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Gear shaping is, currently, the most prominent method for machining internal gears, which are a major component in planetary gearboxes. However, there are few reported studies on the mechanics of the process. This paper presents a comprehensive model of gear shaping that includes the kinematics, cutter–workpiece engagement (CWE), and cutting forces. To predict the cutting forces, the CWE is calculated at discrete time steps using a tridexel discrete solid modeler. From the CWE in tridexel form, the two-dimensional (2D) chip geometry is reconstructed using Delaunay triangulation (DT) and alpha shape reconstruction. This in turn is used to determine the undeformed chip geometry along the cutting edge. The cutting edge is discretized into nodes with varying cutting force directions (tangential, feed, and radial), inclination angles, and rake angles. If engaged in the cut during a particular time-step, each node contributes an incremental force vector calculated with the oblique cutting force model. Using a three-axis dynamometer on a Liebherr LSE500 gear shaping machine tool, the cutting force prediction algorithm was experimentally verified on a variety of processes and gears, which included an internal spur gear, external spur gear, and external helical gear. The simulated and measured force profiles correlate closely with about 3–10% RMS error.

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This project was intended to serve two purposes, the first was to design an origami fold pattern which would be able to fold in a way that mimics an existing shape-morphing mechanism’s kinematics. The second purpose was to demonstrate a general method for using origami to morph one triangle into another triangle. Our method was to design the footprint (the outer edges) based on comparing the dimensions in the two triangles. Fold lines were then added to the footprint that were necessary to morph the larger dimensions into the smaller ones. Finally, fold lines were added to the interior of the design to satisfy the rules of origami i.e. to permit the morphing motion to happen. The result is an origami fold pattern that can fold to each of the required triangles.