New modeling method of spiral bevel gears with spherical involute based on CATIA

Transcription

1 New modeling method of spiral bevel gears with spherical involute based on CATIA HONG Zhaobin, YANG Zhaojun, ZHANG Xuecheng, WANG Yankun College of Mechanical Science and Engineering, Jilin University, Changchun, China ABSTRACT Based on the generating principle of generating line of spiral bevel gears with spherical involute, a new method for building the model of spiral bevel gears with spherical involute is presented by utilizing the modules of part design, assembly design and kinematic simulation in CATIA. In the part design module of CATIA, the models of base cone, tangent surface, and generating line are built respectively. And the built models are assembled in the assembly environment; the simulation of pure rolling is accomplished in the environment of kinematics; and the model of the gear surface is built using the function of tracks. Then the solid model of the spiral bevel gear is built through the functions of transferring the model s file format, multi-section sweep and solid fill. Finally, the analysis to the performance of the bevel gear transmission is conducted. In the environment of the assembly design, the solid model of the gearwheel and the pinion are inserted respectively, and then the assembly is accomplished by applying the constraints. And in the environment of kinematics, the meshing simulation of the bevel gear pair is accomplished by applying angle drive. The research indicates that this method can build the model of spiral bevel gears accurately and rapidly, and the model can illustrate the gearing correctly. Keywords: CATIA, spiral bevel gear, spherical involute, modeling, meshing simulation 1. INTRODUCTION Spiral bevel gears with spherical involute have characteristics such as high contact ratio, powerful load capacity and steady transmission and so on, thus they are widely used in the equipments of aviation, navigation and national defense, and mechanical products such as vehicles, machine tools, engineering mechanics and mining mechanics. [1-4] Meanwhile, the formation mechanism of tooth surface of the spiral bevel gear is comparatively complicated. In order to illustrate the spiral bevel gear from the three-dimensional viewpoint, and implement analysis of strength and stress and implement loaded tooth contact analysis (LTCA), it is especially important to build the model of the spiral bevel gear. Until nowadays, a lot of studies have been carried out for building the model of spiral bevel gears home and overseas. Zhang Hua [5] implemented the design of processing parameters using local synthesis and tooth contact analysis (TCA), and accomplished the three-dimensional model by computer simulation. J.AI-Daccak [6] built the model of the bevel gear by sweeping the tooth profile of spherical involute. From the viewpoint of the fundamental law of gearing, Ji Zhenhai [7] proposed a modeling method for the spiral bevel gear which is independent of the machining method. Li Jingcai [8] proposed a modeling method for the spiral bevel gear based on the machining method and the engagement theory. The methods above build the model mainly in two forms: one is the geometrical simulation based on the machining process of spiral bevel gears, the other is accomplished by building mathematical model of the tooth surface. This paper proposed a method for three-dimensional modeling which utilizes the functions in CATIA such as tracks, multi-section sweep and junction based on the generation principle of generating line. This method can build the solid model of the spiral bevel gear with spherical involute rapidly and accurately, and explore a new way to build the more complex models. hongzhaobin@163.com Fourth International Seminar on Modern Cutting and Measurement Engineering, edited by Jiezhi Xin, Lianqing Zhu, Zhongyu Wang, Proc. of SPIE Vol. 7997, 79970Y 2011 SPIE CCC code: X/11/$18 doi: / Proc. of SPIE Vol Y-1

