Simulation of generation process for asymmetric involute gear tooth shape with and without profile correction

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  • 1. Innovative Systems Design and Engineering www.iiste.orgISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)Vol 3, No 6, 2012 Simulation of Generation Process for Asymmetric Involute Gear Tooth Shape with and without Profile Correction Mohammad Qasim Abdullah* Department of Mechanical Engineering, College of Engineering, University of Baghdad, Baghdad, Iraq *E-mail of the corresponding author: mohq1969@yahoo.com Muhsin Jabir Jweeg Department of Mechanical Engineering, College of Engineering, Al-Nahrain University, Baghdad, Iraq E-mail: muhsinjj@yahoo.comAbstractIn order to achieve any design study on a gear tooth or to carry out any type of analysis on a complete gear drive, thefirst step is the representation of the actual form of this tooth under consideration. In this work, a mathematicalsimulation of generation process for the symmetric involute gear teeth shapes based on the principle of the gearshaping process with a rack-shaped cutter has been developed to take into account the effect of asymmetric toothprofiles and the use of profile correction on each side of tooth with different design parameters for each side of tooth.As a result of this work, a computer program based on this mathematical simulation has been built to representgraphically step-by-step the actual form of symmetric and asymmetric gear teeth shapes with and without profilecorrection for different gear design parameters. Keywords: Involute Gear Teeth, Symmetric, Asymmetric, Profile Correction, Mathematical Simulation, GenerationProcess.1. IntroductionIn this work, the principle of gear shaping process with a rack-shaped cutter which is one of the gear generationmethods [1], has been adopted to simulate mathematically the generation process of involute gear teeth shapes. Bythis process the tooth profile obtained is composed of two well defined parts, the involute part and root fillet part.The involute part is generated by the straight sides of the rack cutter, whereas the root fillet part is formed by thecutter nose. In this mathematical simulation, two cases have been considered, the first case is for the generation ofsymmetric tooth shape and the second case has been developed to generate asymmetric involute gear teeth shapeswith and without profile correction (profile shift) for different design parameters of module and number of teeth ongear, as well as pressure angles, root fillet radii, top edged radii and correction factors for loaded and unloaded toothsides.2. Generation of Symmetric ToothIn the generation process of symmetric tooth shape, the generating rack cutter and gear blank roll in contact in timedrelation to each other simultaneously, they are also given a feed relative to each other to remove the metal from theblank. In this process, the cutting tool (rack cutter) which shown in (Figure 1) is given a shape conjugate with theform of the tooth to be cut. Therefore; the geometrical representation of the rack cutter with its rolling simulation areimperative.2.1 Geometrical Representation of Rack CutterThe basic rack cutter which is shown in (Figure 1) has been adopted as a cutting tool for this generation process [2].Fifteen points are used to define the outer shape of this rack cutter. The Cartesian coor coordinates of these points aremeasured with respect to a reference point which is located on the pitch line .Referring to (Figure 1), it can be that:from : (1) 53
  • 2. Innovative Systems Design and Engineering www.iiste.orgISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)Vol 3, No 6, 2012also, from : (2)and (3)now, (4)By substituting equation (1) and equation (2) into equation (4), yields: (5)Therefore; (6) (7)In the same way, it can be proved that: (8) (9) (10)and (11)Also, from the same figure, it can be shown that:from : (12)also, from : (13)and (14)now, (15) 54
  • 3. Innovative Systems Design and Engineering www.iiste.orgISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)Vol 3, No 6, 2012By substituting equation (12) and equation (13) into equation (15), yields: (16)Therefore; (17) (18)In the same way, it can be proved that: (19) (20) (21)and (22)Now, by referring to (Figure 1) again, the Cartesian coordinates of the geometrical points for the rack cutter can befound as shown below: , , , , , , , 55
