Geometrical design of variable ratio tooth profile based on Boolean subtraction operation and a novel modification method

Authors

  • Ziru Niu 1. College of Mechanical and Electronic Engineering, Shandong Agricultural University, Tai’an 271018, Shandong, China; 2. Shandong Provincial Key Laboratory of Horticultural Machinery and Equipment, Tai’an 271018, Shandong, China
  • Jianlei Li 1. College of Mechanical and Electronic Engineering, Shandong Agricultural University, Tai’an 271018, Shandong, China
  • Shuai Xin 1. College of Mechanical and Electronic Engineering, Shandong Agricultural University, Tai’an 271018, Shandong, China
  • Liangliang Zou 1. College of Mechanical and Electronic Engineering, Shandong Agricultural University, Tai’an 271018, Shandong, China; 2. Shandong Provincial Key Laboratory of Horticultural Machinery and Equipment, Tai’an 271018, Shandong, China
  • Yuhua Li 1. College of Mechanical and Electronic Engineering, Shandong Agricultural University, Tai’an 271018, Shandong, China; 2. Shandong Provincial Key Laboratory of Horticultural Machinery and Equipment, Tai’an 271018, Shandong, China
  • Jialin Hou 1. College of Mechanical and Electronic Engineering, Shandong Agricultural University, Tai’an 271018, Shandong, China; 2. Shandong Provincial Key Laboratory of Horticultural Machinery and Equipment, Tai’an 271018, Shandong, China
  • Ming Tu 3. College of Engineering, Huazhong Agricultural University, Wuhan 430070, China; 4. Key Laboratory of Agricultural Equipment in Mid-lower Reaches of the Yangtze River, Ministry of Agriculture and Rural Affairs, Wuhan 430070, China
  • Ting Sun 5. Shandong Huasheng Pesticide Machinery Co., Ltd, Linyi 276017, Shandong, China

Keywords:

rice transplanter, steering, variable ratio tooth profile, variable ratio curve, Boolean subtraction operation, transmission stability

Abstract

Variable transmission ratio racks show great potential in rice transplanters as a key component of variable transmission ratio steering to balance steering portability and sensitivity. The objective of this study was to develop a novel geometrical design method to achieve quick, high-quality modeling of the free curvilinear tooth profile of a variable transmission ratio rack. First, a discrete envelope motion 3D model was established between the pinion-sector and the variable transmission ratio rack blank based on the mapping relationship between the rotation angle of the pinion-sector and the displacement of the rack, according to the variable transmission ratio function. Based on the loop Boolean subtraction operation, which removed the pinion-sector from the rack blank during all moments of the discrete motion process, the final complex changing tooth shape of the variable transmission ratio rack was enveloped. Then, since Boolean cutting residues made the variable ratio tooth surface fluctuant and eventually affected the precision of the model, this study proposed a modification method for establishing a smooth and continuous tooth profile. First, a novel fitting algorithm used approximate variable ratio tooth profile points extracted from the Boolean cutting marks and generated a series of variable ratio tooth profiles by utilizing B-spline with different orders. Next, based on a transmission stability simulation, the variable ratio tooth profile with optimal dynamic performance was selected as the final design. Finally, tests contrasting the transmission stability of the machining samples of the initial variable ratio tooth profile and the final variable ratio tooth profile were conducted. The results indicated that the final variable ratio tooth profile is more effective than the initial variable ratio tooth profile. Therefore, the proposed variable ratio tooth profile modeling and modification method for eliminating Boolean cutting residues and improving surface accuracy is proved to be feasible. Keywords: rice transplanter, steering, variable ratio tooth profile, variable ratio curve, Boolean subtraction operation, transmission stability DOI: 10.25165/j.ijabe.20201305.4884 Citation: Niu Z R, Li J L, Xin S, Zou L L, Li Y H, Hou J L, et al. Geometrical design of variable ratio tooth profile based on Boolean subtraction operation and a novel modification method. Int J Agric & Biol Eng, 2020; 13(5): 125–133.

