An Analytical Approximation of the Stress Function for Conical Flywheels

The current paper addresses the lack of explicit analytical solutions for stress evaluations in variable-thickness flywheels by proposing an approximate formulation for conical profiles, where thickness varies linearly along the radius. The main objective was to develop a compact and practical expression to estimate radial and tangential stresses without relying on finite element analysis. Starting from a stress function, the model was simplified under the assumption of a small-thickness gradient, allowing the derivation of a closed-form solution. The resulting expression explicitly relates stresses to geometric and material parameters. To validate the approximation, stress distributions were computed for various outer-to-inner thickness ratios and compared with results obtained through FEA. The comparison, evaluated using the coefficient of determination, mean absolute percentage error, root mean squared error, normalized root mean squared error, and stress ratios, demonstrated strong agreement, especially for moderate-thickness ratios ($1\leqq t_o/t_i\leqq 4.5$). The method was more accurate for radial stress than tangential stress, particularly at higher gradients. The results confirmed that the proposed analytical approach provides a reliable and efficient alternative to numerical methods in the design and optimization of conical flywheels, offering practical value for early-stage engineering analysis and reducing reliance on time-intensive simulations.