  • 4. Innovative Systems Design and Engineering www.iiste.orgISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)Vol 3, No 6, 2012 , , , , , , , , , , , , , andWhere : is the pressure angle of symmetric tooth, it’s the same for loaded and unloaded tooth sides, & : are the root fillet and top edged radii of loaded tooth side, & : are the root fillet and top edged radii of unloaded tooth side, & : are the addendum and ded ndum dedendum heig of tooth, m heights : is the circular pitch, and : is the module.2.2 Rolling Simulation of Rack CutterFor the tooth profiles generation, the essential requirement is that the pitch line which is shown in (Figure 1)rolls on the pitch circle of blank w nk without slipping. In (Figure 2), the pitch ccircle with a center has been keptstationary while the pitch line rolls on it on either side of pitch point . Also, in order to achiev the tooth achieve othprofile correction, the pitch line must be withdrawn from the pitch circle by amount “ ” whererepresents the correction factor [3]. According to this profile correction, new addendum and dedendum heights willbe obtained, therefore: (23) 56
  • 5. Innovative Systems Design and Engineering www.iiste.orgISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)Vol 3, No 6, 2012 (24) fThe generation process is carried out by the rolling of a whole rack cutter to the right and left directions from theinitial position without slipping by incremental angles ( ’s) until a whole tooth shape is generated, then from (Figure2), it can be shown that:Firstly, according to the principle of involute construction, [4]: (25)and (26)Also, from the same figure, (27)and (28)where : represents the radius of pitch circle, and represents the number of teeth on a gear.Therefore; the rolling simulation of a rack cutter can be achieved in the following procedure.2.2.1 Rolling Simulation of A Whole Rack Cutter to Right DirectionReferring to Fig.(3.5), it can be found the Cartesian coordinates of rack points for the right direction( ) are as follows: , , , , , , , , 57
  • 6. Innovative Systems Design and Engineering www.iiste.orgISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)Vol 3, No 6, 2012 , , , , , , , , , , , , , , ,2.2.2 Rolling Simulation of A Whole Rack Cutter to Left DirectionIn order to find the coordinates of rack points for the left direction( ), the same prev s e previous procedure of rollingsimulation for the right direction can be done but with replacing the angle by .3. Generation of Asymmetric Tooth ide rentIn this case, there are different profiles on each side of tooth, therefore; there are two different pressure angles forboth sides of tooth, the first is for loaded side “ ” and the other is for unloaded side “ ”. In this case, thegeneration process for the symmetric tooth must be developed to take into account this asymmetry of the toothprofiles. Therefore; the generation process of tooth must be carried out into two stages, the first for loaded side part 58
  • 7. Innovative Systems Design and Engineering www.iiste.orgISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)Vol 3, No 6, 2012and the other for unloaded side part.3.1 Generation Process for Loaded Side Part of Tooth ckBy this stage, the rolling simulation of a half rack cutter (right half) to the right and left directions from the initial de parposition without slipping by incremental angles ( ’s) is carried out until a half tooth shape (loaded side part of thetooth) would be generated. Also, to achieve the toot profile correction of this side, the pitch line tooth ile must bewithdrawn from the pitch circle by amount “ ” where represents the correction factor for loaded side oftooth as shown in (Figure 3). Therefore; the new addendum and dedendum heights for the loaded side of tooth are: (29) (30)Now, from the principle of involute construction for unloaded side part of tooth, it can be shown that: (31) (32) (33)and (34)where : represents the pressure angle of loaded side of tooth.Then, the rolling simulation of the loaded side part of tooth can be achieved in the following procedure.3.1.1 Rolling Simulation of Right Half Rack to Right DirectionReferring to (Figure 3), it can be found that: , , , , , , , , 59
  • 8. Innovative Systems Design and Engineering www.iiste.orgISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)Vol 3, No 6, 2012 , , , ,3.1.2 Rolling Simulation of Right Half Rack to Left DirectionAlso, by referring to (Figure 3), it can be found that: , , , , , , , , , , , ,3.2 Generation Process for Unloaded Side Part of ToothBy this stage, the rolling simulations of the second half rack cutter (left half of rack cutter) to the right and leftdirections from the initial position without slipping by incremental angles ( ’s) until the another half tooth shape(loa(loaded side part of the tooth) is generated and to achieve the tooth ad e addendum m um modification of this side, the pitch line must be withdrawn from the pitch circle by amount “ ” where represents the correction factor forunloaded side of tooth as shown in (Figure 4). Therefore; the new addendum and dedendum heights for unloadedside of tooth are: 60
