fixedfree

Fixedfree, often written fixed-free, is a boundary condition used in structural mechanics and vibration analysis to describe a slender element that is fixed at one end and free at the other. The configuration is also known as a cantilever boundary condition and is central to the analysis of bending, deflection, and dynamic response. This arrangement is commonly contrasted with fixed-fixed or fixed-pinned configurations in structural design and analysis.

In the Euler-Bernoulli beam model, the fixed end enforces zero transverse displacement and zero slope (y(0) = 0 and y'(0) = 0). The free end enforces zero bending moment and zero shear force (M(L) = 0 and V(L) = 0), meaning the beam experiences no internal bending moment or shear at the tip in the absence of applied loads.

Fixedfree beams have characteristic natural frequencies and mode shapes that depend on material properties, cross-section, length, and density. The fundamental mode has a single antinode at the free end. The exact frequency values are determined numerically, with well-known dimensionless constants βn for cantilever beams (β1 ≈ 1.875, β2 ≈ 4.694, etc.).

Applications span mechanical design and engineering analysis, including cantilevered shafts, robotic arms, and MEMS devices where a beam or lever is anchored at one end. The fixedfree condition also serves as a baseline in modal testing, vibration isolation, and structural health monitoring to compare against other boundary conditions.

See also: cantilever, boundary condition, Euler-Bernoulli beam theory, modal analysis.