Vibrations in Semi-Infinite Solids Due to Periodic Surface Loading

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Abstract
This paper is an analytical study of the behavior of an elastic foundation that is subjected to a periodic loading on a portion of its top surface. The foundation is regarded as an elastic, isotropic, and homogeneous semi-infinite solid, or half-space. The periodic pressure forces, distributed axial-symmetrically over a circular region of the surface, represent the action of a mechanical oscillator which rests on the foundation. This study is essentially an extension of a previous analysis by E. Reissner (8), but the additional complication arising from a change in pressure distribution at the oscillator base is also considered. Three assumptions were made which defined the distribution of pressure as (a) uniform, (b) parabolic, and (c) that produced by a rigid base in the static case, at the surface of the foundation. The resonant frequency, the amplitude of oscillation, and the input power requirement were shown to depend upon the pressure distribution as well as upon the characteristics of the oscillator-foundation system. These characteristics are determined as functions of the radius of the loading area, the static weight of the oscillator, the material constants of the foundation, and the radius of rotation as well as the weight of the rotating mass. The effects of these variables are shown by a series of curves. By use of these curves, it is possible to evaluate the elastic constants of a given foundation by means of suitable tests. These elastic constants may then be used as a basis for computing the critical frequency, the amplitude of oscillation, and the power requirement for a wide range of oscillator-foundation combinations.

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