A method of analysis is presented for the dynamic response of large machine-foundation systems. The general case of several machines on a common flexible mat is considered. The origin and types of machine exciting forces are discussed. Particular emphasis is given to the unbalanced inertial forces in reciprocating gas compressors. It is with this type of machine that most serious foundation problems occur. The soil is assumed to be homogeneous and isotropic with both inertial and elastic properties. While this simplification is not compatible with the observed properties of most soils, it gives satisfactory results for this application. An outline of a theoretical method of vibration analysis is presented which can be adapted for use with a computer. The method is applied to the general machine-foundation system. An example analysis is made of a single engine driven gas compressor and its foundation. The natural frequencies and amplitudes of motion are calculated. Recommendations are made for the practical application of the analytical procedure based on the present limitations on the accuracy of dynamic soil data.