Trochoidal paths of rack cutter with a fully rounded-tip for constant clearance 

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... graphs of generating and generated surfaces can be obtained by using a programming language and graphic processor. In this study codes are developed by using GW-BASIC language to obtain the coordinates of the surfaces. GRAPHER 2-D Graphing System is used for displaying computer graphs of the cutters and gears. Also the ANSYS Preprocessor module is used for displaying gear generating process. Illustrative examples are given for both rack- and pinion-type cutters for different types of tool tip geometries. For rack-type generation, types of tip fillet geometry are selected from the study proposed by Alipiev (Alipiev, 2009, 2011) and the related geometries displayed in the table are adopted to the present mathematical model. Table 1 displays the variation of tip geometry of the rack cutters. As illustrated in Table 1, the rack cutter of type-1a has different clearances at its different sides. The side with a higher pressure angle has a lower radius of rounding and a lower clearance. The tooth semi-thicknesses at pitch line of the cutter are different from each other. Design parameters are selected as module m  2 . 5 mm , number of teeth z  24 , left side pressure angle  1  20  , right side pressure angle  2  15  , left side radius of rounding  1  0 . 2  m and right side radius of rounding  2  0 . 3  m . Figure 8 displays the generating cutter of type-1a , generated surface and trochoidal paths of the tip. As illustrated in Fig. 2. and classifed type-1b in Table 1, the cutter has a constant clearance for its all sides. The side with a higher pressure angle has a higher radius of rounding. The tooth semi-thicknesses at pitch line of the cutter are same. This type of cutter is adopted from the standard generating rack to asymmetric gearing. The relation ship between left and right side roundings is  1 ( 1  sin  1 )   2 ( 1  sin  2 ) . Design parameters are selected as module m  2 . 5 mm , number of teeth z  24 , left side pressure angle  1  20  , right side pressure angle  2  15  , left side radius of rounding  1  0 . 38  m and right side radius of rounding  2  0 . 33  m . Generating and generated surfaces and trochoidal paths are illustrated in Fig 9. Rack cutters with asymmetric teeth can also be designed with full rounded tips. The rack cutter of type-2a has a single rounded edge. The side with a higher pressure angle has a lower radius of rounding and a lower clearance. As depicted in Table 1 the centers of the rounded tip are at the center line of the cutter tooth. The tooth semi-thicknesses at pitch line of the cutter are same. Design parameters are selected as module m  2 . 5 mm , number of teeth z  24 , left side pressure angle  1  22 . 5  , right side pressure angle  2  15  , left side radius of rounding  1  0 . 4  m and right side radius of rounding  2  0 . 587  m . Figure 10 displays the generating cutter of type-1a, generated surface and trochoidal paths of the tip. For visual clearity, only the corresponding halves (of secondary trochoids) that contribute to final formation of the generated tooth shape are shown. As classifed type-2b in Table 1, the cutter has a constant clearance for its all sides. The side with a higher pressure angle has a higher radius of rounding. The tooth semi-thicknesses at pitch line of the cutter are different. The relation ship between left and right side roundings is  1 ( 1  sin  1 )   2 ( 1  sin  2 ) . Design parameters are selected as module m  2 . 5 mm , number of teeth z  24 , left side pressure angle  1  22 . 5  , right side pressure angle  2  15  , left side radius of rounding  1  0 . 514  m and right side radius of rounding  2  0 . 428  m . Generating and generated surfaces and trochoidal paths are illustrated in Fig. 11. For visual clearity, only the corresponding halves (of secondary trochoids) that contribute to final formation of the generated tooth shape are shown. The geometric varieties of the rounded corner of pinion-type cutter tooth for generating symmetric and asymmetric involute gear teeth profiles can also be investigated. Illustrated examples for pinion-type generation were given by the present author (Fetvaci, 2011). Table 2 displays possible tip geometries of pinion-type shaper cutters for standard tooth height. As illustrated in Fig. 3. and classifed type-1b in Table 2, the cutter has a constant clearance for its all sides. The side with a higher pressure angle has a higher radius of rounding. The relationship between left and right side roundings is  1 ( 1  sin  1 )   2 ( 1  sin  2 ) . Design parameters are selected as module m  3 mm , number of teeth z  20 , left side pressure angle  1  20  , right side pressure angle  2  15  , left side radius of rounding  1  0 . 25  m and right side radius of rounding  2  0 . 222  m . Generating and generated surfaces and trochoidal paths are illustrated in Fig 13. The shaper cutter of type-2a has a single rounded edge. The side with a higher pressure angle has a lower radius of rounding and a lower clearance. As depicted in Table 2 the centers of the rounded tip are at the center line of the cutter tooth. Design parameters are selected as module m  3 mm , number of teeth z  20 , left side pressure angle  1  20  , right side pressure angle  2  15  , left side radius of rounding  1  0 . 373  m and right side radius of rounding  2  0 . 449  m . Figure 14 displays the generating cutter of type-2a , generated surface and trochoidal paths of the tip. For visual clearity, only the corresponding halves (of secondary trochoids) that contribute to final formation of the generated tooth shape are shown. The shaper cutter with asymmetric involute teeth and with a single rounded edge can not be designed for constant clearance in case of standard tooth height. As illustrated in Fig. 3., the center of the rounding should be on the pressure line of the cutter. As a result, the geometric varieties of pinion-type tool tip is limited for indirect generation. The relative positions of the cutter during generating process can be visualized by using the mathematical of generating surfaces and transformation matrices. The present author used the locus equations of the cutters and obtained illustrations displaying simulated motion path of the cutter during generation by manipulating rolling parameter as   / 4   1   / 4 in the developed code. Each gear gap is produced through successive penetrations of the tool teeth into the workpiece, in the individual generating positions. This simulation can be used to determine the chip geometry (Bouzakis et al., 2008). Figure 15 displays the work gear and simulated motion path of the generating rack cutter with asymmetric teeth. Similiarly, Fig. 16. displays the work gear and simulated motion path of the generating pinion cutter. Figure 17 displays relative positions of the pinion cutter with symmetric involute teeth and a fully-rounded tip. The trochoidal curves exhibits symmetry according to center line of gear tooth space. Generating with a sharp-edge pinion cutter is depicted in Fig.18. In this case, primary trochoids determine the shape of the generated tooth fillet. The secondary trochoids do not exist. Video files displaying generating positions of the cutter can be obtained with a proper software. In this study, ANSYS Parametric Design Language (APDL) is also used for obtaining graphic outputs and animation files displaying the simulated motion path of the generating cutters (ANSYS, 2009). Video files can be seen in the author’s web page: In this study, computerized tooth profile generation of involute gears manufactured by rack- and pinion-type cutters are studied based on Litvin’s vector method. Based on Yang’s application mathematical model of rack cutter with asymmetric involute teeth is given. Trochoidal paths of the rack tool tip are investigated. For pinion-type generation Asymmetric involute teeth is adopted to Chang and Tsay’s application. The developed computer program provides the investigation of the effect of tool parameters on the generated tool profile before manufactured. Trochoidal paths traced by the generating tool tip are investigated. It has been seen that geometric varieties of the rounded corner of pinion-type cutter determines the position of trochoidal paths relative to the center line of tooth space of the generated gear. Because of the position of the center of the tip rounding, there is a limitation on the geometric varieties of pinion-type cutter tip. Based on the given mathematical models, the simulated motion path of the generating cutters are also investigated. The relative position of the cutter to the workpiece has been illustrated. The simulation of shaper ...