Analytical stiffness evolution of involute spline couplings with gap-dependent tooth engagement under combined errors

Combined clearance, manufacturing errors, and misalignment make tooth engagement in involute spline couplings strongly non-uniform and induce pronounced nonlinear stiffness evolution. Existing slice-based formulations usually activate contact once the nominal approach becomes positive and evaluate the meshing geometry near the nominal pitch radius, which obscures the actual gap-closure sequence and weakens robustness near contact-state switching. This work develops an analytical stiffness framework with gap-dependent tooth engagement for involute spline couplings under combined errors. Tooth-dependent initial gaps are introduced to describe staggered contact onset, while the real meshing radius is used to update the working pressure angle, tooth compliance, and moment-arm distribution during deformation. A differentiable contact regularization is further adopted to stabilize Newton iterations in repeated equilibrium solves. Three-dimensional finite-element models generated from a parameterized structured-mesh workflow are used for verification. The analytical predictions agree well with the FE results for both lateral and angular stiffness, with mean relative deviations below about 2.2% for the representative cases considered. The results show that the tolerance-induced gap distribution governs the early-stage nonlinearity and load-sharing heterogeneity, whereas transmitted torque and engagement length increase the global stiffness level and misalignment softens the response. The proposed framework provides an efficient mechanics-based tool for evaluating stiffness evolution and error sensitivity in spline couplings.