Effect of warping on natural frequencies of symmetrical cross-ply laminated composite non-cylindrical helical springs

Free vibration problem of symmetrical cross-ply laminated composite non-cylindrical helical springs with rectangular cross-section is investigated. The effect of the warping deformation of wire cross-section on natural frequencies is first considered in the formulation. The differential equations of motion for the springs, which consist of 14 first-order partial differential equations with variable coefficients, are derived using naturally curved and twisted anisotropic beam theory. The natural frequencies of the springs are found from improved Riccati transfer matrix by iteration. The element transfer matrix is calculated by the use of the Scaling and Squaring method and Pad´e approximations. Three examples are presented for different types of springs with rectangular cross-section under clamped-clamped boundary condition. The accuracy of the proposed method has been compared with the FE-results using three-dimensional solid layered element (Solid 46) in ANSYS code. Numerical results reveal that the warping deformation has a significant influence on the natural frequencies, which should be considered in the free vibration analysis of the springs. Finally, the effects of various parameters and different stacking sequences on the natural frequencies for symmetrically laminated barrel springs with rectangular cross-section have also been studied.