Kinematic Analysis of 3D Printing Mechanisms

The paper presents a novel technique for the kinematic analysis of a geared mechanism manufactured through 3D printing technology. In the syllabus of the undergraduate 3D Printing course, one of the student projects is to manufacture a spur Norton type gear box which includes a planetary mechanism for engaging its three planets, thus three output velocities for the gear box are obtained. The connected elements (carriers, gears, planets) are fabricated without any involvement of assembly operation. At the prototype stage, a CAD model of the mechanism is created and a search for abilities and limitations of the mechanism are required. The analysis of abilities of motion includes determination of mechanism mobility (DOF). A step by step simple technique is presented for determination of the rank for the matrix of coefficients from the kinematic equations. Thus, the DOF is the difference between the number columns and the rank previously determined. The steps required in the analysis are: numbering of links and joints, graph attached to mechanism, matrix of incidence cycles-nodes in graph, and determination of its rank by using Mathematica commands. For the set of base cycles are automatically generated independent scalar equations. The matrix, denoted velocity matroidal, has the coefficients for the mechanism’s unknown absolute angular velocities-determined based on an analogy to a system of parallel forces from static equilibrium, angular velocities being considered analog to the “forces“. The relations between the input-output links’ absolute angular velocities are determined. The coefficients in the kinematic equations for velocities are written as function of gear ratios. The number of gear teeth can be selected also for the desired input-output speed ratios.