The design of beams of cantilever form carrying and end inertia so as to minimize the total mass subject to the constraint that one, two or three of its torsional natural frequencies are fixed at specified values is considered. The problem is stated in variational form with the constraints introduced through Lagrange multipliers. The problem is taken analytically as far as possible and reduces to a set of first order non-linear differential equations. These are integrated numerically. The known solutions for less constrained problems are used as a basis from which to iterate to the required solutions. Optimum profiles and tables of numerical data computed for various examples are given.