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Technical Report on Vibration-Damping Materials and Their Performance Measurement 8

22. About the window functions used in the measurement of loss coefficients

It is expected that the frequency response function will change when measuring loss coefficients using window functions. The figure below shows the lower limit values of the loss coefficients when each window function is applied.

From the previous figure, the following can be said.

  1. FLAT-TOP windows are not suitable for loss coefficient measurement.

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

  3. Traditionally, it has been said that "when the signal is asynchronous, use the Hanning window," but for measuring loss coefficients (when determining η values greater than -13 dB from the peak), the rectangular window can measure accurately up to the smallest loss coefficient. However, if using a rectangular window results in decreased coherence, a tapered window should be used.

  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 a Hanning window without zooming, the measurement limit 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, and (2) performing a frequency zoom, thus confirming the necessity of zoom analysis.

  6. Another utility of zoom analysis is that it extends the time frame, which reduces the distortion of impulse response. Furthermore, since zoom analysis increases frequency resolution, a reduction in errors in this area can also be expected. It is also believed that increasing the number of FFT points can have a similar effect. Recently, there have been developments in calculating spectra with 65,536 points of FFT for 25,600 lines, making it easier to use methods that do not require zoom analysis when the loss coefficient is relatively high (η: around 0.01).

23. Calculation of Young's modulus, etc.

This text presents the formulas for calculating Young's modulus from measurements of pure materials, as well as the use of the RKU (Ross, Kerwin, Unger) equation to derive the loss coefficient and Young's modulus from measurements of composite materials (2-layer and 3-layer) to determine the loss coefficient and Young's modulus of a single damping material.

solid materials

Let the resonant frequency be f n (Hz), the half-width 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/m 3).

If the damping is small, ignore 1/8 Δ f n2.

Also, the loss coefficient η is

here

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

number of times 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

 

the following + π

However, in the case of the cantilever beam method, the formula (1)["img-damp-6-13"]Input the length excluding the grip length from the total length. If using central vibration anti-resonance, refer to formula (1).["img-damp-6-13"]to["img-damp-6-13"]Enter 2.

When using resonance with central vibration, determine the resonance frequency and use the following value for θ n.

number of times 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

the following + π

["img-damp-6-13"]Enter the total length.

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

number of times 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

 

the following + π

["img-damp-6-13"]Enter the total length.

Also, at a higher level, the difference between θ n+1 and θ n almost coincides with π.

The equation, in which the factor θ n of the neighboring order and the density ρ have been eliminated, is useful when the order is unknown.

In the case of a two-layer composite board.

Loss coefficient of composite test piece: η c
Resonant frequency: f c［Hz］

Length of free length: ℓ［m］

Loss coefficient of base material: η 1＝ 0

Resonant frequency: f i［Hz］

Thickness: d 1［m］

Density: ρ 1［kg/m 3］

Young's modulus: E 1［N/m 2］

Loss coefficient of damping material alone: η 2

Thickness: d 2［m］

Density: ρ 2［kg/m 3］

Young's modulus: E 2［N/m 2]

Furthermore, if we denote the Young's modulus ratio E 2 / E 1 = M, the thickness ratio d 2 / d 1 = T, and the density ratio ρ 2 / ρ 1 = D, the loss coefficient and Young's modulus of the damping material can be calculated using the following formula from the data of the base material itself and the data of the composite test specimen.

However, here

It is.

However, it holds when α ≧ 1.1.

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

Loss coefficient of composite test piece: η c
Resonant frequency: f c［Hz］

Length of free length: ℓ［m］

Loss coefficient of base material: η 1＝ 0

Resonant frequency: f i［Hz］

Thickness: d 1［m］

Density: ρ 1［kg/m 3］

Young's modulus: E 1［N/m 2］

Loss coefficient of damping material alone: η 2

Thickness: d 2［m］

Density: ρ 2［kg/m 3］

Young's modulus: E 2［N/m 2]

Let the thickness ratio be d 2 / d 1 = T, and the density ratio be ρ 2 / ρ 1 = D. Then, the loss factor and Young's modulus of the damping material can be calculated using the following formula from the data of the base material itself and the composite test specimen.

Here,img-8-13as written

However, here

holds true at the time of.

Shear Modulus and Loss Factor of Sandwich-Type Composite Board (Damping Steel Plate)

Loss coefficient of composite test piece: η c
Resonant frequency: f c［Hz］

Length of free length: ℓ［m］

Loss coefficient of substrate: η 1＝ 0 assumed

Resonant frequency: f i［Hz］

Thickness: d 1［m］

Density: ρ 1［kg/m 3］

Young's modulus: E 1［N/m 2］

Loss coefficient of damping material alone: η 2

Thickness: d 2［m］

Density: ρ 2［kg/m 3］

Shear modulus: G［N/m 2］

Let the thickness ratio be d 2 / d 1 = T, and the density ratio be ρ 2 / ρ 1 = D. Then, the loss factor and shear modulus of the damping material can be calculated from the data of the base material itself and the data of the composite test specimen using the following formula.

here

However

is limited to the time.