Axial Static Load Dependence Free Vibration Analysis of Helical Springs Based on the Theory of Spatially Curved Bars

Abstract

This work addresses an accurate and detailed axial static load dependence linearly elastic free vibration analysis of cylindrical helical springs based on the theory of spatially curved bars and the transfer matrix method. For a continuous system, governing equations comprise coupled vibration modes namely transverse vibrations in two orthogonal planes, torsional and axial vibrations. The axial and shear deformation effects together with the rotatory inertia effects are all considered based on the first order shear deformation theory and their effects on the frequencies are investigated. The effects of the initial stress resultants on the frequencies are also studied. After buckling, forward-shifting phenomenon of higher frequencies is noticeably demonstrated. It is also revealed that a free/forced vibration analysis with an axial static load should not be performed individually without checking buckling loads.

The General Dispersion Relation for the Vibration Modes of Helical Springs

“…Yildirim [19] demonstrates that Equation (1), which is usually used to determine the spring stiffness, is only valid for spring angles α ≤ 10 • . Figure 3a shows the spring geometry indicating the position of the angle α. Yildirim [19] demonstrates that Equation (1), which is usually used to determine the spring stiffness, is only valid for spring angles α ≤ 10°.…”

“…Yildirim [19] demonstrates that Equation (1), which is usually used to determine the spring stiffness, is only valid for spring angles α ≤ 10 • . Figure 3a shows the spring geometry indicating the position of the angle α. Yildirim [19] demonstrates that Equation (1), which is usually used to determine the spring stiffness, is only valid for spring angles α ≤ 10°. Figure 3a shows the spring geometry indicating the position of the angle α. Yildirim [19] indicates that Equations ( 1) and ( 2) only take into account the effect of the torsion of the spring section when compressing it.…”

“…Figure 3a shows the spring geometry indicating the position of the angle α. Yildirim [19] demonstrates that Equation (1), which is usually used to determine the spring stiffness, is only valid for spring angles α ≤ 10°. Figure 3a shows the spring geometry indicating the position of the angle α. Yildirim [19] indicates that Equations ( 1) and ( 2) only take into account the effect of the torsion of the spring section when compressing it. However, as Yildirim [19] points out when the axial compression force acts, both stresses due to torsion and bending moment and forces normal and shear to the section appear.…”

“…Figure 3a shows the spring geometry indicating the position of the angle α. Yildirim [19] indicates that Equations ( 1) and ( 2) only take into account the effect of the torsion of the spring section when compressing it. However, as Yildirim [19] points out when the axial compression force acts, both stresses due to torsion and bending moment and forces normal and shear to the section appear. These last three effects are negligible for spring angles α ≤ 10° where the torsional effect dominates.…”

Buckling Analysis of Rectangular Beams Having Ceramic Liners at Its Top and Bottom Surfaces with the help of the Exact Transfer Matrix

“…Therefore in the design of columns, determination of the critical buckling loads becomes an inevitable stage. method [26][27][28], the homotopy perturbation method [28][29][30][31][32][33], Adomian decomposition method [28,34], the transfer matrix method [35][36][37][38][39], the stiffness matrix method [39], the fictitious load method [40], the modified vibration modes [41] and much more [42][43][44][45][46]. In the solution of more complex problems, some of the solution methods mentioned above may also be used in a combined manner.…”