Radiation impedance of torsionally vibrating seismic sources

Abstract

With impedance measurements over a sufficiently broad frequency range, it may be able to use the half-space In this paper the study of the radiation impedance function is used to develop an understanding of the interraction of a vibrator and the medium to which it is bonded in terms of physical phenomena occurring in the vicinity of the baseplate.From this analysis methods are suggested for using source related measurements to determine some physical parameters of the medium on which the or rests,The radiation impedance may be thought of as the load the vibrator experiences as it radiates into the medium.An analogy may be drawn between the radiation impedance and the impedance of an electric t.. c t impedance, the load on the current source, is a function of the components in the circuit. The seismic radiation impe the load on vibrator, depends on the components its 11 " , which includes the bas e and the medium on which it rests. 2The impedance of a torsional source is considered in detail in this paper because such a vibrator on the surface of a plane~layered half-space produces only horizontally polarized shear (SH) waves. As a result, the theory is somewhat simplified, and the impedance curves are simpler in form and more easily interpreted than those due to a vertical or horizontal vibrator. The observations made concerning the relationship between the vibrator impedance function and the properties of the medium can be extended to the vertical and horizontal impedance functions.Most of the research on the impedance of a dynamically vibrating torsional source has been directed toward solving the mixed boundary value problem for a rigid source (displacement boundary condition at the source). Relatively little published literature is available on the solution of the simpler boundary value problem that involves a stress condition at the source.The earliest solution for the radiation impedance of an oscillating rigid source was obtained by Sagoci (1944) Collins (1962), Awojobi and Grootenhuis (1965) and Robertson (1967) each derived the same power s s expansion for the problem of a rigid baseplate vibrating on the surface of an elastic half-space valid for low frequencies, using different methods. Robertson (1967) also suggested a method of obtaining the shear modulus of the elastic half-space using the impedance function. Stallybrass (1962, 1967) used variational techniques and asymptotic approxima~ tions to derive an expansion for the rigid source over a half-space valid at low frequencies, and a second expansion for high frequencies. Another high frequency solution has been presented by Thomas (1968), Miller and Pursey (1954) et al., 1955;Bycroft, 1956;Awojobi and Grootenhuis, 1965). Although there appear to be no published radiation impedance calculations for a torsional vibrator over a layered medium, there have been several studies of the response of vertical and horizontal vibrators over layered media. Prominent among these are those by Luco (1974Luco ( , 1976, Kashio (1970) and Wei (1971).Since mos...

Measurement of rheological properties of soft biological tissue with a novel torsional resonator device

“…Here, we use the same measurement principle for characterizing the complex shear modulus of soft biological tissue, using a novel torsional resonator device (TRD). Complex shear modulus is derived from the experimental results using the analytical model of a torsional radiating source on a semi-infinite space (Sagoci 1944;Sneddon 1966;Robertson 1967;Dorn 1980). Thereby, the soft tissue is modeled as linear viscoelastic, homogenous and isotropic.…”

Dynamic Measurements of Soft Tissue Viscoelastic Properties with a Torsional Resonator Device

“…where R is the radius of the contact area. The solution of the boundary value problem with mixed (kinematic and kinetic) boundary conditions of a radiating torsional source, known as Reissner-Sagoci problem [12,13], was derived by Dorn [14], using Hankel Transform methods. The torsional radiation pattern and the torsional mechanical impedance of the medium can be determined for given values of the material parameters G 1 and G 2 for each excitation frequency.…”

“…The soft tissue is modelled analytically as a semi-infinite, homogenous, isotropic medium; a suitable kinematic boundary condition is applied in correspondence of the contact with the resonator. The analytical model consists of a torsional radiating source on a semi infinite space [12,13,14] and is used to extract the material parameters. A mapping procedure and reference tables enable real-time parameter extraction.…”

Analytical study of a disk-type torsional shear-wave radiator immersed in an elastic medium

“…Some reported values of ␣ are as follows: ␣ϭ1.6977 for ϭϪ0.5 case; ␣ϭ1.3030 for ϭ0 case; ␣ϭ0.9463 for ϭ1 case. 16 When comparing the magnitudes of torsional radiation impedance for a given disk driven at various frequencies, this is simply a scaling factor. The magnitude of the rigidity factor decreases as the disk becomes less rigid.…”

“…16. Torsional radiation pattern as a function of distance between the source and the receiver, which is characterized by ϭ0 ͑k s aϭ5.0, circle-R/aϭ10; diamond-R/aϭ5; triangle-R/aϭ2͒.…”