Closed-Form Optimal Profiling for Maximizing the Longitudinal Fundamental Frequency of Rods under Area Constraints

Abstract Maximizing the longitudinal fundamental frequency of rod-like structures is a critical objective for broadening the operational bandwidth of high-frequency transmission systems. This paper derives a closed-form analytical solution for the globally optimal cross-sectional profile of a rod subject to prescribed length and area constraints (Amin and Amax). Utilizing a variational formulation based on Rayleigh's principle, the optimal profile is identified as a novel multi-segment configuration comprising an exponential transition bridged by uniform segments at the geometric limits. A pivotal theoretical finding is that the maximum fundamental frequency scales linearly with the logarithm of the area ratio implying that the frequency enhancement is mathematically unbounded given sufficient geometric freedom. The analysis is further extended to complex boundary conditions, revealing the existence of critical tip-mass thresholds that discretely simplify the optimal topology into two-segment or uniform profiles. To validate engineering applicability, this theory is implemented in the design of an electrodynamic shaker's moving coil. Experimental harmonic response tests confirm that the optimized geometry achieves a 42.4% increase in the first natural frequency compared to traditional designs. This work provides a generalized theoretical framework and a rigorous design methodology for maximizing the dynamic performance of variable cross-section components.