2 2. GEOMETRICAL PARAMETERS OF BEVEL GEAR MODEL The geometrical parameters for building a spiral bevel gear model are divided into initial parameters and computing parameters. The initial parameters are shown in Tab. 1. Table 1: Initial parameters of the model Symbolic names Symbols Number of the pinion teeth Z 1 Number of the gearwheel teeth Z 2 Transverse normal module m Mean spiral angle β m Normal pressure angle Other parameters needed for modeling the bevel gear can be calculated based on the initial parameters in Tab. 1. And they are shown in Tab. 2. Table 2: Computing parameters of the model Symbolic names Symbols Computing expressions Pitch angle δ arctan (Z 1 /Z 2 ) Outer cone distance R e m Z 1 /2 sinδ 1 Face width b R e /3 Tooth trace radius R 5R e /6 Dedendum angle θ f arctan [(h * a +C * ) m/r e ] Addendum angle θ a arctan (h * a m/r e ) Base cone angle δ b arctan {(sinδ 1 ) 2 /[(tanα p ) 2 +(cosδ 1 ) 2 ]} 1/2 Root angle δ f δ θ f Face angle δ a δ +θ a α n 3. MODELING OF TOOTH SURFACE When relative motion of the base cone and its tangent surface is pure rolling, the involute helicoid generated by the arc tooth trace s intertwining around the base cone on the tangent surface is the tooth surface of the spiral bevel gear with spherical involute. Based on the formation theory of tooth surface, in the assembly environment of CATIA, the pure rolling model between the base cone and its tangent surface is built, then the movement of tooth trace is swept using the function of tracks, finally the complete tooth surface of bevel gear is formatted through the function of fragmentation. This paper uses an example to illustrate the process of modeling the tooth surface concretely. The initial parameters of the spiral bevel gear are shown in Tab. 3. Table 3: Initial parameters of the spiral bevel gear Z 1 m β m /( ) α n /( ) Hand of spiral Left-hand The tooth number of gearwheel which meshes with the pinion is Z 2 =60, and other parameters needed for modeling can be calculated based on the initial parameters above. They are presented in Tab. 4. Table 4: Computing parameters of the spiral bevel gear Pitch angle δ/( ) Outer cone distance R e /mm Face width b/mm Face angle δ a /( ) Dedendum angle θ f /mm Addendum angle θ a /( ) Tooth trace radius R/mm Root angle δ f /( ) Base cone angle δ b /( ) Since δ b >δ f, the value of the root angle is chosen as δ f =21.19 [9]. Based on the data above, the 3D models of the base cone and its tangent surface--surface (Q) are built. And in the assembly design module, the assembly of pure rolling is implemented. As shown in Fig. 1, the base cone and the surface (Q) are tangent to line OA, meanwhile the apex and surface (Q) coincide at point O. Proc. of SPIE Vol Y-2

3 On the surface (Q), the arc generating line CD is built with the radius R= and circle center O 1, which is shown in Fig. 2. The center distance is calculated to be OO 1 = based on the spiral angle β=30. A Base cone O Surface (Q) O 1 R O C Figure 1: Assembly drawing of pure rolling D Figure 2: Position of generating line Based on the preparing work above, append the command of scroll curve joining to make arc CD which is on the surface (Q) roll purely on the base cone. Then the tracks of the arc are swept by the function of tracks, which is shown in Fig. 3. The tracks function of CATIA can only sweep points and straight lines, thus the arc CD is substituted by several straight lines which is close to the arc. The maximum geometric deviation between them can be controlled within 0.065mm through analysis. In the module of wireframe and surface design, the operation of multi-section sweep is applied on the movement tracks of the arc CD. And then the surface between the root cone and the face cone is sectioned by the function of fragmentation. The convex side model of this spiral bevel gear is shown in Fig. 4. The direction of pure rolling of surface (Q) on base cone is reversed by appending angle drive once more. And then build the concave side model of the spiral bevel gear using the same method. The model is shown in Fig. 5. Figure 3: Tracks of the arc CD Figure 4: Convex side model Figure 5: Concave side model 4. BUILDING OF THE BEVEL GEAR MODEL In the assembly environment of the base cone and its tangent surface, the convex side model and the concave side model are inserted respectively. And then the assembly is implemented. As shown in Fig. 6. Through calculating, the normal circular thickness at the middle point of pitch cone s face width is S mm = (0.5π cosβ m +2xtanα n +x t ) m m mm (1) where x and x t are the modification coefficients in the radial direction and the axial direction, and m m is the normal module at the middle point of pitch cone s face width. By the function of angle constraint, the relative position between the convex side and the concave side is determined based on the calculated value of the normal circular thickness at the middle point of pitch cone s face width, which is shown in Fig. 7. And the length of the arc EF is equal to the value of the thickness. Proc. of SPIE Vol Y-3