Author Biographies

Ziru Niu, 1. College of Mechanical and Electronic Engineering, Shandong Agricultural University, Tai’an 271018, Shandong, China; 2. Shandong Provincial Key Laboratory of Horticultural Machinery and Equipment, Tai’an 271018, Shandong, China

Associate Professor

Jianlei Li, 1. College of Mechanical and Electronic Engineering, Shandong Agricultural University, Tai’an 271018, Shandong, China

Master Degree Candidate

Shuai Xin, 1. College of Mechanical and Electronic Engineering, Shandong Agricultural University, Tai’an 271018, Shandong, China

Assistant Professor

Liangliang Zou, 1. College of Mechanical and Electronic Engineering, Shandong Agricultural University, Tai’an 271018, Shandong, China; 2. Shandong Provincial Key Laboratory of Horticultural Machinery and Equipment, Tai’an 271018, Shandong, China

Associate Professor

Yuhua Li, 1. College of Mechanical and Electronic Engineering, Shandong Agricultural University, Tai’an 271018, Shandong, China; 2. Shandong Provincial Key Laboratory of Horticultural Machinery and Equipment, Tai’an 271018, Shandong, China

PhD Candidate

Jialin Hou, 1. College of Mechanical and Electronic Engineering, Shandong Agricultural University, Tai’an 271018, Shandong, China; 2. Shandong Provincial Key Laboratory of Horticultural Machinery and Equipment, Tai’an 271018, Shandong, China

Undergraduate

Ming Tu, 3. College of Engineering, Huazhong Agricultural University, Wuhan 430070, China; 4. Key Laboratory of Agricultural Equipment in Mid-lower Reaches of the Yangtze River, Ministry of Agriculture and Rural Affairs, Wuhan 430070, China

Assistant Professor

Ting Sun, 5. Shandong Huasheng Pesticide Machinery Co., Ltd, Linyi 276017, Shandong, China

Professor

References

Alexandru P, Macaveiu D, Alexandru C. A gear with translational wheel for a variable transmission ratio and applications to steering box. Mechanism and Machine Theory, 2012; 52: 267–276.

Alexandru P. The rack-pinion gears for steering gear box with variable transmission ratio. Proceedings of the 12th IFToMM World Congress, Besancon, 2007; pp.353–358.

Lee J, Yi K, Lee D, Jang B, Kim M, Hwang S. Haptic control of steer-by-wire systems for tracking of target steering feedback torque. Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, 2020; 234(5): 1389–1401.

Huang W, Wong P K, Wong K I, Vong C M, Zhao J. Adaptive neural control of vehicle yaw stability with active front steering using an improved random projection neural network. Vehicle System Dynamics, 2020; 58: 1–19.

Zhang N, Wang M. Dynamic modeling of hydraulic power steering system with variable ratio rack and pinion gear. JSME International Journal Series C-Mechanical System Machine Elements and Manufacturing, 2005; 48(2): 251–260.

Alexandru P, Ceausescu B, Diaconescu D. Important aspects of geometry of the pinion-rack gear with variable ratio. Proceedings of the 8th IFToMM International Symposium on Science of Mechanisms and Machines-SYROM, Bucharest, 2001; pp.13–18.

Mao J H, Li L C, Wu X T, Zhang Z. The Numerical computation method of variable ratio rack conjugate surface. Chinese Journal of Mechanical Engineering, 2002; 5(5): 52–55. (in Chinese)

Hu J, Fan W, Niu Z R, Li G Y. Digital design method of variable ratio gear for automobile recirculating ball variable ratio steering gear based on meshing theory. Journal of the Balkan Tribological Association, 2016; 22(1): 171–183.

Hu J, Dong K K, Niu Z R, Li G Y. Digital design method of variable ratio gear pair for automobile mechanical variable ratio steering gear based on enveloping simulation. Journal of the Balkan Tribological Association, 2016; 22(1): 184–199.

Xu M, Han X H, Hua L, Zheng F Y. Modeling and methods for gear shaping process and cutting force prediction of variable transmission ratio rack. International Journal of Mechanical Sciences, 2020; 171: 1389–1401.

Zschippang H A, Weikert S, Küçük K A, Wegener K. Face-gear drive: Geometry generation and tooth contact analysis. Mechanism and Machine Theory, 2019; 142: 103576. doi: 10.1016/j.mechmachtheory.2019.103576.

Feng G S, Xie Z F, Zhou M. Geometric design and analysis of face-gear drive with involute helical pinion. Mechanism and Machine Theory, 2019; 134: 169–196.

Guo H, Gonzalez-Perez I, Fuentes-Aznar A. Computerized generation and meshing simulation of face gear drives manufactured by circular cutters. Mechanism and Machine Theory, 2019; 133: 44–63.

Xiang T M, Gu L Z, Xiao L L. Accurate modeling of logarithmic spiral bevel gear based on the tooth flank formation and Boolean addition operation. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, 2016; 230(9): 1650–1658.

