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Technical Report: Vibration Damping Materials and Their Performance Measurement 10

25. Conversion Frequency Nomogram

The measured values of the complex elastic modulus at various temperatures and frequencies are organized as a composite curve with the converted frequency f r on the horizontal axis (logarithmic scale) and the complex elastic modulus on the left vertical axis. Next, by plotting a straight line (f r = αT ･f) with the shift factor αT as the proportionality constant at representative temperatures ..., T-2, T-1, T0, T1, T2, ... on the same figure, with the converted frequency f r on the horizontal axis and the frequency f on the right vertical axis, the values of the complex elastic modulus and loss coefficient at any temperature [T] and frequency [f] can be directly read from the composite curve. A figure organized in this way is called a converted frequency nomogram. The converted frequency nomogram is recommended in the international standard ISO 10112 as a graphical representation of the complex elastic modulus of vibration-damping materials.

  • Figure 4: An example of a converted frequency nomogram

1. Creation of a converted frequency nomogram

(1) Elimination of the influence of the substrate:
In two-layer beams with damping material attached to one side, the measured loss coefficient represents the properties of the composite structure consisting of the damping material and the base material. The properties of the individual materials (loss coefficient, elastic modulus) necessary for nomogram analysis are calculated by removing the influence of the base material using the "RKU fundamental equations" (see calculations for Young's modulus, etc.), which are the theory of beams.

(2) Application of the temperature-frequency conversion rule:
The damping properties (dynamic properties) of damping materials depend on both temperature and frequency, so a three-dimensional representation is necessary to express the elastic modulus and loss coefficient in terms of both parameters. The temperature-frequency conversion rule is crucial here. This rule holds well for viscoelastic materials (especially in the glass transition region), and by replacing temperature changes with frequency changes (converted frequency), the dynamic properties of damping materials can be expressed in two dimensions. Taking the reference temperature T0 as the glass transition temperature Tg, the WLF equation (Equation 1) yields approximately C1 = 17.44 and C2 = 51.6. WLF holds well for amorphous polymers. On the other hand, the presence of crystalline molecules or fillers increases the error, and the error is said to be large below the glass transition temperature Tg.

(3) Master curve (composite curve):
By applying the temperature-frequency conversion rule, multiple temperature data points are combined into a single continuous master curve. The converted frequency nomogram is then completed by plotting the loss coefficient and Young's modulus of the resulting damping material on the converted frequency axis. Furthermore, to make these complex characteristics easier to handle, the curve is sometimes fitted to a pre-prepared viscoelastic model equation, and the viscoelastic properties are expressed using several parameters.

2. Procedure for creating a nomogram

Figure 5 shows the modulus of elasticity (Young's modulus) of the damping characteristics obtained using the aforementioned RKU formula at a specific temperature range (T-2 to T +2), and Figure 6 shows the loss coefficient as an example.

  • Figure 5 Frequency vs. Elastic Modulus
  • Figure 6 Frequency vs. Loss Factor
  • Figure 7. Converted frequency versus elastic modulus and loss coefficient.

Here, we consider the temperature-dependent behavior of the viscoelastic material shown in Figures 5 and 6. The glass transition temperature is appropriately estimated from these figures, and the reference temperature (T0) on the temperature-frequency conversion side is determined as a vicinity of Tg, as follows.

  1. Determine a specific temperature T1 (the measurement temperature of one test specimen).

  2. Calculate αT.

  3. The resonant frequency of the test specimen measured in the experiment (when the temperature is T1) is multiplied by αT to calculate the converted frequency f r.

  4. If we plot the converted frequency on the horizontal axis and the dynamic characteristics (loss coefficient, Young's modulus) of the damping material itself on the vertical axis, we obtain a portion of Figure 7 at T1.

  5. Repeating this operation for other measurement temperatures T1, T2, etc., will complete Figure 7. If the data in this figure is not smoothly arranged, change the reference temperature T0 and create Figure 7 again, repeating this operation until the data plots are smoothly connected.

  6. Finally, plotting the (resonant) frequency on the right-hand vertical axis and temperature on the upper horizontal axis completes the converted frequency nomogram.

3. Points to note when creating a nomogram

When creating a nomogram, the following points should be kept in mind.

  1. Since the target is a two-layer Euler-Bernoulli beam, the thickness ratio of the test specimens should not be increased (within 3 times).

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

  3. The calculation formula does not take into account the effect of growth.

  4. Since the loss factors for iron and aluminum are less than 0.001, we assume that the loss factor for the base material is 0 (zero).

  5. Measurement errors and material errors have a significant impact during the calculation process.

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

  • When calculating the Young's modulus of a substrate, the thickness is affected by the square of the length, and the length by the fourth power.

Furthermore, when calculating the converted frequency nomogram, measurement data, especially at high temperatures, is often unusable. This is because, when calculating the nomogram of a material, the square root in the RKU calculation formula (the formula shown above in "Calculation of Young's Modulus, etc., in the case of a two-layer composite panel" in section 23) becomes extremely small or negative. For this reason, a judgment condition of α 1.1 is introduced to ensure calculation accuracy. This judgment condition necessitates discarding measurement data when creating the nomogram. Therefore, one method of increasing α (a countermeasure) is to ensure that the loss coefficient of the test specimen is as close to η > 0.01 as possible.

4. How to read a converted frequency nomogram

Figure 8 below shows an example of a converted frequency nomogram. The left vertical axis represents the loss coefficient (η, ● (red circle)), storage modulus (Young's modulus E', ×(blue cross)), and loss modulus E'', ● (green circle)), while the right vertical axis represents the frequency f. The lower horizontal axis is the converted frequency fr, which includes temperature conditions, although it has no physical significance. The upper horizontal axis represents temperature (°C).

  • Figure 8 Example of a converted frequency nomogram

Next, we will show the procedure for determining the loss coefficient and Young's modulus of a single damping material at a frequency of 100 Hz and a temperature of 20 °C.

  1. Look horizontally at the 100 Hz line on the right-hand vertical axis (frequency axis).

  2. Find point A, the intersection of this line and the diagonal line on the upper horizontal axis (temperature axis) where the temperature is 20°C.

  3. By drawing a perpendicular line from intersection point A, and shifting the intersection point B with the loss coefficient curve-fit line horizontally, we obtain approximately η = 0.3 by reading the value C on the loss coefficient axis.

  4. By drawing a perpendicular line from intersection point A and shifting the intersection point D with the Young's modulus curve-fit line horizontally, and reading the value E on the Young's modulus axis, we obtain approximately E' = 1.8E10 (N/ m²).

  5. By drawing a perpendicular line from intersection point A and shifting the intersection point F with the curve-fit line of the loss modulus horizontally, and reading the value G on the loss modulus axis, we obtain approximately E″ = 5.2E9 (N/ m²).