Modulus attenuation

Identify what modulus attenuation is required to provide an aeeeptable phase margin and lienee determine the spaeing between 1/72 and l/T i (i.e. 6dB attenuation requires a one oetave spaeing, 12 dB attenuation needs a two oetave spaeing, ete.). 

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

Assuming that a number of NMR data sets (e.g., 2-D or 3-D maps of displacement vectors resulting from an external periodic excitation) from an object are acquired, the remaining difficulty is their reconstruction into viscoelastic parameters. As written in Section 2 the basic physical equation is a partial differential equation (PDE, Eq. (3)) relating the displacement vector to the density, the attenuation, Young s modulus and Poisson s ratio of the medium. The reconstruction problem is indeed two-fold ... 

The attenuation may be expressed by making the wavenumber complex (this would be k — ia in eqn (6.12)), and the velocity (= w/k) may also be written as a complex quantity. This in turn corresponds to a complex modulus, so that the relationship v - /(B/p) is preserved indeed the acoustic wave equation may be written as a complex-valued equation, without the need for the extra term in (6.11). Complex-valued elastic moduli are frequency-dependent, and the frequency-dependent attenuation and the velocity dispersion are linked by a causal Kramers-Kronig relationship (Lee et al. 1990). 

Thus a measurement of the ultrasonic properties can provide valuable information about the bulk physical properties of a material. The elastic modulus and density of a material measured in an ultrasonic experiment are generally complex and frequency dependent and may have values which are significantly different from the same quantities measured in a static experiment. For materials where the attenuation is not large (i.e., a ca/c) the difference is negligible and can usually be ignored. This is true for most homogeneous materials encountered in the food industry, e.g., water, oils, solutions. 

The ring-opening metathesis polymerization of dicyclopentadiene was monitored by ultrasonic spectroscopy.16 The thermoset poly(dicyclopentadiene) is formed by ringopening and cross-linking in a reaction injection molding system. A reaction cell with a plastic window was constructed for use with pulse echo ultrasonic spectroscopy. Realtime measurements of density, longitudinal velocity, acoustic modulus and attenuation were monitored. Reaction kinetics were successfully determined and monitored using this technique. 

Another limitation on acoustic properties is expressed by the Kramers-Kronig (KK) relations, which are general relations between the real and imaginary parts of a complex function. These relations were originally derived for optics but can be applied in many other areas as well. The essence of the relations is that the real and imaginary parts of the function are not independent of each other but one may be calculated from an integral of the other. As applied to complex modulus, the specific form of the relations is given elsewhere in this book (J. Jarzynski, A Review of the Mechanisms of Sound Attenuation in Materials). 

In each case the attenuation of sound can be formally represented by defining a complex wavenumber, where the sound attenuation coefficient is the imaginary part of the wavenumber. The complex wavenumber also leads to the definition of a complex sound speed and a complex dynamic elastic modulus. 

Eg.8 has the same form as the relation between the (real) elastic modulus and sound speed in a lossless medium. From Eqs.6-8 it follows that the sound attenuation coefficient is related to the complex elastic modulus,... 

From Eq.21 it follows that the imaginary party of the effective bulk modulus K is much larger than the imaginary part of K. Therefore, sound waves are strongly attenuated in this type of material. The viscoelastic polymer-air microbubbles composites are particularly useful in the design of broadband, transition type anechoic coatings for underwater... 

While the design of such coatings is straightforward, selection of appropriate materials is not. Usually materials with the properties required for a particular application are not readily available, and some custom laboratory fabrication is necessary. This usually involves selecting a polymer composite which somewhat approximates the required physical properties. Then minor alterations to the chemical constituents or fillers are used on a trial basis and the acoustic properties (some combination of Young s Modulus and damping factor, sound speed, attenuation, density, and front-face reflectivity) of these sample formulations are measured. This continues until a suitable formulation is achieved. 

The second problem facing the researcher, and the one which is the subject of this paper, is to determine which, if any, effective modulus theory accurately predicts the acoustic wave velocity and attenuation in a microscopically or macroscopically inhomogeneous material. 

Modulus is calculated (2) from the attenuation coefficient per echo by the relation. 

In materials which are highly attenuating, the particle velocity and particle displacement are out of phase, so the elastic modulus and density of the material are complex and dynamic ( .e. frequency dependent). For many materials, the attenuation coefficient is fairly small ( .e. a co/c), so the particle velocity and displacement are in phase and Eq. 9.4 can be replaced with ... 

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