Logarithmic decrement equation

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. 

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 ... 

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 this equation, the logarithmic decrement (assuming that A, P ) will be... 

By applying free damped vibrations in torsion pendulum testing the mechanical vibration damping can be expressed in terms of the logarithmical decrement A which can be expressed as the ratio of two sequent amplitudes according to equation 3.13. 

Here is supposed that gas in the bubble follows polytropic process with exponent X This equation was solved in linear approximation by operational method with the aim to analyze small amplitude, natural oscillations of the constant mass bubble in relaxing liquid. It was taken R = Rq + AR, AR/Ro 1, AR exp(ht) with h being the complex natural frequency. Logarithmic decrement. A, and dimensionless frequency, [t, of the oscillations are defined according to formulas... 

If the internal frictions are not dependent on the amplitude, then the plot of the Napierian logarithm of the amplitudes versus the number of cycles is linear with a slope equal to the logarithmic decrement. Moreover, during damping of a sound wave, the anelastic behavior leads to a lag between the stress and strain, and the phase angle, 8, between the two waves is then related to the Napierian logarithmic decrement by the simple equation ... 

The logarithmic decrement can then be related to the specific loss (Equation (4.33)) and hence tan 5. 

It is the main objective of this example to show that some d5mamic characteristics of a structure with dampers, modelled using the above mentioned models is practically identical if both models possess approximately equal possibilities to dissipate energy. This conclusion is supported by the results presented in Table 4. This table contains logarithmic decrements of damping and the peak values of displacements and accelerations of the fourth-floor, calculated from the obtained solutions to the equations of motion for both damper models. 

As already mentioned, the value of the viscosity (obtained in a relative way), is derived fl om measurements of the logarithmic decrement A and tiie period of oscillation T. The working equation for the viscometer reduces to (Kestin etal, 1978a, 1980)... 

FIGURE 9.5 Typical analysis of torsion pendulum, (a) Oscillating trace made by the tip of moment arm gives a period of 1.67 s corresponding to a frequency of 0.600 Hz or 1.20 n rad/s. (b) Plot according to Equation 9.34 gives a logarithmic decrement of 0.23. 

For the record, there are constancies in these equations. The Gompertz curve implies a constant rate of decline in the increments to the logarithm of the observed value. Logistic growth connotes a constant rate of dechne in the decrements to the reciprocal of the observed value. These features become evident when the equations are converted to the form of a modified exponential curve. 

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