  • 9. Innovative Systems Design and Engineering www.iiste.orgISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)Vol 3, No 6, 2012 (35) (36)Also, from the principle of involute construction of the unloaded side part of tooth, it can be shown that: (37) (38) (39)and (40)where : represents the pressure angle of unloaded side of tooth.Then, the rolling simulation of the unloaded side part of tooth can be achieved in the following procedure.3.2.1 Rolling Simulation of Left Half Rack to Right DirectionReferring to (Figure 4), it can be found that: , , , , , , , , , , 61
  • 10. Innovative Systems Design and Engineering www.iiste.orgISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)Vol 3, No 6, 2012 , ,3.2.2 Rolling Simulation of Left Half Rack to Left DirectionAlso, by referring to (Figure 4), it can be found that: , , , , , , , , , , , ,Finally, in order to represent the final shape of asymmetric tooth, the two generated parts of loaded and unloadedsides of tooth are joined toge oined together with taking into account the vertical shift difference between these two generatedparts with amount ( ), (i.e. the Y-coordinate of each point on th generated unloaded side must be shifted e. thedownward by this amount when and upward when .4. ResultsA computer program based on the above mathematical simulation using has been built using MicrosoftQuick-BASIC (ver. 4.5) to represent graphically the symmetric and asymmetric gear teeth shapes with and withoutprofile correction for different gear design parameters of module and number of teeth on gear, addendum anddedendum heights as well as pressure angles, root fillet radii, top edged radii and correction factors for loaded andunloaded tooth sides. (Figure 5) and (Figure 6) shows selective samples for symmetric and asymmetric involute gearteeth shapes respectively, which are generated by this program. 62
  • 11. Innovative Systems Design and Engineering www.iiste.orgISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)Vol 3, No 6, 20125. ConclusionsIn this work, a developed mathematical simulation of generation process for asymmetric involute gear teeth shapeswith and without profile correction for different gear design parameters of loaded and unloaded tooth sides has beenpresented. Also, a computer program based on this mathematical simulation has been built to represent graphicallystep-by-step the actual form of these gear teeth shapes.References[1] Marinov, V. (2006). Manufacturing Technology. (5th ed.). Pearson Education Inc., USA.[2] Litvin, F. L. (1997). Development of Gear Technology and Theory of Gearing. NASA Reference Publication 1406,USA.[3] Litvin, F. L., and Fuentes, A. (2004). Gear Geometry and Applied Theory. (2nd ed.). Cambridge University Press,New York, USA.[4] Maitra, G. M. (1994). Handbook of Gear Design. (2nd ed.). Tata McGraw-Hill Publishing Company Limited, NewDelhi. 63
  • 12. Innovative Systems Design and Engineering www.iiste.orgISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)Vol 3, No 6, 2012 64
  • 13. Innovative Systems Design and Engineering www.iiste.orgISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)Vol 3, No 6, 2012 65
  • 14. Innovative Systems Design and Engineering www.iiste.orgISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)Vol 3, No 6, 2012 66
  • 15. Innovative Systems Design and Engineering www.iiste.orgISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)Vol 3, No 6, 2012 67
  • 16. This academic article was published by The International Institute for Science,Technology and Education (IISTE). The IISTE is a pioneer in the Open AccessPublishing service based in the U.S. and Europe. The aim of the institute isAccelerating Global Knowledge Sharing.More information about the publisher can be found in the IISTE’s homepage:http://www.iiste.orgThe IISTE is currently hosting more than 30 peer-reviewed academic journals andcollaborating with academic institutions around the world. Prospective authors ofIISTE journals can find the submission instruction on the following page:http://www.iiste.org/Journals/The IISTE editorial team promises to the review and publish all the qualifiedsubmissions in a fast manner. All the journals articles are available online to thereaders all over the world without financial, legal, or technical barriers other thanthose inseparable from gaining access to the internet itself. Printed version of thejournals is also available upon request of readers and authors.IISTE Knowledge Sharing PartnersEBSCO, Index Copernicus, Ulrichs Periodicals Directory, JournalTOCS, PKP OpenArchives Harvester, Bielefeld Academic Search Engine, ElektronischeZeitschriftenbibliothek EZB, Open J-Gate, OCLC WorldCat, Universe DigtialLibrary , NewJour, Google Scholar