This paper presents a detailed discussion of the development of the matrix equations used in the vibration analysis of a lumped-mass approximation of a multi-anchor piping system or foundation structure. The investigation is presented in two main sections, the first of which presents a formulation of the matrix eigenvalue problem for small oscillations. The development of the stiffness matrix is presented in the second section. The coefficients derived by the M. W. Kellogg Co. for the solution of pipe stress problems are utilized here, as well as the matrix transformation methods developed by J. E. Brock in “A Matrix Method For Flexibility Analysis of Piping Systems”. A sample four-anchor foundation was analyzed and the results were in close agreement with measured results published by V. H. Neubert and W. H. Ezell in “Dynamic Behavior of a Foundation-Like Structure”. The procedures presented in this paper will theoretically apply to any piping system of ai;ty degree of complexity, but practical limitations are imposed by the size of presently available digital computers. / Master of Science