406 D.C. Fyler et al. Fig. 37.1 A schematical representation of a double planet planetary gear set Even though, there are a number of studies that develop design methodologies based upon a mesh-by-mesh approach, the literature on the design methodologies for double-planet gearsets based upon a system level approach is sparse. Kwon et al. developed an automated design search for single and double planet planetary gear sets utilizing a loaded tooth contact analysis tool and a mesh by mesh approach [3]. Harianto and Houser studied the effects of various micro-geometries on gear set performance [4]. Houser et al. also developed a method which was applied to optimize the design of helical gears [5]. Methods for using genetic algorithms and finite element analysis were used by Padmanabhan et al. to determine optimal spur gear pair designs [6]. Jeong developed an initial gear design tool using an artificial neural network [7]. In this study, the well-known statistical technique, Design of Experiments (DOE), is used in order to systematically select the right combination of computational experiments and later to create a Response Surface (RS) of all potential design configurations within a prescribed design space [10]. These limited numbers of designs are then modeled using Planetary2D, a deformable body contact model capable of analyzing planetary gear sets in two dimensions [11]. Outputs including root mean square (RMS) transmission error (TE), first harmonics of the transmission error, and gear tooth root maximum bending stresses (Von-Mises stresses used) from these models will be extracted to assess the strengths and weaknesses of potential design candidates in an automated and comparative fashion. Finally, equations are fit to the resulting data to predict the response of intermittent designs that were not modeled. An optimization method which balances the responses of the gearset is applied using a predicted relationship between design factors and system operating parameters. 37.2 Selection of the DOE Technique Various Design of Experiment (DOE) techniques exist in order to develop a set of experiments that accurately explore a given design space and it is critical that the proper one is chosen for each distinctive application. Factorial designs are generally used as a screening DOE method used to determine which factors from a given set have a significant impact on outputs based upon their corresponding sensitivities with respect to the input variables [10]. Previous studies indicate that all the factors being adjusted in this study have a considerable impact on the performance of gear sets so screening with a factorial design is not necessary in this case. Response surface designs are used in succession to factorial designs in order to refine models and identify curvature in the response surface. Central Composite designs are a subset of response surface designs which are capable of fitting a full quadratic model. Figure 37.2 shows a visualization of a central composite design in three factors. The cube in Fig. 37.2 represents a design space, where each black point represents a corner point and the corresponding green points and red points represent axial and center points respectively. It serves as a visual representation of a response surface design with three factors. By offsetting the axial points from the surface of the cube by some distance ˛, the design space is expanded to sphere. For this study a value of ˛ D2:8284is used. This increases the levels of each factor from three to five and allows for the model to capture cubic behavior in the responses [10]. The black points are referred to as corner points and observe the effects of combinations of different factor adjustments. Lastly the red points are the center points where all the factors are kept at their base levels. In this study a central composite full design (CCFD) is used to map the
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