An efficient digital computer method is presented for determining the natural frequencies and modes of undamped free vibration of plane single-branch structures. The structural formulation is based on use of the stiffness matrix, and the resulting eigenvalue problem is solved by a Sturm-sequence procedure. This approach requires considerably fewer arithmetic operations than other standard methods. A general computer program has been written which has proved effective for the solution of a range of problems of practical interest, including the vibration of a multistorey tower, a tall chimney and a long-span arched bridge. Numerical results are presented for the latter structure.