FFT points, and when the loss factor is somewhat large (η: around 0.01), programs that do not require zoom analysis are easy to use.

23. Calculation of Young's modulus, etc.

This document presents formulas for determining Young's modulus from measurements of solid materials, and formulas for determining the loss coefficient and Young's modulus of individual damping materials from the loss coefficient and Young's modulus measured in composite types (2-layer and 3-layer types) using the RKU (Ross, Kerwin, Unger) equation.

Solid wood

共振周波数を fn (Hz)、半値幅を Δ fn (Hz)、試料片の長さを ℓ (m)、試料片の厚さ ｈ (m)、試料片の平均密度を ρ (kg/m3) とすると。

ダンピングが小さい場合は 1/8 Δ fn2 を無視し

Furthermore, the loss coefficient η is

Here

In the case of cantilever beam method and central excitation method using anti-resonance

order n θn θn4
1 1.87510 12.36
2 4.69409 485.5
3 7.85476 3806.6
4 10.99554 14617.3
5 14.13717 39943.8
6 17.27876 89135.4
7 20.42035 173881.2
8 23.56194 308208.2

 

Less than or equal to + π

However, in the case of the cantilever beam method, use equation (1)img-damp-6-13Enter the length obtained by subtracting the gripping allowance from the total length. If using central excitation anti-resonance, use equation (1).img-damp-6-13toimg-damp-6-13Enter /2.

中央加振で共振を使用する場合は共振周波数を求めて、θ n に下記値を使用する。

order n θn θn4
1 4.73004 500.56
2 10.99561 14617.6
3 17.27876 89135.4
4 23.56194 308208.2
5 29.84513 793403.1
6 36.12831 1703690.0
7 42.41150 3235448.8
8 48.69468 5622456.0

Less than or equal to + π

img-damp-6-13Enter the total length.

In the case of a two-point suspension (support) method with both ends free.

order n θn θn4
1 4.73004 500.56
2 7.85320 3803.5
3 10.99561 14617.6
4 14.13717 33943.8
5 17.27876 89135.4
6 20.42035 173881.2
7 23.56194 308208.2

 

Less than or equal to + π

img-damp-6-13Enter the total length.

また高次では θn+1 と θn の差はほとんど π と一致することから

となり次数のファクター θn および密度 ρ が消去されたこの式が、次数がわからない場合に有用な式である。

In the case of a two-layer composite panel

複合試験片の損失係数： η c
共振周波数：ｆc［Hz］

自由長の長さ：ℓ［m］

基材の損失係数： η1 ＝ 0 と置く

共振周波数：ｆi［Hz］

厚み：ｄ1［m］

密度：ρ 1［kg/m3］

ヤング率：Ｅ1［N/m2］

制振材単品の損失係数： η 2

厚み：ｄ2［m］

密度：ρ 2［kg/m3］

ヤング率：Ｅ2 ［N/m2］

さらにヤング率比 E2 / E1 ＝ M、厚み比 d2 / d1 ＝ T、密度比 ρ2 / ρ 1 ＝ D と置くと、制振材単品の損失係数およびヤング率は、基材自身のデータと複合試験片のデータから次式により計算できる。

However, here

That is the case.

However, this holds true only when α ≥ 1.1.

Double-sided composite panel (using the same vibration damping material on both sides)

複合試験片の損失係数： η c
共振周波数：ｆc［Hz］

自由長の長さ：ℓ［m］

基材の損失係数： η1 ＝ 0 と置く

共振周波数：ｆi［Hz］

厚み：ｄ1［m］

密度：ρ 1［kg/m3］

ヤング率：Ｅ1［N/m2］

制振材単品の損失係数： η 2

厚み：ｄ2［m］

密度：ρ 2［kg/m3］

ヤング率：Ｅ2 ［N/m2］

さらに厚み比 d2 / d1 ＝ T、密度比 ρ2 / ρ1 ＝ D と置くと、制振材単品の損失係数およびヤング率は、基材自身のデータと複合試験片のデータから次式により計算できる。

Here,img-8-13It is written

However, here

It is true in that case.

Shear modulus and loss coefficient of sandwich-type composite panels (vibration-damping steel plates)

複合試験片の損失係数： η c
共振周波数：ｆc［Hz］

自由長の長さ：ℓ［m］

基材の損失係数： η1 ＝ 0 と置く

共振周波数：ｆi［Hz］

厚み：ｄ1［m］

密度：ρ 1［kg/m3］

ヤング率：Ｅ1［N/m2］

制振材単品の損失係数： η 2

厚み：ｄ2［m］

密度：ρ 2［kg/m3］

剪断弾性率： Ｇ［N/m2］

さらに厚み比 ｄ2 / ｄ1 ＝ T、密度比 ρ2 / ρ1 ＝ D と置くと、制振材単品の損失係数および剪弾性率は、基材自身のデータと複合試験片のデータから次式により計算できる。

Here

however

Only in that case.