Abstract:The differential equations of motion for unidirectional composite non-cylindrical helical Springs with rectangular cross-section, which include the warping deformation of wire cross-section, are first derived using natural curved and twisted anisotropic beam theory. They consist of 14 first-order partial differential equations with variable coefficients. An explicit analytical expression for the warping function of Saint-Venant’s torsion of unidirectional composite bars with rectangular cross-section is also obtained. The natural frequencies and vibration mode shapes of the springs are determined by the use of improved Riccati transfer matrix method. The element transfer matrix used in the solution is calculated using the Scaling and Squaring method and Pad´e approximations. Numerical results show that the warping deformation has a significant influence on the natural frequencies of such springs, which should be considered in the free vibration analysis. Finally, the effects of various parameters on the natural frequencies of unidirectional composite conical springs with rectangular cross-section have been investigated.