Akın Oktav
The ratio of a structure's first two natural frequencies depends on its boundary conditions but not on its size, material density, or absolute stiffness, which multiply every frequency equally and cancel in the quotient. We use this to identify the boundary conditions of a small-scale beam or plate from two ratio measurements, without knowing its exact dimensions or material. The paper first establishes how the ratio $r=ω_2/ω_1$, and the fractional shift $Δr$ it undergoes when a boundary condition changes, respond to the two effects that matter at small scale: transverse shear deformation, through Timoshenko beam and Mindlin plate theory, and small-scale elasticity, through Eringen's nonlocal Timoshenko model. Using a locking-free Rayleigh-Ritz beam solver and a finite-element Mindlin plate solver, each checked against benchmarks, we find an asymmetry. The absolute ratio moves appreciably with thickness, by 12 to 14 per cent over the shear-deformable range, and with orthotropy, by 6 to 9 per cent, but only weakly with nonlocality; the shift $Δr$, by contrast, varies by less than 0.2 per cent over the full nonlocal range and by a few per cent under thickness change. Because $Δr$ is largely independent of these details, a measured shift identifies which boundary-condition transition a deliberate change of support has produced, with a resolution we quantify and verify against molecular-dynamics data. Two spring continua then extend the method to estimating support compliance, with the non-uniqueness of the tip-support case stated explicitly. The method needs no value of the small-scale parameter or the material constants.