Numerical determination of minimum mass structures with specified natural frequencies

The problem of the axial vibration of a cantilever beam is investigated both analytically and numerically. The mass distribution that minimizes the total mass for a given value of the frequency parameter β is determined using both the sequential ordinary gradient‐restoration algorithm (SOGRA) and the modified quasilinearization algorithm (MQA). Concerning the minimum value of the mass, SOGRA leads to a solution precise to at least 4 significant digits and MQA leads to a solution precise to at least 6 significant digits. Comparison of the optimal beam (a variable‐section beam) with a reference beam (a constant‐section beam) shows that the weight reduction depends strongly on the frequency parameter β. This weight reduction is negligible for β → 0, is 11.3 per cent for β = 1, is 55.3 per cent for β = 1.4, and approaches 100 per cent for β → π/2.