Free vibration analysis of non-cylindrical helices with, variable cross-section by using mixed FEM

The objective of the study is to present a mixed finite element (FE) formulation for the free vibration analysis of non-cylindrical helices with variable cross-section. Non-cylindrical helix geometry was well approximated through the variable arc lengths and linear variations in curvatures via shape functions. The mixed FE model for non-cylindrical helices was developed by using the exact cylindrical helix geometry and corresponding field equations. The element and consistent mass matrices are attained under these assumptions. The element matrix is derived based on the Timoshenko beam theory and the effects of rotary inertia are involved into the mass matrix. Although the formulation is quite simple, the generated element is capable to provide highly accurate solutions for the conical, barrel and hyperboloidal geometries of non-cylindrical helices. The results of presented FE model based on an approximated geometry approach are in a good agreement with the other studies in the actual literature. Some original examples were generated and solved for the literature as well.