4 Convex side Concave side E F Base cone Figure 6: Assembly of tooth surfaces Figure 7: Completed assembly drawing of tooth surfaces The assembly file is saved as a igs format file, that is to transfer the surface s assembly drawing into the surface s part drawing. Then in the module of wireframe and surface design, we open the file and accomplish the closed surface modeling of the bevel gear s single tooth by supplementary function. As shown in Fig. 8. In order to fill the closed surface model into a solid model, the first step is to join the 6 surfaces which will form the tooth into a whole closed surface model using junction function. Then fill the surface model into a solid model using solid fill function. As shown in Fig. 9. Figure 8: Surface model of the tooth Figure 9: Solid model of the tooth Based on the tooth number Z 1 =25, the operation of circular array is applied on the solid model of the single tooth. And the reference element of the array is the central line of the base cone. And thus all the tooth models are built. As shown in Fig. 10. In the axial cross plane of the tooth model, the profile curve of the root cone and the gear shaft are drawn. And the whole model of the bevel gear is made by the function of revolved body. As shown in Fig. 11. Figure 10: Solid models of all the teeth Figure 11: CATIA model of the pinion 5. MESHING SIMULATION OF THE GEAR PAIR Using the modeling method in this paper, the gearwheel model which meshes with the pinion is built. As shown in Fig.12. And the modeling parameters of the gearwheel is presented in Tab. 5. Table 5: Modeling parameters of the gearwheel Modeling parameters Numeric values Tooth number Z 2 60 Transverse normal module m 2 5 Mean spiral angle β m2 /( ) 30 Normal pressure angle α n2 /( ) 20 Pitch angle δ 2 /( ) Face width b 2 /mm Base cone angle δ b2 /( ) Root angle δ f2 /( ) Face angle δ a2 /( ) Proc. of SPIE Vol Y-4

5 In the assembly design module of CATIA, the solid model of the gearwheel and the pinion are inserted respectively. By the constraint of the cone apex and the transmission angle, the assembly of the gear pair is accomplished. As shown in Fig. 13. Then in the kinematics module, an angle drive is applied on the gear pair using the function of gear joints. And thus the meshing simulation of the gear pair is conducted with the transmission ratio of 1:2.4. Figure 12: CATIA model of the gearwheel Figure 13: Assembly drawing of the gear pair In order to carry out the tooth contact analysis in the conjugate transmission process of the bevel gear pair, we hide the solid parts except the teeth. As shown in Fig. 14. And then the tooth contact analysis of the meshing simulation process is carried out. The analysis result indicates that: the solid model of the spiral bevel gear built in this paper can achieve conjugate transmission in the form of line contact in the state of zero load and zero distortion. And the contact line is a part of arc on the surface (Q), which is the internal common plane of the base cones of the gearwheel and the pinion. The arc AB in Fig. 14 is a contact line at a certain moment in the process of gear pair s conjugate transmission. Surface (Q) B A Figure 14: Contact analysis of the gear pair 6. CONCLUSIONS A new modeling method of spiral bevel gears with spherical involute is proposed in this paper, which is based on CATIA. The method can simulate the generating motion of the bevel gear s convex side and concave side correctly, therefore building the exact model of the bevel gear. In order to validate the model which has been built, this paper carries out the simulation of the gear assembly and the meshing transmission. And a tooth contact analysis is implemented during the meshing simulation process of the gear pair. The result indicates that the spiral bevel gear model built in this paper can implement conjugate transmission correctly in the form of line contact. ACKNOWLEDGMENTS This project is supported by Research Fund for the Doctoral Program of Higher Education of China (Grant No ) and Key Program of Technology Development of Jilin Province (Grant No ). Proc. of SPIE Vol Y-5

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