Nonlinear modelling of helical springs

The nonlinear response of helical springs can be very rich and challenging to be predicted, due to the mechanics of the coiled wire, the effect of large motion and the possibility of contact events. Moreover, there are some situations where the springs are purposely loaded beyond the elastic limit. In those cases, simulation techniques available in the literature and adopted in industrial practice show some limitations, either because they cannot include all those effects at once, or either because the computational cost would be very high. The aim of this research is to provide and validate an alternative approach, addressing separately the nonlinearities induced by geometry and contact, and those due to the material properties. The combined effect of three-dimensional large displacements and contact between coils is considered, deriving an innovative lumped model, based on the equivalent beam approach and multi-mass model. This formulation is particularly advantageous in all those cases where springs undergo large coupled longitudinal - transverse motion and coils are eventually clashing. Several nonlinear static and transient dynamic analysis are run and results are compared to nonlinear finite elements, to literature data and to analytical solutions, when available. The efficient formulations of tangent stiffness and contact, together with the few elements required, make the lumped model faster to run than its finite elements counterpart, indicating it as a valid option for the implementation in multi-body environment. The experimental tests that have been conducted confirmed the strength of the proposed model, outlining a major difficulty well-known in the literature: the modelling of the effects due to end turns in the springs. However, due to its characteristics, the proposed model can be easily modified to incorporate those effects, eventually implementing a model update procedure, that would not be practical for more expensive simulation techniques. Nonlinear material response of compression and extension helical springs is also studied, to predict the permanent deflection of components loaded beyond the elastic limit. An important experimental campaign supported this investigation. It is found that, with some care, the permanent deformation can be predicted considering an elasto-plastic loading phase and an elastic unloading. Under this statement the problem reduces to the determination of the elasto-plastic response to an applied deflection. This is achieved implementing a kinematic model for cross-section deformation, considering rotation due to torsion moment and shearing due to direct shear, together with the effect of wire curvature on the length of material fibre. The elasto-plastic response of the model is validated with solid finite elements and with tests on real springs. When comparing measured and predicted permanent deflections, it is found that the knowledge of material curve is crucial for the accuracy of the results.