In order to analyze the rectangular machine foundations subjected to a vertical harmonic force, a simple mathematical model based on variational principle is developed. A surface footing is resting on a layer of soil deposit with a non-compliant rock or rock-like material at the base. Governing equations of the problem are obtained through the minimization of energy using Hamilton's principle. Equations are nondimensionalized and written in terms of nondimensional parameters which are very useful in engineering practice. Some of the graphs which are very useful and necessary for the design are generated for the first time to be used by the engineers.