Analytical prediction of formed geometry in multi-stage single point incremental forming

Abstract

Single Point Incremental Forming (SPIF) is a die-less forming process that can be economically used for low volume production of sheet metal components. One of the limitations of SPIF is the maximum wall angle that can be formed in a single stage. To overcome this limitation, Multi-stage Single Point Incremental Forming (MSPIF) is used to form components with large wall angles. When the tool is moved from out-to-in during any stage, material present ahead of it (towards the centre of the component) moves down rigidly. If this rigid body displacement is not considered during tool path generation for MSPIF, it leads to stepped/unwanted features. Predicting the component geometry after each stage helps in monitoring the shape being developed and in turn is useful in designing intermediate stages to form required final geometry with desired accuracy. In the present work, a simple methodology is proposed to predict rigid body displacement based on tool path and process parameters (tool diameter, incremental depth, sheet thickness) used. Tool and sheet deflections due to forming force are also considered to predict final geometry of the component. Proposed methodology is validated by comparing predicted profiles with experimentally measured profiles of high wall angle axisymmetric components formed using different materials and sheet thicknesses. Predicted profiles are in good agreement with experimental results.

Appendix A. Rigid body displacement prediction proposed by Malhotra et al. [9] and Xu et al. [13]

In this model the rigid body displacement for OI pass is given by:

$$ {Z}_{RBD-OI}=\varDelta y-\left(\frac{L}{\gamma}\right)\left(1-\frac{2\left[E(R)-E\left({\varphi}_0,R\right)\right]}{K(R)-F\left({\varphi}_0,R\right)}\right) $$

(7)

where ∆y is the distance between contact point in n + 1th stage and its projection on to the profile after n^th stage (Fig. 14b), L is the length of profile in n^th stage from the projected point to the base (Fig. 14a), γ is a constant calibrated using FEA, K (R), E (R) are the complete elliptic integrals of first and second kind respectively and F (φ ₀, R), E (φ ₀, R) are the incomplete elliptic integrals of first and second kind respectively, φ ₀, R are the functions of θ ₁ , θ ₂ , S where θ ₂ is the wall angle at the contact point in n + 1th stage, θ ₁ is the wall angle at the projection of contact pint on to the n^th stage profile (Fig. 14b) and S is the distance from component opening to the contact point along n + 1th profile (Fig. 14b).