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Technical Report on Vibration-Damping Materials and Their Performance Measurement 10

Conversion Nomogram for Frequency

The measured values of the complex elastic coefficients at various temperatures and frequencies are organized as a composite curve with the logarithmic converted frequency f r on the horizontal axis and the complex elastic coefficient on the left vertical axis. Next, a straight line representing the shift factors α T at representative temperatures T-2, T-1, T 0, T 1, T 2, ... is drawn in the same figure, with the horizontal axis as the converted frequency f r and the right vertical axis as the frequency f (f r = α T ･f). This allows for the direct reading of the values of the complex elastic coefficient and loss coefficient at any temperature [T] and frequency [f] from the composite curve. The diagram organized in this way is called a converted frequency nomogram. The converted frequency nomogram is recommended as a graphical representation of the complex elastic modulus of damping materials in the international standard ISO 10112.

  • Figure 4: An example of a conversion frequency nomogram.

1. Creation of a frequency nomogram.

(1) Removal of the influence of the substrate:
In the case of a two-layer beam with damping material applied to one side, the measured loss coefficient becomes the characteristic of the composite consisting of the damping material and the substrate. The properties of the material in isolation (loss coefficient, elastic modulus) necessary for nomogram analysis are calculated to remove the influence of the substrate using the theoretical "RKU fundamental equation" of the beam (Refer to calculations of Young's modulus, etc.).

(2) Temperature - Application of Frequency Conversion Law:
Since the damping characteristics of damping materials (dynamic characteristics) depend on both temperature and frequency, a three-dimensional representation is necessary when trying to express elastic modulus and loss coefficient concerning these two parameters. What is important here is the temperature-frequency conversion law. The temperature-frequency conversion law holds well for viscoelastic materials (especially in the glass transition region) and allows the dynamic characteristics of damping materials to be expressed in two dimensions by replacing changes in temperature with changes in frequency (conversion frequency). Taking the reference temperature T 0 as the glass transition temperature T g, the WLF equation (Equation 1) results in approximately C 1 = 17.44, C 2 = 51.6. WLF is well established for amorphous polymers. On the other hand, when crystalline molecules or fillers are present, the error increases, and it is said that the error is significant below the glass transition temperature T g.

(3) Master Curve (Synthetic Curve):
Applying the temperature-frequency conversion rule, multiple temperature data are synthesized into a continuous single master curve. The conversion frequency nomogram is completed by plotting the loss coefficient and Young's modulus of the obtained viscoelastic material on the conversion frequency axis. Additionally, to make this complex characteristic easier to handle, curve fitting is performed to a pre-prepared model equation of viscoelastic properties, representing the viscoelastic characteristics with several parameters.

2. Procedure for Creating a Nomogram

The elastic modulus (Young's modulus) at a specific temperature (T-2 to T +2) of the damping characteristics obtained using the aforementioned RKU method is shown in Figure 5, and the loss coefficient is shown in Figure 6 as an example.

  • Figure 5 Frequency vs. Elastic Modulus
  • Figure 6 Frequency vs. Loss Coefficient
  • Figure 7 Conversion Frequency vs. Elastic Modulus and Loss Coefficient

Here, we consider the temperature behavior of the viscoelastic materials shown in Figures 5 and 6. From this figure, we appropriately estimate the glass transition temperature and determine the reference temperature on the temperature-frequency conversion side (T 0) in the vicinity of T g as follows.

  1. Determine a specific temperature T 1 (the measurement temperature of one specimen).

  2. Calculate α T.

  3. Calculate the converted frequency f r by multiplying the resonance frequency of the specimen measured in the experiment (at temperature T 1) by α T.

  4. Taking the conversion frequency on the horizontal axis and the dynamic characteristics of the damping material alone (loss coefficient, Young's modulus) on the vertical axis, a part of Figure 7 at T 1 can be obtained.

  5. Repeating this operation for other measurement temperatures T 1 and T 2, etc., will complete Figure 7. If the arrangement of the data in this figure is not smooth, change the reference temperature T 0 and recreate Figure 7, repeating this operation until the data plot connects smoothly.

  6. Finally, by taking the (resonance) frequency on the vertical axis on the right and the temperature on the horizontal axis at the top, the conversion frequency nomogram is completed.

3. Points to note when creating a nomogram.

When creating a nomogram, it is necessary to pay attention to the points shown below.

  1. Since the subject is a two-layer type of Euler-Bernoulli beam, the thickness ratio of the test specimen should not exceed three times.

  2. In the case of a two-layer type, care should be taken to prevent shear (sliding) deformation.

  3. The calculation formula does not take into account the effects of elongation.

  4. The loss coefficients of iron and aluminum are less than 0.001, so the loss coefficient of the substrate is assumed to be zero.

  5. The influence of measurement errors and material errors becomes significant during the calculation process.

  • When calculating the Young's modulus, frequency has a squared effect.

  • When calculating the Young's modulus of the substrate, thickness affects it to the power of two, and length affects it to the power of four.

When seeking the conversion frequency nomogram, it is often the case that measurement data at high temperatures cannot be used, especially. The reason for this is that when determining the material's nomogram, the value inside the RKU calculation formula (23. Calculation of Young's modulus, etc. "For two-layer composite plates" above formula) becomes very small or negative due to the square root. Therefore, to ensure calculation accuracy, a criterion of α 1.1 is established. Due to this judgment condition, it is necessary to reject measurement data when creating the nomogram. As one of the methods for expanding α (countermeasures), the loss coefficient of the test specimen should be made as much as possible to ensure that η > 0.01.

4. How to read the frequency conversion nomogram.

An example of a conversion frequency nomogram is shown in Figure 8. The left vertical axis represents the loss coefficient (η, ● (red circle)) and the storage modulus (Young's modulus E’, ×(blue cross)), as well as the loss modulus E”, ● (green circle). The right vertical axis represents the frequency f. The lower horizontal axis is the conversion frequency f r, which has no physical meaning but includes temperature conditions. The upper horizontal axis represents temperature (°C).

  • Figure 8 Example of Conversion Frequency Nomogram

Next, I will outline the procedure to determine the loss factor and Young's modulus of a damping material at a frequency of 100 Hz and a temperature of 20 °C.

  1. Look horizontally at the 100 Hz line on the vertical axis on the right side.

  2. Find the intersection point A of this line and the diagonal line on the upper horizontal axis (temperature axis) at a temperature of 20 ℃.

  3. Draw a perpendicular line from the intersection point A, move horizontally to the intersection point B of the loss coefficient curve fit line, and read the value C on the loss coefficient axis to obtain approximately η = 0.3.

  4. Draw a perpendicular line from the intersection point A, move horizontally to the intersection point D of the Young's modulus curve fit line, and read the value E on the Young's modulus axis, which gives approximately E’ = 1.8E10 (N/m 2).

  5. Draw a perpendicular line from the intersection point A, move horizontally to the intersection point F with the curve fit line of the loss modulus, and read the value G on the loss modulus axis to obtain approximately E″= 5.2E9 (N/m 2).