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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

Let the resonant frequency be f n (Hz), the full width at half maximum be Δ f n (Hz), the length of the sample piece be ℓ (m), the thickness of the sample piece be h (m), and the average density of the sample piece be ρ (kg/ m³).

If the damping is small, ignore 1/8 Δfn².​

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.

When using resonance with central excitation, determine the resonant frequency and use the following value for θn.

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.

Furthermore, in higher orders, the difference between θn +1 and θn is almost equal to π.

This equation, in which the degree factor θn and density ρ are eliminated, is a useful equation when the degree is unknown.

In the case of a two-layer composite panel

Loss factor of composite test specimen: η c
Resonance frequency: f c [Hz]

Free length: ℓ [m]

Substrate loss coefficient: Set η = 1 = 0

Resonance frequency: f i [Hz]

Thickness: d 1 [m]

Density: ρ 1 [kg/m 3]

Young's modulus: E 1 [N/ m²]

Loss coefficient of a single damping material: η²

Thickness: d 2 [m]

Density: ρ 2 [kg/m 3]

Young's modulus: E² [N/ m²]

Furthermore, if we set the Young's modulus ratio E² / E¹ = M, the thickness ratio d² / d¹ = T, and the density ratio ρ² / ρ¹ = D, the loss coefficient and Young's modulus of the damping material alone can be calculated from the data of the substrate itself and the data of the composite test piece using the following formula.

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)

Loss factor of composite test specimen: η c
Resonance frequency: f c [Hz]

Free length: ℓ [m]

Substrate loss coefficient: Set η = 1 = 0

Resonance frequency: f i [Hz]

Thickness: d 1 [m]

Density: ρ 1 [kg/m 3]

Young's modulus: E 1 [N/ m²]

Loss coefficient of a single damping material: η²

Thickness: d 2 [m]

Density: ρ 2 [kg/m 3]

Young's modulus: E² [N/ m²]

Furthermore, if we set the thickness ratio d² / d¹ = T and the density ratio ρ² / ρ¹ = D, the loss coefficient and Young's modulus of the damping material alone can be calculated from the data of the substrate itself and the data of the composite test piece using the following formula.

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)

Loss factor of composite test specimen: η c
Resonance frequency: f c [Hz]

Free length: ℓ [m]

Substrate loss coefficient: Set η = 1 = 0

Resonance frequency: f i [Hz]

Thickness: d 1 [m]

Density: ρ 1 [kg/m 3]

Young's modulus: E 1 [N/ m²]

Loss coefficient of a single damping material: η²

Thickness: d 2 [m]

Density: ρ 2 [kg/m 3]

Shear modulus: G [N/ m²]

Furthermore, if we set the thickness ratio d² / d¹ = T and the density ratio ρ² / ρ¹ = D, the loss coefficient and shear modulus of the damping material alone can be calculated from the data of the substrate itself and the data of the composite test piece using the following formula.

Here

however

Only in that case.