Logarithm decrement

Fig. 20. Logarithmic decrement (related to tan 5 and loss modulus) vs temperature for a fluorocarbon dibenzoxazole (148). After drying up to 200°C, the...

Method (h) Logarithmic decrement. Consider the ratio of successive peaks a and 02... 

Tim, temperature of the highest logarithmic decrement X Tg (dso temperature of the inflection point in the DSC diagram... 

A body of a cylindrical or spherical shape is suspended in a melt and oscillating rotational motion is fed to it. A schematic drawing of a viscometer is shown in Fig. 23. This initial oscillation is gradually attenuated by the viscosity resistance. The viscosity is obtained as an absolute value from the logarithmic decrement of the swings of the pendulum s oscillation. Since the sample melt can be completely closed in this method, this is the best method for a melt of high temperature. 

The logarithmic decrement of the oscillations of a pendulum consisting of a crucible containing a test liquid is measured by the oscillating cup method. A schematic drawing of the oscillating cup (vessel) viscometer is shown in Fig. 24. 

Sato et al. " measured the viscosities of some binary and ternary alkali carbonates. Since melt creep must be prevented, a highly sintered alumina crucible was used instead of a gold-plated nickel crucible. Homogeneity of a mixture sample was achieved by gas bubbling. A laser beam is combined with a computer-assisted time counter to obtain the logarithmic decrement. Roscoe s equationi3i has been used for calculation of the viscosity, while it has been claimed by Abe et al. that the viscosities calculated from Roscoe s equation are 0.6-1.5% lower than those from more rigorous equations. 

Damping is often expressed in terms of quantities conveniently obtained with the type of instrument used. Since there are so many kinds of instruments, there are many damping terms in common use, such as the logarithmic decrement A, the half-width of a resonance peak, the half-power width of a resonance peak, the Q factor, specific damping capacity i <, the resilience R, and decibels of damping dB. 

Note 2 Damping factor, is related to logarithmic decrement, 8, by the equation ... 

Note 2 The logarithmic decrement can be used to evaluate the decay constant, p. From the equation for the damping curve of a Voigt-Kelvin solid. 

The damping, expressed as the logarithmic decrement, A, is calculated from the rate at which the amplitude of the oscillation decreases according to the equation ... 

A = (natural) logarithmic decrement of oscillations with film... 

Frequency Dependence of the Logarithmic Decrement. The frequency dependence of 5 can now be found. Putting Eqs. 8.67 and 8.69 into Eq. 8.92 and equating the real and imaginary parts yields two equations which can be solved for ey and 2 in the forms... 

In a study of internal friction in polytetrafiuoroethylene, McCrum (1959a) investigated samples varying from 48% to 92% crystallinity. The dependence of the logarithmic decrement and the torsion modulus... 

Samples C8 and D13 differ in Tg in fact, the Tg (extrapolated) of C8 is about 10°C higher than that of D13. For the blends, a clearly defined maximum of the logarithmic decrement A is observed at about 85°C—i.e.y at the glass transition temperature of the homopolymer PVC. This indicates incompatibility between PVC and CPVC (II). The in-... 

Figure 6.2 Torsional braid analysis (TBA) during isothermal reaction of a diepoxy, DGEBA, with a diamine, bis(p-aminocyclohexyl) methane, from -15°C to 220°C (a) relative rigidity (b) logarithmic decrement (Tg0 =-19°C, Tgoo = 170°C, and Tgge = 49°C). (Enns and Gillham, 1983 -Copyright 2001 - Reprinted by permission of John Wiley Sons, Inc.)...

The loss factor, tan 8, can be measured with the aid of dynamic-mechanical experiments (such as the torsion pendulum). The deformation in such a test varies as indicated in Figure 7.13 the damping follows from the logarithmic decrement , A, it can be easily shown that... 

Yano et al. [53] studied acoustic properties of acetylated Sitka spruce by specific dynamic Young s modulus and by logarithmic decrement. For oven-dried specimens, both the modulus and the decrement have been found to increase. Meanwhile, mechanical properties are generally unchanged and adhesive strength is reduced by acetylation [2]. Furthermore, creep deformation of wood under humidity change is remarkably reduced by acetylation [54]. 

Kevlar aramid fiber has good vibration damping characteristics. Dynamic (commonly sinusoidal) perturbations are used to study the damping behavior of a material. The material is subjected to an oscillatory strain. We can characterize the damping behavior in terms of a quantity called the logarithmic decrement, A, which is defined as the natural logarithm of the ratio of amplitudes of successive vibrations, i.e. 

In order to construct a TTT cure diagram, a series of isothermal cures are performed at different temperatures, and the relative rigidity and logarithmic decrement are recorded vs. time. From each damped wave 1/P and A are extracted the numerical methods used have been described... 

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