Free vibration analysis of cylindrical helical springs with noncircular cross-sections

The free vibration analysis of cylindrical helical springs with noncircular cross-sections is carried out by means of an analytical study. In the governing equations of motion of a spring, all displacement functions and a generalized warping coordinate are defined at the centroidal principal axis. The effects of the rotational inertia, axial and shear deformations, including torsion-related warping deformations, are also considered in the formulations. Explicit analytical expressions that give the vibrating mode shapes are derived by rigorous application of the symbolic computing package MATHEMATICA, and the Muller root search method is used to determine the natural frequencies. Numerical examples are provided for springs with elliptic, rectangular and equilateral triangular cross-sections, and subjected to clamped–clamped and clamped–free boundary conditions. The natural frequencies are presented for a range of geometric parameters. In the case of elliptical wires, results are presented for the aspect ratio λ=a/b   ranging from 3/5 to 5/3, the helix pitch angle α¯ from 5 to 12.5, the number of active turns n from 6 to 12, and the ratio of cylinder radius to minor axis R/2a ranging from 20/3 to 50/3. Validation of the proposed model has been achieved through comparison with a finite element model using three-dimensional solid elements and the results available in published literature, which in these cases indicates a good correlation.