A General Modeling Methodology for the Quasistatic Behavior of Spiral Torsion Springs

Due to their robustness, compactness, simplicity, and the possibility of nonlinear torque curves, spiral springs are being increasingly contemplated for industrial application. Recent manufacturing technologies and materials allow for the creation of spiral springs of various shapes and geometries able to provide the required torque curves. Modeling the behavior of this kind of springs is highly complex due to the strong nonlinearities arisen from large deflections and the possibility of coiling of strip length around the spring barrel or arbor. For this reason, up to our knowledge, existing models only provide design features such as deflection and torque curve for the simplest strip geometries, and fewer models supply, only if no strip coiling occurs, reactions at the strip-barrel and strip-arbor clampings. In addition, to our knowledge, just semi-empirical models for strip-barrel, -arbor and -strip contact forces or friction torques were available. In this work, we introduce a novel general, an analytical quasistatic model for the calculation of all the above spring characteristics for any length-dependent strip material and initial geometry and strip cross-sectional shape and for any barrel and arbor radii. The strip deflection curvatures are calculated minimizing the sum of elastic and gravitational potential energies under geometrical constraints associated with eventual strip coiling. Once the curvatures are calculated, the spring internal, contact, and reaction forces can be straightforwardly calculated by solving the elastica differential equations. Friction is taken into account by evaluating the contact conditions at the strip coiled sections.