The nonlinearity of helical springs: An energy-based approach

TL;DR

This work develops a general energy-based description for helical springs, accounting for two end degrees of freedom: the spring height via the pitch parameter $χ$ and the end rotation via the number of turns $N$. The authors derive the total elastic energy as $U(χ,N) = \frac{(d/2)^4 π^3}{2L} [ 2G (Nχ - N_0χ_0)^2 + E (N√{1-χ^2} - N_0√{1-χ_0^2})^2 ]$, with separate expressions for torsion and bending energies, and consider both fixed and freely rotating ends. They obtain the force and torque from this energy, including a closed-form expression for the unwinding condition $N_{free}(χ)$ and the corresponding energy $U_{free}(χ)$, demonstrating nonlinear corrections to Hooke's law. Experimental verification with steel springs confirms the free-end model and yields realistic elastic moduli ($G ≈ 82$ GPa, $E ≈ 201$ GPa), validating the energy-based approach and its potential for advanced mechanics education and precise dynamic analyses.

Abstract

A general expression for the elastic potential energy of a helical spring is derived using the basic concepts of elasticity theory and geometry. Both the translational and the rotational displacement of the spring's moving ends are considered. The resulting expression is employed to derive the general relation between force and spring height, and corrections to Hooke's law are discussed. The results can be used to estimate the magnitude of non-linear effects in realistic situations, to predict how springs unwind under load, and to determine the spring constant when spring ends are allowed to rotate freely. This energy-based approach is suitable for inclusion in advanced mechanics courses. The results are validated via measurements on steel springs using a simple experimental setup appropriate for use in physics laboratories at all levels.

Figures (12)

• Figure 1: The upper photo shows the spring in its maximally compressed state with a chalk line marking parts of its outer surface. In the lower photo the same spring has been stretched and twisted by $4\pi$. Note that the chalk line remains on the outer surface of the spring indicating the validity of our assumption 5.
• Figure 2: Wire element of length $L$ and diameter $d$ subject to torsion. Left: Side view of the wire. Right: Cross-section of the wire. The points initially located along the dashed line in the relaxed wire are displaced into a helix (solid curve) when the wire is under torsion. The opposite directions of rotation of the two wire ends are indicated by arrows. The wire torsion angle $\gamma_{\mathrm{torsion}}$ and the wire surface shear angle $\delta^{\circ}_{\mathrm{shear}}$ are related by $\gamma_{\mathrm{torsion}} = (2L/d) \delta^\circ_{\mathrm{shear}}$. The shaded region of the cross-section indicates the surface element used in the integration of the potential energy due to torsion, see Appendix \ref{['app:torsion']}.
• Figure 3: Element of a curved wire with diameter $d$, center line length $L_0$, and center line curvature radius $R$: The center line length $L_0$ is assumed to be invariant upon bending, that is under changes in $R$, while the lengths of the innermost and the outermost arcs $L_\pm$ change with $R$. The circle to the right is the cross-section of the wire, the shaded region indicates the surface element used in the integration of potential energy due to bending, see Appendix (\ref{['app:bending']}).
• Figure 4: Diagram clarifying the notations used in this paper: The hypotenuse of the small triangle represents one coil of the spring, $\ell^2 = (D\pi)^2 + h^2$. The hypotenuse of the large triangle (not to scale, dashed) represents the whole spring.
• Figure 5: (Color online) Longitudinal cross-section of a helical spring with wire of diameter $d$ and length of wire per turn $\ell$. Black circles (which are actually slightly elliptical in b) and c)) are the intersections of the wire with the plane of the drawing. Blue dotted curves are the projections of the wire center line onto the plane of the drawing. Gray dashed lines are the longitudinal axes of the spring. a) Spring in the physically inaccessible state with overlapping turns of wire, $\chi=0$. b) Spring in the maximally compressed state (${\mathrm{mcs}}$). c) Spring in a realistic state with $\chi=1/10$. Gray ellipses are the projections of the wire cross-sections onto the plane of the drawing.