Modulus crossover frequency

The open-loop transfer function is third-order type 2, and is unstable for all values of open-loop gain K, as can be seen from the Nichols chart in Figure 6.33. From Figure 6.33 it can be seen that the zero modulus crossover occurs at a frequency of 1.9 rad/s, with a phase margin of —21°. A lead compensator should therefore have its maximum phase advance 0m at this frequency. Flowever, inserting the lead compensator in the loop will change (increase) the modulus crossover frequency. 

Place ujm at the modulus crossover frequency of 2 rad/s and position the compensator corner frequencies an octave below, and an octave above this frequency. Set the compensator gain to unity. Flence... 

Figure 6.35 shows the Bode gain and phase for both compensated and uncompensated systems. From Figure 6.35, it can be seen that by reducing the open-loop gain by 5.4dB, the original modulus crossover frequency, where the phase advance is a maximum, can be attained. 

Required modulus attenuation is 12 dB. This reduces the modulus crossover frequency from 1.4 to 0.6 rad/s. 

Figure 38 Master curves of elastic storage (S, ) and viscous loss (S", o) linear viscoelastic moduli of the 12-arm 12 828 (a) and 64-am 6430 (b) star-PBd polymers in the temperature range from 150 up to -103°C, with reference temperature-83 °C. Solid arrows represent the various transitions and corresponding crossover frequencies (cos. glass to Rouse-like transition cof.. transition to rubber plateau

Below to , the storage modulus G (cu) is (1) independent of frequency (to leading order) and (2) vanishes as the square root of the distance to jamming (Equation 12.44), while the dynamic viscosity is (1) independent of frequency and (2) diverges as the square root of A(j) (Equation 12.45). The question is how do these functions depend on cu and A(]) above the crossover frequency ... 

There are also some far-fetched proposals for the LST a maximum in tan S [151] or a maximum in G" [152] at LST. However, these expectations are not consistent with the observed behavior. The G" maximum seems to occur much beyond the gel point. It also has been proposed that the gel point may be reached when the storage modulus equals the loss modulus, G = G" [153,154], but this is contradicted by the observation that the G — G" crossover depends on the specific choice of frequency [154], Obviously, the gel point cannot depend on the probing frequency. Chambon and Winter [5, 6], however, showed that there is one exception for the special group of materials with a relaxation exponent value n = 0.5, the loss tangent becomes unity, tan Sc = 1, and the G — G" crossover coincides with the gel point. This shows that the crossover G = G" does not in general coincide with the LST. 

Now these expressions describe the frequency dependence of the stress with respect to the strain. It is normal to represent these as two moduli which determine the component of stress in phase with the applied strain (storage modulus) and the component out of phase by 90°. The functions have some identifying features. As the frequency increases, the loss modulus at first increases from zero to G/2 and then reduces to zero giving the bell-shaped curve in Figure 4.7. The maximum in the curve and crossover point between storage and loss moduli occurs at im. 

The polyethenes prepared with catalyst 2 (Fig. 3a) have greatly elevated elastic modulus G values due to LCB compared to the linear polymers shown in Fig. 3b. LCB also shifts the crossover point to lower frequencies and modulus values. The measured complex viscosities of branched polymers (see also Table 2) are more than an order of magnitude higher than calculated zero shear viscosities of polymers having the same molecular weight but a linear structure. The linear polymers have, in turn, t] (0.02 radvs)... 

The two asymptotic temporal power-laws of MCT also affect the frequency dependence of G" in the minimum region. The scaling function Q describes the minimum as crossover between two power laws in frequency. The approximation for the modulus around tlie minimum in the quiescent fluid becomes [38]... 

In the post-gel region, p>pc the polymer exhibits a finite equilibrium modulus of goo. A characteristic frequency cf, can be defined again(see Figure 4) for respective p values. At high frequencies above some crossover of, the sample shows the behaviour of critical gel state and at frequency below it, the sample behaves like a typical viscoelastic solid (G = goo, G"°CQ)at co- O). ra value increases with increasing the extent of cross-linking. 

Zeichner and coworkers [33,34] developed a measure of the breadth of the molecular weight distribution that is based on the curves of storage and loss moduli versus frequency. Based on data for a series of polypropylenes made by Ziegler-Natta catalysts and degraded by random chain scission, they found that the polydispersity index, i.e., the ratio Af /M , was related to the crossover modulus, G, as shown in Eq. 5.8... 

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