SOME OPTIMIZATION PROBLEMS IN TORSIONAL VIBRATION

The design of beams to maximize a torsional natural frequency for a given total mass is considered. The beams analyzed are in the main cantilevered and of rectangular cross-section but the theory is easily extended to other end conditions and section shapes. A variety of cases is considered involving the inclusion of upper or lower bounds on the section dimensions or the addition of a concentrated end inertia. In each case the problem is stated in variational form with the introduction of constraints through Lagrange multipliers. In most instances the beam has tapered regions and uniform section regions with continuity conditions at the corners between them. Development of the analysis results in non-linear equations for the unknowns; these are solved numerically. The results of the analysis of one of the cases is compared with experimental results for the first torsional natural frequency of a series of beams having the same length, weight and upper and lower bounds on the section width. The optimum profile beam was found to have the highest frequency. © 1978.