Analytical and numerical optimization of broaching tool tooth distribution

This paper presents a detailed mathematical analysis of the effect of tooth distribution on the stability of broaching operations. Analytic and numeric techniques are employed to analyse a simplified one-degree-of-freedom mechanical model of variable pitch broaching, to assess its stability, and to find the optimal tooth distances maximizing robustness against harmful self-excited chatter vibrations. A novel modelling approach, considering a theoretical infinitely long broaching tool, draws parallels with variable pitch milling, and analytic formulas for tuning milling cutters are extended to broaching tools. For a further increase of robustness, and to take feasibility constraints into account, a goal function based on the semi-discretization and spectral collocation techniques is implemented in a direct numeric optimisation framework. A new, detailed derivation of the corresponding parameter gradients is presented to enable the use of gradient descent techniques. Since broaching is inherently a time-limited process, optimal parameters found in this manner are then validated via time domain simulations to show the desirable transient behaviours achievable by this ideal tuning of the tool geometry.