Liu S Y, Song C S, Zhu C C, Ni G X. Effect of tooth modifications on mesh characteristics of crossed beveloid gear pair with small shaft angle. Mechanism and Machine Theory, 2018; 119: 142–160.

Fu X Z, Fang Z D, Cui Y M, Hou X Y, Li J H. Modelling, design and analysis of offset, non-orthogonal and profile-shifted face gear drives. Advances in Mechanical Engineering, 2018; 10(9): 1–12.

Zheng F Y, Han X H, Hua L, Zhang M D, Zhang W Q. Design and manufacture of new type of non-circular cylindrical gear generated by face-milling method. Mechanism and Machine Theory, 2018; 122: 326–346.

Luo S M, Liao L X, Wang J, Wang Y, Yi J X. Study on inspection and avoidance of interferences in five-axis end milling of cycloidal gears. The International Journal of Advanced Manufacturing Technology, 2017; 91(9-12): 3307–3314.

Litvin F L, Feng P H. Computerized design, generation, and simulation of meshing of rotors of screw compressor. Mechanism and Machine Theory, 1997; 32(2): 137–160.

Litvin F L, Fuentes A, Gonzalez-perez L, Carvenali L, Kawasaki K, Handschuh R F. Modified involute helical gears: computerized design, simulation of meshing and stress analysis. Comput. Methods Appl. Mech. Engrg., 2003; 192(33-34): 3619–3665.

Litvin F L, Gonzalez-Perez I, Fuentes A, Hayasaka K, Yukishima K. Topology of modified surfaces of involute helical gears with line contact developed for improvement of bearing contact, reduction of transmission errors, and stress analysis. Mathematical and Computer Modelling, 2005; 42(9-10): 1063–1078.

Litvin F L, Gonzalez-Perez I, Yukishima K, Fuentes A, Hayasaka K. Generation of planar and helical elliptical gears by application of rack-cutter, hob, and shaper. Computer Methods in Applied Mechanics and Engineering, 2007; 196(41): 4321–4336.

Litvin F L, Gonzalez-Perez I, Fuentes A, Hayasaka K. Design and investigation of gear drives with non-circular gears applied for speed variation and generation of functions. Comput. Methods Appl. Mech. Engrg., 2008; 197: 3783–3802.

Liu D W, Ren T Z, Jin X. Geometrical model and tooth analysis of undulating face gear. Mechanism and Machine Theory, 2015; 86: 140–155.

Wu Y H, Zhou Y S, Zhou Z Y, Tang J Y, Ouyang H W. An advanced CAD/CAE integration method for the generative design of face gears. Advances in Engineering Software, 2018; 126: 90–99.

Zhou Y S, Wang S H, Wang L M, Tang J Y, Chen Z Z. CNC milling of face gears with a novel geometric analysis. Mechanism and Machine Theory, 2019; 139: 46–65.

Mazak J, Klocke F, Bergs T, Brecher C, Löpenhaus C. Simulation-based process analysis for discontinuous cutting of generated bevel gears. Proceedings of the Institution of Mechanical Engineers, 2019; 233(21-22): 7378–7390.

Guo E K, Hong R J, Huang X D, Fang C G. Research on the design of skiving tool for machining involute gears. Journal of Mechanical Science and Technology, 2014; 28(12): 5107–5115.

Lin C, Xia X G, Li P L. Geometric design and kinematics analysis of coplanar double internal meshing non-circular planetary gear train. Advances in Mechanical Engineering, 2018; 10(12): 1–12.

De Boor C. A Practical Guide to Splines. New York: Springer-Verlag, 1978; 346p.

Downloads

Published

2020-10-13

How to Cite

Niu, Z., Li, J., Xin, S., Zou, L., Li, Y., Hou, J., … Sun, T. (2020). Geometrical design of variable ratio tooth profile based on Boolean subtraction operation and a novel modification method. International Journal of Agricultural and Biological Engineering, 13(5), 125–133. Retrieved from https://ijabe.migration.pkpps06.publicknowledgeproject.org/index.php/ijabe/article/view/4884

Issue

Vol. 13 No. 5 (2020): IJABE

Section

Power and Machinery Systems

License

IJABE is an international peer reviewed, open access journal, adopting Creative Commons Copyright Notices as follows.

Authors who publish with this journal agree to the following terms:

  1. Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
  2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
  3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).