Approximation of exponential curves for CNC machining of toolholders used in ultrasonic machining

In ultrasonic machining (USM), a transducer converts electrical energy to high-frequency mechanical motion which is transmitted to a tool via a mechanical coupler known as the horn or toolholder. The toolholders have tapering cross section designed to increase the amplitude of the tool stroke, resulting in higher material removal rates. Different tapering toolholders have been designed according to the type of curves governing the variation in the cross section along the axis of the toolholder. Common among these are conical, stepped or multi-step cylindrical, and exponential toolholders. The resonant length of the exponential toolholder for a required amplification ratio and toolholder material is the simplest to compute. In terms of the section area ratio, maximum strain, and bending stiffness, it is also among the most feasible with high magnification factor. With advancement in NC technology, it is now feasible to produce the exponential toolholder by CNC turning. This paper presents an approach to approximate the exponential curve required for CNC machining of the exponential toolholder. Fast algorithms have been developed to approximate the exponential curve as a composite curve of linear or circular segments, for a given tolerance. FEM analysis has also been used to determine the minimum tolerance required to approximate the exponential curve to achieve the best resonance for the given amplification ratio and exponential profile. Computed results have been compared with actual prototypes produced.