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This paper examines the eigenvalues and eigenvectors of a massive rigid square block bonded to an elastic medium. Uncoupled vertical and coupled sliding-rocking modes of vibration are considered. The supporting medium is modeled as an elastic homogeneous or layered, inertial or massless half-space. For a geometrically damped system, the eigenvalues .are complex-valued, while they are purely imaginary in an un-damped system. Numerical results are presented to illustrate the effects of inertial interaction. The data is computed by a procedure that takes into account the mixed boundary conditions at the surface of the half-space.

In this paper a numerical model for the minimum-weight design of a rectangular machine foundation under a harmonic vertical load is presented. The analysis of the dynamics of foundation-soil interaction is based on frequency-dependent dynamic properties of a semi-infinite supporting medium and includes the shape of the foundation plan, the embedment of the foundation into the soil, and hysteretic material damping of the soil. Dimensions of the concrete block are assumed as design variables. Constraints are placed on resonant frequency, vertical displacement amplitude, stresses in the soil and dimensions of the foundation concrete block. A sequential programming method with variable move limits is used to obtain the optimal solution, which is affected by inertia properties of the machine-foundation-soil system, damping from dynamic soil-foundation interaction and local soil conditions. Numerical examples are given to demonstrate the applications of the proposed approach.