Vibration analysis of cylindrical helical springs considering warping deformation effect

The free vibrational behavior of cylindrical helicalsprings with rectangular cross sections is analytically investigated in thispaper based on spatial curved beam theory. In the differential equations ofmotion of the springs, all displacement functions and a generalized warpingcoordinate are defined at the centroid principal axes and the warping effectupon the natural frequencies is also considered in present study. Explicitanalytical expressions which give the vibrating mode shapes are derived byrigorous application of the symbolic computing package MATHEMATICA and aprocess of searching is used to determine the exact natural frequencies.Numerical examples are provided for the springs with the rectangularcross-section and clamped-clamped and clamped-free boundary conditions. Thefree vibrational parameters are chosen as the ratio of the width to height(a / b = 0.6 - 1.7) for a rectangular cross section, the number of activeturns (n = 6 - 12), the helix pitch angle (\bar \alpha = 5 - 12.5^\circ ) and the radial of cylinder (R = 4 - 10mm) in a wide range.Validation of the proposed model has been achieved through comparison with afinite element model using three-dimensional solid elements (Solid 45) andthe available literature, showing a good agreement among them.