Modulus curve

Relaxations of a-PVDF have been investigated by various methods including dielectric, dynamic mechanical, nmr, dilatometric, and piezoelectric and reviewed (3). Significant relaxation ranges are seen in the loss-modulus curve of the dynamic mechanical spectmm for a-PVDF at about 100°C (a ), 50°C (a ), —38° C (P), and —70° C (y). PVDF relaxation temperatures are rather complex because the behavior of PVDF varies with thermal or mechanical history and with the testing methodology (131). 

The excellent low temperature properties of FZ have been iadicated ia Table 1. Modulus curves were obtained usiag dynamic mechanical spectroscopy to compare several elastomer types at a constant 75 durometer hardness. These curves iadicate the low temperature flexibiUty of FZ is similar to fluorosihcone and ia great contrast to that of a fluorocarbon elastomer (vinyUdene fluoride copolymer) (Fig. 3) (15). 

It may be seen from Fig. 2.59 that the two modulus curves for temperatures T1 and T 2 are separated by a uniform distance (log aj). Thus, if the material behaviour is known at Ti, in order to get the modulus at time, t, and temperature... 

The exceptionally strong influence of calcium-ions on pectin solutions especially made with HM citrus pectins can be shown by a frequency sweep. The addition of calcium leads to an increase of the complex viscosity. Additionally we can observe a stable trapping of air bubbles in the solution. This effect can not be caused by the increase of viscosity. The frequency sweeps of the solutions give the answer. The storage modulus curves show the significant increase of the elastic shares caused by the addition of calcium-ions. 

Fig. 22 The strength versus the modulus curves of PpPTA fibres calculated with Eq. 58 for three different values of the critical shear strain...

Fig. 24 The strength versus the modulus curves for PBO fibres calculated for three different critical shear stress values and the observed strength of PBO (Zylon) given by the manufacturer... 

Figure 8.9 Third stage of SIM. The sections of the creep modulus curve are shifted parallel to the time axis to produce a single continuous curve. Small corrections are applied to allow for fibre shrinkage and for the thermal history of the material.

Figure 8.10 Fourth stage of SIM. The composite modulus curve of Figure 8.9 is inverted to yield the predicted creep curve for the reference temperature.

Figure 2.2 Illustrative plot of the Lennard-Jones-Devonshire interatomic potential showing the force and the modulus curve for the pair interaction. Positive values indicate repulsion and negative values indicate attraction...

In some case, however, only a flattening of the osmotic modulus curve is observed. Such a case is found with star-branched macromolecules. This observation has rather comprehensively been investigated by Roovers et al. with stars of 64 and 128 arms [172]. The authors give the following explanation. At the point of coil overlap and at somewhat higher concentrations the stars feel the interaction as a quasi colloidal particle. Hence, a steeper increase of the osmotic mod-... 

Both the phase angle and log modulus curves are horizontal lines on a Bode ni as shown in Fig-1Z14. 

Note the very unique shape of the log modulus curves in Fig. 12.19. The lower the damping coefficient, the higher the peak in the L curve. A damping coefficient of about 0.4 gives a peak of about +2 dB, We will use this property extensively in our tuning of feedback controllers. We will adjust the controller gain to give a maximum peak of +2 dB in the log modulus curve for the closedloop servo transfer function X/X. ... 

G. GENERAL TRANSFER FUNCTIONS IN SERIES. The historical reason for the widespread use of Bode plots is that, before the use of computers, they made it possible to handle complex processes fairly easily. A complex transfer function can be broken down into its simple elements leads, lags, gains, deadtimes, etc. Then each of these is plotted on the same Bode plots. Finally the total complex transfer function is obtained by adding the individual log modulus curves and the individual phase curves at each value of frequency. 

Therefore the log modulus curves and phase-angle curves of the individual components arc simply added at each value of frequency to get the total L and d curves for the complex transfer function. 

Equation (13.30) says that we want the output to track the setpoint perfectly for all frequencies, and we want the output to be unaffected by the load disturbance for ail frequencies. Log modulus curves for these ideal (but unattainable) closed-loop systems are shown in Fig. 13.10b. 

In most systems, the closedloop servo log modulus curves move out to higher frequencies as the gain of the feedback controller is increased. This is desirable since it means a faster closedloop system. Remember, the breakpoint frequency is the reciprocal of the closedloop time constant. 

A commonly used maximum closedloop log modulus specification is 4 2 dB. The controller parameters are adjusted to give a maximum peak in the closedloop servo log modulus curve of -1-2 dB. This corresponds to a magnitude ratio of 1.3 and is approximately equivalent to an underdamped system with a damping coefficient of 0.4,... 

A proportional controller merely multiplies the magnitude of at every frequency by a constant. On a Bode plot, this means a proportional controller raises the log modulus curve by 20 logiQ decibels but has no effect on the phase-angle curve. See Fig. 13.13n. 

On a Bode plot (Fig, 13.16), the log modulus curve of G B must pass through the 0-dB point when the phase-angle curve is at —135°. This occurs at tu = 1 radian per minute. The log modulus curve for K,. = 8 must be raised - -9 dB (gain 2,82). Therefore the controller gain must be (8X2.82) = 22.6. 

Once the log modulus curve has been adequately fitted by an approximate transfer function G(J ), the phase angle of G( a) is compared with the experimental phase-angle curve. The difference is usually the contribution of deadtime. The procedure is illustrated in Fig. 14.2. 

A. SCALAR SISO SYSTEMS. Remember in the scalar SISO case we looked at the closedloop servo transfer function G B/ll + GuB). The peak in this curve, the maximum closedloop log modulus L (as shown in Fig. 16.9a), is a measure of the damping coefficient of the system. The higher the peak, the more underdamped die system and the less margin for changes in parameter values. Thus, in SISO systems the peak in the closedloop log modulus curve is a measure of robustness. 

The molecular weight between crosslinks (Me) was determined for each epoxy/amine ratio of the neat resin from the rubbery plateau region of the modulus curve following the Tg region. This can be seen in Figure 13 for several epoxy/amine ratios. The Me values were calculated from the following equation ... 

The value of the modulus and the shape of the modulus curve allow deductions concerning not only the state of aggregation but also the structure of polymers. Thus, by means of torsion-oscillation measurements, one can determine the proportions of amorphous and crystalline regions, crosslinking and chemical non-uniformity, and can distinguish random copolymers from block copolymers. This procedure is also very suitable for the investigation of plasticized or filled polymers, as well as for the characterization of mixtures of different polymers (polymer blends). 

The modulus curves of three blends of PEO E4000 (75, 50, and 25 wt. %) with PVN are shown in Figure 1 along with that for pure PVN. The 25% PEO blend... 

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