An analytical method is developed for determining the natural frequencies of a composite system which consists of a uniform beam with a concentrated mass, spring, and dashpot attached at any point along the length of the beam. The method makes use of a series expansion in terms of the set of orthogonal functions which represent the normal modes of vibration of the beam alone. Numerical examples are given for a beam with a combined concentrated mass and spring, for a beam with two concentrated masses, and for a beam with an attached dashpot.