Reduced-frequency nomograph

FIGURE 14.3 Reduced frequency nomograph of simulated data. (Adapted from Corsaro, R. D. andL. H. Sperling, eds., Sound and Vibration Damping with Polymers, ACS Symposium Series 424, P. T. Weissman and R. P. Chartoff, p. 115, 1990.) 

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Figure 2. Illustrative Viscoelastic Master Curves Represented on Reduced Frequency Nomograph, Using Simulated Data.

Figure 10. Master Curves of PYC Acoustical Material Represented on Reduced Frequency Nomograph. Enlargement of Area A in Figure 7. Empirical WLF Equation Obtained from Shift of E Curves.

The master curve is then represented on a reduced frequency nomograph which allows direct reading of modulus (E or E") or loss tangent as functions of either frequency or temperature. 

As illustrated in Figure 10.14, the master curve allows the extrapolation of data over broad temperature and time ranges. Similar master curves can be constructed with frequency as the variable, instead of time. More elegant still is the reduced frequency nomograph, which permits both reduced frequency and temperature simultaneously see Figure 10.15 (42). 

The reduced frequency nomograph (42,43) is constructed as follows. First, the storage and loss modulus (or tan S) are plotted versus reduced frequency. 

Figure 10.15 The reduced frequency nomograph. Here, the quantity ij s tan 5 (42).

Applications of the reduced frequency nomograph include sound and vibration damping (see Section 8.12) and earthquake damage control, inter-... 

In vibration and acoustic analysis the viscoelastic spectrum are often represented as a master curve known as the reduced frequency nomograph. Figure 14.3 is an example of a nomograph constructed with simulated data. In addition to the thermomechanical information that is usually presented, both absolute temperature and frequency axes are included in the nomograph. The information presented by the inclusion of the temperature and frequency axes is already contained in the master curve of the viscoelastic data. However, it is convoluted within the master curve. In order to obtain frequency data at other tanperatures, a new master curve must be created from the raw data or transformed from the original master curve. 

The nomograph is created by plotting the viscoelastic function vs. reduced frequency. Reduced frequency is defined as where... 

The test results of a material damping test are most useful when placed on a reduced temperature nomograph, which plots the limited number of test results to a graph from which one can obtain the damping properties (modulus and loss factor) at any given combination of temperature and frequency. The WLF equation (51 is used to obtain a nomograph for the results of each test. 

Figure 2 is a reduced temperature nomograph which demonstrates the procedure for reading the nomograph as follows Select a combination of temperature and frequency, for example, 1000 Hz and 75 C. Find the point for 1000 Hz on the right-hand frequency axis. Proceed horizontally to the temperature line for 75 C. At this intersection, draw a vertical line. Then, read the modulus and loss factor values from the appropriate data curve, at the point of intersection with the vertical line. In this example, modulus G(1000 Hz, 75 C) = 8 x 10° U/wr and loss factor (1000 Hz, 75 C) = 1.96. 

Using a computerized data reduction scheme that incorporates a generalized WLF equation, dynamic mechanical data for two different polymers were correlated on master curves. The data then were related to the vibration damping behavior of each material over a broad range of frequencies and temperatures. The master curves are represented on a novel reduced temperature nomograph which presents the storage modulus and loss tangent plots simultaneously as functions of frequency and temperature. ... 

Construction of the nomograph can be computerized so that plots can be generated directly from dynamic mechanical data files (Weissman and Chartoff 1990). As an example of the use of the nomograph, master curves of E and E" for a poly(methyl methacrylate) sample are shown in Fig. 5.58. The data were reduced using the universal WLF constants. From the master curve one can also generate isochronal (fixed frequency) data such as that shown for a frequency of 0.1 Hz in Rg. 5.59. 

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